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+DRAFT VERSION JANUARY 13, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+The JWST Resolved Stellar Populations Early Release Science Program II.
+Survey Overview
+DANIEL R. WEISZ,1 KRISTEN B. W. MCQUINN,2 ALESSANDRO SAVINO,1 NITYA KALLIVAYALIL,3 JAY ANDERSON,4
+MARTHA L. BOYER,4 MATTEO CORRENTI,5, 6 MARLA C. GEHA,7 ANDREW E. DOLPHIN,8, 9 KARIN M. SANDSTROM,10
+ANDREW A. COLE,11 BENJAMIN F. WILLIAMS,12 EVAN D. SKILLMAN,13 ROGER E. COHEN,2 MAX J. B. NEWMAN,2
+RACHAEL BEATON,4, 14, 15 ALESSANDRO BRESSAN,16 ALBERTO BOLATTO,17, 18 MICHAEL BOYLAN-KOLCHIN,19
+ALYSON M. BROOKS,2, 20 JAMES S. BULLOCK,21 CHARLIE CONROY,22 M. C. COOPER,21 JULIANNE J. DALCANTON,12, 20
+AARON L. DOTTER,23 TOBIAS K. FRITZ,3 CHRISTOPHER T. GARLING,3 MARIO GENNARO,24, 25 KAROLINE M. GILBERT,24, 25
+L´EO GIRARDI,26 BENJAMIN D. JOHNSON,22 L. CLIFTON JOHNSON,27 JASON S. KALIRAI,28 EVAN N. KIRBY,29 DUSTIN LANG,30
+PAOLA MARIGO,31 HANNAH RICHSTEIN,3 EDWARD F. SCHLAFLY,24 JUDY SCHMIDT,32 ERIK J. TOLLERUD,4 JACK T. WARFIELD,3
+AND ANDREW WETZEL33
+1Department of Astronomy, University of California, Berkeley, CA 94720, USA
+2Department of Physics and Astronomy, Rutgers, the State University of New Jersey, 136 Frelinghuysen Road, Piscataway, NJ 08854, USA
+3Department of Astronomy, University of Virginia, 530 McCormick Road, Charlottesville, VA 22904, USA
+4Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA
+5INAF Osservatorio Astronomico di Roma, Via Frascati 33, 00078, Monteporzio Catone, Rome, Italy
+6ASI-Space Science Data Center, Via del Politecnico, I-00133, Rome, Italy
+7Department of Astronomy, Yale University, New Haven, CT 06520, USA
+8Raytheon Technologies, 1151 E. Hermans Road, Tucson, AZ 85756, USA
+9Steward Observatory, University of Arizona, 933 N. Cherry Avenue, Tucson, AZ 85719, USA
+10Center for Astrophysics and Space Sciences, Department of Physics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA
+11School of Natural Sciences, University of Tasmania, Private Bag 37, Hobart, Tasmania 7001, Australia
+12Department of Astronomy, University of Washington, Box 351580, U.W., Seattle, WA 98195-1580, USA
+13University of Minnesota, Minnesota Institute for Astrophysics, School of Physics and Astronomy, 116 Church Street, S.E., Minneapolis, MN 55455, USA
+14Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA
+15The Observatories of the Carnegie Institution for Science, 813 Santa Barbara St., Pasadena, CA 91101, USA
+16SISSA, Via Bonomea 265, 34136 Trieste, Italy
+17Department of Astronomy, University of Maryland, College Park, MD 20742, USA
+18Joint Space-Science Institute, University of Maryland, College Park, MD 20742, USA
+19Department of Astronomy, The University of Texas at Austin, 2515 Speedway, Stop C1400, Austin, TX 78712-1205, USA
+20Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA
+21Department of Physics and Astronomy, University of California, Irvine, CA 92697 USA
+22Center for Astrophysics — Harvard & Smithsonian, Cambridge, MA, 02138, USA
+23Department of Physics and Astronomy, Dartmouth College, 6127 Wilder Laboratory, Hanover, NH 03755, USA
+24Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA
+25The William H. Miller III Department of Physics & Astronomy, Bloomberg Center for Physics and Astronomy, Johns Hopkins University, 3400 N. Charles
+Street, Baltimore, MD 21218, USA
+26Padova Astronomical Observatory, Vicolo dell’Osservatorio 5, Padova, Italy
+27Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and Department of Physics and Astronomy, Northwestern University, 1800
+Sherman Avenue, Evanston, IL 60201, USA
+28John Hopkins Applied Physics Laboratory, 11100 Johns Hopkins Road, Laurel, MD 20723, USA
+29Department of Physics, University of Notre Dame, Notre Dame, IN 46556, USA
+30Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada
+31Department of Physics and Astronomy G. Galilei, University of Padova, Vicolo dell’Osservatorio 3, I-35122, Padova, Italy
+322817 Rudge Pl, Modesto, CA 95355, USA
+33Department of Physics and Astronomy, University of California, Davis, CA 95616, USA
+ABSTRACT
+Corresponding author: Daniel R. Weisz
+dan.weisz@berkeley.edu
+arXiv:2301.04659v1  [astro-ph.GA]  11 Jan 2023
+
+2
+WEISZ ET AL.
+We present the JWST Resolved Stellar Populations Early Release Science (ERS) science program. We ob-
+tained 27.5 hours of NIRCam and NIRISS imaging of three targets in the Local Group (Milky Way globular
+cluster M92, ultra-faint dwarf galaxy Draco II, star-forming dwarf galaxy WLM), which span factors of ∼ 105
+in luminosity, ∼ 104 in distance, and ∼ 105 in surface brightness. We describe the survey strategy, scientific and
+technical goals, implementation details, present select NIRCam color-magnitude diagrams (CMDs), and vali-
+date the NIRCam exposure time calculator (ETC). Our CMDs are among the deepest in existence for each class
+of target. They touch the theoretical hydrogen burning limit in M92 (< 0.08 M⊙; SNR ∼ 5 at mF 090W ∼ 28.2;
+MF 090W ∼ +13.6), include the lowest-mass stars observed outside the Milky Way in Draco II (0.09 M⊙;
+SNR = 10 at mF 090W ∼ 29; MF 090W ∼ +12.1), and reach ∼ 1.5 magnitudes below the oldest main se-
+quence turnoff in WLM (SNR = 10 at mF 090W ∼ 29.5; MF 090W ∼ +4.6). The PARSEC stellar models
+provide a good qualitative match to the NIRCam CMDs, though are ∼ 0.05 mag too blue compared to M92
+F090W−F150W data. The NIRCam ETC (v2.0) matches the SNRs based on photon noise from DOLPHOT
+stellar photometry in uncrowded fields, but the ETC may not be accurate in more crowded fields, similar to what
+is known for HST. We release beta versions of DOLPHOT NIRCam and NIRISS modules to the community.
+Results from this ERS program will establish JWST as the premier instrument for resolved stellar populations
+studies for decades to come.
+Keywords: Stellar photometry (1620), Local Group (929), Stellar populations (1622), Hertzsprung Russell dia-
+gram (725), JWST (2291)
+1. INTRODUCTION
+The resolved stellar populations of nearby galaxies are
+central to a wide range of astrophysics. The observed col-
+ors, luminosities, and spectral features of resolved stars in
+galaxies within the Local Volume (LV) anchor our knowl-
+edge of star formation (e.g., star cluster formation, the ini-
+tial mass function, the importance of binarity; Massey 2003;
+McKee & Ostriker 2007; Sarajedini et al. 2007; Bastian et al.
+2010; Sana et al. 2012; Kroupa et al. 2013; Krumholz 2014;
+Piotto et al. 2015; Krumholz et al. 2019), stellar feedback
+(e.g., the interplay between stars and their immediate sur-
+roundings; e.g., Oey 1996; Dohm-Palmer et al. 1998; Stin-
+son et al. 2006, 2007; Governato et al. 2010; Ostriker et al.
+2010; Lopez et al. 2011; Pellegrini et al. 2011; Lopez et al.
+2014; El-Badry et al. 2016; McQuinn et al. 2018, 2019b),
+dust production and characteristics (e.g., Gordon et al. 2003;
+Boyer et al. 2006; Boyer 2010; Meixner et al. 2010; Dal-
+canton et al. 2015; Schlafly et al. 2016; Green et al. 2019;
+Gordon et al. 2021; Yanchulova Merica-Jones et al. 2021),
+and stellar chemistry and kinematics across a wide range of
+environments (e.g., Venn et al. 2004; Simon & Geha 2007;
+Geha et al. 2009; Kirby et al. 2011; Collins et al. 2013; Var-
+gas et al. 2013; Gilbert et al. 2014; Vargas et al. 2014; Ji et al.
+2016; Gilbert et al. 2019b; Escala et al. 2020; Kirby et al.
+2020; Gilbert et al. 2022). Resolved stars are the basis for
+the local distance ladder (e.g., Freedman et al. 2001; Riess
+et al. 2011; Beaton et al. 2016; Riess et al. 2016; McQuinn
+et al. 2017a, 2019a; Freedman et al. 2020; Riess et al. 2021),
+which provides constraints on the expansion of the Universe
+and the nature of dark energy (e.g., Di Valentino et al. 2021).
+They anchor our knowledge of the stellar evolution models
+that are used to interpret the light of distant galaxies (e.g.,
+Dotter et al. 2008; Ekstr¨om et al. 2012; Girardi et al. 2010;
+Bressan et al. 2012; VandenBerg et al. 2012; Choi et al. 2016;
+Eldridge et al. 2017; Conroy et al. 2018; Hidalgo et al. 2018;
+Eldridge & Stanway 2022) and provide detailed insight into
+the dynamic assembly of our Galactic neighborhood, cos-
+mic reionization, the first stars, near-field cosmology, and
+the nature of dark matter on the smallest scales (e.g., Ma-
+teo 1998; Tolstoy et al. 2009; Brown et al. 2014; Weisz et al.
+2014a; Boylan-Kolchin et al. 2015; Gallart et al. 2015; Mc-
+Quinn et al. 2015; Frebel & Norris 2015; Wetzel et al. 2016;
+Bullock & Boylan-Kolchin 2017; Starkenburg et al. 2017b;
+McConnachie et al. 2018; Kallivayalil et al. 2018; Conroy
+et al. 2019; Simon 2019; Patel et al. 2020; Boylan-Kolchin &
+Weisz 2021; Sacchi et al. 2021; Pearson et al. 2022).
+Over the past ∼ 30 years, much of this science has been
+enabled by the Hubble Space Telescope (HST). Since its first
+images of resolved stars in the local Universe (e.g., Paresce
+et al. 1991; Campbell et al. 1992; Guhathakurta et al. 1992;
+Freedman et al. 1994; Hunter et al. 1995), HST’s exquisite
+sensitivity, angular resolution, and broad wavelength cover-
+age have transformed our knowledge of the Universe by ob-
+serving hundreds of nearby galaxies for thousands of hours
+(e.g., Freedman et al. 2001; Brown et al. 2006; Holtzman
+et al. 2006; Dalcanton et al. 2009; Brown et al. 2012; Dal-
+canton et al. 2012a; Gallart et al. 2015; Riess et al. 2016;
+Skillman et al. 2017; Tully et al. 2019; Williams et al. 2021),
+including the Panchromatic Hubble Andromeda Treasury
+(PHAT) program, which resolved 100 million stars across the
+disk of M31 (Dalcanton et al. 2012b; Williams et al. 2014).
+However, while HST continues to catalyze new astrophys-
+ical insights in nearby galaxies, it has only scratched the sur-
+face of science enabled by infrared (IR) observations. Com-
+pared to the UV and optical, HST’s IR camera has coarser an-
+gular resolution, which limits it to brighter stars due to stellar
+crowding, and it can only observe a small portion of the IR
+spectrum, which limits the types of stellar populations it can
+study.
+JWST will be transformative for resolved stellar popula-
+tions in the IR. Compared to any other facility, JWST will re-
+
+JWST RESOLVED STELLAR POPULATIONS I
+3
+solve individual stars at larger distances, to fainter luminosi-
+ties, over wider color baselines, in more crowded areas, and
+in regions of higher extinction. JWST can provide the first
+main-sequence turnoff-based (MSTO) star formation histo-
+ries (SFHs) of galaxies beyond the Local Group (LG; e.g.,
+Weisz & Boylan-Kolchin 2019), systematically measure the
+sub-Solar mass stellar IMF directly from star counts as a
+function of environment (e.g., Geha et al. 2013; Kalirai et al.
+2013; El-Badry et al. 2017; Gennaro et al. 2018a; Filion et al.
+2020; Gennaro & Robberto 2020), determine proper motions
+and orbital histories for dozens of galaxies outside our imme-
+diate Galactic neighborhood (e.g., van der Marel et al. 2012;
+Sohn et al. 2013; Kallivayalil et al. 2013; Zivick et al. 2018;
+Sohn et al. 2020; Warfield et al. 2023), construct parsec-scale
+maps of the interstellar medium (ISM) in galaxies out to sev-
+eral Mpc (e.g., Dalcanton et al. 2015; Gordon et al. 2016;
+Yanchulova Merica-Jones et al. 2017, 2021), establish a new
+anchor to the physics of the evolved stars that dominate the
+rest-frame near-IR light of distant galaxies (e.g., Maraston
+2006; Melbourne et al. 2012; Boyer et al. 2015, 2019), pro-
+vide high fidelity distances to galaxies throughout the Lo-
+cal Volume (e.g., Beaton et al. 2016; McQuinn et al. 2019a;
+Tully et al. 2019; Freedman et al. 2020; Riess et al. 2021),
+and much more.
+With these remarkable capabilities in mind, we have under-
+taken the JWST Resolved Stellar Populations Early Release
+Science (ERS) Program (DD-1334; PI D. Weisz) to establish
+JWST as the premier facility for the study of resolved stellar
+populations in the IR such that it can match and exceed HST’s
+successes in the local Universe. To realize this goal, our ERS
+program has acquired deep multi-band NIRCam and NIRISS
+imaging of three targets in the Local Group (LG): one Milky
+Way globular cluster (GC; M92), one ultra-faint dwarf galaxy
+(UFD; Draco II), and one distant star-forming dwarf galaxy
+(WLM). These diverse targets showcase a broad range of the
+science described above and enable the development and test-
+ing of JWST-specific modules for the widely used crowded
+field stellar photometry package DOLPHOT (Dolphin 2000,
+2016).
+In this paper, we summarize the design of our ERS
+program, illustrate the new JWST-specific capabilities of
+DOLPHOT, outline the photometric reduction process,
+present a first look at JWST observations of our targets, and
+undertake select comparisons with stellar models and the cur-
+rent JWST exposure time calculator (ETC). Papers in prepa-
+ration by our team will provide a detailed overview of the
+new NIRCam and NIRISS modules for DOLPHOT and will
+focus on a wide variety of science results enabled by the ERS
+data beyond what is described here.
+This paper is organized as follows.
+We summarize the
+program’s overarching science and technical aims and tar-
+get selection in §2.
+We then describe how we translated
+these goals into an observational strategy in §3. We detail
+the actual ERS observations in §4 and summarize the ap-
+plication of DOLPHOT in §5. In §6 we present NIRCam
+color-magnitude diagrams (CMDs) and compare them to se-
+lect stellar models and evaluate the performance of NIRCam
+ETC.
+2. PROGRAM GOALS
+Our team developed a set of main science and technical
+goals based on anticipated common community use cases of
+JWST for resolved stellar populations. For simplicity, we
+limited our considerations to science cases based on imaging
+with NIRCam, which is considered the “workhorse” camera
+of JWST, as well as NIRISS imaging, which we used in par-
+allel. This setup is analogous to the commonly used mode of
+HST in which ACS/WFC operates as the primary instrument
+with WFC3/UVIS acquiring imaging in parallel (e.g., Dal-
+canton et al. 2012a; Gallart et al. 2015; Skillman et al. 2017;
+Albers et al. 2019; Williams et al. 2021).
+Science based on imaging of resolved stars often requires
+stellar photometry in crowded fields. Because of that, re-
+solved stellar population studies are technically daunting,
+requiring highly optimized observations and sophisticated
+analysis tools that have been developed and refined over the
+past ∼ 40 years (e.g., Buonanno et al. 1979; Tody 1980; Stet-
+son 1987; Schechter et al. 1993; Stetson 1994; Anderson &
+King 2000; Dalcanton et al. 2012b; Williams et al. 2014). A
+main technical goal of our program is to develop and release
+NIRCam and NIRISS modules for DOLPHOT, along with
+practical recommendations and demonstrations for applying
+DOLPHOT to NIRCam and NIRISS imaging. Here, we sum-
+marize our main science goals, technical goals, and science
+“deliverables” which guide our ERS program.
+2.1. Scientific Goals
+Our team identified six main science themes that guided
+the construction of our ERS program. They are:
+1. Star Formation Histories. A galaxy’s resolved stel-
+lar content encodes its star formation history (SFH),
+which can be reconstructed by fitting CMDs with stel-
+lar evolution models (e.g., Tosi et al. 1989; Tolstoy
+1996; Harris & Zaritsky 2001; Dolphin 2002; Hidalgo
+et al. 2009). These SFHs are particularly robust when
+CMDs extend below the oldest main sequence turnoff
+(MSTO; e.g., Gallart et al. 2005). The faintness of this
+feature in the optical (MV ∼ +4) has limited current
+‘gold standard’ SFHs to galaxies within the LG. How-
+ever, the relatively low effective temperatures of these
+stars, combined with the decreased sky background in
+the near-IR and JWST’s excellent sensitivity and angu-
+lar resolution, will enable it to measure the first SFHs
+based on the oldest MSTOs for galaxies outside the
+LG (e.g., Weisz & Boylan-Kolchin 2019; JWST-GO-
+1617 PI K. McQuinn) from which outstanding ques-
+tions (e.g., the effects of reionization and/or environ-
+ment on galaxy formation) can be uniquely addressed
+(e.g., Bullock & Boylan-Kolchin 2017; Simon 2019).
+Our JWST program will showcase JWST’s ability to
+measure robust SFHs.
+
+4
+WEISZ ET AL.
+2. The Sub-Solar Mass IMF. Resolved star counts shows
+that lowest-mass galaxies appear to have sub-Solar
+IMF slopes which deviate from the Galactic value.
+(e.g., Geha et al. 2013; Kalirai et al. 2013; Gen-
+naro et al. 2018b). However, even with HST, it has
+proven challenging to acquire sufficiently deep data
+(down to ∼ 0.2 M⊙ El-Badry et al. 2017; Gennaro
+et al. 2018b,a) to unambiguously confirm these puta-
+tive IMF variations. Our ERS program will illustrate
+JWST’s capabilities for definitively measuring the sub-
+Solar IMF in a ultra-faint MW satellite, paving the way
+for a systematic study of the low-mass IMF and star
+formation in extreme environments.
+3. Proper Motions.
+High-precision astrometry enables
+the measurement of proper motions (PMs) throughout
+the LG. Gaia has been transformative for objects in
+the MW halo, while HST has laid the foundation for
+fainter, more distant systems. JWST is the future of
+precision astrometry for faint and/or more distant ob-
+jects. On its own, and in tandem with Gaia, HST, and
+Roman, JWST imaging will provide measurements of
+total masses, dark matter profiles, and orbital histories
+for ∼ 100 galaxies in and around the LG (e.g., Bullock
+& Boylan-Kolchin 2017; Kallivayalil et al. 2015; Fritz
+et al. 2018; Gilbert et al. 2019a; Battaglia et al. 2022;
+Warfield et al. 2023). Our ERS program will showcase
+the PM measurements capabilities of JWST.
+4. Evolved Stars. Cool evolved stars such as red super-
+giants and asympotic giant branch (AGB) stars are re-
+sponsible for 20–70% of the rest-frame near-IR lumi-
+nosity of star-forming galaxies (e.g., Maraston 2006;
+Melbourne et al. 2012) and are sites of dust production
+(e.g., Ventura 2001). However, the rapid evolution of
+dusty AGB stars is challenging to model (e.g., Maras-
+ton 2006; Girardi et al. 2010; Conroy 2013; Marigo
+et al. 2017), which has only begun to be alleviated
+by recent observations (e.g., Boyer et al. 2015, 2017).
+JWST’s expansive IR filter set will reveal elusive dust-
+enshrouded populations of AGB stars (e.g., oxygen-
+rich M stars and carbon-rich C stars) across a wide
+range of galactic environments (e.g., Hjort et al. 2016;
+Jones et al. 2017; Marini et al. 2020). Our ERS pro-
+gram will demonstrate JWST’s capacity to study IR-
+bright, evolved stars.
+5. Extinction Mapping.
+In the LG, Spitzer and Her-
+schel have mapped dust emission at ∼ 10 − 40′′ and
+∼ 7 − 12′′ resolution, respectively (10 pc for the
+Magellanic Clouds; 100 pc for M31 and M33; Draine
+2007; Gordon 2014; Chastenet et al. 2019; Utomo et al.
+2019). JWST can map the cold ISM at significantly
+higher spatial resolution by inferring dust content from
+its impact on stellar spectral energy distributions (e.g.,
+Dalcanton et al. 2015; Gordon et al. 2016). Our ERS
+observations will demonstrate JWST’s ability to map
+dust extinction and relate it to properties of the cold
+ISM.
+6. Ages of Globular Clusters. Accurate ages of the oldest
+GCs are particularly important for connecting the stel-
+lar fossil record to events in the early Universe includ-
+ing cosmic reionization and the age of the Universe it-
+self (e.g., Chaboyer et al. 1996; Grebel & Gallagher
+2004; Ricotti & Gnedin 2005; Monelli et al. 2010;
+Brown et al. 2014; Weisz et al. 2014b; Boylan-Kolchin
+et al. 2015). Current age estimates are typically limited
+to ∼ 1 Gyr precision (i.e., twice as long as reionization
+lasted) due to the age-metallicity degeneracies at the
+MSTO (see Boylan-Kolchin & Weisz 2021 and refer-
+ences therein). JWST observations of the ‘kink’ on the
+lower main sequence (MS) can yield more precise esti-
+mates of cluster ages (e.g., Sarajedini et al. 2009; Bono
+et al. 2010; Kalirai et al. 2012; Correnti et al. 2018).
+Our ERS data will showcase the powerful capabilities
+of JWST for precise GC age-dating.
+Beyond enabling our main science goals, we sought to
+identify observations that would make our ERS program
+rich for archival pursuits. Examples include measuring ex-
+tragalactic distances in JWST bands (e.g., TRGB, variable
+stars; Beaton et al. 2016; Madore et al. 2018; McQuinn et al.
+2019a), identifying rare stars from low-mass metal-poor stars
+to luminous red supergiants (e.g., Schlaufman & Casey 2014;
+Casey & Schlaufman 2015; Levesque 2018), searching for
+dust production among red giant branch stars (RGB; e.g.,
+Boyer et al. 2006; Boyer 2010), and examining the nature of
+dark matter using wide binaries (e.g., Pe˜narrubia et al. 2016).
+2.2. Technical Goals
+The main technical goal of our ERS program is to enable
+resolved star science by the broader community. At the heart
+of this goal is the addition of NIRCam and NIRISS modules
+to DOLPHOT. This process includes the technical develop-
+ment of NIRCam and NIRISS modules for DOLPHOT, test-
+ing their performance on real data, releasing data products
+that immediately enable science (e.g., stellar catalogs), and
+providing guidance to the community on best use practices
+of DOLPHOT for future applications. Here, we broadly de-
+scribe each of these technical goals and how they influenced
+the observational strategy of our ERS program.
+As with previous updates to DOLPHOT (e.g., Dalcanton
+et al. 2012a; Dolphin 2016, and many unpublished updates),
+the core functionality of the code remains the same as de-
+scribed in Dolphin (2000), but certain aspects have been up-
+dated for NIRCam and NIRISS.
+The DOLPHOT modules for NIRCam and NIRISS each
+feature their own pre-processing routines that apply masks
+(e.g., of reference, saturated, and other unusable pixels) to the
+images based on the data quality flags provided by the STScI
+pipeline. They also apply the pixel area maps appropriate
+to each camera. Other updates include the use of photomet-
+ric calibrations provided in the image metadata, conversions
+
+JWST RESOLVED STELLAR POPULATIONS I
+5
+to VEGAMAG, and camera-specific PSF models with corre-
+sponding encircled energy corrections.
+For testing DOLPHOT on real data, we identified several
+observational scenarios that we anticipate to be common for
+NIRCam and NIRISS studies of resolved stars. They are:
+1. Targets with various levels of crowding. This includes
+images that are completely uncrowded (e.g., in which
+aperture vs PSF photometry can be compared), images
+with highly variable amounts of crowding (e.g., due
+to surface brightness variations), and highly crowded
+images (i.e., the photometric depth is primarily limited
+by crowding).
+2. Targets that include stars spanning a large dynamic
+range in brightness in the same image. An example
+would be a GC, in which there are very bright red gi-
+ants and extremely faint dwarfs. This enables a variety
+of tests, including the ability to recover faint sources
+next to very bright objects.
+3. Targets with bright, saturated stars. JWST is extremely
+sensitive. Understanding the degree to which saturated
+stars affect the photometry of fainter objects will be
+important to a variety of science goals.
+4. Targets that demonstrate the ability of using the higher
+angular resolution short-wavelength (SW) images to
+increase the accuracy of the long-wavelength (LW)
+photometry. PHAT showed that joint reduction of HST
+optical and IR data produced IR photometry that pro-
+vides significantly sharper CMDs compared to reduc-
+ing IR data alone (e.g., Williams et al. 2014). Similar
+gains should be possible with NIRCam.
+5. Targets that enable the simultaneous reduction of HST
+and JWST imaging. To date, DOLPHOT has produced
+wonderful cross-camera results for HST (e.g., Dalcan-
+ton et al. 2012b; Williams et al. 2014, 2021), but it
+needs to be vetted and optimized for cross-facility use.
+2.3. Deliverables
+Our program is in the process of providing several “deliv-
+erables” to the community that can be found on our team
+website1.
+A primary deliverable is the public release of
+DOLPHOT with NIRCam and NIRISS specific modules for
+which “beta” versions can be found on the main DOLPHOT
+website2. This software enables crowded field stellar pho-
+tometry for a diverse range of science in the local Universe.
+Along with the software release we will provide extensive
+documentation of how to use DOLPHOT and examples of
+it applied to our ERS observations. Following careful cali-
+bration and testing, we will release high level science prod-
+ucts including the output of our team DOLPHOT runs on
+1 https://ers-stars.github.io
+2 http://americano.dolphinsim.com/dolphot/
+ERS data (e.g., diagnostic plots and files), and NIRCam and
+NIRISS stellar catalogs for each target along with artificial
+star tests. These data products will be refined as our un-
+derstanding of JWST improves (e.g., due to updated PSF
+models) and will eventually include examples of how to use
+DOLPHOT for simultaneous reduction of HST and JWST
+imaging.
+3. STRATEGY
+3.1. Filters
+The diversity of our science cases required careful consid-
+eration of filter selection. Several of our science goals are
+centered around maximizing depth, color baseline, and astro-
+metric precision. Accordingly, we primarily focused on SW
+filter selection, which has better sensitivity (for most stars)
+and angular resolving power than the long-wavelength chan-
+nel.
+Using an ancient,
+metal-poor isochrone (12.5 Gyr,
+[Fe/H]=−2.0) from the MIST stellar models (Choi et al.
+2016), we examined the expected performance for the SW
+wide filter (F070W, F090W, F115W, F150W, F200W) per-
+mutations at three different CMD locations: the blue HB
+(Teff ∼ 7000 K), the MSTO (Teff ∼ 6000 K), and the lower
+MS (∼ 0.2 M⊙; Teff ∼ 4000 K). At each point, we used the
+pre-commissioning JWST ETC (v1.1.1) to compute the ex-
+posure time required to reach a SNR= 10 for the “scene” in
+the ETC.
+The best performing filters for our areas of consideration
+are F090W, F115W, and F150W. They all exhibit compa-
+rable performance at the HB and MSTO. However, F090W
+requires 2.5 times more exposure time to achieve the same
+SNR for a 0.2 M⊙ star as either F115W or F150W. Never-
+theless, we opted for F090W over F115W because compared
+to F115W−F150W, F090W−F150W provides superior color
+information for most stars and F090W has the potential for
+higher angular resolution (if dithered appropriately), which
+is critical for astrometry.
+Finally, the similarity between
+F090W and HST/F814W (or Johnson I-band) provides use-
+ful features such as matching catalogs between facilities and
+TRGB distance determinations (e.g., McQuinn et al. 2019a).
+F070W and F200W provide the largest color baseline, but
+each filter is less sensitive to stars far from their effective
+wavelength. For example, F070W required 4 times more in-
+tegration time for a 0.2 M⊙ star than the next bluest filter,
+F090W. F200W requires twice as much exposure time for a
+HB star than F150W.
+We opted to use the same F090W−F150W filter combi-
+nation for all targets to provide for an empirical comparison
+between the GC and UFD (e.g., Brown et al. 2012) and for
+good sampling of the oldest MSTO in the distant dwarf. We
+considered more than two SW filters, but the cost of acquir-
+ing extra data outweighed the scientific utility. We selected
+simultaneously observed LW filters on a per target basis, as
+they enable secondary science unique to each object. Finally,
+we selected F090W and F150W for parallel NIRISS imaging
+for consistency with NIRCam.
+
+6
+WEISZ ET AL.
+Figure 1. The locations of our NIRcam and NIRISS observations (plotted in red) for each ERS target, overplotted on a Pan-STARRS optical
+image. The orange dotted lines indicate: (a) 2 and 5 half-light radii (rh), (b) 1 and 2 rh and (c) 1 rh of each target. We show additional
+pointings for each system: (a) M92: select HST optical (green; HST-GO-10775; Sarajedini et al. 2007) and IR (pink; HST-GO-11664; e.g.,
+Brown et al. 2010). (b) Draco II: HST/ACS optical data (HST-GO-14734; PI Kallivayalil) are shown in green. (c) WLM: An exhaustive,
+though not complete set of HST observations including HST/WFPC2 UV and optical imaging in blue (HST-GO-11079; Bianchi et al. 2012),
+HST/WFC3 UVIS UV imaging in purple (HST-GO-15275 PI Gilbert), HST/ACS and HST/UVIS optical imaging in green (HST-GO-13768;
+PI Weisz, Albers et al. 2019), and HST/WFC3 IR imaging in pink (HST-GO-16162, PI Boyer). We opted not to undertake large dithers to fill
+the NIRCam chip and module gaps which would have substantially increased the program time while only marginally enhancing our science
+goals.
+We emphasize that while our filter combinations represent
+a good compromise across the CMD for our program goals,
+they may not be optimal for all science cases. We encourage
+exploration tailored to a program’s particular science aims.
+3.2. Target Selection
+We selected targets by first considering all known GCs in
+the MW (Harris 2010) and galaxies within ∼ 1 Mpc (Mc-
+Connachie 2012), including updates to both catalogs and
+discoveries through 2017 (e.g., Laevens et al. 2014, 2015a;
+Bechtol et al. 2015; Drlica-Wagner et al. 2015; Koposov et al.
+2015). The limiting distance was selected to ensure we could
+reach the oldest MSTO with SNR= 10 in the most distant
+system in a reasonable amount of time based on previous ex-
+perience with HST (e.g., Cole et al. 2014; Albers et al. 2019)
+and results from the JWST exposure time calculator (ETC).
+We required that each target have extensive HST imag-
+ing (e.g., to enable combined HST and JWST proper mo-
+tion studies, create panchromatic stellar catalogs) and have
+a good sampling of ground-based spectra (e.g., for full phase
+space information, comparing stellar properties from spec-
+tra and photometry, incorporating stellar abundance patterns
+into various analyses).
+We then identified a minimum set of targets that could
+be used to achieve our science and technical goals: one
+MW GC, one UFD, and one more distant star-forming dwarf
+galaxy. We then sought to maximize observational efficiency
+by focusing on some of the nearest examples of these classes.
+We eliminated targets that were not visible during the nomi-
+nal ERS window.
+This selection process yielded three targets: MW GC M92,
+MW satellite UFD Draco II, and star-forming dwarf galaxy
+WLM. Basic observational characteristics of these targets are
+Table 1. Basic observational properties of the three ERS targets.
+Properties for M92 have been taken from the updated MW GC cat-
+alog of Harris (2010), while those of Draco II and WLM are from
+the updated LG galaxy catalog of McConnachie (2012). Note that
+µ0 is the effective surface brightness and rh is the half-light radius.
+M92
+Draco II
+WLM
+RA (J2000)
+17h17m07.27s
+15h52m47.60s
+00h01m58.16s
+Dec (J2000)
++43d08m11.5s
++64d33m55.0s
+−15d27m39.34s
+MV (mag)
+−8.2
+−0.8
+−14.2
+E(B-V) (mag)
+0.02
+0.01
+0.03
+(m − M)0 (mag)
+14.6
+16.9
+24.9
+µ0 (mag arcsec−2)
+15.5
+28.1
+24.8
+rh (′)
+1.0
+2.7
+7.8
+listed in Table 1. We detail the observational strategy for each
+target in the following sections.
+3.3. M92
+M92 (NGC 6341) is a well-studied, metal-poor GC in the
+MW that is often used as a benchmark for extragalactic stel-
+lar population studies and for photometric calibration (e.g.,
+to verify zero points; e.g., Dalcanton et al. 2009; Brown et al.
+2014; Gallart et al. 2015). Imaging this system satisfies sev-
+eral science and technical goals including GC ages, individ-
+ual star proper motions, the present day mass function, test-
+ing DOLPHOT over a large dynamic range of stellar bright-
+ness and spatially varying stellar density, and gauging the ef-
+fects of bright saturated stars on the photometric process.
+
+(a) M92
+(b) Draco II
+(c) WLM
+HST/AWFPC2
+HST/ACS
+HST/ACS
+UWST
+HST/ACS
+JWST
+JWST
+HST/WFC3 IR
+HST/WFC3UVIS
+'NIRCam & NIRISS
+HSTWFC3 UVIS
+NIRCam & NIRISS
+HST/WFC3IR
+NIRCam & NIRISSJWST RESOLVED STELLAR POPULATIONS I
+7
+Figure 2. NIRCam color composite images for our 3 ERS targets.
+In each RGB image, F090W was used as the blue channel, F150W as the
+green and a combination of the two LW filters as the red channel.
+
+(a) M92
+(b) Draco il
+(c) WLM8
+WEISZ ET AL.
+As illustrated in Figure 1, we placed the NIRCam field
+near the center of M92, with the aim of maximizing NIRCam
+spatial overlap with a wealth of multi-band HST imaging of
+M92. The parallel NIRISS field is located at ∼ 5 half-light
+radii. We constrained the orientation such that the NIRISS
+field had a modest probability of overlapping at least some
+HST data in the outer regions. However, orientations allowed
+by the final ERS window did not result in overlap between
+NIRISS and HST imaging.
+We chose the F090W, F150W, F277W, and F444W for
+our NIRCam imaging of M92.
+We selected F277W and
+F444W for their broad scientific utility including studying
+the lower MS kink at long wavelengths (e.g., Sarajedini et al.
+2009; Bono et al. 2010), searching for dust production at
+low-metallicities (e.g., Boyer et al. 2006; Boyer 2010), and
+exploring multiple populations in the IR (e.g., Milone et al.
+2012, 2014; Correnti et al. 2016; Milone et al. 2017).
+We aimed to reach SNR= 10 at 0.1 M⊙ in F090W and
+F150W, which we estimate to be mF 090W ≈ 26 (MF 090W ≈
++11.4) and mF 150W ≈ 25.8 (MF 150W ≈ +11.2) based on
+the MIST isochrones and the characteristics of M92 listed in
+Table 1. The 2017 versions of the ETC (v1.1.1) and APT
+(v25.1.1) yielded exposure times of 1288s in each filter. The
+anticipated SNRs in the LW filters at mF 090W ≈ 26 were
+21.1 in F277W and 6.4 in F444W. We estimated NIRISS
+imaging to be marginally shallower than NIRCam with in-
+tegration times of 1074s in each of F090W and F150W. The
+difference in duration is set by facility overheads as calcu-
+lated by APT.
+3.4. Draco II
+At ∼20 kpc, Draco II is one of the nearest examples of a
+MW satellite (Laevens et al. 2015b; Longeard et al. 2018).
+At the time of program design, Draco II was designated as a
+UFD (i.e., it has dark matter; Willman & Strader 2012; Mar-
+tin et al. 2016). In the interim, there has been some debate
+in the literature over its status as a UFD vs GC (e.g., whether
+or not it has dark matter, a metallicity spread, the slope of its
+sub-Solar stellar mass function; e.g., Longeard et al. 2018;
+Baumgardt et al. 2022), an issue our JWST data should help
+resolve.
+Draco II’s close proximity provides for efficient JWST
+imaging that reaches far down the lower main sequence
+(≲ 0.2 M⊙). Our deep Draco II data satisfies several of our
+science goals including measuring the low-mass stellar IMF
+(which can shed insight into its status as a GC or UFD), de-
+termining the SFH of an ancient sparsely populated system,
+measuring proper motions of individual faint stars using HST
+and JWST, and exploring our ability to distinguish between
+faint stars and unresolved background galaxies.
+We placed the NIRCam field on the center of Draco II. The
+field overlaps with archival HST imaging obtained in March
+2017 (GO-14734; PI N. Kallivayalil), Keck spectroscopy,
+and is well-matched to the half-light radius. To maximize
+scheduling opportunities, we did not constrain the orienta-
+tion. The NIRISS field is unlikely to contain many Draco II
+member stars, so its exact placement was not crucial. The
+primary use of the NIRISS data will be to aid with modeling
+contamination (e.g., foreground stars and background galax-
+ies) in the F090W−F150W CMD, e.g., as part of measuring
+the low-mass IMF.
+We selected the F090W, F150W, F360M, and F480M for
+our NIRCam imaging of Draco II. The rationale for the two
+SW filters is described in §3.3. The LW filters are located
+near metallicity sensitive molecular features in the mid-IR,
+and they may be suitable for measuring photometric metal-
+licities of metal-poor stars (e.g., Schlaufman & Casey 2014;
+Casey & Schlaufman 2015) similar to what is possible in
+the optical using, for example, the Calcium H&K lines (e.g.,
+Starkenburg et al. 2017b; Fu et al. 2022). For NIRISS, we
+used the F090W and F150W filters.
+Our target depth is set by low-mass IMF science. Tightly
+constraining the low-mass IMF in Draco II requires reaching
+stars ≲ 0.2 M⊙ (e.g., El-Badry et al. 2017) with SNR= 10
+in F090W and F150W. Using the MIST stellar models and
+the observational properites of Draco II listed in Table 1, our
+target depths are SNR∼ 10 at mF 090W = 27 (MF 090W =
++10.3) and mF 150W = 26.8 (MF 150W = +10.1). Using
+the 2017 versions of the ETC (v1.1.1) and APT (v25.1.1)
+we found exposure times of 12798s in F090W and 6399s
+in F150W would each these depths. We opted on SW/LW
+combinations of F090W/F480M and F150W/F360M, which
+provided for SNRs of ∼ 7 (F360M) and ∼ 4 (F480M)
+at F150W=25. We estimated the NIRISS imaging to have
+12540s in F090W and 6270s in F150W, which will result in
+marginally shallower CMDs than NIRCam.
+3.5. WLM
+WLM (Wolf 1909; Melotte 1926) is a metal-poor ([Fe/H]=
+−1.2; Leaman et al. 2009) star-forming dwarf galaxy at
+∼ 0.9 Mpc (e.g., Albers et al. 2019). Though slightly closer
+objects of this class exist (e.g., IC 1613), WLM is the nearest
+example of a low-metallicity environment in which resolved
+CO clouds have been detected (Rubio et al. 2015). It has a
+sufficiently high star formation rate in the past several Gyr
+that it should host a sizable population of AGB stars (e.g.,
+Dolphin 2000; Weisz et al. 2014a; McQuinn et al. 2017b;
+Albers et al. 2019). In terms of science, our JWST imaging
+of WLM will allow us to measure its SFH from the ancient
+MSTO and compare it to the existing HST-based SFH, mea-
+sure its bulk PM using archival HST imaging, explore the
+stellar populations associated with it CO clouds, construct
+parsec-scale extinction using IR-only techniques and com-
+bined UV-optical-IR HST and JWST stellar SEDs (e.g., Dal-
+canton et al. 2015; Gordon et al. 2016). On the technical side,
+our observations of WLM allow us to test DOLPHOT in a
+regime of faint, crowded stars, that is typical of more distant
+systems, and also test its capabilities for simultaneously mea-
+suring stellar photometry across facilities (HST and JWST)
+and instruments (WFPC2, ACS, UVIS, NIRCam).
+We required that one module of the NIRCam observa-
+tions overlap UV-optical-IR HST observations as well as the
+ALMA-detected CO clouds (Rubio et al. 2015). The other
+NIRCam module overlaps with the deep optical HST/ACS
+
+JWST RESOLVED STELLAR POPULATIONS I
+9
+imaging presented in Albers et al. (2019). We enforced this
+configuration by requiring orientations of 70–110◦ or 250–
+290◦. This orientation placed the NIRISS field in the stellar
+halo of WLM, providing an expansion on the areal cover-
+age to build on the population gradient studies in WLM (e.g.,
+Leaman et al. 2013; Albers et al. 2019). We used artificial
+star tests associated with the deep HST imaging of Albers
+et al. (2019) to ensure that our NIRCam observations would
+only be modestly affected by stellar crowding.
+We selected the F090W, F150W, F250M, and F430M for
+our NIRCam imaging of WLM. As discussed in §3.3, the
+SW filter combination F090W−F150W is likely to be widely
+used for SFHs measured from the ancient MSTO. Medium
+band filters in the near-IR have proven remarkably efficient
+for photometric identification and classification of AGB stars
+(e.g., Boyer et al. 2017). Our simulations, based on these
+studies, suggest that the F250M and F430M filters should
+work well for similar science. For NIRISS, we selected the
+F090W and F150W filters.
+In the optical, measuring a well-constrained SFH for dis-
+tant dwarf galaxies requires a CMD that reaches a SNR∼5–
+10 at the oldest MSTO (e.g., Cole et al. 2007; Monelli et al.
+2010; Cole et al. 2014; Skillman et al. 2014; Gallart et al.
+2015; Skillman et al. 2017; Albers et al. 2019). Using the
+MIST stellar models, we find that the ancient MSTO for
+metal-poor stellar population has MF 090W
+= +3.4 and
+MF 150W = +3.1. Using the parameters for WLM in Table
+1 and v1.1.1 of the ETC, we estimated that 33241s in F090W
+and 18724s in F150W would reach the required depths of
+mF 090W = 28.3 and mF 150W = 28.1 with SNR=10 in each
+filter.
+3.6. Program Updates Since 2017
+Our program has only had minor changes since it was
+first approved in November 2017.
+In 2021, we changed
+the DEEP2 readout patterns for WLM and Draco II to
+MEDIUM8, following updated advice from a STScI tech-
+nical review. The number of groups and integrations was
+changed accordingly. In 2021, STScI increased our allocated
+time from 27.35 hr to 27.5 hr to reflect updated overhead ac-
+counting.
+Following commissioning in spring 2022, STScI staff
+changed the range of aperture PA ranges of Draco II from
+unconstrained to be have a value of 47–118◦, 143–280◦, or
+354–24◦ in order to avoid the “claws” feature3 which is due
+to stray light from the bright source that is outside the field
+of view of NIRCam (jdo 2016; Rigby et al. 2022).
+4. OBSERVATIONS
+We acquired NIRCam and NIRISS imaging of our three
+targets in July and August of 2022. Figure 1 shows the NIR-
+Cam and NIRISS footprints for each of our targets overlaid
+on ground-based images and Table 1 lists our observational
+3 urlhttps://jwst-docs.stsci.edu/jwst-near-infrared-camera/nircam-features-
+and-caveats/nircam-claws-and-wisps
+configurations. Full details on implementation can be viewed
+by retrieving proposal 1334 in the Astronomer’s Proposal
+Tool (APT)4.
+We followed the same fundamental observing strategy for
+all three targets. For each target, the primary instrument,
+NIRCam, imaged a single central field in SW and LW fil-
+ters as described in §4. NIRISS obtained imaging in parallel
+in the two filters F090W and F150W.
+All observations were taken with the 4 point subpixel
+dither pattern 4-POINT-MEDIUM-WITH-NIRISS. This
+pattern ensured adequate PSF sampling for both cameras, as
+well as improved rejection of cosmic rays, hot pixels, etc.
+We opted against primary dithers. The main advantage of
+the primary dithers is to fill the gaps between the dectectors
+and/or modules. However, the inter-gap imaging is generally
+shallower than the rest of the data and requires more JWST
+time to acquire. For our particular science cases, including
+primary dithers added a modest amount of time to the pro-
+gram but would not substantially enhance the data for our
+main science goals. Other science cases (e.g., covering a
+large region such as PHAT did) may benefit from primary
+dithers and filling gaps.
+For M92, we used the SHALLOW4 readout pattern for NIR-
+Cam and the NIS readout pattern for NIRISS. The orienta-
+tion was constrained to an Aperture PA range of 156 to 226◦
+in order to maximize the possibility of overlap between the
+NIRISS field and existing HST imaging, while also allow-
+ing for reasonable schedulability very early in the lifetime
+of JWST. The central location of the NIRCam field ensures
+it will overlap with existing HST imaging. Including over-
+heads, the total time charged for observing M92 was 7037s.
+The ratio of science to charged time was ∼ 0.35 (counting the
+primary NIRCam imaging only) or ∼ 0.7 (for both NIRCam
+and NIRISS imaging). The data volume was well within the
+allowable range.
+For Draco II, we used the MEDIUM8 readout pattern for
+NIRCam and the NIS readout pattern for NIRISS. The NIR-
+Cam field was centered on the galaxy and will largely overlap
+with existing HST imaging and Keck spectroscopic data. The
+large angular separation of the NIRISS and NIRCam fields
+compared to the size of Draco II means that the NIRISS
+field will contain few, if any, bona fide members of Draco II.
+Including overheads, the total time charged for observing
+Draco II was 24539s. The ratio of science to charged time
+was ∼ 0.71 (counting the primary NIRCam imaging only) or
+∼ 1.36 (for both NIRCam and NIRISS imaging). The data
+volume was well within the allowable range.
+For WLM, we used the MEDIUM8 readout pattern for NIR-
+Cam and the NIS readout pattern for NIRISS. The NIRCam
+field was placed in the center of WLM in order to overlap
+the low-metallicity molecular clouds discovered by ALMA
+as well as deep archival optical HST imaging. Subsequent
+UV and near-IR HST imaging of WLM obtained by mem-
+4 apt.stsci.edu
+
+10
+WEISZ ET AL.
+Table 2. A summary of our JWST ERS observations taken in 2022.
+Target
+Date
+Camera
+Filter
+texp [s]
+Groups
+Integrations
+Dithers
+M92
+June 20–21
+NIRCam
+F090W/F277W
+1245.465
+6
+1
+4
+NIRCam
+F150W/F444W
+1245.465
+6
+1
+4
+NIRISS
+F090W
+1245.465
+7
+1
+4
+NIRISS
+F150W
+1245.465
+7
+1
+4
+Draco II
+July 3
+NIRCam
+F090W/F480M
+11810.447
+7
+4
+4
+NIRCam
+F150W/F360M
+5883.75
+7
+2
+4
+NIRISS
+F090W
+11123.294
+9
+7
+4
+NIRISS
+F150W
+5883.75
+10
+3
+4
+WLM
+July 23–24
+NIRCam
+F090W/F430M
+30492.427
+8
+9
+4
+NIRCam
+F150W/F250M
+23706.788
+8
+7
+4
+NIRISS
+F090W
+26670.137
+17
+9
+4
+NIRISS
+F150W
+19841.551
+19
+6
+4
+bers of our team were placed to maximize the chances of
+overlap with our JWST observations of WLM.
+Including overheads, the total time charged for observing
+WLM was 66884s.
+The ratio of science to charged time
+was ∼ 0.8 (counting the primary NIRCam imaging only) or
+∼ 1.50 (for both NIRCam and NIRISS imaging). The large
+data volume (19.962 GB) for WLM generated a “Data Ex-
+cess over Lower Threshold” warning in APT. Though this
+level of warning is only a recommendation to mitigate data
+volume excess, we nevertheless consulted with STScI about
+mitigation strategies. However, we were unable to identify
+a way to reduce data volume without compromising the sci-
+ence goals and no changes were made. We caution that even
+longer integrations (e.g., that may be needed for deep CMDs
+outside the LG) may require careful planning to avoid data
+volume limitations.
+Because our WLM observations span several continuous
+hours (Table 2), our observations of WLM should allow us
+to recover the light curves of short period variables (e.g., RR
+Lyrae). Generation of the light curves requires performing
+photometry on calibrated images at each integration. At the
+time of this paper’s writing, due to the large data volume,
+the STScI JWST reduction pipeline generates integration-
+level calibrated images only when the time series observa-
+tion (TSO) mode is used. However, we were unable to use
+the TSO mode of JWST as it does not permit dithers nor
+parallel observations. Instead, producing the necessary im-
+ages to generate light curves requires running the reduction
+pipeline locally and creating custom time series analysis soft-
+ware, which is beyond the scope of this paper.
+In total, all of our science observations with NIRCam total
+20.45 hr and observations with NIRISS total 17.85 hr, indi-
+cating a fairly high ratio of science-to-charged time.
+5. PHOTOMETRY
+We perform photometric reductions of our observations us-
+ing the newly developed NIRCam module for DOLPHOT.
+The detailed description of how DOLPHOT works is well-
+documented in the literature (e.g., Dolphin 2000; Dalcanton
+et al. 2012a; Dolphin 2016) and the functionality of the new
+NIRCam module are summarized in § 2.2. A more detailed
+write up of the DOLPHOT NIRCam and NIRISS modules
+are the subjects of an upcoming paper from our team. For this
+paper, we only focus on the NIRCam data. The WebbPSF5
+NIRISS PSF models appear to concentrate light significantly
+more than what we have observed in the ERS images. Con-
+sequently, our NIRISS photometry is not yet reliable and fur-
+ther updates await improvements to the PSF models.
+We first acquired all images from MAST. Per their
+FITS headers,
+versions of all JWST images used in
+this paper have the following JWST pipeline version-
+ing information CAL VER=1.7.2,CRDS VER=11.16.11, and
+CRDS CTX=jwst p1009.pmap.
+Next,
+we
+performed
+DOLPHOT reductions on the level 2b crf frames and use
+the level 3 i2d F150W drizzled image as the astrometric
+reference frame. The use of this reference image ensures
+excellent internal alignment of our image stack and ties the
+absolute astrometry to Gaia DR2. We perform photomery
+in all four bands simultaneously. Since there is no spatial
+overlap between the footprint of the two NIRCam modules,
+we photometer them independently and merge the catalogs a
+posteriori.
+We have not included Frame 0 in the photomeric reduc-
+tions for this paper. As of this paper’s writing, the JWST
+pipeline does not automatically provide Frame 0, requiring
+these images be generated locally. Even when created with
+our own execution of the JWST pipeline, we have yet to
+fully resolve how 1/f noise issue (e.g., Schlawin et al. 2020;
+Bagley et al. 2022) can be resolved in a manner that ensures
+self-consistent photometry with DOLPHOT for all frames.
+This issue will be addressed in more detail in our forthcom-
+ing photometry paper. In the meantime, without Frame 0
+data, our photometry saturates at fainter magnitudes that we
+5 https://webbpsf.readthedocs.io/en/latest/
+
+JWST RESOLVED STELLAR POPULATIONS I
+11
+Figure 3. CMDs from all 8 NIRCam chips for each of our targets. Panel (a): M92 extends from our bright saturation limit without Frame 0
+data (F090W∼ 19) just below the MSTO and reaches just fainter than the hydrogen burning limit (F090W > 28.2). The inflection point of
+the MS at F090W ∼ 21 is the MS kink. Panel (b): The CMD of Draco II includes the MS kink at F090W ∼ 23 and reaches the bottom
+of the stellar sequence provided by the standard PARSEC models (0.09 M⊙). Panel (c): The CMD of WLM displays a wide variety of stellar
+sequences that span a range of ages including young stars (e.g., the upper main sequence, red core helium burning stars), intermediate age stars
+(e.g., AGB and RGB stars), and ancient stars that span a wide range of colors and magnitude (e.g., RGB, SGB, oMSTO, lower MS). This CMD
+of WLM (SNR=10 at MF 090W = +4.6) is the deepest taken for a galaxy outside the immediate vicinity of the MW.
+expect to recover once the Frame 0 data in included in our
+DOLPHOT runs.
+For our photometric reductions, we adopt the DOLPHOT
+parameter setup recommended by PHAT (Williams et al.
+2014). We use the parameters recommend for ACS from
+PHAT for the SW images and the WFC3/IR parameters for
+the LW images. Subsequent data releases will make use of
+a set of parameters that we are tailoring to NIRCam and
+NIRISS observations of resolved stars.
+We make use of NIRCam PSF models generated with
+WebbPSF (Perrin et al. 2014). We use the Optical Path De-
+lay maps from July 24th, 2022 (the best matching file to our
+WLM observation epoch). Inspection of JWST’s wavefront
+field at the epochs of our three observations show little vari-
+ation in the optical performance of the telescope, justifying
+our choice of a common PSF library. We are currently work-
+ing in quantifying the full effect of JWST time-dependent
+PSF variations on DOLPHOT photometry.
+The final products of the DOLPHOT photometric run are
+catalogs with positions, VEGAmag magnitudes (calibrated
+Table 3. Quality-metric criteria used to cull our DOLPHOT photo-
+metric catalogs.
+Band
+SNR
+Sharp2
+Crowd
+Flag
+Object Type
+F090W
+≥ 4
+≤ 0.01
+≤ 0.5
+≤ 2
+≤ 1
+F150W
+≥ 4
+≤ 0.01
+≤ 0.5
+≤ 2
+≤ 1
+using the latests NIRCam zero-points6), uncertainties based
+on photon noise, and a set of quality metrics related to the
+goodness of point-source photometry (e.g., SNR, χ2 of the
+PSF fit, angular extent of the source, crowding level). At this
+stage, metrics such as the photometric error and the SNR are
+based on the Poissonian treatment of photon noise. While in
+many cases this is sufficient for rough estimation, there are
+6 https://jwst-docs.stsci.edu/jwst-near-infrared-camera/
+nircam-performance/nircam-absolute-flux-calibration-and-zeropoints
+
+(a) M92
+(b) Draco Il
+(c) WLM
+18
+20 
+SNR=500
+22 
+SNR=500
+F090W
+100
+24
+SNR=500
+50
+100
+26
+100
+10
+50
+28
+10
+30 
+0.0
+0.5
+1.0
+1.5
+0.0
+0.5
+1.0
+1.5
+-0.5
+0.0
+0.5
+1.0
+1.5
+F090W-F150W
+F090W-F150W
+F090W-F150W12
+WEISZ ET AL.
+caveats associated with this approximation, especially when
+measuring stars in very crowded fields, or close to the limit-
+ing magnitude. We will provide a more thorough discussion
+on SNRs estimation in our upcoming JWST DOLPHOT pho-
+tometry paper.
+The catalogs provided by DOLPHOT are subsequently in-
+spected and culled to remove contaminants (e.g., artifacts,
+cosmic rays, extended sources) while aiming to retain the
+largest number of bona fide stars. We identify a set of quality-
+metric cuts, listed in Table 3, that provide a reasonable trade-
+off between completeness and purity of the stellar sample;
+though for this initial presentation we erred on the side of pu-
+rity. The selection criteria need to be satisfied in the F090W
+and F150W bands simultaneously.
+We only use the SW
+bands as they are shared by all three targets, allowing a com-
+mon set of culling criteria. Our initial exploration suggests
+that the LW photometry may improve star–galaxy separation
+at faint magnitude. This important topic is being further in-
+vestigated by members of our team.
+Full characterization of uncertainties in resolved stel-
+lar populations studies required artificial star tests (ASTs).
+ASTs consist of adding mock stars of known properties into
+each frame and recovering them using the same photometric
+procedure that is applied to the real data. For the purposes
+of this paper focused on survey description, we have not in-
+cluded results of the ASTs. The large data volume and multi-
+filter nature of the data make running ASTs computationally
+challenging. We will present full AST results and analyses in
+the upcoming JWST DOLPHOT photometry paper.
+6. DISCUSSION
+6.1. Color-Magnitude Diagrams
+Figure 3 shows the NIRCam SW CMDs for all three tar-
+gets over the same magnitude range. The juxtaposition of
+these CMDs illustrates the quality and diversity of science
+possibilities provided by our program. In each panel, we
+overplot select SNRs reported by DOLPHOT. As discussed
+in §5, these SNRs are solely based on photon noise and do
+not account for the effects of crowding and incompletness.
+We discuss SNRs further in §6.2.
+The SNRs for each target are remarkable, with SNRs rang-
+ing from 500 near the MS kink to 10 for the lowest mass stars
+in M92 and Draco II. WLM has a photon noise-based SNR
+of ∼ 50 at the oldest MSTO, making it the highest fidelity
+resolved stars observation of a distant dwarf in existence. We
+now discuss the multi-band CMDs for each of our targets in
+more detail.
+Figures 4–6 show illustrative NIRCam CMDs in a selec-
+tion of filter combinations for each ERS target. In all cases,
+we apply the catalog culling parameters described in §5 and
+listed in Table 3.
+For guidance, we overplot a selection of stellar isochrones
+from the PARSEC v1.2S stellar libraries (Bressan et al. 2012;
+Chen et al. 2015). These models span the full range of metal-
+licities and ages needed to characterize our datasets.
+We
+have adjusted these isochrones to the distances and redden-
+ings listed in Table 1.
+6.1.1. M92
+Figure 4 shows select NIRCam CMDs for M92, along with
+select HST-based CMDs for comparison. The HST CMDs
+were reduced using DOLPHOT and the parameters recom-
+mended in Williams et al. (2014).
+We overplot select PARSEC isochrones at a fixed age of
+13 Gyr with varying metallicities, which we discuss below.
+Though stars brighter than F090W ∼ 19 are omitted due to
+saturation effects (see §5), key CMD features for faint, low-
+mass stars are clearly visible. Notably, the MS kink, which
+is due to opacity effects in M dwarfs, the region in which
+is in panels (a) - (c) at F090W ∼ 21 and F150W ∼ 20.
+The MS kink exhibits the sharpest inflection point in the
+F150W−F277W filter combination. The kink is much less
+pronounced in the F277W−F444W CMD (panel d). This
+is partially due to the lower SNRs of the LW observations
+as well as the LW filters being far into the Rayleigh–Jeans
+tail of the stars’ spectral energy distributions (SEDs), which
+makes them only weakly sensitive to stellar temperature, thus
+resulting in similar colors and a less obvious kink.
+Our NIRCam observations have produced the deepest
+CMD of M92 to date. The data extend fainter than the lowest
+stellar mass (0.09 M⊙) available from the standard PARSEC
+stellar library in the SW filters and to slightly higher masses
+(0.12 M⊙) in the LW filters.
+The faintest objects in the F090W−F150W CMD fall into
+the bright end of the expected hydrogen burning sequence.
+The exact mass at which a star-like object cannot sustain
+hydrogen fusion has long been debated (e.g., Hayashi &
+Nakano 1963; Kumar 1963; Chabrier et al. 2000). For this
+analysis, we computed custom PARSEC models with a mass
+resolution of 0.002 M⊙ and find that that the minimum mass
+for hydrogen burning is 0.078 ≤ M < 0.08 M⊙ for a metal-
+licity of [Fe/H]= −1.7 dex and an age of 13 Gyr. This trans-
+lates to a magnitude range of 28.2 < mF 090W ≤ 29.5 mag
+in M92. This depth is quite remarkable considering our ob-
+servations only consistent of ∼ 1050s in each filter. In com-
+parison, the faintest stars in the HST WFC3/IR CMD (panel
+e) are a few magnitudes brighter despite ∼ 1200s of inte-
+gration time in each filter.
+To date, comparably deep IR
+studies of metal-poor, extremely low-mass stars with HST
+have been limited to the nearest GCs (e.g., M4; Dieball et al.
+2016, 2019). As our M92 data shows, the superior sensitivity
+of NIRCam will make such studies possible throughout the
+MW.
+The PARSEC models over-plotted in Figure 4 are nom-
+inally higher than the metallicity of M92 ([Fe/H] =
+−2.23 dex) derived from high-resolution APOGEE spec-
+troscopy (M´esz´aros et al. 2020). This discrepancy is because
+the current version of the PARSEC models are Solar-scaled,
+whereas M92 is highly α-enhanced with [α/Fe] ∼ 0.5 dex.
+Well-established corrections can be applied to match Solar-
+scaled models with α-enhanced populations (e.g., Salaris
+et al. 1993). For M92, the corrective factor for the PAR-
+SEC models results in values of [Fe/H]
+∼ 0.5 dex higher
+than derived from spectroscopy. PARSEC models with α-
+
+JWST RESOLVED STELLAR POPULATIONS I
+13
+Figure 4. Select NIRCam CMDs of M92, along with HST WFC3/IR (HST-GO-11664) and ACS/WFC (GO-9453, GO-10775, GO-12116,
+GO-16298) CMDS. Overploted are Solar-scaled PARSEC stellar models at fixed age (13 Gyr) over a select range of [Fe/H] values, which
+have been corrected to match the known α-enhancement of M92 (M´esz´aros et al. 2020). The SW NIRCam CMD extends from our saturation
+limit (F090W
+∼ 19) to below the lowest-mass star from the standard PARSEC models (M = 0.09 M⊙) and into the expected hydrogen
+burning limit regime (0.078 ≤ M < 0.08 M⊙). The LW CMDs reach slightly higher limits (M = 0.12 M⊙). These are among the deepest
+CMDs of a GC in existence and highlight how JWST will easily enable the study of prominent features for low-mass stars such as the MS-kink
+and lowest-mass stars. While the stellar models are in excellent agreement with the more luminous stellar evolutionary phases (e.g., RGB,
+MSTO) in the HST CMDs, they are ∼ 0.05 mag too blue for lower-mass stars in the SW JWST filters. This offset could be due to the complex
+atmospheres of low-mass stars.
+enhancements are under construction and will mitigate the
+need to apply such corrections.
+Overall,
+the
+PARSEC
+models
+are
+in
+reasonably
+good agreement with the NIRCam CMDs.
+For the
+F150W−F444W and F277W−F444W CMDs, the models
+trace the locus of the data quite well.
+However, for the
+F090W−F150W and F150W−F277W CMDs, the models
+are systematically too blue by ∼ 0.05 mag. The source of
+this offset is not due to distance, reddening, age, or metal-
+licity as these same models are well-matched to the MSTO
+in the brighter HST CMDs (panels e and f of Figure 4). In
+general, stars above the MS kink are well-matched by the
+models in the LW NIRCam and HST filters, limiting the
+offsets to only SWs. One possible source of the offset is the
+presence of poorly modeled absorption features (e.g., TiO)
+in the atmospheres of very cool, low-mass stars. A detailed
+exploration of this offset is beyond the scope of this paper,
+but we note that deep JWST imaging of a larger set of GCs
+in several filters has the potential to help elucidate the exact
+nature of this issue.
+The NIRCam CMDs also exhibit scatter in color that is
+larger than photon noise and variations in age or metallicity.
+The most likely explanation is the presence of multiple chem-
+ically distinct populations.
+Like many luminous Galactic
+GCs, M92 exhibits multiple populations with distinct abun-
+dance patterns (e.g., M´esz´aros et al. 2020). Other GCs with
+deep CMDs and similar abundance patterns show a broad-
+ening of stellar sequences at and below the MS kink. This
+broadening has been attributed to the persistence of these
+chemically distinct sequences to the lowest stellar masses
+(e.g., Milone et al. 2012, 2014; Correnti et al. 2016; Milone
+et al. 2017), which is thought to be driven by oxygen varia-
+tions expressed via water lines (e.g., Dotter et al. 2015; Van-
+denBerg et al. 2022). Finally, we note that some of the scatter
+
+18
+[Fe/H]=-2.0 
+18
+(b)
+(a)
+(c)
+[Fe/H]=-1.7
+18
+20
+[Fe/H]=-1.4
+20
+22
+F090W
+L50W
+20
+.50W
+24
+22
+22
+26
+24
+24 
+28
+26
+26
+0.5
+1.0
+1.5
+0.00
+0.25
+0.50
+0.0
+0.5
+F090W-F150W
+F150W-F277W
+F150W-F444W
+10
+(d)
+(e) HST WFC3/IR
+(f) HST/ACS
+18
+10
+15
+277W
+20
+F160W
+814W
+15
+20
+20 
+24 
+25
+26
+25
+-0.25
+0.00
+0.25
+0.50
+0.00
+0.25
+0.50
+0
+1
+2
+3
+F277W-F444W
+F110W-F160W
+F475W-F814W14
+WEISZ ET AL.
+Figure 5. Select NIRCam CMDs of Draco II along with the deepest optical CMD which is based on HST/ACS imaging (GO-14734; panel
+b). Overploted are PARSEC stellar models at fixed age (13 Gyr) for the same [Fe/H] values selected for M92. The SW CMD extends to the
+lowest-mass stellar model (M = 0.09 M⊙), making it the deepest CMD of a MW satellite galaxy to date. The exquisite depth of our data
+indicate how JWST enables a variety of science including constraining the low-mass IMF and quantifying low-mass star features (e.g,. the MS
+kink and objects near the hydrogen burning limit) outside the MW.
+could be due to NIRCam zero point calibrations which will
+not be finalized with uncertainties until at least the end of
+Cycle 1 (Gordon et al. 2022).
+6.1.2. Draco II
+Figure 5 shows select NIRcam CMDs of Draco II, along
+with the deepest existing optical HST CMD for reference
+(GO-14734; PI N. Kallivayalil).
+Like M92, the brightest
+stars in Draco II suffer from saturation and are not included
+in our current photometric reduction. We include the same
+PARSEC isochrones as shown in M92 (Figure 4) as Draco II
+hosts a comparably ancient (13 Gy), metal-poor, and likely
+α-enhanced stellar population (e.g., Longeard et al. 2018; Si-
+mon 2019). The selected isochrones provide a good fit of the
+
+(a)
+[Fe/H]=-2.0
+(b) HST/ACS
+18
+18
+[Fe/H]=-1.7
+[Fe/H]=-1.4
+20
+20 
+F090W
+22
+F814W
+22
+24
+24
+26
+26
+28
+28
+30
+30
+0
+1
+2
+0
+1
+2
+F090W-F150W
+F606W-F814W
+(c)
+(d)
+18
+18
+20
+20
+50W
+22
+60M
+22
+5
+24
+3
+出
+26
+24
+28
+26
+30
+0
+1
+2
+-0.5
+0.0
+0.5
+F150W-F360M
+F360M-F480MJWST RESOLVED STELLAR POPULATIONS I
+15
+Figure 6. Select NIRCam CMDs of WLM along with the deepest available HST/ACS optical CMD from Albers et al. (2019). Overplotted
+are isochrones from the PARSEC Solar-scaled stellar models at a fixed metallicity of [Fe/H]= −1.2 dex for a variety of indicated ages. Panel
+(a) is the deepest CMD of an isolated dwarf galaxy to date, extending ∼ 1.5 mag below the oldest MSTO; it is deeper than the HST CMD
+despite ∼ 7000s less integration time. The isochrones indicate the variety of stellar ages present in WLM. The LW CMD shown in panel (d)
+extends ∼ 2 magnitudes below the red clump. Such deep CMDs show that JWST will provide for a variety of science at different cosmic epochs
+including exquisite lifetime SFHs, the study of evolved red stars, TRGB distances, and very young stars in galaxies outside the LG.
+MSTO in the HST CMD (panel b) suggesting the adopted pa-
+rameters (i.e., age, metallicity, distance, extinction) are rea-
+sonable, but as with M92, the models are slightly too blue in
+the NIRCam SW CMDs of Draco II.
+Our F090W−F150W CMD of Draco II is the deepest
+CMD of a galaxy outside the MW and has imaged the lowest-
+mass stars outside the MW (0.09 M⊙).
+Previously, the deepest CMD in an external galaxy was
+from Gennaro et al. (2018b), which used HST WFC3/IR data
+of MW UFD Coma Berenicies to study its low-mass IMF.
+From 32780s of integration time in each filter, their F110W
+and F160W photometry reached a usable low-mass limit of
+0.17 M⊙, whereas our data extend to 0.09 M⊙. The large dy-
+namic range of stellar masses in Draco II provides excellent
+leverage for a low-mass IMF measurement. Tight constraints
+
+20
+20
+(a)
+(b)
+HST/
+22
+ACS
+22
+F090W
+.4W
+24
+24
+13 Gyr
+5 Gyr
+0.5 Gyr
+1
+26
+26
+0.05 Gyr
+8
+F
+28
+28 
+30 
+30
+0.0
+1.0
+2.2
+-1
+0
+1
+2
+F090W-F150W
+F475W-F814W
+20
+18
+(c)
+(d)
+22
+20 
+F090W
+24
+F250M
+22
+26
+24
+28
+26
+30 -
+-1
+0
+1
+2
+-0.25
+0.00
+0.25
+0.50
+F090W-F250M
+F250M-F430M16
+WEISZ ET AL.
+on the IMF in Draco II could provide a new means of distin-
+guishing whether faint stellar systems are dark-matter domi-
+nated dwarf galaxies or GCs (e.g., Willman & Strader 2012;
+Baumgardt et al. 2022), as well as insight into star formation
+in extreme environments (e.g., Geha et al. 2013; Krumholz
+et al. 2019).
+The width of the lower MS is in excess of photometric
+noise.
+Metallicities derived from more luminous stars in
+Draco II suggest a spread of σ ∼ 0.5 dex (e.g., Li et al.
+2017; Longeard et al. 2018; Fu et al. 2022), which may con-
+tribute to this scatter. Background galaxies are also a source
+of contamination and likely contribute to the scatter, partic-
+ularly at the faintest magnitudes. The combination of very
+deep imaging and the sparsity of Draco II’s stellar popula-
+tion mean that background galaxies are a large source of con-
+tamination. Our preliminary investigations indicate that the
+multi-color NIRCam photometry may be efficient for star-
+galaxy separation (Warfield et al. in prep).
+The F150−F360M CMD (panel c) extends to a compara-
+bly low stellar mass as the SW CMD, albeit at lower SNR.
+The LW CMD (panel d) is much shallower; though the pri-
+mary purpose of these filters is to explore their potential as
+photometric metallicity indicators akin to the Calcium H &
+K filters being used in the optical (e.g., Starkenburg et al.
+2017a; Longeard et al. 2018; Fu et al. 2022).
+As with M92, the stellar isochrones provide a good quali-
+tative match to the data. The shape and magnitude of the MS
+kink appears to track the data well. However, as discussed in
+the context of M92 the models are modestly too blue com-
+pared to the data.
+6.1.3. WLM
+Figure 6 shows select NIRCam CMDs for WLM, along
+with the deepest optical CMD of WLM taken with HST. Due
+to its large distance, few stars in WLM are affected by satu-
+ration.
+The CMDs of WLM exhibit a wide variety of stellar se-
+quences that span a range of ages and phases of evolution.
+Examples include the young MS, the RGB and AGB, the HB,
+and the oldest MSTO. The relative positions of these features
+are generally similar to what is known for optical CMDs with
+subtle changes due to the shift to IR wavelengths (e.g., Dal-
+canton et al. 2012b; Williams et al. 2014; Gull et al. 2022).
+For example, the HB slopes to fainter values at bluer wave-
+lengths as hot blue HB stars are less luminous at IR wave-
+lengths. Similarly, red stars (e.g., RGB, AGB) become more
+luminous as the IR wavelengths are closer to their peak tem-
+peratures compared to optical wavelengths.
+Figure 6 shows that our NIRCam SW CMD of WLM is at
+∼ 1 mag deeper than the HST/ACS CMD despite similar in-
+tegration times (∼ 54200s for NIRCam versus ∼ 61400s for
+ACS). This increase owes primarily to the increased sensitiv-
+ity of JWST at these wavelengths. McQuinn et al. (in prep.)
+is deriving the SFHs from both datasets to quantify the capa-
+bilites of JWST imaging for detailed SFH determinations.
+More broadly, our CMDs of WLM are the deepest in ex-
+istence for an isolated dwarf galaxy. Prior to our program,
+HST/ACS observations of Leo A from Cole et al. (2007)
+extended to the lowest stellar masses in an galaxy outside
+the immediate vicinity of the MW. The HST observations of
+Leo A reach nearly as deep, but the larger distance of WLM
+(0.5 mag farther) means that our measurements actually ex-
+tend to less luminous stars on the MS.
+Though not as deep as the SW data, the LW is remark-
+ably deep for medium bands.
+The F090W−F250M data
+extends below the oldest MSTO, making it the deepest
+medium band data available for an isolated dwarf galaxy. The
+F250M−F430M (panel d) CMD extends well below the red
+clump. This data is expected to provide an excellent means of
+identifying AGB stars and helping constrain their underlying
+physics.
+The over-plotted isochrones provide a reasonable match to
+the data. In this case, we have selected a single metallic-
+ity of [Fe/H] = −1.2 dex matched to RGB spectroscopic
+abundances (Leaman et al. 2009), and plotted select ages that
+range from 50 Myr to 13 Gyr. Visually, the PARSEC models
+provide a good qualitative match to the data. The level of
+agreement will be formally quantified in McQuinn et al. (in
+prep.).
+6.2. Comparisons with the Exposure Time Calculator
+Our photometry provides an opportunity to gauge the ac-
+curacy of the NIRCam ETC in practice.
+In Figure 7, we consider the SNRs for stars in M92 as re-
+ported by DOLPHOT against what is reported by the NIR-
+Cam ETC. M92 is the best of our three targets for this ex-
+ploratory exercise as it is not particularly crowded and its
+CMD is well-populated. The lack of crowding means that
+the SNRs reported by DOLPHOT are a reasonable proxy for
+real noise, whereas ASTs are necessary to accurately assess
+the noise in even moderately crowded images.
+We compute SNRs as a function of F090W and F150W
+by only considering stars that pass a stricter version of the
+culling criteria listed in §5.
+Specifically, we require that
+each star has crowd ≤ 0.1, which eliminates all but the
+least crowded stars. We also only consider stars with 0.2 <
+F090W − F150W < 1.5, which isolates the MS in the SW
+filters and removes much of the contamination (e.g., back-
+ground galaxies, diffraction spike artifacts) from our analy-
+sis. Finally, we exclude the brightest stars as they may be af-
+fected by (partial) saturation. We specifically only consider
+stars fainter than F090W = 20 and 150W = 19 for their
+respective SNR calculations.
+From stars that pass these cuts, we compute the 50th, 16th,
+and 84th percentiles of the F090W and F150W SNR distri-
+butions in 0.25 mag bins over the entire magnitude ranges
+considered.
+For the expected SNRs, we use v2.0 of the JWST ETC to
+compute SNR as a function of F090W and F150W. In the
+ETC, the detector strategies are set to match our F090W and
+F150W observational set up for M92 as listed in Table 2 and
+described in §3.3. We verified that the integration times in
+the ETC are identical to what our program acquired.
+
+JWST RESOLVED STELLAR POPULATIONS I
+17
+Figure 7. Comparisons between the photon noise-based SNRs from the DOLPHOT F090W and F150W photometry of M92 (grey shaded
+region) and the expected SNRs from v2.0 of the NIRCam ETC (black lines). Over most of the dynamic ranges in magnitude the SNRs reported
+by DOLPHOT and the ETC are consistent within scatter (∼ 20%) of the data. Deviations at the bright end may be due to the presence of
+saturated pixels, while PSF shape, non-stellar objects (e.g., background galaxies), and incompleteness may contribute to a slight increase in the
+ratio at the faintest F150W magnitudes.
+For the ETC scene, we used a K5V star (Teff = 4250 K,
+log(g) = 4.5 dex) from the Phoenix stellar models. Though
+the stellar type varies over the color and luminosity range
+considered, we found that reasonable changes in the choice
+of stellar atmosphere only affected our findings at the ∼ 5%
+level. For simplicity, we adopted a single stellar atmosphere
+model for this calculation.
+We adopted an extinction of AV = 0.06 mag and a MW
+extinction curve. We set the background model to the central
+coordinates of M92 on June 20th, 2022, the date of our obser-
+vations. We computed the SNR in the F090W and F150Ws
+filter in 0.5 mag steps from 19 to 30 mag in each filter, renor-
+malizing after extinction was applied.
+The result is a smooth variation in SNR as a function of
+magnitude. We interpolated the results onto a finer magni-
+tude grid for clearer comparison with the DOLPHOT results.
+Interpolation errors are < 1%.
+To compute the SNR, we used the default aperture pho-
+tometry setup in the NIRCam ETC. Specifically, this uses
+an aperture radius of 0.1′′ and performs background subtrac-
+tion using an annulus 0.22 to 0.4′′ from the source. For both
+F090W and F150W, this radius within the aperture radius
+range of 2-3× the PSF FWHM specified in JDOX7 as rec-
+ommended by the JWST help desk. We will explore more
+filter SNRs and variations in the ETC photometric set p in a
+future paper.
+Figure 7 shows a comparison between the SNRs reported
+by DOLPHOT (grey shaded regions) and the NIRCam ETC
+(black lines) as a function of F090W and F150W magnitude.
+The bottom panels show the ratios of the DOLPHOT and
+ETC SNRs. Both visually and quantitatively, the expected
+ETCs agree quite well. For most of the magnitude ranges,
+the DOLPHOT and ETC SNRs agree within ∼ 20%, which is
+within the bounds of our uncertainty range. The small struc-
+tures in the residuals over this range are due to finite numbers
+of real stars in each bin. There are some noticeable devia-
+tions from unity at bright magnitudes for both F090W and
+F150W. We believe that these may be due to saturation ef-
+fects that might be mitigated by improved data quality masks
+and/or the use of Frame 0 data. Both will be explored in our
+forthcoming DOLPHOT NIRcam module paper. Similarly,
+the increased ratio at the faintest F150W magnitudes is not
+7 https://jwst-docs.stsci.edu/jwst-near-infrared-camera/
+nircam-performance/nircam-point-spread-functions
+
+F090W ETC
+F150W ETC
+DOLPHOT
+DOLPHOT
+800
+800
+600
+600
+NR
+S400
+400
+200
+200
+0
+DOLPHOT / ETC
+Z
+2
+1
+1
+0
+0
+20
+22
+24
+26
+28
+18
+20
+22
+24
+26
+F090W
+F150W18
+WEISZ ET AL.
+overly concerning as small variations in the PSF shape (Dol-
+phin 2000) and the presence of non-stellar artifacts at the very
+bottom of the CMD can affect the photon noise-based SNRs.
+The uptick could also be caused by the removal of objects
+that don’t meet our culling criteria; formally a correct cal-
+culation requires factoring in completeness as determined by
+ASTs.
+Overall, this comparison provides preliminary indications
+that v2.0 of the ETC provides reasonable SNR estimates for
+fairly uncrowded stars imaged with NIRCam. Of course, in
+practice, many resolved stellar systems that will be targeted
+by NIRCam will be more affected by crowding than M92,
+which can lead to larger discrepancies in the expected versus
+recovered SNR. For HST, this effect is partially mitigated by
+the optimal SNR reported by its ETC8. This number reflects
+that expected SNR for an isolated point source recovered by
+PSF fitting and is generally a factor of 1.5–2 higher than the
+regular SNR reported by the HST ETC. Our initial analysis of
+M92 suggests that the baseline SNRs from the NIRCam ETC
+may not be off by as large a factor. However, further explo-
+ration in a variety of images with variable crowding, stellar
+type, etc. Ultimately, ASTs will aid in calculation of SNRs
+seen in the data over a range of stellar densities. We will
+carry out such an exploration in the context of our NIRCam
+and NIRISS DOLPHOT photometry paper.
+7. SUMMARY
+We have undertaken the JWST resolved stellar populations
+Early Release Science program in order to establish JWST
+as a the premier facility for resolved stellar populations early
+JWST’s lifetime. In this paper, we have described the motiva-
+tion, planning, implementation, execution, and present NIR-
+Cam CMDs from preliminary photometric reductions with
+DOLPHOT. Some key takeaways from our survey include:
+• Our 27.5 hr program obtained NIRCam (primary) and
+NIRISS (parallel) imaging of 3 diverse targets: Milky
+Way globular cluster M92, satellite ultra-faint galaxy
+Draco II, and more distant (0.9 Mpc) star-forming
+galaxy WLM. A summary of their properties are listed
+in Table 1 while a summary of our JWST observations
+for each target are listed in Table 2. These targets were
+selected in order to enable a variety of science and
+technical goals related to resolved stellar populations
+analysis as described in §2.
+• This ERS program facilitated the development of NIR-
+Cam and NIRISS modules for DOLPHOT, a widely
+used stellar crowded field photometry package. We
+used our ERS targets to test these modules for a va-
+riety of image properties (e.g., various filter combina-
+tions, over a large dynamic range in stellar crowding).
+We describe the application of DOLPHOT to our ERS
+data in §5. Beta versions of these DOLPHOT modules,
+8 https://etc.stsci.edu/etcstatic/users guide/1 3 imaging.html
+along with theoretical PSF models for all NIRCam and
+NIRISS filters are publicly available on our team web-
+site and on the DOLPHOT website.
+• We presented preliminary NIRCam CMDs in select
+SW and LW filter combinations from a first pass
+DOLPHOT reduction. The CMDs are among deep-
+est CMDs in existence for each class of object. The
+F090W−F150W CMD of M92 touches the hydrogen
+burning limit (F090W > 28.2; (M < 0.08 M⊙).
+The F090W−F150W CMD of Draco II reaches the the
+bottom of the stellar sequence (0.09 M⊙) in the stan-
+dard PARSEC models. The F090W−F150W CMD of
+WLM extends ∼1.5 mag below the oldest MSTOs in
+WLM.
+• We compare our NIRCam CMDs to select age and
+metallicity isochrones from the PARSEC models. We
+find that the models are in generally good agreement
+with all JWST CMDs, though we find them to be ∼
+0.05 mag systematically bluer of the lower MS in M92
+and Draco II. We posit that this color offset may be due
+to the complexity of stellar atmospheres in extremely
+low-mass stars that is currently not well-captured in
+theoretical stellar atmospheres. A notable example in-
+cludes the known sensitivity of color to oxygen abun-
+dance (e.g., VandenBerg et al. 2022)
+• We compare the photon-noise based SNRs for the
+F090W and F150W reported by DOLPHOT for stars in
+M92 with expectations from v2.0 of the NIRCam ETC.
+We find they agree within ∼ 20% over most of the
+magnitude range, with slightly larger deviations at the
+very bright and very faint limits. The differences may
+be due to saturation effects at the bright end and selec-
+tion effects and/or subtle mismatches between theoret-
+ical and observed PSFs at the faint end. We caution
+that this preliminary comparison does not capture ef-
+fects such as crowding, which is important in distant
+dwarf galaxies such as WLM.
+• We are in the process of optimizing DOLPHOT for
+use with NIRCam and NIRISS. All technical details
+of the DOLPHOT modules and their application to our
+ERS data the subject of an upcoming publication on
+crowded field photometry.
+
+JWST RESOLVED STELLAR POPULATIONS I
+19
+The authors would like to thank David W. Hogg for his
+input on the program and paper.
+This work is based on
+observations made with the NASA/ESA/CSA James Webb
+Space Telescope. The data were obtained from the Mikulski
+Archive for Space Telescopes at the Space Telescope Science
+Institute, which is operated by the Association of Universi-
+ties for Research in Astronomy, Inc., under NASA contract
+NAS 5-03127 for JWST. These observations are associated
+with program DD-ERS-1334.
+This program also benefits
+from recent DOLPHOT development work based on obser-
+vations made with the NASA/ESA Hubble Space Telescope
+obtained from the Space Telescope Science Institute, which
+is operated by the Association of Universities for Research in
+Astronomy, Inc., under NASA contract NAS 5–26555. These
+observations are associated with program HST-GO-15902.
+Facilities: JWST(NIRCAM), JWST(NIRISS)
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+page_content=' Cherry Avenue, Tucson, AZ 85719, USA  10Center for Astrophysics and Space Sciences, Department of Physics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093, USA  11School of Natural Sciences, University of Tasmania, Private Bag 37, Hobart, Tasmania 7001, Australia  12Department of Astronomy, University of Washington, Box 351580, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=', Minneapolis, MN 55455, USA  14Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA  15The Observatories of the Carnegie Institution for Science, 813 Santa Barbara St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=' 2515 Speedway,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=' Flatiron Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=' Dartmouth College,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 6127 Wilder Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=' Miller III Department of Physics & Astronomy, Bloomberg Center for Physics and Astronomy, Johns Hopkins University, 3400 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=' Northwestern University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 1800  Sherman Avenue,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=' USA  28John Hopkins Applied Physics Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 11100 Johns Hopkins Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=' University of Notre Dame,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=' USA  30Perimeter Institute for Theoretical Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Waterloo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=' Canada  31Department of Physics and Astronomy G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Galilei, University of Padova, Vicolo dell’Osservatorio 3, I-35122, Padova, Italy  322817 Rudge Pl, Modesto, CA 95355, USA  33Department of Physics and Astronomy, University of California, Davis, CA 95616, USA  ABSTRACT  Corresponding author: Daniel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Weisz  dan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='weisz@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='edu  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='04659v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='GA]  11 Jan 2023  2  WEISZ ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We present the JWST Resolved Stellar Populations Early Release Science (ERS) science program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We ob-  tained 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5 hours of NIRCam and NIRISS imaging of three targets in the Local Group (Milky Way globular  cluster M92, ultra-faint dwarf galaxy Draco II, star-forming dwarf galaxy WLM), which span factors of ∼ 105  in luminosity, ∼ 104 in distance, and ∼ 105 in surface brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We describe the survey strategy, scientific and  technical goals, implementation details, present select NIRCam color-magnitude diagrams (CMDs), and vali-  date the NIRCam exposure time calculator (ETC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Our CMDs are among the deepest in existence for each class  of target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' They touch the theoretical hydrogen burning limit in M92 (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='08 M⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' SNR ∼ 5 at mF 090W ∼ 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  MF 090W ∼ +13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='6), include the lowest-mass stars observed outside the Milky Way in Draco II (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='09 M⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  SNR = 10 at mF 090W ∼ 29;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' MF 090W ∼ +12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1), and reach ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5 magnitudes below the oldest main se-  quence turnoff in WLM (SNR = 10 at mF 090W ∼ 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' MF 090W ∼ +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The PARSEC stellar models  provide a good qualitative match to the NIRCam CMDs, though are ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='05 mag too blue compared to M92  F090W−F150W data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The NIRCam ETC (v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0) matches the SNRs based on photon noise from DOLPHOT  stellar photometry in uncrowded fields, but the ETC may not be accurate in more crowded fields, similar to what  is known for HST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We release beta versions of DOLPHOT NIRCam and NIRISS modules to the community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Results from this ERS program will establish JWST as the premier instrument for resolved stellar populations  studies for decades to come.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Keywords: Stellar photometry (1620), Local Group (929), Stellar populations (1622), Hertzsprung Russell dia-  gram (725), JWST (2291)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' INTRODUCTION  The resolved stellar populations of nearby galaxies are  central to a wide range of astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The observed col-  ors, luminosities, and spectral features of resolved stars in  galaxies within the Local Volume (LV) anchor our knowl-  edge of star formation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', star cluster formation, the ini-  tial mass function, the importance of binarity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Massey 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  McKee & Ostriker 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Sarajedini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Bastian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Sana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Kroupa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Krumholz 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Piotto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Krumholz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2019), stellar feedback  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', the interplay between stars and their immediate sur-  roundings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Oey 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Dohm-Palmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Stin-  son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2006, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Governato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Ostriker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Lopez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Pellegrini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Lopez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' El-Badry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2018, 2019b),  dust production and characteristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Gordon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Boyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Boyer 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Meixner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Dal-  canton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Schlafly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Green et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Gordon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Yanchulova Merica-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2021),  and stellar chemistry and kinematics across a wide range of  environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Venn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Simon & Geha 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Geha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Kirby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Collins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Var-  gas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Gilbert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Vargas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Ji et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Gilbert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Escala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Kirby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Gilbert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Resolved stars are the basis for  the local distance ladder (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Freedman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Riess  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Beaton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' McQuinn  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2017a, 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Freedman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2021),  which provides constraints on the expansion of the Universe  and the nature of dark energy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Di Valentino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  They anchor our knowledge of the stellar evolution models  that are used to interpret the light of distant galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=',  Dotter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Ekstr¨om et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Girardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Bressan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' VandenBerg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Eldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Conroy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Hidalgo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Eldridge & Stanway 2022) and provide detailed insight into  the dynamic assembly of our Galactic neighborhood, cos-  mic reionization, the first stars, near-field cosmology, and  the nature of dark matter on the smallest scales (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Ma-  teo 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Tolstoy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Weisz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2014a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Boylan-Kolchin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Gallart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Mc-  Quinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Frebel & Norris 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Wetzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Bullock & Boylan-Kolchin 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Starkenburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2017b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  McConnachie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Kallivayalil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Conroy  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Simon 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Boylan-Kolchin &  Weisz 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Sacchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Pearson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Over the past ∼ 30 years, much of this science has been  enabled by the Hubble Space Telescope (HST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Since its first  images of resolved stars in the local Universe (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Paresce  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Campbell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Guhathakurta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Freedman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Hunter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 1995), HST’s exquisite  sensitivity, angular resolution, and broad wavelength cover-  age have transformed our knowledge of the Universe by ob-  serving hundreds of nearby galaxies for thousands of hours  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Freedman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Holtzman  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Dalcanton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Dal-  canton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Gallart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Skillman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Tully et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2021),  including the Panchromatic Hubble Andromeda Treasury  (PHAT) program, which resolved 100 million stars across the  disk of M31 (Dalcanton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  However, while HST continues to catalyze new astrophys-  ical insights in nearby galaxies, it has only scratched the sur-  face of science enabled by infrared (IR) observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Com-  pared to the UV and optical, HST’s IR camera has coarser an-  gular resolution, which limits it to brighter stars due to stellar  crowding, and it can only observe a small portion of the IR  spectrum, which limits the types of stellar populations it can  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  JWST will be transformative for resolved stellar popula-  tions in the IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Compared to any other facility, JWST will re-  JWST RESOLVED STELLAR POPULATIONS I  3  solve individual stars at larger distances, to fainter luminosi-  ties, over wider color baselines, in more crowded areas, and  in regions of higher extinction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' JWST can provide the first  main-sequence turnoff-based (MSTO) star formation histo-  ries (SFHs) of galaxies beyond the Local Group (LG;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=',  Weisz & Boylan-Kolchin 2019), systematically measure the  sub-Solar mass stellar IMF directly from star counts as a  function of environment (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Geha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Kalirai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' El-Badry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Gennaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Filion et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Gennaro & Robberto 2020), determine proper motions  and orbital histories for dozens of galaxies outside our imme-  diate Galactic neighborhood (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', van der Marel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Sohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Kallivayalil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Zivick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Sohn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Warfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2023), construct parsec-scale  maps of the interstellar medium (ISM) in galaxies out to sev-  eral Mpc (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Dalcanton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Gordon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Yanchulova Merica-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2017, 2021), establish a new  anchor to the physics of the evolved stars that dominate the  rest-frame near-IR light of distant galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Maraston  2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Melbourne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Boyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015, 2019), pro-  vide high fidelity distances to galaxies throughout the Lo-  cal Volume (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Beaton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Tully et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Freedman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Riess et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2021),  and much more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  With these remarkable capabilities in mind, we have under-  taken the JWST Resolved Stellar Populations Early Release  Science (ERS) Program (DD-1334;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' PI D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Weisz) to establish  JWST as the premier facility for the study of resolved stellar  populations in the IR such that it can match and exceed HST’s  successes in the local Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' To realize this goal, our ERS  program has acquired deep multi-band NIRCam and NIRISS  imaging of three targets in the Local Group (LG): one Milky  Way globular cluster (GC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' M92), one ultra-faint dwarf galaxy  (UFD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Draco II), and one distant star-forming dwarf galaxy  (WLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' These diverse targets showcase a broad range of the  science described above and enable the development and test-  ing of JWST-specific modules for the widely used crowded  field stellar photometry package DOLPHOT (Dolphin 2000,  2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  In this paper, we summarize the design of our ERS  program, illustrate the new JWST-specific capabilities of  DOLPHOT, outline the photometric reduction process,  present a first look at JWST observations of our targets, and  undertake select comparisons with stellar models and the cur-  rent JWST exposure time calculator (ETC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Papers in prepa-  ration by our team will provide a detailed overview of the  new NIRCam and NIRISS modules for DOLPHOT and will  focus on a wide variety of science results enabled by the ERS  data beyond what is described here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We summarize the  program’s overarching science and technical aims and tar-  get selection in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We then describe how we translated  these goals into an observational strategy in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We detail  the actual ERS observations in §4 and summarize the ap-  plication of DOLPHOT in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' In §6 we present NIRCam  color-magnitude diagrams (CMDs) and compare them to se-  lect stellar models and evaluate the performance of NIRCam  ETC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' PROGRAM GOALS  Our team developed a set of main science and technical  goals based on anticipated common community use cases of  JWST for resolved stellar populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' For simplicity, we  limited our considerations to science cases based on imaging  with NIRCam, which is considered the “workhorse” camera  of JWST, as well as NIRISS imaging, which we used in par-  allel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' This setup is analogous to the commonly used mode of  HST in which ACS/WFC operates as the primary instrument  with WFC3/UVIS acquiring imaging in parallel (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Dal-  canton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Gallart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Skillman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Albers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Science based on imaging of resolved stars often requires  stellar photometry in crowded fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Because of that, re-  solved stellar population studies are technically daunting,  requiring highly optimized observations and sophisticated  analysis tools that have been developed and refined over the  past ∼ 40 years (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Buonanno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Tody 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Stet-  son 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Schechter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Stetson 1994;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Anderson &  King 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Dalcanton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' A  main technical goal of our program is to develop and release  NIRCam and NIRISS modules for DOLPHOT, along with  practical recommendations and demonstrations for applying  DOLPHOT to NIRCam and NIRISS imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Here, we sum-  marize our main science goals, technical goals, and science  “deliverables” which guide our ERS program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Scientific Goals  Our team identified six main science themes that guided  the construction of our ERS program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' They are:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Star Formation Histories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' A galaxy’s resolved stel-  lar content encodes its star formation history (SFH),  which can be reconstructed by fitting CMDs with stel-  lar evolution models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Tosi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Tolstoy  1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Harris & Zaritsky 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Dolphin 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Hidalgo  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' These SFHs are particularly robust when  CMDs extend below the oldest main sequence turnoff  (MSTO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Gallart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The faintness of this  feature in the optical (MV ∼ +4) has limited current  ‘gold standard’ SFHs to galaxies within the LG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' How-  ever, the relatively low effective temperatures of these  stars, combined with the decreased sky background in  the near-IR and JWST’s excellent sensitivity and angu-  lar resolution, will enable it to measure the first SFHs  based on the oldest MSTOs for galaxies outside the  LG (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Weisz & Boylan-Kolchin 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' JWST-GO-  1617 PI K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' McQuinn) from which outstanding ques-  tions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', the effects of reionization and/or environ-  ment on galaxy formation) can be uniquely addressed  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Bullock & Boylan-Kolchin 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Simon 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Our JWST program will showcase JWST’s ability to  measure robust SFHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  4  WEISZ ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The Sub-Solar Mass IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Resolved star counts shows  that lowest-mass galaxies appear to have sub-Solar  IMF slopes which deviate from the Galactic value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Geha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Kalirai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Gen-  naro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2018b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' However, even with HST, it has  proven challenging to acquire sufficiently deep data  (down to ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2 M⊙ El-Badry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Gennaro  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2018b,a) to unambiguously confirm these puta-  tive IMF variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Our ERS program will illustrate  JWST’s capabilities for definitively measuring the sub-  Solar IMF in a ultra-faint MW satellite, paving the way  for a systematic study of the low-mass IMF and star  formation in extreme environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Proper Motions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  High-precision astrometry enables  the measurement of proper motions (PMs) throughout  the LG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Gaia has been transformative for objects in  the MW halo, while HST has laid the foundation for  fainter, more distant systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' JWST is the future of  precision astrometry for faint and/or more distant ob-  jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' On its own, and in tandem with Gaia, HST, and  Roman, JWST imaging will provide measurements of  total masses, dark matter profiles, and orbital histories  for ∼ 100 galaxies in and around the LG (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Bullock  & Boylan-Kolchin 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Kallivayalil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Fritz  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Gilbert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Battaglia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Warfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Our ERS program will showcase  the PM measurements capabilities of JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Evolved Stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Cool evolved stars such as red super-  giants and asympotic giant branch (AGB) stars are re-  sponsible for 20–70% of the rest-frame near-IR lumi-  nosity of star-forming galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Maraston 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Melbourne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012) and are sites of dust production  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Ventura 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' However, the rapid evolution of  dusty AGB stars is challenging to model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Maras-  ton 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Girardi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Conroy 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Marigo  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2017), which has only begun to be alleviated  by recent observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Boyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  JWST’s expansive IR filter set will reveal elusive dust-  enshrouded populations of AGB stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', oxygen-  rich M stars and carbon-rich C stars) across a wide  range of galactic environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Hjort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Marini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Our ERS pro-  gram will demonstrate JWST’s capacity to study IR-  bright, evolved stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Extinction Mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  In the LG, Spitzer and Her-  schel have mapped dust emission at ∼ 10 − 40′′ and  ∼ 7 − 12′′ resolution, respectively (10 pc for the  Magellanic Clouds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 100 pc for M31 and M33;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Draine  2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Gordon 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Chastenet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Utomo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' JWST can map the cold ISM at significantly  higher spatial resolution by inferring dust content from  its impact on stellar spectral energy distributions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=',  Dalcanton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Gordon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Our ERS  observations will demonstrate JWST’s ability to map  dust extinction and relate it to properties of the cold  ISM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Ages of Globular Clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Accurate ages of the oldest  GCs are particularly important for connecting the stel-  lar fossil record to events in the early Universe includ-  ing cosmic reionization and the age of the Universe it-  self (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Chaboyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Grebel & Gallagher  2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Ricotti & Gnedin 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Monelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Weisz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2014b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Boylan-Kolchin  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Current age estimates are typically limited  to ∼ 1 Gyr precision (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', twice as long as reionization  lasted) due to the age-metallicity degeneracies at the  MSTO (see Boylan-Kolchin & Weisz 2021 and refer-  ences therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' JWST observations of the ‘kink’ on the  lower main sequence (MS) can yield more precise esti-  mates of cluster ages (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Sarajedini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Bono  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Kalirai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Correnti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Our ERS data will showcase the powerful capabilities  of JWST for precise GC age-dating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Beyond enabling our main science goals, we sought to  identify observations that would make our ERS program  rich for archival pursuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Examples include measuring ex-  tragalactic distances in JWST bands (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', TRGB, variable  stars;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Beaton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Madore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2019a), identifying rare stars from low-mass metal-poor stars  to luminous red supergiants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Schlaufman & Casey 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Casey & Schlaufman 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Levesque 2018), searching for  dust production among red giant branch stars (RGB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=',  Boyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Boyer 2010), and examining the nature of  dark matter using wide binaries (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Pe˜narrubia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Technical Goals  The main technical goal of our ERS program is to enable  resolved star science by the broader community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' At the heart  of this goal is the addition of NIRCam and NIRISS modules  to DOLPHOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' This process includes the technical develop-  ment of NIRCam and NIRISS modules for DOLPHOT, test-  ing their performance on real data, releasing data products  that immediately enable science (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', stellar catalogs), and  providing guidance to the community on best use practices  of DOLPHOT for future applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Here, we broadly de-  scribe each of these technical goals and how they influenced  the observational strategy of our ERS program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  As with previous updates to DOLPHOT (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Dalcanton  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Dolphin 2016, and many unpublished updates),  the core functionality of the code remains the same as de-  scribed in Dolphin (2000), but certain aspects have been up-  dated for NIRCam and NIRISS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The DOLPHOT modules for NIRCam and NIRISS each  feature their own pre-processing routines that apply masks  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', of reference, saturated, and other unusable pixels) to the  images based on the data quality flags provided by the STScI  pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' They also apply the pixel area maps appropriate  to each camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Other updates include the use of photomet-  ric calibrations provided in the image metadata, conversions  JWST RESOLVED STELLAR POPULATIONS I  5  to VEGAMAG, and camera-specific PSF models with corre-  sponding encircled energy corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  For testing DOLPHOT on real data, we identified several  observational scenarios that we anticipate to be common for  NIRCam and NIRISS studies of resolved stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' They are:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Targets with various levels of crowding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' This includes  images that are completely uncrowded (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', in which  aperture vs PSF photometry can be compared), images  with highly variable amounts of crowding (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', due  to surface brightness variations), and highly crowded  images (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', the photometric depth is primarily limited  by crowding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Targets that include stars spanning a large dynamic  range in brightness in the same image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' An example  would be a GC, in which there are very bright red gi-  ants and extremely faint dwarfs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' This enables a variety  of tests, including the ability to recover faint sources  next to very bright objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Targets with bright, saturated stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' JWST is extremely  sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Understanding the degree to which saturated  stars affect the photometry of fainter objects will be  important to a variety of science goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Targets that demonstrate the ability of using the higher  angular resolution short-wavelength (SW) images to  increase the accuracy of the long-wavelength (LW)  photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' PHAT showed that joint reduction of HST  optical and IR data produced IR photometry that pro-  vides significantly sharper CMDs compared to reduc-  ing IR data alone (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Similar  gains should be possible with NIRCam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Targets that enable the simultaneous reduction of HST  and JWST imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' To date, DOLPHOT has produced  wonderful cross-camera results for HST (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Dalcan-  ton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2014, 2021), but it  needs to be vetted and optimized for cross-facility use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Deliverables  Our program is in the process of providing several “deliv-  erables” to the community that can be found on our team  website1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  A primary deliverable is the public release of  DOLPHOT with NIRCam and NIRISS specific modules for  which “beta” versions can be found on the main DOLPHOT  website2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' This software enables crowded field stellar pho-  tometry for a diverse range of science in the local Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Along with the software release we will provide extensive  documentation of how to use DOLPHOT and examples of  it applied to our ERS observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Following careful cali-  bration and testing, we will release high level science prod-  ucts including the output of our team DOLPHOT runs on  1 https://ers-stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='io  2 http://americano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='dolphinsim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='com/dolphot/  ERS data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', diagnostic plots and files), and NIRCam and  NIRISS stellar catalogs for each target along with artificial  star tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' These data products will be refined as our un-  derstanding of JWST improves (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', due to updated PSF  models) and will eventually include examples of how to use  DOLPHOT for simultaneous reduction of HST and JWST  imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' STRATEGY  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Filters  The diversity of our science cases required careful consid-  eration of filter selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Several of our science goals are  centered around maximizing depth, color baseline, and astro-  metric precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Accordingly, we primarily focused on SW  filter selection, which has better sensitivity (for most stars)  and angular resolving power than the long-wavelength chan-  nel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Using an ancient,  metal-poor isochrone (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5 Gyr,  [Fe/H]=−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0) from the MIST stellar models (Choi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2016), we examined the expected performance for the SW  wide filter (F070W, F090W, F115W, F150W, F200W) per-  mutations at three different CMD locations: the blue HB  (Teff ∼ 7000 K), the MSTO (Teff ∼ 6000 K), and the lower  MS (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2 M⊙;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Teff ∼ 4000 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' At each point, we used the  pre-commissioning JWST ETC (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1) to compute the ex-  posure time required to reach a SNR= 10 for the “scene” in  the ETC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The best performing filters for our areas of consideration  are F090W, F115W, and F150W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' They all exhibit compa-  rable performance at the HB and MSTO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' However, F090W  requires 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5 times more exposure time to achieve the same  SNR for a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2 M⊙ star as either F115W or F150W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Never-  theless, we opted for F090W over F115W because compared  to F115W−F150W, F090W−F150W provides superior color  information for most stars and F090W has the potential for  higher angular resolution (if dithered appropriately), which  is critical for astrometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Finally, the similarity between  F090W and HST/F814W (or Johnson I-band) provides use-  ful features such as matching catalogs between facilities and  TRGB distance determinations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  F070W and F200W provide the largest color baseline, but  each filter is less sensitive to stars far from their effective  wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' For example, F070W required 4 times more in-  tegration time for a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2 M⊙ star than the next bluest filter,  F090W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' F200W requires twice as much exposure time for a  HB star than F150W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We opted to use the same F090W−F150W filter combi-  nation for all targets to provide for an empirical comparison  between the GC and UFD (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012) and for  good sampling of the oldest MSTO in the distant dwarf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We  considered more than two SW filters, but the cost of acquir-  ing extra data outweighed the scientific utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We selected  simultaneously observed LW filters on a per target basis, as  they enable secondary science unique to each object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Finally,  we selected F090W and F150W for parallel NIRISS imaging  for consistency with NIRCam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  6  WEISZ ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The locations of our NIRcam and NIRISS observations (plotted in red) for each ERS target, overplotted on a Pan-STARRS optical  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The orange dotted lines indicate: (a) 2 and 5 half-light radii (rh), (b) 1 and 2 rh and (c) 1 rh of each target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We show additional  pointings for each system: (a) M92: select HST optical (green;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' HST-GO-10775;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Sarajedini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2007) and IR (pink;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' HST-GO-11664;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=',  Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' (b) Draco II: HST/ACS optical data (HST-GO-14734;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' PI Kallivayalil) are shown in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' (c) WLM: An exhaustive,  though not complete set of HST observations including HST/WFPC2 UV and optical imaging in blue (HST-GO-11079;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Bianchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012),  HST/WFC3 UVIS UV imaging in purple (HST-GO-15275 PI Gilbert), HST/ACS and HST/UVIS optical imaging in green (HST-GO-13768;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  PI Weisz, Albers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2019), and HST/WFC3 IR imaging in pink (HST-GO-16162, PI Boyer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We opted not to undertake large dithers to fill  the NIRCam chip and module gaps which would have substantially increased the program time while only marginally enhancing our science  goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We emphasize that while our filter combinations represent  a good compromise across the CMD for our program goals,  they may not be optimal for all science cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We encourage  exploration tailored to a program’s particular science aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Target Selection  We selected targets by first considering all known GCs in  the MW (Harris 2010) and galaxies within ∼ 1 Mpc (Mc-  Connachie 2012), including updates to both catalogs and  discoveries through 2017 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Laevens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2014, 2015a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Bechtol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Drlica-Wagner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Koposov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The limiting distance was selected to ensure we could  reach the oldest MSTO with SNR= 10 in the most distant  system in a reasonable amount of time based on previous ex-  perience with HST (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Cole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Albers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2019)  and results from the JWST exposure time calculator (ETC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We required that each target have extensive HST imag-  ing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', to enable combined HST and JWST proper mo-  tion studies, create panchromatic stellar catalogs) and have  a good sampling of ground-based spectra (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', for full phase  space information, comparing stellar properties from spec-  tra and photometry, incorporating stellar abundance patterns  into various analyses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We then identified a minimum set of targets that could  be used to achieve our science and technical goals: one  MW GC, one UFD, and one more distant star-forming dwarf  galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We then sought to maximize observational efficiency  by focusing on some of the nearest examples of these classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We eliminated targets that were not visible during the nomi-  nal ERS window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  This selection process yielded three targets: MW GC M92,  MW satellite UFD Draco II, and star-forming dwarf galaxy  WLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Basic observational characteristics of these targets are  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Basic observational properties of the three ERS targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Properties for M92 have been taken from the updated MW GC cat-  alog of Harris (2010), while those of Draco II and WLM are from  the updated LG galaxy catalog of McConnachie (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Note that  µ0 is the effective surface brightness and rh is the half-light radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  M92  Draco II  WLM  RA (J2000)  17h17m07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='27s  15h52m47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='60s  00h01m58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='16s  Dec (J2000)  +43d08m11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5s  +64d33m55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0s  −15d27m39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='34s  MV (mag)  −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='8  −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2  E(B-V) (mag)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='03  (m − M)0 (mag)  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='6  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='9  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='9  µ0 (mag arcsec−2)  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='8  rh (′)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='7  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='8  listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We detail the observational strategy for each  target in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' M92  M92 (NGC 6341) is a well-studied, metal-poor GC in the  MW that is often used as a benchmark for extragalactic stel-  lar population studies and for photometric calibration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=',  to verify zero points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Dalcanton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Gallart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Imaging this system satisfies sev-  eral science and technical goals including GC ages, individ-  ual star proper motions, the present day mass function, test-  ing DOLPHOT over a large dynamic range of stellar bright-  ness and spatially varying stellar density, and gauging the ef-  fects of bright saturated stars on the photometric process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content="  (a) M92  (b) Draco II  (c) WLM  HST/AWFPC2  HST/ACS  HST/ACS  UWST  HST/ACS  JWST  JWST  HST/WFC3 IR  HST/WFC3UVIS  'NIRCam & NIRISS  HSTWFC3 UVIS  NIRCam & NIRISS  HST/WFC3IR  NIRCam & NIRISSJWST RESOLVED STELLAR POPULATIONS I  7  Figure 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' NIRCam color composite images for our 3 ERS targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  In each RGB image, F090W was used as the blue channel, F150W as the  green and a combination of the two LW filters as the red channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  (a) M92  (b) Draco il  (c) WLM8  WEISZ ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  As illustrated in Figure 1, we placed the NIRCam field  near the center of M92, with the aim of maximizing NIRCam  spatial overlap with a wealth of multi-band HST imaging of  M92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The parallel NIRISS field is located at ∼ 5 half-light  radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We constrained the orientation such that the NIRISS  field had a modest probability of overlapping at least some  HST data in the outer regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' However, orientations allowed  by the final ERS window did not result in overlap between  NIRISS and HST imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We chose the F090W, F150W, F277W, and F444W for  our NIRCam imaging of M92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We selected F277W and  F444W for their broad scientific utility including studying  the lower MS kink at long wavelengths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Sarajedini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Bono et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2010), searching for dust production at  low-metallicities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Boyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Boyer 2010), and  exploring multiple populations in the IR (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2012, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Correnti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We aimed to reach SNR= 10 at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1 M⊙ in F090W and  F150W, which we estimate to be mF 090W ≈ 26 (MF 090W ≈  +11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='4) and mF 150W ≈ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='8 (MF 150W ≈ +11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2) based on  the MIST isochrones and the characteristics of M92 listed in  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The 2017 versions of the ETC (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1) and APT  (v25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1) yielded exposure times of 1288s in each filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The  anticipated SNRs in the LW filters at mF 090W ≈ 26 were  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1 in F277W and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='4 in F444W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We estimated NIRISS  imaging to be marginally shallower than NIRCam with in-  tegration times of 1074s in each of F090W and F150W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The  difference in duration is set by facility overheads as calcu-  lated by APT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Draco II  At ∼20 kpc, Draco II is one of the nearest examples of a  MW satellite (Laevens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Longeard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  At the time of program design, Draco II was designated as a  UFD (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', it has dark matter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Willman & Strader 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Mar-  tin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' In the interim, there has been some debate  in the literature over its status as a UFD vs GC (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', whether  or not it has dark matter, a metallicity spread, the slope of its  sub-Solar stellar mass function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Longeard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Baumgardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2022), an issue our JWST data should help  resolve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Draco II’s close proximity provides for efficient JWST  imaging that reaches far down the lower main sequence  (≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Our deep Draco II data satisfies several of our  science goals including measuring the low-mass stellar IMF  (which can shed insight into its status as a GC or UFD), de-  termining the SFH of an ancient sparsely populated system,  measuring proper motions of individual faint stars using HST  and JWST, and exploring our ability to distinguish between  faint stars and unresolved background galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We placed the NIRCam field on the center of Draco II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The  field overlaps with archival HST imaging obtained in March  2017 (GO-14734;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' PI N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Kallivayalil), Keck spectroscopy,  and is well-matched to the half-light radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' To maximize  scheduling opportunities, we did not constrain the orienta-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The NIRISS field is unlikely to contain many Draco II  member stars, so its exact placement was not crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The  primary use of the NIRISS data will be to aid with modeling  contamination (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', foreground stars and background galax-  ies) in the F090W−F150W CMD, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', as part of measuring  the low-mass IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We selected the F090W, F150W, F360M, and F480M for  our NIRCam imaging of Draco II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The rationale for the two  SW filters is described in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The LW filters are located  near metallicity sensitive molecular features in the mid-IR,  and they may be suitable for measuring photometric metal-  licities of metal-poor stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Schlaufman & Casey 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Casey & Schlaufman 2015) similar to what is possible in  the optical using, for example, the Calcium H&K lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=',  Starkenburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2017b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' For NIRISS, we  used the F090W and F150W filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Our target depth is set by low-mass IMF science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Tightly  constraining the low-mass IMF in Draco II requires reaching  stars ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2 M⊙ (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', El-Badry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2017) with SNR= 10  in F090W and F150W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Using the MIST stellar models and  the observational properites of Draco II listed in Table 1, our  target depths are SNR∼ 10 at mF 090W = 27 (MF 090W =  +10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='3) and mF 150W = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='8 (MF 150W = +10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Using  the 2017 versions of the ETC (v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1) and APT (v25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1)  we found exposure times of 12798s in F090W and 6399s  in F150W would each these depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We opted on SW/LW  combinations of F090W/F480M and F150W/F360M, which  provided for SNRs of ∼ 7 (F360M) and ∼ 4 (F480M)  at F150W=25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We estimated the NIRISS imaging to have  12540s in F090W and 6270s in F150W, which will result in  marginally shallower CMDs than NIRCam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' WLM  WLM (Wolf 1909;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Melotte 1926) is a metal-poor ([Fe/H]=  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Leaman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2009) star-forming dwarf galaxy at  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='9 Mpc (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Albers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Though slightly closer  objects of this class exist (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', IC 1613), WLM is the nearest  example of a low-metallicity environment in which resolved  CO clouds have been detected (Rubio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' It has a  sufficiently high star formation rate in the past several Gyr  that it should host a sizable population of AGB stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=',  Dolphin 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Weisz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2014a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2017b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Albers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' In terms of science, our JWST imaging  of WLM will allow us to measure its SFH from the ancient  MSTO and compare it to the existing HST-based SFH, mea-  sure its bulk PM using archival HST imaging, explore the  stellar populations associated with it CO clouds, construct  parsec-scale extinction using IR-only techniques and com-  bined UV-optical-IR HST and JWST stellar SEDs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Dal-  canton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Gordon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' On the technical side,  our observations of WLM allow us to test DOLPHOT in a  regime of faint, crowded stars, that is typical of more distant  systems, and also test its capabilities for simultaneously mea-  suring stellar photometry across facilities (HST and JWST)  and instruments (WFPC2, ACS, UVIS, NIRCam).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We required that one module of the NIRCam observa-  tions overlap UV-optical-IR HST observations as well as the  ALMA-detected CO clouds (Rubio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The other  NIRCam module overlaps with the deep optical HST/ACS  JWST RESOLVED STELLAR POPULATIONS I  9  imaging presented in Albers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We enforced this  configuration by requiring orientations of 70–110◦ or 250–  290◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' This orientation placed the NIRISS field in the stellar  halo of WLM, providing an expansion on the areal cover-  age to build on the population gradient studies in WLM (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=',  Leaman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Albers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We used artificial  star tests associated with the deep HST imaging of Albers  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' (2019) to ensure that our NIRCam observations would  only be modestly affected by stellar crowding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We selected the F090W, F150W, F250M, and F430M for  our NIRCam imaging of WLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' As discussed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='3, the  SW filter combination F090W−F150W is likely to be widely  used for SFHs measured from the ancient MSTO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Medium  band filters in the near-IR have proven remarkably efficient  for photometric identification and classification of AGB stars  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Boyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Our simulations, based on these  studies, suggest that the F250M and F430M filters should  work well for similar science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' For NIRISS, we selected the  F090W and F150W filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  In the optical, measuring a well-constrained SFH for dis-  tant dwarf galaxies requires a CMD that reaches a SNR∼5–  10 at the oldest MSTO (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Cole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Monelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Cole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Skillman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Gallart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Skillman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Albers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Using the  MIST stellar models, we find that the ancient MSTO for  metal-poor stellar population has MF 090W  = +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='4 and  MF 150W = +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Using the parameters for WLM in Table  1 and v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1 of the ETC, we estimated that 33241s in F090W  and 18724s in F150W would reach the required depths of  mF 090W = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='3 and mF 150W = 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1 with SNR=10 in each  filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Program Updates Since 2017  Our program has only had minor changes since it was  first approved in November 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  In 2021, we changed  the DEEP2 readout patterns for WLM and Draco II to  MEDIUM8, following updated advice from a STScI tech-  nical review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The number of groups and integrations was  changed accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' In 2021, STScI increased our allocated  time from 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='35 hr to 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5 hr to reflect updated overhead ac-  counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Following commissioning in spring 2022, STScI staff  changed the range of aperture PA ranges of Draco II from  unconstrained to be have a value of 47–118◦, 143–280◦, or  354–24◦ in order to avoid the “claws” feature3 which is due  to stray light from the bright source that is outside the field  of view of NIRCam (jdo 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Rigby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' OBSERVATIONS  We acquired NIRCam and NIRISS imaging of our three  targets in July and August of 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Figure 1 shows the NIR-  Cam and NIRISS footprints for each of our targets overlaid  on ground-based images and Table 1 lists our observational  3 urlhttps://jwst-docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='edu/jwst-near-infrared-camera/nircam-features-  and-caveats/nircam-claws-and-wisps  configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Full details on implementation can be viewed  by retrieving proposal 1334 in the Astronomer’s Proposal  Tool (APT)4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We followed the same fundamental observing strategy for  all three targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' For each target, the primary instrument,  NIRCam, imaged a single central field in SW and LW fil-  ters as described in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' NIRISS obtained imaging in parallel  in the two filters F090W and F150W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  All observations were taken with the 4 point subpixel  dither pattern 4-POINT-MEDIUM-WITH-NIRISS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' This  pattern ensured adequate PSF sampling for both cameras, as  well as improved rejection of cosmic rays, hot pixels, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We opted against primary dithers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The main advantage of  the primary dithers is to fill the gaps between the dectectors  and/or modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' However, the inter-gap imaging is generally  shallower than the rest of the data and requires more JWST  time to acquire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' For our particular science cases, including  primary dithers added a modest amount of time to the pro-  gram but would not substantially enhance the data for our  main science goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Other science cases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', covering a  large region such as PHAT did) may benefit from primary  dithers and filling gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  For M92, we used the SHALLOW4 readout pattern for NIR-  Cam and the NIS readout pattern for NIRISS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The orienta-  tion was constrained to an Aperture PA range of 156 to 226◦  in order to maximize the possibility of overlap between the  NIRISS field and existing HST imaging, while also allow-  ing for reasonable schedulability very early in the lifetime  of JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The central location of the NIRCam field ensures  it will overlap with existing HST imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Including over-  heads, the total time charged for observing M92 was 7037s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The ratio of science to charged time was ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='35 (counting the  primary NIRCam imaging only) or ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='7 (for both NIRCam  and NIRISS imaging).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The data volume was well within the  allowable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  For Draco II, we used the MEDIUM8 readout pattern for  NIRCam and the NIS readout pattern for NIRISS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The NIR-  Cam field was centered on the galaxy and will largely overlap  with existing HST imaging and Keck spectroscopic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The  large angular separation of the NIRISS and NIRCam fields  compared to the size of Draco II means that the NIRISS  field will contain few, if any, bona fide members of Draco II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Including overheads, the total time charged for observing  Draco II was 24539s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The ratio of science to charged time  was ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='71 (counting the primary NIRCam imaging only) or  ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='36 (for both NIRCam and NIRISS imaging).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The data  volume was well within the allowable range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  For WLM, we used the MEDIUM8 readout pattern for NIR-  Cam and the NIS readout pattern for NIRISS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The NIRCam  field was placed in the center of WLM in order to overlap  the low-metallicity molecular clouds discovered by ALMA  as well as deep archival optical HST imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Subsequent  UV and near-IR HST imaging of WLM obtained by mem-  4 apt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='edu  10  WEISZ ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' A summary of our JWST ERS observations taken in 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Target  Date  Camera  Filter  texp [s]  Groups  Integrations  Dithers  M92  June 20–21  NIRCam  F090W/F277W  1245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='465  6  1  4  NIRCam  F150W/F444W  1245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='465  6  1  4  NIRISS  F090W  1245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='465  7  1  4  NIRISS  F150W  1245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='465  7  1  4  Draco II  July 3  NIRCam  F090W/F480M  11810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='447  7  4  4  NIRCam  F150W/F360M  5883.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='75  7  2  4  NIRISS  F090W  11123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='294  9  7  4  NIRISS  F150W  5883.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='75  10  3  4  WLM  July 23–24  NIRCam  F090W/F430M  30492.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='427  8  9  4  NIRCam  F150W/F250M  23706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='788  8  7  4  NIRISS  F090W  26670.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='137  17  9  4  NIRISS  F150W  19841.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='551  19  6  4  bers of our team were placed to maximize the chances of  overlap with our JWST observations of WLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Including overheads, the total time charged for observing  WLM was 66884s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The ratio of science to charged time  was ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='8 (counting the primary NIRCam imaging only) or  ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='50 (for both NIRCam and NIRISS imaging).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The large  data volume (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='962 GB) for WLM generated a “Data Ex-  cess over Lower Threshold” warning in APT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Though this  level of warning is only a recommendation to mitigate data  volume excess, we nevertheless consulted with STScI about  mitigation strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' However, we were unable to identify  a way to reduce data volume without compromising the sci-  ence goals and no changes were made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We caution that even  longer integrations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', that may be needed for deep CMDs  outside the LG) may require careful planning to avoid data  volume limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Because our WLM observations span several continuous  hours (Table 2), our observations of WLM should allow us  to recover the light curves of short period variables (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', RR  Lyrae).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Generation of the light curves requires performing  photometry on calibrated images at each integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' At the  time of this paper’s writing, due to the large data volume,  the STScI JWST reduction pipeline generates integration-  level calibrated images only when the time series observa-  tion (TSO) mode is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' However, we were unable to use  the TSO mode of JWST as it does not permit dithers nor  parallel observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Instead, producing the necessary im-  ages to generate light curves requires running the reduction  pipeline locally and creating custom time series analysis soft-  ware, which is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  In total, all of our science observations with NIRCam total  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='45 hr and observations with NIRISS total 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='85 hr, indi-  cating a fairly high ratio of science-to-charged time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' PHOTOMETRY  We perform photometric reductions of our observations us-  ing the newly developed NIRCam module for DOLPHOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The detailed description of how DOLPHOT works is well-  documented in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Dolphin 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Dalcanton  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Dolphin 2016) and the functionality of the new  NIRCam module are summarized in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' A more detailed  write up of the DOLPHOT NIRCam and NIRISS modules  are the subjects of an upcoming paper from our team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' For this  paper, we only focus on the NIRCam data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The WebbPSF5  NIRISS PSF models appear to concentrate light significantly  more than what we have observed in the ERS images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Con-  sequently, our NIRISS photometry is not yet reliable and fur-  ther updates await improvements to the PSF models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We first acquired all images from MAST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Per their  FITS headers,  versions of all JWST images used in  this paper have the following JWST pipeline version-  ing information CAL VER=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2,CRDS VER=11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='11, and  CRDS CTX=jwst p1009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='pmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Next,  we  performed  DOLPHOT reductions on the level 2b crf frames and use  the level 3 i2d F150W drizzled image as the astrometric  reference frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The use of this reference image ensures  excellent internal alignment of our image stack and ties the  absolute astrometry to Gaia DR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We perform photomery  in all four bands simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Since there is no spatial  overlap between the footprint of the two NIRCam modules,  we photometer them independently and merge the catalogs a  posteriori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We have not included Frame 0 in the photomeric reduc-  tions for this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' As of this paper’s writing, the JWST  pipeline does not automatically provide Frame 0, requiring  these images be generated locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Even when created with  our own execution of the JWST pipeline, we have yet to  fully resolve how 1/f noise issue (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Schlawin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Bagley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2022) can be resolved in a manner that ensures  self-consistent photometry with DOLPHOT for all frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  This issue will be addressed in more detail in our forthcom-  ing photometry paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' In the meantime, without Frame 0  data, our photometry saturates at fainter magnitudes that we  5 https://webbpsf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='io/en/latest/  JWST RESOLVED STELLAR POPULATIONS I  11  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' CMDs from all 8 NIRCam chips for each of our targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Panel (a): M92 extends from our bright saturation limit without Frame 0  data (F090W∼ 19) just below the MSTO and reaches just fainter than the hydrogen burning limit (F090W > 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The inflection point of  the MS at F090W ∼ 21 is the MS kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Panel (b): The CMD of Draco II includes the MS kink at F090W ∼ 23 and reaches the bottom  of the stellar sequence provided by the standard PARSEC models (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='09 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Panel (c): The CMD of WLM displays a wide variety of stellar  sequences that span a range of ages including young stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', the upper main sequence, red core helium burning stars), intermediate age stars  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', AGB and RGB stars), and ancient stars that span a wide range of colors and magnitude (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', RGB, SGB, oMSTO, lower MS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' This CMD  of WLM (SNR=10 at MF 090W = +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='6) is the deepest taken for a galaxy outside the immediate vicinity of the MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  expect to recover once the Frame 0 data in included in our  DOLPHOT runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  For our photometric reductions, we adopt the DOLPHOT  parameter setup recommended by PHAT (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We use the parameters recommend for ACS from  PHAT for the SW images and the WFC3/IR parameters for  the LW images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Subsequent data releases will make use of  a set of parameters that we are tailoring to NIRCam and  NIRISS observations of resolved stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We make use of NIRCam PSF models generated with  WebbPSF (Perrin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We use the Optical Path De-  lay maps from July 24th, 2022 (the best matching file to our  WLM observation epoch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Inspection of JWST’s wavefront  field at the epochs of our three observations show little vari-  ation in the optical performance of the telescope, justifying  our choice of a common PSF library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We are currently work-  ing in quantifying the full effect of JWST time-dependent  PSF variations on DOLPHOT photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The final products of the DOLPHOT photometric run are  catalogs with positions, VEGAmag magnitudes (calibrated  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Quality-metric criteria used to cull our DOLPHOT photo-  metric catalogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Band  SNR  Sharp2  Crowd  Flag  Object Type  F090W  ≥ 4  ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='01  ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5  ≤ 2  ≤ 1  F150W  ≥ 4  ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='01  ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5  ≤ 2  ≤ 1  using the latests NIRCam zero-points6), uncertainties based  on photon noise, and a set of quality metrics related to the  goodness of point-source photometry (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', SNR, χ2 of the  PSF fit, angular extent of the source, crowding level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' At this  stage, metrics such as the photometric error and the SNR are  based on the Poissonian treatment of photon noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' While in  many cases this is sufficient for rough estimation, there are  6 https://jwst-docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='edu/jwst-near-infrared-camera/  nircam-performance/nircam-absolute-flux-calibration-and-zeropoints  (a) M92  (b) Draco Il  (c) WLM  18  20  SNR=500  22  SNR=500  F090W  100  24  SNR=500  50  100  26  100  10  50  28  10  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5  F090W-F150W  F090W-F150W  F090W-F150W12  WEISZ ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  caveats associated with this approximation, especially when  measuring stars in very crowded fields, or close to the limit-  ing magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We will provide a more thorough discussion  on SNRs estimation in our upcoming JWST DOLPHOT pho-  tometry paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The catalogs provided by DOLPHOT are subsequently in-  spected and culled to remove contaminants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', artifacts,  cosmic rays, extended sources) while aiming to retain the  largest number of bona fide stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We identify a set of quality-  metric cuts, listed in Table 3, that provide a reasonable trade-  off between completeness and purity of the stellar sample;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  though for this initial presentation we erred on the side of pu-  rity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The selection criteria need to be satisfied in the F090W  and F150W bands simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We only use the SW  bands as they are shared by all three targets, allowing a com-  mon set of culling criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Our initial exploration suggests  that the LW photometry may improve star–galaxy separation  at faint magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' This important topic is being further in-  vestigated by members of our team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Full characterization of uncertainties in resolved stel-  lar populations studies required artificial star tests (ASTs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  ASTs consist of adding mock stars of known properties into  each frame and recovering them using the same photometric  procedure that is applied to the real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' For the purposes  of this paper focused on survey description, we have not in-  cluded results of the ASTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The large data volume and multi-  filter nature of the data make running ASTs computationally  challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We will present full AST results and analyses in  the upcoming JWST DOLPHOT photometry paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' DISCUSSION  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Color-Magnitude Diagrams  Figure 3 shows the NIRCam SW CMDs for all three tar-  gets over the same magnitude range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The juxtaposition of  these CMDs illustrates the quality and diversity of science  possibilities provided by our program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' In each panel, we  overplot select SNRs reported by DOLPHOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' As discussed  in §5, these SNRs are solely based on photon noise and do  not account for the effects of crowding and incompletness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We discuss SNRs further in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The SNRs for each target are remarkable, with SNRs rang-  ing from 500 near the MS kink to 10 for the lowest mass stars  in M92 and Draco II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' WLM has a photon noise-based SNR  of ∼ 50 at the oldest MSTO, making it the highest fidelity  resolved stars observation of a distant dwarf in existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We  now discuss the multi-band CMDs for each of our targets in  more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Figures 4–6 show illustrative NIRCam CMDs in a selec-  tion of filter combinations for each ERS target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' In all cases,  we apply the catalog culling parameters described in §5 and  listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  For guidance, we overplot a selection of stellar isochrones  from the PARSEC v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2S stellar libraries (Bressan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' These models span the full range of metal-  licities and ages needed to characterize our datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We  have adjusted these isochrones to the distances and redden-  ings listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' M92  Figure 4 shows select NIRCam CMDs for M92, along with  select HST-based CMDs for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The HST CMDs  were reduced using DOLPHOT and the parameters recom-  mended in Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We overplot select PARSEC isochrones at a fixed age of  13 Gyr with varying metallicities, which we discuss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Though stars brighter than F090W ∼ 19 are omitted due to  saturation effects (see §5), key CMD features for faint, low-  mass stars are clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Notably, the MS kink, which  is due to opacity effects in M dwarfs, the region in which  is in panels (a) - (c) at F090W ∼ 21 and F150W ∼ 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The MS kink exhibits the sharpest inflection point in the  F150W−F277W filter combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The kink is much less  pronounced in the F277W−F444W CMD (panel d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' This  is partially due to the lower SNRs of the LW observations  as well as the LW filters being far into the Rayleigh–Jeans  tail of the stars’ spectral energy distributions (SEDs), which  makes them only weakly sensitive to stellar temperature, thus  resulting in similar colors and a less obvious kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Our NIRCam observations have produced the deepest  CMD of M92 to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The data extend fainter than the lowest  stellar mass (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='09 M⊙) available from the standard PARSEC  stellar library in the SW filters and to slightly higher masses  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='12 M⊙) in the LW filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The faintest objects in the F090W−F150W CMD fall into  the bright end of the expected hydrogen burning sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The exact mass at which a star-like object cannot sustain  hydrogen fusion has long been debated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Hayashi &  Nakano 1963;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Kumar 1963;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Chabrier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' For this  analysis, we computed custom PARSEC models with a mass  resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='002 M⊙ and find that that the minimum mass  for hydrogen burning is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='078 ≤ M < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='08 M⊙ for a metal-  licity of [Fe/H]= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='7 dex and an age of 13 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' This trans-  lates to a magnitude range of 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2 < mF 090W ≤ 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5 mag  in M92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' This depth is quite remarkable considering our ob-  servations only consistent of ∼ 1050s in each filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' In com-  parison, the faintest stars in the HST WFC3/IR CMD (panel  e) are a few magnitudes brighter despite ∼ 1200s of inte-  gration time in each filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  To date, comparably deep IR  studies of metal-poor, extremely low-mass stars with HST  have been limited to the nearest GCs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', M4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Dieball et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2016, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' As our M92 data shows, the superior sensitivity  of NIRCam will make such studies possible throughout the  MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The PARSEC models over-plotted in Figure 4 are nom-  inally higher than the metallicity of M92 ([Fe/H] =  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='23 dex) derived from high-resolution APOGEE spec-  troscopy (M´esz´aros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' This discrepancy is because  the current version of the PARSEC models are Solar-scaled,  whereas M92 is highly α-enhanced with [α/Fe] ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Well-established corrections can be applied to match Solar-  scaled models with α-enhanced populations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Salaris  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' For M92, the corrective factor for the PAR-  SEC models results in values of [Fe/H]  ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5 dex higher  than derived from spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' PARSEC models with α-  JWST RESOLVED STELLAR POPULATIONS I  13  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Select NIRCam CMDs of M92, along with HST WFC3/IR (HST-GO-11664) and ACS/WFC (GO-9453, GO-10775, GO-12116,  GO-16298) CMDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Overploted are Solar-scaled PARSEC stellar models at fixed age (13 Gyr) over a select range of [Fe/H] values, which  have been corrected to match the known α-enhancement of M92 (M´esz´aros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The SW NIRCam CMD extends from our saturation  limit (F090W  ∼ 19) to below the lowest-mass star from the standard PARSEC models (M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='09 M⊙) and into the expected hydrogen  burning limit regime (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='078 ≤ M < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='08 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The LW CMDs reach slightly higher limits (M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='12 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' These are among the deepest  CMDs of a GC in existence and highlight how JWST will easily enable the study of prominent features for low-mass stars such as the MS-kink  and lowest-mass stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' While the stellar models are in excellent agreement with the more luminous stellar evolutionary phases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', RGB,  MSTO) in the HST CMDs, they are ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='05 mag too blue for lower-mass stars in the SW JWST filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' This offset could be due to the complex  atmospheres of low-mass stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  enhancements are under construction and will mitigate the  need to apply such corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Overall,  the  PARSEC  models  are  in  reasonably  good agreement with the NIRCam CMDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  For the  F150W−F444W and F277W−F444W CMDs, the models  trace the locus of the data quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  However, for the  F090W−F150W and F150W−F277W CMDs, the models  are systematically too blue by ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='05 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The source of  this offset is not due to distance, reddening, age, or metal-  licity as these same models are well-matched to the MSTO  in the brighter HST CMDs (panels e and f of Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' In  general, stars above the MS kink are well-matched by the  models in the LW NIRCam and HST filters, limiting the  offsets to only SWs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' One possible source of the offset is the  presence of poorly modeled absorption features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', TiO)  in the atmospheres of very cool, low-mass stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' A detailed  exploration of this offset is beyond the scope of this paper,  but we note that deep JWST imaging of a larger set of GCs  in several filters has the potential to help elucidate the exact  nature of this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The NIRCam CMDs also exhibit scatter in color that is  larger than photon noise and variations in age or metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The most likely explanation is the presence of multiple chem-  ically distinct populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Like many luminous Galactic  GCs, M92 exhibits multiple populations with distinct abun-  dance patterns (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', M´esz´aros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Other GCs with  deep CMDs and similar abundance patterns show a broad-  ening of stellar sequences at and below the MS kink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' This  broadening has been attributed to the persistence of these  chemically distinct sequences to the lowest stellar masses  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Milone et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Correnti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Milone  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2017), which is thought to be driven by oxygen varia-  tions expressed via water lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Dotter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Van-  denBerg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Finally, we note that some of the scatter  18  [Fe/H]=-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0  18  (b)  (a)  (c)  [Fe/H]=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='7  18  20  [Fe/H]=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='4  20  22  F090W  L50W  20  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='50W  24  22  22  26  24  24  28  26  26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5  F090W-F150W  F150W-F277W  F150W-F444W  10  (d)  (e) HST WFC3/IR  (f) HST/ACS  18  10  15  277W  20  F160W  814W  15  20  20  24  25  26  25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='50  0  1  2  3  F277W-F444W  F110W-F160W  F475W-F814W14  WEISZ ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Select NIRCam CMDs of Draco II along with the deepest optical CMD which is based on HST/ACS imaging (GO-14734;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' panel  b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Overploted are PARSEC stellar models at fixed age (13 Gyr) for the same [Fe/H] values selected for M92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The SW CMD extends to the  lowest-mass stellar model (M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='09 M⊙), making it the deepest CMD of a MW satellite galaxy to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The exquisite depth of our data  indicate how JWST enables a variety of science including constraining the low-mass IMF and quantifying low-mass star features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' the MS  kink and objects near the hydrogen burning limit) outside the MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  could be due to NIRCam zero point calibrations which will  not be finalized with uncertainties until at least the end of  Cycle 1 (Gordon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Draco II  Figure 5 shows select NIRcam CMDs of Draco II, along  with the deepest existing optical HST CMD for reference  (GO-14734;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' PI N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Kallivayalil).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Like M92, the brightest  stars in Draco II suffer from saturation and are not included  in our current photometric reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We include the same  PARSEC isochrones as shown in M92 (Figure 4) as Draco II  hosts a comparably ancient (13 Gy), metal-poor, and likely  α-enhanced stellar population (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Longeard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Si-  mon 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The selected isochrones provide a good fit of the  (a)  [Fe/H]=-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0  (b) HST/ACS  18  18  [Fe/H]=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='7  [Fe/H]=-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='4  20  20  F090W  22  F814W  22  24  24  26  26  28  28  30  30  0  1  2  0  1  2  F090W-F150W  F606W-F814W  (c)  (d)  18  18  20  20  50W  22  60M  22  5  24  3  出  26  24  28  26  30  0  1  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5  F150W-F360M  F360M-F480MJWST RESOLVED STELLAR POPULATIONS I  15  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Select NIRCam CMDs of WLM along with the deepest available HST/ACS optical CMD from Albers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Overplotted  are isochrones from the PARSEC Solar-scaled stellar models at a fixed metallicity of [Fe/H]= −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2 dex for a variety of indicated ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Panel  (a) is the deepest CMD of an isolated dwarf galaxy to date, extending ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5 mag below the oldest MSTO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' it is deeper than the HST CMD  despite ∼ 7000s less integration time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The isochrones indicate the variety of stellar ages present in WLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The LW CMD shown in panel (d)  extends ∼ 2 magnitudes below the red clump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Such deep CMDs show that JWST will provide for a variety of science at different cosmic epochs  including exquisite lifetime SFHs, the study of evolved red stars, TRGB distances, and very young stars in galaxies outside the LG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  MSTO in the HST CMD (panel b) suggesting the adopted pa-  rameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', age, metallicity, distance, extinction) are rea-  sonable, but as with M92, the models are slightly too blue in  the NIRCam SW CMDs of Draco II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Our F090W−F150W CMD of Draco II is the deepest  CMD of a galaxy outside the MW and has imaged the lowest-  mass stars outside the MW (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='09 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Previously, the deepest CMD in an external galaxy was  from Gennaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' (2018b), which used HST WFC3/IR data  of MW UFD Coma Berenicies to study its low-mass IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  From 32780s of integration time in each filter, their F110W  and F160W photometry reached a usable low-mass limit of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='17 M⊙, whereas our data extend to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='09 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The large dy-  namic range of stellar masses in Draco II provides excellent  leverage for a low-mass IMF measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Tight constraints  20  20  (a)  (b)  HST/  22  ACS  22  F090W  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='4W  24  24  13 Gyr  5 Gyr  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5 Gyr  1  26  26  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='05 Gyr  8  F  28  28  30  30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2  1  0  1  2  F090W-F150W  F475W-F814W  20  18  (c)  (d)  22  20  F090W  24  F250M  22  26  24  28  26  30 -  1  0  1  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='50  F090W-F250M  F250M-F430M16  WEISZ ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  on the IMF in Draco II could provide a new means of distin-  guishing whether faint stellar systems are dark-matter domi-  nated dwarf galaxies or GCs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Willman & Strader 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Baumgardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2022), as well as insight into star formation  in extreme environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Geha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Krumholz  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The width of the lower MS is in excess of photometric  noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Metallicities derived from more luminous stars in  Draco II suggest a spread of σ ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5 dex (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Longeard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2022), which may con-  tribute to this scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Background galaxies are also a source  of contamination and likely contribute to the scatter, partic-  ularly at the faintest magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The combination of very  deep imaging and the sparsity of Draco II’s stellar popula-  tion mean that background galaxies are a large source of con-  tamination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Our preliminary investigations indicate that the  multi-color NIRCam photometry may be efficient for star-  galaxy separation (Warfield et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' in prep).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The F150−F360M CMD (panel c) extends to a compara-  bly low stellar mass as the SW CMD, albeit at lower SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The LW CMD (panel d) is much shallower;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' though the pri-  mary purpose of these filters is to explore their potential as  photometric metallicity indicators akin to the Calcium H &  K filters being used in the optical (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Starkenburg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Longeard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Fu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  As with M92, the stellar isochrones provide a good quali-  tative match to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The shape and magnitude of the MS  kink appears to track the data well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' However, as discussed in  the context of M92 the models are modestly too blue com-  pared to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' WLM  Figure 6 shows select NIRCam CMDs for WLM, along  with the deepest optical CMD of WLM taken with HST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Due  to its large distance, few stars in WLM are affected by satu-  ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The CMDs of WLM exhibit a wide variety of stellar se-  quences that span a range of ages and phases of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Examples include the young MS, the RGB and AGB, the HB,  and the oldest MSTO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The relative positions of these features  are generally similar to what is known for optical CMDs with  subtle changes due to the shift to IR wavelengths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Dal-  canton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2012b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Gull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  For example, the HB slopes to fainter values at bluer wave-  lengths as hot blue HB stars are less luminous at IR wave-  lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Similarly, red stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', RGB, AGB) become more  luminous as the IR wavelengths are closer to their peak tem-  peratures compared to optical wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Figure 6 shows that our NIRCam SW CMD of WLM is at  ∼ 1 mag deeper than the HST/ACS CMD despite similar in-  tegration times (∼ 54200s for NIRCam versus ∼ 61400s for  ACS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' This increase owes primarily to the increased sensitiv-  ity of JWST at these wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' (in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=')  is deriving the SFHs from both datasets to quantify the capa-  bilites of JWST imaging for detailed SFH determinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  More broadly, our CMDs of WLM are the deepest in ex-  istence for an isolated dwarf galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Prior to our program,  HST/ACS observations of Leo A from Cole et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' (2007)  extended to the lowest stellar masses in an galaxy outside  the immediate vicinity of the MW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The HST observations of  Leo A reach nearly as deep, but the larger distance of WLM  (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5 mag farther) means that our measurements actually ex-  tend to less luminous stars on the MS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Though not as deep as the SW data, the LW is remark-  ably deep for medium bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The F090W−F250M data  extends below the oldest MSTO, making it the deepest  medium band data available for an isolated dwarf galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The  F250M−F430M (panel d) CMD extends well below the red  clump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' This data is expected to provide an excellent means of  identifying AGB stars and helping constrain their underlying  physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The over-plotted isochrones provide a reasonable match to  the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' In this case, we have selected a single metallic-  ity of [Fe/H] = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2 dex matched to RGB spectroscopic  abundances (Leaman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2009), and plotted select ages that  range from 50 Myr to 13 Gyr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Visually, the PARSEC models  provide a good qualitative match to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The level of  agreement will be formally quantified in McQuinn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' (in  prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Comparisons with the Exposure Time Calculator  Our photometry provides an opportunity to gauge the ac-  curacy of the NIRCam ETC in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  In Figure 7, we consider the SNRs for stars in M92 as re-  ported by DOLPHOT against what is reported by the NIR-  Cam ETC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' M92 is the best of our three targets for this ex-  ploratory exercise as it is not particularly crowded and its  CMD is well-populated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The lack of crowding means that  the SNRs reported by DOLPHOT are a reasonable proxy for  real noise, whereas ASTs are necessary to accurately assess  the noise in even moderately crowded images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We compute SNRs as a function of F090W and F150W  by only considering stars that pass a stricter version of the  culling criteria listed in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Specifically, we require that  each star has crowd ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1, which eliminates all but the  least crowded stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We also only consider stars with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2 <  F090W − F150W < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5, which isolates the MS in the SW  filters and removes much of the contamination (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', back-  ground galaxies, diffraction spike artifacts) from our analy-  sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Finally, we exclude the brightest stars as they may be af-  fected by (partial) saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We specifically only consider  stars fainter than F090W = 20 and 150W = 19 for their  respective SNR calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  From stars that pass these cuts, we compute the 50th, 16th,  and 84th percentiles of the F090W and F150W SNR distri-  butions in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='25 mag bins over the entire magnitude ranges  considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  For the expected SNRs, we use v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0 of the JWST ETC to  compute SNR as a function of F090W and F150W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' In the  ETC, the detector strategies are set to match our F090W and  F150W observational set up for M92 as listed in Table 2 and  described in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We verified that the integration times in  the ETC are identical to what our program acquired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  JWST RESOLVED STELLAR POPULATIONS I  17  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Comparisons between the photon noise-based SNRs from the DOLPHOT F090W and F150W photometry of M92 (grey shaded  region) and the expected SNRs from v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0 of the NIRCam ETC (black lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Over most of the dynamic ranges in magnitude the SNRs reported  by DOLPHOT and the ETC are consistent within scatter (∼ 20%) of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Deviations at the bright end may be due to the presence of  saturated pixels, while PSF shape, non-stellar objects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', background galaxies), and incompleteness may contribute to a slight increase in the  ratio at the faintest F150W magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  For the ETC scene, we used a K5V star (Teff = 4250 K,  log(g) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5 dex) from the Phoenix stellar models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Though  the stellar type varies over the color and luminosity range  considered, we found that reasonable changes in the choice  of stellar atmosphere only affected our findings at the ∼ 5%  level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' For simplicity, we adopted a single stellar atmosphere  model for this calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We adopted an extinction of AV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='06 mag and a MW  extinction curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We set the background model to the central  coordinates of M92 on June 20th, 2022, the date of our obser-  vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We computed the SNR in the F090W and F150Ws  filter in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5 mag steps from 19 to 30 mag in each filter, renor-  malizing after extinction was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The result is a smooth variation in SNR as a function of  magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We interpolated the results onto a finer magni-  tude grid for clearer comparison with the DOLPHOT results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Interpolation errors are < 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  To compute the SNR, we used the default aperture pho-  tometry setup in the NIRCam ETC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Specifically, this uses  an aperture radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1′′ and performs background subtrac-  tion using an annulus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='22 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='4′′ from the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' For both  F090W and F150W, this radius within the aperture radius  range of 2-3× the PSF FWHM specified in JDOX7 as rec-  ommended by the JWST help desk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We will explore more  filter SNRs and variations in the ETC photometric set p in a  future paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Figure 7 shows a comparison between the SNRs reported  by DOLPHOT (grey shaded regions) and the NIRCam ETC  (black lines) as a function of F090W and F150W magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The bottom panels show the ratios of the DOLPHOT and  ETC SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Both visually and quantitatively, the expected  ETCs agree quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' For most of the magnitude ranges,  the DOLPHOT and ETC SNRs agree within ∼ 20%, which is  within the bounds of our uncertainty range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The small struc-  tures in the residuals over this range are due to finite numbers  of real stars in each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' There are some noticeable devia-  tions from unity at bright magnitudes for both F090W and  F150W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We believe that these may be due to saturation ef-  fects that might be mitigated by improved data quality masks  and/or the use of Frame 0 data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Both will be explored in our  forthcoming DOLPHOT NIRcam module paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Similarly,  the increased ratio at the faintest F150W magnitudes is not  7 https://jwst-docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='edu/jwst-near-infrared-camera/  nircam-performance/nircam-point-spread-functions  F090W ETC  F150W ETC  DOLPHOT  DOLPHOT  800  800  600  600  NR  S400  400  200  200  0  DOLPHOT / ETC  Z  2  1  1  0  0  20  22  24  26  28  18  20  22  24  26  F090W  F150W18  WEISZ ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  overly concerning as small variations in the PSF shape (Dol-  phin 2000) and the presence of non-stellar artifacts at the very  bottom of the CMD can affect the photon noise-based SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The uptick could also be caused by the removal of objects  that don’t meet our culling criteria;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' formally a correct cal-  culation requires factoring in completeness as determined by  ASTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  Overall, this comparison provides preliminary indications  that v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0 of the ETC provides reasonable SNR estimates for  fairly uncrowded stars imaged with NIRCam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Of course, in  practice, many resolved stellar systems that will be targeted  by NIRCam will be more affected by crowding than M92,  which can lead to larger discrepancies in the expected versus  recovered SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' For HST, this effect is partially mitigated by  the optimal SNR reported by its ETC8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' This number reflects  that expected SNR for an isolated point source recovered by  PSF fitting and is generally a factor of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5–2 higher than the  regular SNR reported by the HST ETC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Our initial analysis of  M92 suggests that the baseline SNRs from the NIRCam ETC  may not be off by as large a factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' However, further explo-  ration in a variety of images with variable crowding, stellar  type, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Ultimately, ASTs will aid in calculation of SNRs  seen in the data over a range of stellar densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We will  carry out such an exploration in the context of our NIRCam  and NIRISS DOLPHOT photometry paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' SUMMARY  We have undertaken the JWST resolved stellar populations  Early Release Science program in order to establish JWST  as a the premier facility for resolved stellar populations early  JWST’s lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' In this paper, we have described the motiva-  tion, planning, implementation, execution, and present NIR-  Cam CMDs from preliminary photometric reductions with  DOLPHOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Some key takeaways from our survey include:  Our 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5 hr program obtained NIRCam (primary) and  NIRISS (parallel) imaging of 3 diverse targets: Milky  Way globular cluster M92, satellite ultra-faint galaxy  Draco II, and more distant (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='9 Mpc) star-forming  galaxy WLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' A summary of their properties are listed  in Table 1 while a summary of our JWST observations  for each target are listed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' These targets were  selected in order to enable a variety of science and  technical goals related to resolved stellar populations  analysis as described in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  This ERS program facilitated the development of NIR-  Cam and NIRISS modules for DOLPHOT, a widely  used stellar crowded field photometry package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We  used our ERS targets to test these modules for a va-  riety of image properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', various filter combina-  tions, over a large dynamic range in stellar crowding).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We describe the application of DOLPHOT to our ERS  data in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Beta versions of these DOLPHOT modules,  8 https://etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='edu/etcstatic/users guide/1 3 imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='html  along with theoretical PSF models for all NIRCam and  NIRISS filters are publicly available on our team web-  site and on the DOLPHOT website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We presented preliminary NIRCam CMDs in select  SW and LW filter combinations from a first pass  DOLPHOT reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The CMDs are among deep-  est CMDs in existence for each class of object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The  F090W−F150W CMD of M92 touches the hydrogen  burning limit (F090W > 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' (M < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='08 M⊙).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  The F090W−F150W CMD of Draco II reaches the the  bottom of the stellar sequence (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='09 M⊙) in the stan-  dard PARSEC models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The F090W−F150W CMD of  WLM extends ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='5 mag below the oldest MSTOs in  WLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We compare our NIRCam CMDs to select age and  metallicity isochrones from the PARSEC models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We  find that the models are in generally good agreement  with all JWST CMDs, though we find them to be ∼  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='05 mag systematically bluer of the lower MS in M92  and Draco II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We posit that this color offset may be due  to the complexity of stellar atmospheres in extremely  low-mass stars that is currently not well-captured in  theoretical stellar atmospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' A notable example in-  cludes the known sensitivity of color to oxygen abun-  dance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', VandenBerg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2022)  We compare the photon-noise based SNRs for the  F090W and F150W reported by DOLPHOT for stars in  M92 with expectations from v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='0 of the NIRCam ETC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We find they agree within ∼ 20% over most of the  magnitude range, with slightly larger deviations at the  very bright and very faint limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The differences may  be due to saturation effects at the bright end and selec-  tion effects and/or subtle mismatches between theoret-  ical and observed PSFs at the faint end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' We caution  that this preliminary comparison does not capture ef-  fects such as crowding, which is important in distant  dwarf galaxies such as WLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  We are in the process of optimizing DOLPHOT for  use with NIRCam and NIRISS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' All technical details  of the DOLPHOT modules and their application to our  ERS data the subject of an upcoming publication on  crowded field photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  JWST RESOLVED STELLAR POPULATIONS I  19  The authors would like to thank David W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' Hogg for his  input on the program and paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  This work is based on  observations made with the NASA/ESA/CSA James Webb  Space Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' The data were obtained from the Mikulski  Archive for Space Telescopes at the Space Telescope Science  Institute, which is operated by the Association of Universi-  ties for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', under NASA contract  NAS 5-03127 for JWST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' These observations are associated  with program DD-ERS-1334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='  This program also benefits  from recent DOLPHOT development work based on obser-  vations made with the NASA/ESA Hubble Space Telescope  obtained from the Space Telescope Science Institute, which  is operated by the Association of Universities for Research in  Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', under NASA contract NAS 5–26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' These  observations are associated with program HST-GO-15902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=', Geha, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=', Simon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Martin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Sesar, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2014, ApJL, 786,  L3, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='1088/2041-8205/786/1/L3  Laevens, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=', Ibata, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2015a, ApJ,  802, L18, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=' 2015b, ApJ,  813, 44, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=', Cole, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2009, ApJ, 699, 1,  doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=' 2013, ApJ, 767,  131, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=' 2014, ApJ,  795, 121, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
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+page_content=', Johnson, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=',  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2017, ApJ, 847, 102, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='3847/1538-4357/aa8a67  —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2021, ApJ, 907, 50, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='3847/1538-4357/abc48b  Zivick, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', Kallivayalil, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', van der Marel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content=' 2018, ApJ,  864, 55, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
+page_content='3847/1538-4357/aad4b0' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NE3T4oBgHgl3EQfrQru/content/2301.04659v1.pdf'}
diff --git a/-NFQT4oBgHgl3EQf6jZ7/content/tmp_files/2301.13439v1.pdf.txt b/-NFQT4oBgHgl3EQf6jZ7/content/tmp_files/2301.13439v1.pdf.txt
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+arXiv:2301.13439v1  [hep-ph]  31 Jan 2023
+USTC-ICTS/PCFT-23-04
+January 2023
+Rare W-boson decays into a vector meson and lepton pair
+Dao-Neng Gao†
+Interdisciplinary Center for Theoretical Study, University of Science and Technology of China,
+Hefei, Anhui 230026 China
+Peng Huanwu Center for Fundamental Theory, Hefei, Anhui 230026 China
+Abstract
+We have presented a theoretical study of exclusive rare W-boson decays, W → V ℓ¯νℓ
+with V denoting a neutral vector meson and ℓ = e or µ, in the standard model.
+The leading-order contributions to these processes are given by W → γ∗ℓ¯νℓ with the
+subsequent γ∗ → V transition. Branching fractions of these decay modes, for V = ρ,
+ω, φ, and J/Ψ, respectively, have been calculated and predicted around 10−6 ∼ 10−7,
+which are surprisingly larger than those of two-body hadronic radiative decays W ± →
+M±γ with M denoting a pseudoscalar or vector meson. Thus it is expected that rare
+W decays into a neutral vector meson plus lepton pair may be the promising channels
+in future experimental facilities with a large number of W-boson events produced.
+† E-mail address: gaodn@ustc.edu.cn
+
+Exclusive rare W-boson decays, which contain hadronic final states, could provide inter-
+esting probes to increase our understanding of the properties of the fundamental weak gauge
+boson as well as offer some deep insights into quantum chromodynamics [1, 2, 3, 4, 5]. Experi-
+mentally, no such processes have been observed so far, and only upper limits on the branching
+fractions of three exclusive modes: B(W ± → D±
+s γ) < 1.3×10−3, B(W ± → π±γ) < 7×10−6,
+and B(W ± → π+π−π±) < 1.01 × 10−6, were set at 95% confidence level [6]. On the other
+hand, a huge number of W events, about O(1011), will be expectedly accumulated in the
+high-luminosity Large Hadron Collider (LHC) [3]. This may significantly facilitate the ex-
+perimental studies of rare W-boson decay channels, which can be very helpful both to test
+the standard model (SM) and to search for new physics beyond the SM.
+Our main focus in the present paper is on another types of rare W-boson decays: W →
+V ℓ¯νℓ with V denoting the neutral vector particle including heavy quarkonium J/Ψ or light
+mesons ρ, ω, and φ etc.
+ℓ is the lepton with ℓ = e or µ.
+The leading-order Feynman
+diagrams contributing to these processes in the SM have been shown in Figure 1, in which
+the transitions can proceed through W → γ∗ℓ¯νℓ, followed by γ∗ → ¯qq → V . This is similar
+to the case of Z → V ℓ+ℓ+ decays, which have been studied in Refs. [7, 8, 9].
+First let us go into the decay amplitude of W − → V ℓ−¯νℓ. Using the standard vertices
+Wℓ¯νℓ, γq¯q, and γWW, one can carry out the direct calculation for Figure 1, which gives
+M(W − → V ℓ−¯νℓ)
+=
+−e2gQV fV
+2
+√
+2mV
+ǫµ(p)ǫ∗
+ν(q)¯u(k1)
+�2kν
+1γµ + γνq/γµ
+q2 + 2k1 · q
+−(2k + q)νγµ + 2qµγν − 2q/gµν
+q2 + 2k · q
+�
+(1 − γ5)v(k2),
+(1)
+where p, q, k1, and k2 represent the momenta of W − and the final particles including V , ℓ−,
+and ¯νℓ, respectively. k = k1 + k2 denotes the momentum sum of lepton pair. e is the QED
+coupling constant and g is the weak SU(2)L coupling constant. fV is the decay constant of
+the vector meson, which is defined by
+⟨V (p, ǫ)|¯qγνq|0⟩ = fV mV ǫ∗
+ν.
+(2)
+Here ǫ∗
+ν is polarization vector of V , and the value of fV can be extracted from the measured
+V → e+e− width. As shown in Ref. [3], it has been already given that, fρ = 216.3 ± 1.3
+MeV, fω = 194.2 ± 2.1 MeV, fφ = 223.0 ± 1.4 MeV, and fJ/Ψ = 403.3 ± 5.1 MeV. QV is the
+quantity related to the electric charge of the quark inside V with Qρ = 1/
+√
+2, Qω = 1/3
+√
+2,
+Qφ = −1/3, and QJ/Ψ = 2/3. Note that the use of the relation (2) in deriving eq. (1) also
+fulfills the hadronization of the electromagnetic current ¯qγνq into the final state particle V .
+Next, by squaring the decay amplitude (1), summing or averaging the polarizations of
+final or initial particles, the differential decay rate of W − → V ℓ−¯νℓ can be expressed as
+dΓ
+dsV dsℓ
+= mW
+256π3
+1
+3
+�
+spins
+|M(W − → V ℓ−¯νℓ)|2.
+(3)
+Consequently, we get
+dΓ
+dsV dsℓ
+= α2
+emQ2
+V g2f 2
+V
+384πmWr2
+V
+IV ,
+(4)
+1
+
+*
+*
+*
+(a)
+(b)
+W
+W
+W
+V
+V
+�
+�
+�
+�
+��
+�
+�
+_
+_
+Figure 1: The lowest-order Feynman diagrams for W → V ℓ¯νℓ decays.
+where αem = e2/4π, rV = mV /mW, and the lepton mass has been neglected in the calcula-
+tion. The explicit expression of the dimensionless quantity IV is a little tedious, which will
+be shown in the Appendix. The Lorentz invariant dimensionless kinematical variables are
+defined as
+sV ≡ (p − q)2/m2
+W,
+sℓ ≡ (p − k1)2/m2
+W,
+(5)
+and the phase space can be given by
+0 ≤ sV ≤ (1 − sℓ)(1 − r2
+V /sℓ),
+r2
+V ≤ sℓ ≤ 1.
+(6)
+Meanwhile, it is easy to compute the leading-order contribution to the width of pure
+leptonic W-boson decay for ℓ = e or µ, which reads
+Γ(W − → ℓ−¯νℓ) = g2mW
+48π
+= GFm3
+W
+6
+√
+2π ≡ Γ0,
+(7)
+where GF is the Fermi constant given by GF/
+√
+2 = g2/8m2
+W.
+Then one can choose to
+normalize the decay rate of W − → V ℓ−ℓ¯νℓ to Γ0, which leads to
+1
+Γ0
+dΓ
+dsV dsℓ
+= α2
+emQ2
+V f 2
+V
+8m2
+V
+IV .
+(8)
+By further defining
+YV ≡
+�
+IV dsV dsℓ
+(9)
+with the integral bound is given in eq. (6), one can get
+Γ(W − → V ℓ−¯νℓ)
+Γ0
+= α2
+emQ2
+V f 2
+V
+8m2
+V
+YV .
+(10)
+As mentioned above, the decay constants (fV ) of the neutral vector mesons have been
+extracted by the authors of Ref. [3] from the experimental data, and
+Γ(V → e+e−) = 4πQ2
+V f 2
+V
+3mV
+α2
+em(mV )
+(11)
+2
+
+V
+mV (GeV)
+Γ(V → e+e−)(keV)
+YV
+Γ(W − → V ℓ−¯νℓ)/Γ0
+ρ
+0.775
+7.04 ± 0.06
+194.91
+(5.28 ± 0.04) × 10−5
+ω
+0.782
+0.60 ± 0.02
+193.94
+(4.44 ± 0.15) × 10−6
+φ
+1.019
+1.27 ± 0.04
+166.32
+(6.18 ± 0.19) × 10−6
+J/Ψ
+3.097
+5.53 ± 0.10
+74.53
+(3.97 ± 0.07) × 10−6
+Table 1: Decay rates of W − → V ℓ−¯νℓ normalized to Γ(W − → ℓ−¯νℓ) for ℓ = e or µ. The
+values of Γ(V → e+e−) are taken from Ref. [6].
+has been used. Therefore, after integrating over IV in eq. (9) to get YV , one can easily
+predict the decay rates of W → V ℓ¯νℓ for V = ρ, ω, φ, and J/Ψ, respectively.
+On the other hand, note that the scale of the electromagnetic coupling αem in eq. (8)
+should also be at mV since, in Figure 1, this electromagnetic transition is via γ∗ → V .
+Therefore, combing eq. (10) with eq. (11), one will obtain
+Γ(W − → V ℓ−¯νℓ)
+Γ0
+=
+3YV
+32πmV
+Γ(V → e+e−),
+(12)
+which means that we can get Γ(W − → V ℓ−¯νℓ)/Γ0 using the experimental data of Γ(V →
+e+e−) given by Particle Data Group [6] directly. Numerical results have been listed in Table
+1, and the errors of the predictions in the fifth column are due to the uncertainties of the
+measured widths of Γ(V → e+e−) only. To transform them into the branching fractions of
+W → V ℓ¯νℓ, one may use the experimental data of B(W → ℓ¯νℓ), which can be found in Ref.
+[6] that
+B(W − → e−¯νe) = (10.71 ± 0.16)%,
+B(W − → µ−¯νµ) = (10.63 ± 0.15)%.
+(13)
+For our numerical analysis, we take
+B(W − → ℓ−¯νℓ) = (10.67 ± 0.16)%
+(14)
+by simply averaging over the electron and muon modes. Thus, it is straightforward to obtain
+the branching fractions of rare W-boson decays into a vector meson and lepton pair, for ℓ = e
+or µ, which read
+B(W − → ρℓ−¯νℓ) = (5.64 ± 0.10) × 10−6,
+(15)
+B(W − → ωℓ−¯νℓ) = (4.74 ± 0.17) × 10−7,
+(16)
+B(W − → φℓ−¯νℓ) = (6.60 ± 0.23) × 10−7,
+(17)
+B(W − → J/Ψℓ−¯νℓ) = (4.24 ± 0.10) × 10−7.
+(18)
+Here the quoted errors of our theoretical results show the uncertainties from the experimental
+values of Γ(V → e+e−) in the third column of Table 1, and also B(W − → ℓ−¯νℓ) in eq. (14).
+It is found that branching ratios of W → V ℓ¯νℓ decays obtained in the present work are
+quite larger than those of the hadronic radiative decays W ± → M±γ (M is a pseudoscalar
+3
+
+W
+W
+J/�
+�
+�
+�
+_
+c
+s
+c
+_
+*
+Figure 2: The Feynman diagram contributing to W → J/Ψℓ¯νℓ decays via W → J/ΨW ∗
+transition.
+or vector meson such as π, K, ρ, K∗, and Ds etc), which are maximally around 10−8 or even
+smaller, predicted by the authors of Ref. [3]. Naively, one may expect that Γ(W → V ℓ¯νℓ)
+should be smaller than Γ(W ± → M±γ) since the former rate is suppressed by a power of
+αem compared to the latter rate. However, careful observation can tell us this expectation is
+not correct. As given in Ref. [3], we know
+Γ(W ± → M±γ) ∼ αemf 2
+M
+192mW
+.
+(19)
+Comparing with eq. (4), one will find a relevant factor m2
+W/m2
+V in the formula of Γ(W − →
+V ℓ−¯νℓ), which could significantly counteract the suppression of αem if the mass of vector
+meson is very small relative to the W mass. Obviously, the appearance of this factor is
+actually due to the virtual photon propagator of γ∗ → V transition in Figure 1.
+Similar situation also occurs in rare Z-boson decays. In particular, it has been shown in
+Ref. [8] that the dominant contribution to Z → V ℓ+ℓ− comes from Z → γ∗ℓ+ℓ− with the
+subsequent transition γ∗ → V , since, in comparison, the radiative decays Z → V γ are quite
+suppressed. One can thus neglect the contribution from Z → V γ∗ → V ℓ+ℓ− although it is
+of the same order of αem as the dominant part.
+Analogous to Z → V γ∗ → V ℓ+ℓ−, the rare charged weak gauge boson decays considered
+in the present paper could happen through W → V W ∗ → V ℓ¯νℓ. The Feynman diagram
+has been displayed in Figure 2, and we take V = J/Ψ as an explicit example. As a good
+approximation for the leading order calculation, the momenta of the quark (c) and anti-
+quark (¯c) are taken to be one half of J/Ψ momentum q, so the strange quark propagator in
+this diagram is proportional to 1/(k + q
+2)2, which is of order 1/m2
+W. By contrast, the virtual
+photon propagator in the diagrams of Figure 1 is of order 1/m2
+J/Ψ. This means that the
+contribution from Figure 2, relative to that from Figure 1, is strongly suppressed, which can
+be safely neglected.
+Furthermore, recall that the differential decay rate of W − → V ℓ−¯νℓ has been given in
+eq. (4). Now one can rewrite
+sV = 1 + r2
+V − 2EV /mW,
+sℓ = 1 − 2Eℓ/mW,
+(20)
+where EV is the vector meson energy and Eℓ is the lepton energy in the rest frame of W
+4
+
+0
+10
+20
+30
+40
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+EJ (GeV)
+1/
+�
+d
+�
+/dEJ (GeV-1)
+0
+10
+20
+30
+40
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+0.06
+Eℓ
+(GeV)
+1/Γ dΓ/dEℓ (GeV-1)
+Figure 3: The normalized energy spectrum of W → J/Ψℓ−¯νℓ decays with respect to J/Ψ
+energy EJ (left plot), and with respect to the lepton energy Eℓ (right plot).
+boson. In terms of EV and Eℓ, we have
+dΓ
+dEV dEℓ
+= α2
+emQ2
+V g2f 2
+V
+96πm3
+Wr2
+V
+IV .
+(21)
+Thus the energy spectrum of the rare decays can be obtained by integrating over EV or Eℓ.
+The normalized energy distributions of W − → J/Ψℓ−¯νℓ with respect to EJ and Eℓ have
+been plotted in Figure 3, respectively. The peak of the distribution is corresponding to the
+small J/ψ energy or large lepton energy region. Since we have neglected the lepton mass in
+the calculation, the spectrum in left plot does not go to zero for EJ at ∼ mW/2. We are not
+going to display the plots for the differential rate of W − → V ℓ−¯νℓ decays when V is the light
+vector meson (ρ, ω, and φ) because it is believed that one will achieve the similar behavior
+as above.
+To summarize, we have presented the analysis of exclusive rare W-boson decays into a
+vector meson and lepton pair. In the SM, the leading order contributions to these processes
+come from W → γ∗ℓ¯νℓ, followed by γ∗ → V . Using the measured widths of Γ(V → e+e−)
+given in [6], we have determined the branching fractions of W − → V ℓ−¯νℓ for V = ρ, ω,
+φ, and J/Ψ, respectively, as shown in eqs.
+(15) – (18).
+It is surprising that branching
+fractions of these three-body decays, although they are suppressed by a power of αem, are
+quite larger than those of two-body hadronic radiative decays W ± → M±γ, which have been
+predicted by the authors of Ref. [3] already. Furthermore, note that the γWW vertex, as
+shown in Figure 1(b), is involved in the transition, thus both experimental and theoretical
+investigations of W → V ℓ¯νℓ decays may be also helpful to test triple gauge couplings.
+Our experimentalists have been trying to search for exclusive rare W-boson processes
+containing hadronic final states. Unfortunately, so far no such decays have been observed.
+Theoretical predictions on branching fractions of W → V ℓ¯νℓ in the present paper are around
+10−6 ∼ 10−7. Experimentally, the heavy quarkonium J/ψ is in general reconstructed via
+leptonic decays with their rates: B(J/Ψ → ℓ+ℓ−) = (5.971±0.032)% [6]; while for light vector
+mesons, ρ decays almost exclusively to π+π−, ω and φ have a large rate into π+π−π− and
+K+K−, respectively, in the event construction. Therefore, our analysis seems to indicate that
+5
+
+these exclusive rare W decay modes could be the promising candidates in future experimental
+machines, for instance, in the high-luminosity LHC, where large amount of W bosons about
+O(1011) events will be produced. We eagerly await some dedicated searches for such decays
+at these facilities.
+Acknowledgments
+This work was supported in part by the National Natural Science Foundation of China
+under Grants No. 11575175, No. 12047502, and No. 12247103, and by National Research
+and Development Program of China under Contract No. 2020YFA0406400.
+Appendix: Explicit expression of IV
+After squaring the W − → V ℓ−¯νℓ decay amplitude and summing/averaging spins of all par-
+ticles, we get the differential decay rate of eq. (4) as
+dΓ
+dsV dsℓ
+= α2
+emQ2
+V g2f 2
+V
+384πmWr2
+V
+IV .
+The full expression of IV , in terms of dot product of the relevant four-momenta, can be given
+by
+IV = I1 + I2 + I3,
+(A1)
+where
+I1 =
+8
+(q2 + 2k1 · q)2
+�
+(2k1 · qk2 · q − q2k1 · k2) + 2p · k2
+m2
+W
+(2k1 · qp · q − q2k1 · p)
+�
+,
+(A2)
+I2 =
+8
+(q2 + 2k1 · q)(q2 + 2k · q)
+�
+2k1 · k2(4k1 · k2 + 2k · q + 4q2) − 4k1 · qk2 · q
+− 1
+m2
+W
+[(2k1 · q + 4k · q + 3q2)(2k1 · qk2 · q − q2k1 · k2)
+−4p · k2(q2 + 2k2 · q)k1 · k2 + 4p · q(q2 + 2k1 · q)k1 · k2]
+�
+,
+(A3)
+and
+I3 =
+8
+(q2 + 2k · q)2
+�
+12k1 · qk2 · q − ((2k + q)2 + 10q2)k1 · k2
+−
+1
+2m2
+W
+(2k + q)2(2k1 · qk2 · q − q2k1 · k2) + 4(p · q)2
+m2
+W
+�
+.
+(A4)
+On the other hand, one can easily get
+k1 · k2 = m2
+W
+2 sV ,
+p · q = m2
+W
+2 (1 + r2
+V − sV ),
+6
+
+k1 · q = m2
+W
+2 (1 − sV − sℓ),
+k2 · q = m2
+W
+2 (sℓ − r2
+V ),
+k1 · p = m2
+W
+2 (1 − sℓ),
+k2 · p = m2
+W
+2 (sV + sℓ − r2
+V ).
+For on-shell initial and final states particles, we could take p2 = m2
+W, q2 = m2
+V , and k2
+1 =
+k2
+2 = 0 (lepton masses are set to be zero already). This shows that IV can be in terms of the
+kinematical variables sV and sℓ completely.
+References
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+[2] M. Mangano and T. Melia, Eur. Phys. J. C 75, 258 (2015), arXiv:1410.7475 [hep-ph].
+[3] Y. Grossman, M. K¨onig, and M. Neubert, J. High Energy Phys. 04 (2015) 101,
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+7
+
diff --git a/-NFQT4oBgHgl3EQf6jZ7/content/tmp_files/load_file.txt b/-NFQT4oBgHgl3EQf6jZ7/content/tmp_files/load_file.txt
new file mode 100644
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--- /dev/null
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@@ -0,0 +1,261 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf,len=260
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='13439v1  [hep-ph]  31 Jan 2023  USTC-ICTS/PCFT-23-04  January 2023  Rare W-boson decays into a vector meson and lepton pair  Dao-Neng Gao†  Interdisciplinary Center for Theoretical Study,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Anhui 230026 China  Peng Huanwu Center for Fundamental Theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Hefei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Anhui 230026 China  Abstract  We have presented a theoretical study of exclusive rare W-boson decays,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' W → V ℓ¯νℓ  with V denoting a neutral vector meson and ℓ = e or µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' in the standard model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  The leading-order contributions to these processes are given by W → γ∗ℓ¯νℓ with the  subsequent γ∗ → V transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Branching fractions of these decay modes, for V = ρ,  ω, φ, and J/Ψ, respectively, have been calculated and predicted around 10−6 ∼ 10−7,  which are surprisingly larger than those of two-body hadronic radiative decays W ± →  M±γ with M denoting a pseudoscalar or vector meson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Thus it is expected that rare  W decays into a neutral vector meson plus lepton pair may be the promising channels  in future experimental facilities with a large number of W-boson events produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  † E-mail address: gaodn@ustc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='cn  Exclusive rare W-boson decays, which contain hadronic final states, could provide inter-  esting probes to increase our understanding of the properties of the fundamental weak gauge  boson as well as offer some deep insights into quantum chromodynamics [1, 2, 3, 4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Experi-  mentally, no such processes have been observed so far, and only upper limits on the branching  fractions of three exclusive modes: B(W ± → D±  s γ) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='3×10−3, B(W ± → π±γ) < 7×10−6,  and B(W ± → π+π−π±) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='01 × 10−6, were set at 95% confidence level [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' On the other  hand, a huge number of W events, about O(1011), will be expectedly accumulated in the  high-luminosity Large Hadron Collider (LHC) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' This may significantly facilitate the ex-  perimental studies of rare W-boson decay channels, which can be very helpful both to test  the standard model (SM) and to search for new physics beyond the SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Our main focus in the present paper is on another types of rare W-boson decays: W →  V ℓ¯νℓ with V denoting the neutral vector particle including heavy quarkonium J/Ψ or light  mesons ρ, ω, and φ etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  ℓ is the lepton with ℓ = e or µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  The leading-order Feynman  diagrams contributing to these processes in the SM have been shown in Figure 1, in which  the transitions can proceed through W → γ∗ℓ¯νℓ, followed by γ∗ → ¯qq → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' This is similar  to the case of Z → V ℓ+ℓ+ decays, which have been studied in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' [7, 8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  First let us go into the decay amplitude of W − → V ℓ−¯νℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Using the standard vertices  Wℓ¯νℓ, γq¯q, and γWW, one can carry out the direct calculation for Figure 1, which gives  M(W − → V ℓ−¯νℓ)  =  −e2gQV fV  2  √  2mV  ǫµ(p)ǫ∗  ν(q)¯u(k1)  �2kν  1γµ + γνq/γµ  q2 + 2k1 · q  −(2k + q)νγµ + 2qµγν − 2q/gµν  q2 + 2k · q  �  (1 − γ5)v(k2),  (1)  where p, q, k1, and k2 represent the momenta of W − and the final particles including V , ℓ−,  and ¯νℓ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' k = k1 + k2 denotes the momentum sum of lepton pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' e is the QED  coupling constant and g is the weak SU(2)L coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' fV is the decay constant of  the vector meson, which is defined by  ⟨V (p, ǫ)|¯qγνq|0⟩ = fV mV ǫ∗  ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (2)  Here ǫ∗  ν is polarization vector of V , and the value of fV can be extracted from the measured  V → e+e− width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' [3], it has been already given that, fρ = 216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='3  MeV, fω = 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='1 MeV, fφ = 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='4 MeV, and fJ/Ψ = 403.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='3 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='1 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' QV is the  quantity related to the electric charge of the quark inside V with Qρ = 1/  √  2, Qω = 1/3  √  2,  Qφ = −1/3, and QJ/Ψ = 2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Note that the use of the relation (2) in deriving eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (1) also  fulfills the hadronization of the electromagnetic current ¯qγνq into the final state particle V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Next, by squaring the decay amplitude (1), summing or averaging the polarizations of  final or initial particles, the differential decay rate of W − → V ℓ−¯νℓ can be expressed as  dΓ  dsV dsℓ  = mW  256π3  1  3  �  spins  |M(W − → V ℓ−¯νℓ)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (3)  Consequently, we get  dΓ  dsV dsℓ  = α2  emQ2  V g2f 2  V  384πmWr2  V  IV ,  (4)  1 (a)  (b)  W  W  W  V  V  �  �  �  �  ��  �  �  _  _  Figure 1: The lowest-order Feynman diagrams for W → V ℓ¯νℓ decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  where αem = e2/4π, rV = mV /mW, and the lepton mass has been neglected in the calcula-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' The explicit expression of the dimensionless quantity IV is a little tedious, which will  be shown in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' The Lorentz invariant dimensionless kinematical variables are  defined as  sV ≡ (p − q)2/m2  W,  sℓ ≡ (p − k1)2/m2  W,  (5)  and the phase space can be given by  0 ≤ sV ≤ (1 − sℓ)(1 − r2  V /sℓ),  r2  V ≤ sℓ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (6)  Meanwhile, it is easy to compute the leading-order contribution to the width of pure  leptonic W-boson decay for ℓ = e or µ, which reads  Γ(W − → ℓ−¯νℓ) = g2mW  48π  = GFm3  W  6  √  2π ≡ Γ0,  (7)  where GF is the Fermi constant given by GF/  √  2 = g2/8m2  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Then one can choose to  normalize the decay rate of W − → V ℓ−ℓ¯νℓ to Γ0, which leads to  1  Γ0  dΓ  dsV dsℓ  = α2  emQ2  V f 2  V  8m2  V  IV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (8)  By further defining  YV ≡  �  IV dsV dsℓ  (9)  with the integral bound is given in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (6), one can get  Γ(W − → V ℓ−¯νℓ)  Γ0  = α2  emQ2  V f 2  V  8m2  V  YV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (10)  As mentioned above, the decay constants (fV ) of the neutral vector mesons have been  extracted by the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' [3] from the experimental data, and  Γ(V → e+e−) = 4πQ2  V f 2  V  3mV  α2  em(mV )  (11)  2  V  mV (GeV)  Γ(V → e+e−)(keV)  YV  Γ(W − → V ℓ−¯νℓ)/Γ0  ρ  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='775  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='06  194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='91  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='28 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='04) × 10−5  ω  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='782  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='02  193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='94  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='15) × 10−6  φ  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='019  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='27 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='04  166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='32  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='19) × 10−6  J/Ψ  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='097  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='10  74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='53  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='07) × 10−6  Table 1: Decay rates of W − → V ℓ−¯νℓ normalized to Γ(W − → ℓ−¯νℓ) for ℓ = e or µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' The  values of Γ(V → e+e−) are taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Therefore, after integrating over IV in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (9) to get YV , one can easily  predict the decay rates of W → V ℓ¯νℓ for V = ρ, ω, φ, and J/Ψ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  On the other hand, note that the scale of the electromagnetic coupling αem in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (8)  should also be at mV since, in Figure 1, this electromagnetic transition is via γ∗ → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Therefore, combing eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (10) with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (11), one will obtain  Γ(W − → V ℓ−¯νℓ)  Γ0  =  3YV  32πmV  Γ(V → e+e−),  (12)  which means that we can get Γ(W − → V ℓ−¯νℓ)/Γ0 using the experimental data of Γ(V →  e+e−) given by Particle Data Group [6] directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Numerical results have been listed in Table  1, and the errors of the predictions in the fifth column are due to the uncertainties of the  measured widths of Γ(V → e+e−) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' To transform them into the branching fractions of  W → V ℓ¯νℓ, one may use the experimental data of B(W → ℓ¯νℓ), which can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  [6] that  B(W − → e−¯νe) = (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='16)%,  B(W − → µ−¯νµ) = (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='15)%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (13)  For our numerical analysis, we take  B(W − → ℓ−¯νℓ) = (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='16)%  (14)  by simply averaging over the electron and muon modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Thus, it is straightforward to obtain  the branching fractions of rare W-boson decays into a vector meson and lepton pair, for ℓ = e  or µ, which read  B(W − → ρℓ−¯νℓ) = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='10) × 10−6,  (15)  B(W − → ωℓ−¯νℓ) = (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='17) × 10−7,  (16)  B(W − → φℓ−¯νℓ) = (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='23) × 10−7,  (17)  B(W − → J/Ψℓ−¯νℓ) = (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='10) × 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (18)  Here the quoted errors of our theoretical results show the uncertainties from the experimental  values of Γ(V → e+e−) in the third column of Table 1, and also B(W − → ℓ−¯νℓ) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  It is found that branching ratios of W → V ℓ¯νℓ decays obtained in the present work are  quite larger than those of the hadronic radiative decays W ± → M±γ (M is a pseudoscalar  3  W  W  J/�  �  �  �  _  c  s  c  _ Figure 2: The Feynman diagram contributing to W → J/Ψℓ¯νℓ decays via W → J/ΨW ∗  transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  or vector meson such as π, K, ρ, K∗, and Ds etc), which are maximally around 10−8 or even  smaller, predicted by the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Naively, one may expect that Γ(W → V ℓ¯νℓ)  should be smaller than Γ(W ± → M±γ) since the former rate is suppressed by a power of  αem compared to the latter rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' However, careful observation can tell us this expectation is  not correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' As given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' [3], we know  Γ(W ± → M±γ) ∼ αemf 2  M  192mW  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (19)  Comparing with eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (4), one will find a relevant factor m2  W/m2  V in the formula of Γ(W − →  V ℓ−¯νℓ), which could significantly counteract the suppression of αem if the mass of vector  meson is very small relative to the W mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Obviously, the appearance of this factor is  actually due to the virtual photon propagator of γ∗ → V transition in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Similar situation also occurs in rare Z-boson decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' In particular, it has been shown in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' [8] that the dominant contribution to Z → V ℓ+ℓ− comes from Z → γ∗ℓ+ℓ− with the  subsequent transition γ∗ → V , since, in comparison, the radiative decays Z → V γ are quite  suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' One can thus neglect the contribution from Z → V γ∗ → V ℓ+ℓ− although it is  of the same order of αem as the dominant part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Analogous to Z → V γ∗ → V ℓ+ℓ−, the rare charged weak gauge boson decays considered  in the present paper could happen through W → V W ∗ → V ℓ¯νℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' The Feynman diagram  has been displayed in Figure 2, and we take V = J/Ψ as an explicit example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' As a good  approximation for the leading order calculation, the momenta of the quark (c) and anti-  quark (¯c) are taken to be one half of J/Ψ momentum q, so the strange quark propagator in  this diagram is proportional to 1/(k + q  2)2, which is of order 1/m2  W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' By contrast, the virtual  photon propagator in the diagrams of Figure 1 is of order 1/m2  J/Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' This means that the  contribution from Figure 2, relative to that from Figure 1, is strongly suppressed, which can  be safely neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Furthermore, recall that the differential decay rate of W − → V ℓ−¯νℓ has been given in  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Now one can rewrite  sV = 1 + r2  V − 2EV /mW,  sℓ = 1 − 2Eℓ/mW,  (20)  where EV is the vector meson energy and Eℓ is the lepton energy in the rest frame of W  4  0  10  20  30  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='06  EJ (GeV)  1/  �  d  �  /dEJ (GeV-1)  0  10  20  30  40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='06  Eℓ  (GeV)  1/Γ dΓ/dEℓ (GeV-1)  Figure 3: The normalized energy spectrum of W → J/Ψℓ−¯νℓ decays with respect to J/Ψ  energy EJ (left plot), and with respect to the lepton energy Eℓ (right plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  boson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' In terms of EV and Eℓ, we have  dΓ  dEV dEℓ  = α2  emQ2  V g2f 2  V  96πm3  Wr2  V  IV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (21)  Thus the energy spectrum of the rare decays can be obtained by integrating over EV or Eℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  The normalized energy distributions of W − → J/Ψℓ−¯νℓ with respect to EJ and Eℓ have  been plotted in Figure 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' The peak of the distribution is corresponding to the  small J/ψ energy or large lepton energy region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Since we have neglected the lepton mass in  the calculation, the spectrum in left plot does not go to zero for EJ at ∼ mW/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' We are not  going to display the plots for the differential rate of W − → V ℓ−¯νℓ decays when V is the light  vector meson (ρ, ω, and φ) because it is believed that one will achieve the similar behavior  as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  To summarize, we have presented the analysis of exclusive rare W-boson decays into a  vector meson and lepton pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' In the SM, the leading order contributions to these processes  come from W → γ∗ℓ¯νℓ, followed by γ∗ → V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Using the measured widths of Γ(V → e+e−)  given in [6], we have determined the branching fractions of W − → V ℓ−¯νℓ for V = ρ, ω,  φ, and J/Ψ, respectively, as shown in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (15) – (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  It is surprising that branching  fractions of these three-body decays, although they are suppressed by a power of αem, are  quite larger than those of two-body hadronic radiative decays W ± → M±γ, which have been  predicted by the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' [3] already.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Furthermore, note that the γWW vertex, as  shown in Figure 1(b), is involved in the transition, thus both experimental and theoretical  investigations of W → V ℓ¯νℓ decays may be also helpful to test triple gauge couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Our experimentalists have been trying to search for exclusive rare W-boson processes  containing hadronic final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Unfortunately, so far no such decays have been observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Theoretical predictions on branching fractions of W → V ℓ¯νℓ in the present paper are around  10−6 ∼ 10−7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Experimentally, the heavy quarkonium J/ψ is in general reconstructed via  leptonic decays with their rates: B(J/Ψ → ℓ+ℓ−) = (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='971±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='032)% [6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' while for light vector  mesons, ρ decays almost exclusively to π+π−, ω and φ have a large rate into π+π−π− and  K+K−, respectively, in the event construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Therefore, our analysis seems to indicate that  5  these exclusive rare W decay modes could be the promising candidates in future experimental  machines, for instance, in the high-luminosity LHC, where large amount of W bosons about  O(1011) events will be produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' We eagerly await some dedicated searches for such decays  at these facilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Acknowledgments  This work was supported in part by the National Natural Science Foundation of China  under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' 11575175, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' 12047502, and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' 12247103, and by National Research  and Development Program of China under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' 2020YFA0406400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  Appendix: Explicit expression of IV  After squaring the W − → V ℓ−¯νℓ decay amplitude and summing/averaging spins of all par-  ticles, we get the differential decay rate of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (4) as  dΓ  dsV dsℓ  = α2  emQ2  V g2f 2  V  384πmWr2  V  IV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  The full expression of IV ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' in terms of dot product of the relevant four-momenta,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' can be given  by  IV = I1 + I2 + I3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (A1)  where  I1 =  8  (q2 + 2k1 · q)2  �  (2k1 · qk2 · q − q2k1 · k2) + 2p · k2  m2  W  (2k1 · qp · q − q2k1 · p)  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (A2)  I2 =  8  (q2 + 2k1 · q)(q2 + 2k · q)  �  2k1 · k2(4k1 · k2 + 2k · q + 4q2) − 4k1 · qk2 · q  − 1  m2  W  [(2k1 · q + 4k · q + 3q2)(2k1 · qk2 · q − q2k1 · k2)  −4p · k2(q2 + 2k2 · q)k1 · k2 + 4p · q(q2 + 2k1 · q)k1 · k2]  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (A3)  and  I3 =  8  (q2 + 2k · q)2  �  12k1 · qk2 · q − ((2k + q)2 + 10q2)k1 · k2  −  1  2m2  W  (2k + q)2(2k1 · qk2 · q − q2k1 · k2) + 4(p · q)2  m2  W  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  (A4)  On the other hand, one can easily get  k1 · k2 = m2  W  2 sV ,  p · q = m2  W  2 (1 + r2  V − sV ),  6  k1 · q = m2  W  2 (1 − sV − sℓ),  k2 · q = m2  W  2 (sℓ − r2  V ),  k1 · p = m2  W  2 (1 − sℓ),  k2 · p = m2  W  2 (sV + sℓ − r2  V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  For on-shell initial and final states particles, we could take p2 = m2  W, q2 = m2  V , and k2  1 =  k2  2 = 0 (lepton masses are set to be zero already).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' This shows that IV can be in terms of the  kinematical variables sV and sℓ completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  References  [1] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Arnellos, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Marciano, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Parsa, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' B196, 378 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Mangano and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Melia, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' C 75, 258 (2015), arXiv:1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='7475 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  [3] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Grossman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' K¨onig, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Neubert, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' 04 (2015) 101,  arXiv:1501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='06569 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  [4] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Keum and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Pham, Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' A 9, 1545 (1994), hep-ph/9303300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Ishaq, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Jia, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Xiong, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Yang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' D 100, 054027 (2019),  arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='12627 [hep-ph];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Feng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Jia, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Sang, arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='11288 [hep-  ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  [6] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Workman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' (Particle Data Group), Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' 2022, 083C01  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  [7] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Bergstr¨om and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Robinett, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' B 245, 249 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Fleming, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' D 48, R1914 (1993), hep-ph/9304270;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Fleming, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  D 50, 5808 (1994), hep-ph/9403396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  [9] CMS Collaboration, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Sirunyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content=' 121, 141801 (2018),  arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='04213 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
+page_content='  7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NFQT4oBgHgl3EQf6jZ7/content/2301.13439v1.pdf'}
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+arXiv:2301.12143v1  [math.NT]  28 Jan 2023
+The endoscopic classification of representations of non-quasi-split odd
+special orthogonal groups
+Hiroshi Ishimoto∗
+[January 31, 2023]
+Abstract
+In an earlier book of Arthur, the endoscopic classification of representations of quasi-split orthogonal and
+symplectic groups was established. Later Mok gave that of quasi-split unitary groups. After that, Kaletha, Minguez,
+Shin, and White gave that of non-quasi-split unitary groups for generic parameters. In this paper we prove the
+endoscopic classification of representations of non-quasi-split odd special orthogonal groups for generic parameters,
+following Kaletha, Minguez, Shin, and White.
+Contents
+1
+Introduction
+1
+2
+Odd special orthogonal groups
+5
+3
+Notions around endoscopy and parameters, and Main theorem
+9
+4
+Local intertwining relation
+22
+5
+The decomposition into near equivalence classes and the standard model
+49
+6
+Globalizations and the proof of local classification
+54
+7
+The proof of the global theorem
+66
+1
+Introduction
+Background & Main result
+In a 2013 book [7], Arthur established the endoscopic classification of irreducible representations of quasi-split
+special orthogonal and symplectic groups over local fields of characteristic zero and of automorphic representations of
+those groups over global fields of characteristic zero. To local A-parameters he attached local A-packets characterized
+by the endoscopic character relation (which we shall call ECR for short), and proved that the local A-packets for
+generic parameters give the local Langlands correspondence (which we shall call LLC for short), and that automorphic
+representations are classified in terms of automorphic cuspidal representations of general linear groups and local
+A-packets.
+Let us roughly recall his result. Let first F be a local field of characteristic zero, and G a split odd special orthogonal
+group over F for simplicity. A local A-parameter for G is a homomorphism
+ψ : LF × SU(2, R) → LG,
+∗ishimoto.hiroshi.55m@gmail.com
+1
+
+with some conditions, where LF and LG denote the Langlands group of F and the L-group of G, respectively. For every
+A-parameter ψ, Arthur first constructed a multiset Πψ(G) over the set Πunit(G) of equivalence classes of irreducible
+unitary representations of G(F), and a mapping Πψ(G) → Irr(π0(Sψ)), where Sψ denotes the centralizer of ψ in
+the Langlands dual group “
+G of G, and then he proved that they satisfy ECR. An A-parameter is called a bounded
+L-parameter if its restriction to SU(2, R) is trivial. He also proved LLC by showing that if ψ is a bounded L-parameter
+then the packet Πψ(G) gives the L-packet, which has been expected to exist. Let us recall here the basic form of LLC,
+which is a conjecture in general.
+Conjecture 1.1 (LLC). Let G be a connected reductive algebraic group over a local field F. Let Πtemp(G) be the set
+of equivalence classes of irreducible tempered representations of G(F), and Φbdd(G) the set of equivalence classes of
+bounded L-parameters. There exists a canonical map
+LL : Πtemp(G) −→ Φbdd(G),
+such that for each φ ∈ Φbdd(G), the fiber Πφ(G) = LL−1(φ) is a finite set and there is an injective map ιφ : Πφ(G) →
+Irr(π0(Sφ)). These two correspondences satisfy some interesting properties. The finite set Πφ(G) is called the L-packet
+of φ.
+We refer the reader to [17] for details.
+On the other hand let next F be a number field, and G a split odd special orthogonal group over F. A global
+A-parameter for G is a formal finite unordered sum
+ψ =⊞
+i
+πi ⊠ νi,
+with some conditions, where πi is an irreducible cuspidal automorphic representation of a general linear group, and
+νi is an irreducible finite dimensional representation of SU(2, R).
+For all place v of F, we have the localization
+ψv = �
+i φi,v ⊠ νi, where φi,v is the unique L-parameter for a general linear group over Fv corresponding to πi,v
+via LLC. Then ψv is a local A-parameter for Gv over Fv. (Strictly speaking, it may not be an A-parameter, but a
+packet for it is defined.) Arthur determined an appropriate subset Πψ(εψ) of {�
+v πv | πv ∈ Πψv(Gv)} and showed
+the decomposition
+L2
+disc(G(F)\G(AF )) =
+�
+ψ
+L2
+disc,ψ(G(F)\G(AF )),
+L2
+disc,ψ(G(F)\G(AF )) =
+�
+π∈Πψ(εψ)
+π,
+(1.1)
+of the discrete spectrum into near equivalence classes, and then into irreducible automorphic representations. We shall
+call a decomposition like (1.1) ”Arthur’s multiplicity formula”, and abbreviate it as AMF.
+Later, Mok [30] proved ECR, LLC, and AMF for quasi-split unitary groups by the similar argument, and Kaletha-
+Minguez-Shin-White [18] partially proved those for non-quasi-split unitary groups. In this paper, following [18], we
+shall partially prove the analogous classification (ECR, LLC, and AMF) for non-quasi-split odd special orthogonal
+groups by the similar argument. In other words, our main theorem (Theorem 3.14) is the following:
+Theorem 1.2. Over a local field of characteristic zero, LLC holds for any non-quasi-split odd special orthogonal
+group, and the L-packets equipped with mappings ιφ satisfy ECR. Over a global field of characteristic zero, AMF holds
+for any non-quasi-split odd special orthogonal group, except the irreducible decompositions of L2
+disc,ψ(G(F)\G(AF )) for
+non-generic parameters ψ.
+The most important difference between this paper and [18] is the proof of the local intertwining relation. The local
+intertwining relation ([18, Theorem* 2.6.2], Theorem 4.13 in this paper), for which we shall write LIR for short, is a
+key theorem in the proof of the local main theorems ECR and LLC. By the similar idea to [18], we can reduce the
+proof of LIR to that for two special cases: the real special orthogonal groups SO(1, 4) and SO(2, 5) relative to Levi
+subgroups isomorphic to GL1 × SO(0, 3) and GL2 × SO(0, 3) respectively, and the special parameters. In the case of
+SO(1, 4), the special representation of the Levi subgroup under consideration is the trivial representation, and hence
+we can prove LIR by the similar argument to §2.9 in loc.cit. However, in the case of SO(2, 5), the situation is too
+complicated to calculate similarly to loc. cit., since the representation of the Levi subgroup is infinite dimensional.
+For this reason in this paper, we shall prove them by a completely different argument. We choose a test function using
+2
+
+the Iwasawa decomposition, while the relative Bruhat decomposition was used in loc. cit. The argument will appear
+in §4.9.
+We remark that LLC of non-quasi-split odd special orthogonal groups has already studied by Mœglin-Renard [28],
+but their result does not contain LIR. Thus our local theorem is differentiated from their work.
+Application
+In [7, Chapter 9], Arthur formulated the classification for non-quasi-split symmetric and orthogonal groups, and he
+has the intention of proving it. The results of this paper are included in his project, but we have a different motivation,
+the representation theory of the metaplectic groups. It is one of the most important application of this paper. Let us
+recall the results on the metaplectic groups by Adams, Barbasch, Gan, Savin, and Ichino.
+Let F be a local field. The metaplectic group, denoted by Mp2n(F), is a unique nonlinear two-fold cover of Sp2n(F)
+except F = C, in which case Mp2n(C) = Sp2n(C) × {±1}. We identify Mp2n(F) with Sp2n(F) × {±1} as sets, and
+a representation π of Mp2n(F) is said to be genuine if π((1, −1)) is not trivial. Let Πtemp(Mp2n) denote the set of
+equivalence classes of genuine tempered irreducible representations of Mp2n(F). Adams-Barbasch [2, 3], Adams [1]
+(archimedean case), and Gan-Savin [12] (non-archimedean case) showed that the local theta correspondence gives a
+canonical bijection
+Πtemp(Mp2n) ←→
+�
+V
+Πtemp(SO(V )),
+where V runs over all (2n + 1)-dimensional quadratic space of discriminant 1 over F. Thanks to their results, LLC
+for Mp2n is implied by that for all SO(V ). In addition, there is an article [16] which proved LIR for Mp2n assuming
+that for all SO(V ) over a p-adic field.
+Let next F be a number field. The metaplectic group Mp2n(AF ) is a nontrivial two-fold cover of Sp2n(AF ), and there
+is a canonical injective homomorphism Sp2n(F) ֒→ Mp2n(AF ). Hence the notions of ”automorphic representations”
+and ”discrete spectrum” are defined in a canonical way. AMF for the metaplectic group was studied by Gan-Ichino
+[11]. They proved the decomposition of the discrete spectrum of Mp2n into near equivalence classes without any
+assumption, and proved the decomposition into irreducible automorphic representations of the generic part assuming
+that for all non-quasi-split odd special orthogonal groups.
+Those theories will be completed by this paper. Namely, LLC and LIR for the metaplectic groups will hold true,
+and the result of Gan-Ichino [11] on the generic part of AMF will be unconditional.
+Organization
+§2 is the preliminary section, where some notations for odd special orthogonal groups are established.
+In §3 we first recall the notions of endoscopic triple, transfer factor, local and global parameters, and the canonical
+correspondence (e, ψe) ↔ (ψ, s). Next we shall state LLC, ECR, and AMF more precisely. We will recall the result of
+Arthur [7] and state our main theorem (Theorem 3.14).
+§4 is the most important section in this paper. We define the local intertwining operator, state LIR, and reduce
+its proof to the case when the parameter is discrete for M ∗ and elliptic or exceptional for G∗, following [18, 7, 30].
+Then in §4.9 we give a proof of LIR for the special case explained above.
+In §5, we will recall the global theory on the trace formula, and obtain some lemmas. Proofs of some lemmas are
+omitted since they are quite similar to those in [18, §3].
+In §6, we complete the proof of the local main theorem by the argument involving globalizations and trace Paley-
+Wiener theorem.
+In §7, we complete the proof of the global main theorem.
+Convention & Notation
+We marked some theorems, lemmas, and propositions with symbol * to indicate that they are only proved for
+generic parameters in this paper.
+(We omit * when we refer to them.)
+The author expect that their proofs for
+non-generic parameters will be completed by an analogous argument to a sequel of [18].
+We do not use ”S” in this paper. Instead, we shall use ”S” to denote the component groups.
+3
+
+In this paper every field is assumed to be of characteristic zero. In particular, a local field is either R, C, or a finite
+extension of Qp for some prime number p ∈ Z, and a global field is a number field, i.e., a finite extension of Q. For
+a field F, we write F for its algebraic closure, and Γ = ΓF for its absolute Galois group Gal(F/F). For a connected
+reductive algebraic group G over F, we write e(G) for the Kottwitz sign ([21]) of G, and “
+G for the dual group over
+C. If moreover F is a local or global field, we write WF for the absolute Weil group of F, and the Weil form of the
+L-group is defined by LG = “
+G ⋊ WF .
+If F is a number field, we write AF for the ring of adeles of F. We often fix a nontrivial additive character of
+F\AF , which always denoted by ψF . For each place v of F, we write ψF,v for the local component of ψF at v. We
+abbreviate ΓFv as Γv. Following [7] and [18], we do not use a symbol �′
+v for a restricted tensor product. We simply
+write �
+v. Similarly, we write �
+v in place of �′
+v. Unless otherwise specified, �
+v, �
+v, �
+v, and �
+v denote the certain
+operations taken over all places v of F, respectively.
+If F is a local field, the Langlands group is defied as follows.
+LF =
+®WF × SU(2, R),
+if F is non-archimedean,
+WF ,
+if F is archimedean.
+As in the global case, we often fix a nontrivial additive character of F, which always denoted by ψF .
+For an algebraic or abstract group G, its center is denoted by Z(G).
+In addition, when X is a subgroup or
+an element of G, we write Cent(X, G), ZX(G), or Z(X, G) (resp.
+NG(X) or N(X, G)) for the centralizer (resp.
+normalizer) of X in G.
+For a topological group G, its connected component of the identity element is denoted by G◦, and we put π0(G) =
+G/G◦.
+For an algebraic group G over a field F, put X∗(G) = Hom(G, GL1) and X∗(G) = Hom(GL1, G), which are
+equipped with the F-structure. Put moreover
+aG = Hom(X∗(G)F , R),
+a∗
+G = X∗(G)F ⊗Z R,
+aG,C = Hom(X∗(G)F , C),
+a∗
+G,C = X∗(G)F ⊗Z C.
+For any representation π, let π∨ denote its contragredient representation. If π is a representation of a topological
+group G with a Haar measure dg, for a function f on G, let fG(π) denote its character i.e.,
+fG(π) = tr(π(f)),
+where
+π(f) =
+�
+G
+f(g)π(g)dg.
+We also write f(π) if there is no danger of confusion.
+In this paper we shall write Ei,j for the (i, j)-th matrix unit. For a positive integer N, we put
+J = JN =
+
+
+
+
+
+
+
+
+
+1
+−1
+1
+...
+(−1)N−2
+(−1)N−1
+
+
+
+
+
+
+
+
+
+∈ GLN,
+and define an automorphism θN of GLN by θN(g) = J tg−1J−1. Then the standard pinning (TN, BN, {Ei,i+1}N−1
+i=1 ),
+where TN is the maximal torus consisting of diagonal matrices and BN is the Borel subgroup consisting of upper
+triangular matrices, is θN-stable. The dual group for GLN is GL(N, C), and the automorphism �θN of GL(N, C),
+which is dual to θN, is given by �θN(g) = J tg−1J−1. As GLN is split, the Galois action on GL(N, C) is trivial.
+Acknowledgment
+The author would like to thank his doctoral advisor Atsushi Ichino for his helpful advice. He also thanks the
+co-advisor Wen-Wei Li for his helpful comments. In addition, he also thanks Masao Oi and Hirotaka Kakuhama for
+sincere and useful comments. This work was partially supported by JSPS Research Fellowships for Young Scientists
+KAKENHI Grant Number 20J10875 and JSPS KAKENHI Grant Number 22K20333. The author also would like to
+appreciate Naoki Imai for his great support by JSPS KAKENHI Grant Number 22H00093.
+4
+
+2
+Odd special orthogonal groups
+In this section, we establish some notations for the odd special orthogonal groups, and recall the Kottwitz map.
+In the third subsection, we shall describe the real case as a preparation for the proof of Lemma 4.2.
+2.1
+Split odd special orthogonal groups SO2n+1
+Let F be any field of characteristic zero, and n a non-negative integer. We shall write SO2n+1 for the split odd
+special orthogonal group over F of size (2n + 1), which is defined by
+SO2n+1 =
+
+
+ g ∈ GL2n+1
+������
+tg
+Ñ
+1n
+2
+1n
+é
+g =
+Ñ
+1n
+2
+1n
+é 
+
+ .
+We put 2 at the center to make root vectors simple. Its Lie algebra is realized as
+so2n+1 =
+
+
+ X ∈ M2n+1
+������
+tX
+Ñ
+1n
+2
+1n
+é
++
+Ñ
+1n
+2
+1n
+é
+X = 0
+
+
+ ,
+over F. Let us fix the standard Borel subgroup and the standard maximal torus
+B∗ =
+
+
+
+Ñ
+a
+∗
+∗
+1
+∗
+ta−1
+é
+∈ SO2n+1
+������
+a ∈ GLn, upper triangular
+
+
+ ,
+T ∗ = � t = diag(t1, . . . , tn, 1, t−1
+1 , . . . , t−1
+n )
+�� ti ∈ GL1
+� ,
+and let χi denote the element of X∗(T ∗) such that χi(t) = ti, for i = 1, . . . , n. We shall define simple roots αi and
+simple root vectors Xαi so that
+R(T ∗, SO2n+1) = { ±(χi − χj), ±(χi + χj) }1≤i<j≤n ∪ { ±χi }1≤i≤n ,
+∆(B∗) = { α1, . . . , αn }
+= { χ1 − χ2, χ2 − χ3, . . . , χn−1 − χn, χn } ,
+Xαi = Ei,i+1 − En+1+i+1,n+1+i,
+1 ≤ i ≤ n − 1,
+Xαn = 2En,n+1 − En+1,2n+1,
+where R(T ∗, SO2n+1) and ∆(B∗) denote the root system of T ∗ in SO2n+1 and the set of simple roots with respect to
+B∗, respectively. Here we write the group law on X∗(T ∗) additively. Put
+Hi = Ei,i − Ei+1,i+1 − En+1+i,n+1+i + En+1+i+1,n+1+i+1,
+for 1 ≤ i ≤ n − 1,
+Hn = 2En,n − 2E2n+1,2n+1,
+for i = n,
+and
+X−αi = Ei+1,i − En+1+i,n+1+i+1,
+for 1 ≤ i ≤ n − 1,
+X−αn = En+1,n − 2E2n+1,n+1,
+for i = n.
+It can be seen that X−αi is a root vector corresponding to a negative root −αi. Also one has dαi(Hi) = 2 and
+[Xαi, X−αi] = Hi, so Hi is the coroot for αi and X−αi is the opposite root vector to Xαi. In this paper, by the
+standard pinning of SO2n+1, we mean the pinning (T ∗, B∗, {Xα}α∈∆(B∗)).
+The dual group of SO2n+1 is the symplectic group Sp(2n, C). We shall choose a standard pinning of Sp(2n, C)
+dual to the standard pinning of SO2n+1, to be the same as in [16, §7.1].
+5
+
+2.2
+Inner forms of odd special orthogonal groups
+Let F be any field of characteristic zero, and G∗ = SO2n+1 the split special orthogonal group over F. Since any
+automorphism of G∗ is an inner automorphism, the isomorphism classes of inner forms of G∗ and those of inner twists
+of G∗ are bijectively corresponding to each other in the natural way. Moreover, since G∗
+ad = G∗ and Z(G∗) = {1}, the
+isomorphism classes of inner twists, of pure inner twists, and of extended pure inner twists of G∗ are also bijectively
+corresponding to each other in the natural way.
+We refer the reader to [18, §0.3.1] for the definitions and basic
+properties of inner twists and so on.
+Hence there are bijective correspondences between the sets of isomorphism
+classes of inner forms, inner twists, and pure inner twists of G∗, and the Galois cohomology set H1(F, G∗):
+{inner forms} / ≃
+←→
+{inner twists} / ≃
+←→
+{pure inner twists} / ≃
+←→
+H1(F, G∗),
+G
+�→
+ξ : G∗ → G
+�→
+(ξ, z) : G∗ → G
+�→
+z.
+By abuse of terminology we often identify them. We remark that for any (2n + 1)-dimensional quadratic space V over
+F, its special orthogonal group SO(V ) is an inner form of G∗, and every inner form of G∗ is isomorphic to SO(V ) for
+some quadratic space V over F.
+Let G be an inner form of G∗, and M ∗ ⊂ G∗ a Levi subgroup. We say that M ∗ transfers to G if there exists an
+inner twist ξ : G∗ → G such that M := ξ(M ∗) ⊂ G is a Levi subgroup over F and the restriction ξ|M∗ : M ∗ → M is
+an inner twist over F. When M ∗ transfers to G, we will always choose an inner twist ξ as above.
+Let F be a local field. Kottwitz [22, Theorem 1.2] gave a canonical map of pointed sets
+α : H1(F, G∗) → π0(Z(”
+G∗)Γ)D,
+(2.1)
+extending the Tate-Nakayama isomorphism, where D denotes the Pontryagin dual. For any pure inner twist (ξ, z); G∗ →
+G, which we regard as an element of H1(F, G∗), put χG = α(G). We shall also write ⟨−, z⟩ or ⟨−, ξ⟩ for it. Note that
+in our case we have π0(Z(”
+G∗)Γ)D = Z(Sp(2n, C))D ≃ Z/2Z.
+If F is non-archimedean, then there are only two inner forms of G∗. One is non-quasi-split, and the other is split.
+In this case the map α is the unique isomorphism of pointed sets. If F = C, then H1(C, G∗) is singleton, and hence
+α is trivial. Suppose that F = R. For non-negative integers p and q with p + q = 2n + 1, put
+SO(p, q) =
+ß
+g ∈ GL2n+1(R)
+����
+tg
+Å1p
+−1q
+ã
+g =
+Å1p
+−1q
+ã ™
+.
+Then SO(p, q) is an inner form of G∗, and every inner form is isomorphic to SO(p, q) for some (p, q). Since SO(p1, q1)
+and SO(p2, q2) are isomorphic if and only if (p1, q1) = (p2, q2) or (q2, p2), we can naturally identify H1(R, SO2n+1)
+with {(p, q) | p, q ∈ Z≥0, p + q = 2n + 1}/(p, q) ∼ (q, p). Moreover, It is known that the Kottwitz map is characterized
+by
+ker α = { SO(p, q) | p − q ≡ 1, 7
+mod 8 } .
+Let F be a global field. For each place v of F there is a localization map H1(F, G∗) → H1(Fv, G∗). Let αv denote
+the Kottwitz map (2.1) from H1(Fv, G∗) to π0(Z(”
+G∗)Γv)D = π0(Z(”
+G∗)Γ)D. By [22, Proposition 2.6] we have an exact
+sequence
+H1(F, G∗) −→
+�
+v
+H1(Fv, G∗)
+�
+v αv
+−→ π0(Z(”
+G∗)Γ)D.
+(2.2)
+Remark. Such an exact sequence holds for general G∗.
+2.3
+On the real odd special orthogonal groups SO(p, q)
+Let us describe the real special orthogonal groups SO(p, q) and their Lie algebras so(p, q) more explicitly. Let p > q
+be non-negative integers with p + q = 2n + 1, and put r = p − n − 1 = n − q. Put
+S0 =
+Ñ 1n
+1n
+2
+1n
+−1n
+é
+.
+6
+
+Then an assignment g �→ S0gS−1
+0
+determines an isomorphism from SO2n+1 to SO(n + 1, n) over R. Put next
+S′
+p,q =
+Ñ 1n+1
+−i1r
+1q
+é
+.
+Then an assignment g �→ S′
+p,qgS′
+p,q
+−1 determines an isomorphism from SO(n + 1, n) to SO(p, q) over C. We thus
+obtain an inner twist ξ = ξp,q : G∗ = SO2n+1 → SO(p, q) given by g �→ Sp,qgS−1
+p,q, where Sp,q = S′
+p,qS0.
+The standard pinning (T ∗, B∗, {Xα}α) of G∗ gives a Chevalley basis of g∗ = so2n+1:
+[Xβ, Xγ] = ±(b + 1)Xβ+γ,
+where b is the greatest positive integer such that γ − bβ is a root. Explicitly,
+Xχi−χj = Ei,j − En+1+j,n+1+i,
+for 1 ≤ i < j ≤ n,
+Xχi+χj = Ei,n+1+j − Ej,n+1+i,
+for 1 ≤ i < j ≤ n,
+Xχi = 2Ei,n+1 − En+1,n+1+i,
+for 1 ≤ i ≤ n,
+Hχi−χj = Ei,i − Ej,j − En+1+i,n+1+i + En+1+j,n+1+j,
+for 1 ≤ i < j ≤ n,
+Hχi+χj = Ei,i + Ej,j − En+1+i,n+1+i − En+1+j,n+1+j,
+for 1 ≤ i < j ≤ n,
+Hχi = 2Ei,i − 2En+1+i,n+1+i,
+for 1 ≤ i ≤ n,
+X−(χi−χj) = Ej,i − En+1+i,n+1+j,
+for 1 ≤ i < j ≤ n,
+X−(χi+χj) = En+1+j,i − En+1+i,j,
+for 1 ≤ i < j ≤ n,
+X−χi = En+1,i − 2En+1+i,n+1,
+for 1 ≤ i ≤ n.
+As usual the Lie algebra of SO(p, q) has the real structure
+so(p, q)R =
+ß
+X ∈ M2n+1(R)
+����
+tX
+Å1p
+−1q
+ã
++
+Å1p
+−1q
+ã
+X = 0
+™
+,
+and the inner twist ξ = ξp,q : G∗ = SO2n+1 → SO(p, q) gives an isomorphism ξ : g∗ ∋ X �→ Sp,qXS−1
+p,q ∈ so(p, q) of the
+complex Lie algebras.
+The inner twist ξp,q sends an element
+diag(. . . , 1,
+i
+t, 1, . . . , 1,
+n+1+i
+t−1 , 1, . . .) ∈ T ∗,
+to
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+à 1i−1
+cos z
+− sin z
+1n
+sin z
+cos z
+1n−i
+í
+,
+for i ≤ r,
+à 1i−1
+t+t−1
+2
+t−t−1
+2
+1n
+t−t−1
+2
+t+t−1
+2
+1n−i
+í
+,
+for r < i,
+where z is a complex number such that t = e
+√−1z. (i ≤ r is non-split part, i > r is split part.)
+The images of the root vectors and the coroots are given as follows:
+ξ(Xχi−χj) = 1
+2×
+7
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Ä
+Ei,j +
+√
+−1Ei,n+1+j − Ej,i +
+√
+−1Ej,n+1+i
+−
+√
+−1En+1+i,j + En+1+i,n+1+j −
+√
+−1En+1+j,i − En+1+j,n+1+i
+ä
+,
+for i < j ≤ r,
+Ä
+Ei,j + Ei,n+1+j − Ej,i +
+√
+−1Ej,n+1+i
+−
+√
+−1En+1+i,j −
+√
+−1En+1+i,n+1+j + En+1+j,i −
+√
+−1En+1+j,n+1+i
+ä
+,
+for i ≤ r < j,
+(Ei,j + Ei,n+1+j − Ej,i + Ej,n+1+i
++En+1+i,j + En+1+i,n+1+j + En+1+j,i − En+1+j,n+1+i) ,
+for r < i < j,
+ξ(Hχi−χj) =
+
+
+
+
+
+√
+−1 (Ei,n+1+i − Ej,n+1+j − En+1+i,i + Ej,n+1+j) ,
+for i < j ≤ r,
+√
+−1Ei,n+1+i − Ej,n+1+j −
+√
+−1En+1+i,i − Ej,n+1+j,
+for i ≤ r < j,
+Ei,n+1+i − Ej,n+1+j + En+1+i,i − Ej,n+1+j,
+for r < i < j,
+ξ(X−(χi−χj)) = 1
+2×
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Ä
+−Ei,j +
+√
+−1Ei,n+1+j + Ej,i +
+√
+−1Ej,n+1+i
+−
+√
+−1En+1+i,j − En+1+i,n+1+j −
+√
+−1En+1+j,i + En+1+j,n+1+i
+ä
+,
+for i < j ≤ r,
+Ä
+−Ei,j + Ei,n+1+j + Ej,i +
+√
+−1Ej,n+1+i
+−
+√
+−1En+1+i,j +
+√
+−1En+1+i,n+1+j + En+1+j,i +
+√
+−1En+1+j,n+1+i
+ä
+,
+for i ≤ r < j,
+(−Ei,j + Ei,n+1+j + Ej,i + Ej,n+1+i
++En+1+i,j − En+1+i,n+1+j + En+1+j,i + En+1+j,n+1+i) ,
+for r < i < j,
+ξ(Xχi+χj) = 1
+2×
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Ä
+Ei,j −
+√
+−1Ei,n+1+j − Ej,i +
+√
+−1Ej,n+1+i
+−
+√
+−1En+1+i,j − En+1+i,n+1+j +
+√
+−1En+1+j,i + En+1+j,n+1+i
+ä
+,
+for i < j ≤ r,
+Ä
+Ei,j − Ei,n+1+j − Ej,i +
+√
+−1Ej,n+1+i
+−
+√
+−1En+1+i,j +
+√
+−1En+1+i,n+1+j − En+1+j,i +
+√
+−1En+1+j,n+1+i
+ä
+,
+for i ≤ r < j,
+(Ei,j − Ei,n+1+j − Ej,i + Ej,n+1+i
++En+1+i,j − En+1+i,n+1+j − En+1+j,i + En+1+j,n+1+i) ,
+for r < i < j,
+ξ(Hχi+χj) =
+
+
+
+
+
+√
+−1 (Ei,n+1+i + Ej,n+1+j − En+1+i,i − Ej,n+1+j) ,
+for i < j ≤ r,
+√
+−1Ei,n+1+i + Ej,n+1+j −
+√
+−1En+1+i,i + Ej,n+1+j,
+for i ≤ r < j,
+Ei,n+1+i + Ej,n+1+j + En+1+i,i + Ej,n+1+j,
+for r < i < j,
+ξ(X−(χi+χj)) = 1
+2×
+8
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Ä
+−Ei,j −
+√
+−1Ei,n+1+j + Ej,i +
+√
+−1Ej,n+1+i
+−
+√
+−1En+1+i,j + En+1+i,n+1+j +
+√
+−1En+1+j,i − En+1+j,n+1+i
+ä
+,
+for i < j ≤ r,
+Ä
+−Ei,j − Ei,n+1+j + Ej,i +
+√
+−1Ej,n+1+i
+−
+√
+−1En+1+i,j −
+√
+−1En+1+i,n+1+j − En+1+j,i −
+√
+−1En+1+j,n+1+i
+ä
+,
+for i ≤ r < j,
+(−Ei,j − Ei,n+1+j + Ej,i + Ej,n+1+i
++En+1+i,j + En+1+i,n+1+j − En+1+j,i − En+1+j,n+1+i) ,
+for r < i < j,
+ξ(Xχi) =
+®
+Ei,n+1 − En+1,i +
+√
+−1En+1,n+1+i −
+√
+−1En+1+i,n+1,
+for i ≤ r,
+Ei,n+1 − En+1,i + En+1,n+1+i + En+1+i,n+1,
+for r < i,
+ξ(Hχi) = 2 ×
+®√
+−1 (Ei,n+1+i − En+1+i,i) ,
+for i ≤ r,
+(Ei,n+1+i + En+1+i,i) ,
+for r < i,
+ξ(X−χi) =
+®
+−Ei,n+1 + En+1,i +
+√
+−1En+1,n+1+i −
+√
+−1En+1+i,n+1,
+for i ≤ r,
+−Ei,n+1 + En+1,i + En+1,n+1+i + En+1+i,n+1,
+for r < i.
+In particular we have
+tξ(Xα) = ξ(X−α) for any α ∈ R(T ∗, G∗).
+Put T = ξ(T ∗). The real form T (R) consists of the elements of the form ξ(diag(t1, . . . , tr, e
+√−1θr+1, . . . , e
+√−1θn, 1, . . .)),
+where ti ∈ R× and θi ∈ R/2πZ. For an element t ∈ T (R) of such form and a positive root α ∈ R(T, SO(p, q)), which
+is equal to R(T ∗, G∗) as sets, we have
+Ad(t)(ξ(Xα)) = α(t)ξ(Xα),
+where
+α(t) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+e
+√−1(θi±θj),
+for α = χi ± χj, i < j ≤ r,
+e
+√−1θit±1
+j ,
+for α = χi ± χj, i ≤ r < j,
+tit±1
+j ,
+for α = χi ± χj, r < i < j,
+e
+√−1θi,
+for α = χi, i ≤ r,
+ti,
+for α = χi, r < i.
+This means that as roots of T in G, α = ±(χi ± χj) (i < j ≤ r), ±χi (i ≤ r) are imaginary roots (i.e., α = −α),
+α = ±(χi ± χj) (i ≤ r < j) are complex roots (i.e., α ̸= ±α), and α = ±(χi ± χj) (r < i < j), ±χi (r < i) are real
+roots (i.e., α = α).
+3
+Notions around endoscopy and parameters, and Main theorem
+In this section, we first establish some notations for the endoscopy groups, the transfer factors, and L- or A-
+parameters. Afterwards, we shall recall Arthur’s work and state our main theorem.
+3.1
+Endoscopy
+In this subsection, we recall notion of an endoscopic triple and the transfer factor from [18, §1.1].
+3.1.1
+Endoscopic triples
+Let F be a global or local field. Consider a pair (G∗, θ∗) consisting of a connected quasi-split reductive group G∗
+defined over F with a fixed F-pinning, and a pinned automorphism θ∗ of G∗. Here, an automorphism θ∗ is said to
+9
+
+be pinned if it preserves the fixed pinning (i.e., the maximal torus, the Borel subgroup, and the set of simple root
+vectors are θ∗-stable). Then we have an automorphism “
+θ∗ of ”
+G∗, which preserves a Γ-invariant pinning for ”
+G∗. Put
+Lθ∗ = “
+θ∗ ⋊ idWF , which is an L-automorphism of LG∗.
+Definition 3.1. An endoscopic triple for (G∗, θ∗) is a triple e = (Ge, se, ηe) of a connected quasi-split reductive group
+Ge defined over F, a semisimple element se ∈ ”
+G∗, and an L-homomorphism ηe : LGe → LG∗ such that
+• Ad(se) ◦ Lθ∗ ◦ ηe = ηe;
+• ηe(�
+Ge) is the connected component of the subgroup of Ad(se) ◦ “
+θ∗-fixed elements in ”
+G∗.
+When η(Z(�
+Ge)Γ)◦ ⊂ Z(”
+G∗), the endoscopic triple e is said to be elliptic. When θ∗ is trivial (resp. not trivial), every
+endoscopic triple for (G∗, θ∗) is said to be ordinary (resp. twisted).
+Definition 3.2. Two endoscopic triples e1 = (Ge
+1, se
+1, ηe
+1) and e2 = (Ge
+2, se
+2, ηe
+2) for G∗ are said to be isomorphic (resp.
+strictly isomorphic) if there exists an element g ∈ ”
+G∗ such that
+• gηe
+1(LGe
+1)g−1 = ηe
+2(LGe
+2);
+• zgse
+1“
+θ∗(g)−1 = se
+2 for some z ∈ Z(”
+G∗), (resp. gse
+1“
+θ∗(g)−1 = se
+2).
+Then the element g is called an isomorphism (resp. strict isomorphism) from e1 to e2. If g is an isomorphism (resp.
+strict isomorphism) from e to e itself, then it is said to be an automorphism (resp. strict automorphism) of e, and we
+shall write AutG∗(e) (resp. AutG∗(e)) for the set of automorphisms (resp. strict automorphisms) of e.
+Let ξ : G∗ → G be an inner twist, and put θ = ξ ◦ θ∗ ◦ ξ−1. Then θ is an automorphism of G defined over F.
+Definition 3.3. By an endoscopic triple for (G, θ), we mean that for (G∗, θ∗). The notion of elliptic endoscopic triple,
+isomorphism and strict isomorphism of endoscopic triples are also the same as those for (G∗, θ∗).
+We shall write E(G⋊θ) (resp. E(G⋊θ)) for the set of isomorphism (resp. strict isomorphism) classes of endoscopic
+triples for (G, θ). The corresponding subsets of elliptic endoscopic triples will be indicated by the lower right index
+ell:
+E(G ⋊ θ)
+� � E(G ⋊ θ)
+Eell(G ⋊ θ)
+� �
+�
+Eell(G ⋊ θ).
+�
+When θ is trivial, then we may write E(G), E(G), Eell(G), and Eell(G).
+Put �E(N) = E(GLN, θN).
+If e = (Ge, se, ηe) ∈ �E(N), the group Ge is a direct product of finite numbers of
+symplectic or special orthogonal groups. We shall say e is simple if the number of factors is one. Define �Esim(N) to be
+the set of strictly isomorphism classes of simple endoscopic triples of (GLN, θN).
+3.1.2
+Normalized transfer factors
+Next, we shall recall some properties of transfer factors. Let G∗ be a quasi-split connected reductive group over
+a local or global field F with a fixed F-pinning, and θ∗ an automorphism of G∗ preserving the pinning. We assume
+that Z(G∗) is connected. Note that if G∗ is a direct product of a finite number of SO2m+1 or GLm (m ≥ 1), then
+G∗ is quasi-split connected reductive and its center is connected. Let (ξ, z) : G∗ → G be a pure inner twist, and put
+θ = ξ ◦θ∗ ◦ξ−1. Then θ is an automorphism of G defined over F. Not all twisted groups (G, θ) arise in this way, which
+is why an element gθ ∈ G∗ such that θ∗ = Ad(gθ) ◦ ξ−1 ◦ θ ◦ ξ was introduced in [23, §1.2]. In this paper, however, it
+suffices to assume gθ of [23] is 1. We assume that θ∗(z) = z, which is enough for our purpose. Let e = (Ge, se, ηe) be
+an endoscopic triple for (G, θ).
+Let first F be a local field, and ψF : F → C1 a nontrivial additive character. Let further δ ∈ Gθ−srss(F) and
+γ ∈ Ge(F), where Gθ−srss(F) denotes the set of θ-strongly regular and θ-semisimple elements in G(F). Then we have
+the transfer factor:
+∆[e, ξ, z](γ, δ) =
+
+
+
+ǫ(1
+2, V, ψF )∆new
+I
+(γ, δ)−1∆II(γ, δ)∆III(γ, δ)−1∆IV (γ, δ),
+if γ is a norm of δ,
+0,
+otherwise,
+10
+
+where V , ∆new
+I
+, ∆II, ∆III, and ∆IV are defined as in [18, pp.38-39]. Although they treated an extended pure inner
+twist in loc. cit., the similar arguments also works for a pure inner twist. The main difference is that we utilize an
+ordinary cocycle, instead of a basic cocycle. See also [23, §4.3-4.5, Appendices A-C] and [24, §3.4].
+Remark. In the case G∗ = SO2n+1 and θ∗ = 1, the transfer factor ∆[e, ξ, z] is determined by the isomorphism class
+of G, not by the choice of (ξ, z).
+Proposition 3.1. Let γ1, γ2 ∈ Ge(F) be norms of δ1, δ2 ∈ Gθ−srss(F), respectively. Then we have
+∆[e, ξ, z](γ1, δ1)
+∆[e, ξ, z](γ2, δ2) = ∆′(γ1, δ1; γ2, δ2),
+where the right hand side is the relative transfer factor of [24].
+Proof. The proof is similar to that of [18, Proposition 1.1.1].
+Lemma 3.2. Let se denote the image of se in (”
+G∗/”
+G∗der)Γ
+�
+θ∗,free, which is the set of Γ-fixed points in the torsion free
+quotient of the “
+θ∗-covariants of ”
+G∗/”
+G∗der. For x ∈ Z(”
+G∗)Γ and y ∈ Z1(F, (Z(G∗)θ∗)◦), we have
+∆[xe, ξ, z] = ⟨x, z⟩∆[e, ξ, z],
+and
+∆[e, ξ, yz] = ⟨se, y⟩∆[e, ξ, z],
+where xe denotes the endoscopic triple (Ge, xse, ηe).
+Proof. The proof is similar to that of [18, Lemma 1.1.2].
+We shall now recall the notion of matching functions from [23, §5.5]. For δ ∈ Gθ−srss(F) and f ∈ H(G), the orbital
+integral is defined as
+Oδθ(f) =
+�
+Gδθ(F )\G(F )
+f(g−1δθ(g))dg,
+where Gδθ = Centθ(δ, G) = {x ∈ G | δθ(x) = xδ}. If θ is trivial, it may be simply written Oδ(f). For γ ∈ Ge
+srss(F)
+and f e ∈ H(Ge), the stable orbital integral is defined as
+SOγ(f) =
+�
+γ′
+Oγ′(f e),
+where the sum is taken over a set of representatives for the conjugacy classes in the stable conjugacy class of γ.
+Definition 3.4. Two functions f e ∈ H(Ge) and f ∈ H(G) are called ∆[e, ξ, z]-matching (or matching when there is
+no danger of confusion) if for any strongly G-regular semisimple element γ ∈ Ge(F) we have
+SOγ(f e) =
+�
+δ
+∆[e, ξ, z](γ, δ)Oδθ(f),
+where the sum is taken over a set of representatives for the θ-conjugacy classes under G(F) of Gθ−srss(F). We also
+say that f and f e have ∆[e, ξ, z]-matching orbital integrals.
+Remark. If we do not assume θ∗(z) = z, then we need θe on Ge as in [23, §5.4].
+Next, let F be a global field, and ψF : AF /F → C1 a nontrivial additive character. For δ ∈ Gθ−srss(AF ) and
+γ ∈ Ge(AF ), the global absolute transfer factor is defined by
+∆A[e, ξ](γ, δ) =
+
+
+
+�
+v
+∆[ev, ξv, zv](γv, δv),
+if γ is a norm of δ,
+0,
+otherwise.
+Note that the product is well-defined because the almost all factors are 1. Moreover, Lemma 3.2 and the exact sequence
+(2.2) imply that ∆A[e, ξ] is independent of z.
+11
+
+Proposition 3.3. The factor ∆A[e, ξ] coincides with the inverse of the canonical adelic transfer given in [23].
+Proof. The proof is similar to that of [18, Proposition 1.1.3].
+We shall finish this subsubsection stating a lemma regarding local transfer factors and Levi subgroups. Let again
+F be a local field, and ψF : F → C1 a nontrivial additive character. Let M ∗ ⊂ G∗ be a standard Levi subgroup
+such that M = ξ(M ∗) ⊂ G is a Levi subgroup defined over F. Assume that θ is trivial. Let e = (Gese, ηe) and
+eM = (M e, se, ηe|LMe) be endoscopic triples for G and M respectively, such that M e ⊂ Ge is a Levi subgroup.
+Lemma 3.4.
+1. For each δ ∈ M(F) and γ ∈ M e(F), we have
+∆[eM, ξ|M∗, z](γ, δ) = ∆[e, ξ, z](γ, δ)
+Ç
+|DGe
+Me(γ)|
+|DG
+M(δ)|
+å 1
+2
+,
+where DG
+M and DGe
+Me(γ) are the relative Weyl discriminants.
+2. If f ∈ H(G) and f e ∈ H(Ge) are ∆[e, ξ, z]-matching, then their constant terms fM ∈ H(M) and f e
+Me ∈ H(M e)
+are ∆[eM, ξ|M∗, z]-matching.
+Proof. The proof is similar to that of [18, Lemma 1.1.4].
+3.1.3
+Endoscopic triples for odd special orthogonal groups
+Here we will explicate the set of representatives of isomorphism or strict isomorphism classes of ordinary endoscopic
+triples for odd special orthogonal groups. Let F be a local or global field, G∗ = SO2n+1 the split special orthogonal
+group of size 2n + 1 defined over F, and G an inner form of G∗. Then the Galois group Γ = ΓF acts on “
+G = Sp(2n, C)
+trivially, and thus we have LG = Sp(2n, C) × WF and Z(“
+G)Γ = {±1}.
+Every semisimple element s ∈ Sp(2n, C) is Sp(2n, C)-conjugate to an element of the form
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+a11m1
+...
+ar1mr
+1n1
+−1n2
+a−1
+1 1m1
+...
+a−1
+r 1mr
+1n1
+−1n2
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+,
+(3.1)
+where r ≥ 0, m1, . . . , mr ≥ 1, and n1, n2 ≥ 0 are integers such that m1 + · · · mr + n1 + n2 = n, and a1, . . . , ar ∈ C \
+{0, 1, −1} are complex numbers other than 0, 1, −1 that are different to each other. Thus its centralizer Cent(s, Sp(2n, C))
+is isomorphic to
+GL(m1, C) × · · · × GL(mr, C) × Sp(2n1, C) × Sp(2n2, C),
+(3.2)
+which is the dual group of
+GLm1 × · · · × GLmr × SO2n1+1 × SO2n2+1 .
+(3.3)
+Therefore, each endoscopic triple is isomorphic to a triple of a group (3.3), a semisimple element (3.1), and a natural
+inclusion map. Any description of complete systems of representatives of isomorphism or strict isomorphism classes
+of them is so complicated that we do not write down it here. Elliptic ones are simpler.
+For two nonnegative integers n1 and n2 such that n1 + n2 = n, put en1,n2 = (Ge
+n1,n2, sn1,n2, ηn1,n2), where
+• Ge
+n1,n2 = SO2n1+1 × SO2n2+1,
+12
+
+• sn1,n2 =
+Ü 1n1
+−1n2
+1n1
+−1n2
+ê
+,
+• ηn1,n2 denotes the direct product of the inclusion map Sp(2n1, C) × Sp(n2, C) ֒→ Sp(2n, C) given by
+ÅÅ A1
+B1
+C1
+D1
+ã
+,
+Å A2
+B2
+C2
+D2
+ãã
+�→
+Ü
+A1
+B1
+A2
+B2
+C1
+D1
+C2
+D2
+ê
+,
+and the identity map of WF .
+Then en1,n2 is an elliptic endoscopic triple for G, and four triples ±en1,n2, ±en2,n1 are isomorphic to each other. Here
+−(Ge, se, ηe) denotes (Ge, −se, ηe). Therefore, a set
+{ en1,n2 | n1 ≥ n2 ≥ 0, n1 + n2 = n }
+is a complete system of representatives of isomorphism classes of elliptic endoscopic triples for G. The set Eell(G) may
+be identified with it. In addition, en1,n2 is strictly isomorphic to −en2,n1, but not strictly isomorphic to en2,n1 unless
+n1 = n2. Therefore, a set
+{ en1,n2 | n1, n2 ≥ 0, n1 + n2 = n }
+is a complete system of representatives of strictly isomorphism classes of elliptic endoscopic triples for G. The set
+Eell(G) may be identified with it.
+3.2
+Local parameters
+In this subsection, we recall the notion of local parameters. Let F be a local field. Let G be a quasi-split connected
+reductive group over F. We say that a continuous homomorphism φ : LF → LG is an L-parameter for G if it commutes
+with the canonical projections LF ։ WF and LG ։ WF , and sends semisimple elements to semisimple elements.
+Two L-parameters are said to be equivalent if they are conjugate by an element in “
+G. We write Φ(G) for the set
+of equivalence classes of L-parameters for G. An L-parameter φ is called bounded (or tempered) if the image of the
+projection of φ(LF ) to “
+G is bounded. We write Φbdd(G) for the subset of Φ(G) consisting of bounded ones.
+We say that a continuous homomorphism ψ : LF × SU(2, R) → LG is a local A-parameter for G if the restriction
+ψ|LF to LF is a bounded L-parameter. Two local A-parameters are said to be equivalent if they are conjugate by an
+element in “
+G. We write Ψ(G) for the set of equivalence classes of local A-parameters for G. Let Ψ+(G) denote the
+set of equivalence classes of a continuous homomorphism ψ : LF × SU(2, R) → LG whose restriction ψ|LF to LF is an
+L-parameter. A parameter ψ ∈ Ψ+(G) is called generic if the restriction ψ|SU(2,R) (to the second SU(2, R)) is trivial.
+The subset of Ψ+(G) (resp. Ψ(G)) consisting of generic elements can be identified with Φ(G) (resp. Φbdd(G)). We say
+that ψ ∈ Ψ+(G) is discrete if there is no proper parabolic subgroup of LG which contains the image of ψ, and write
+Ψ+
+2 (G) for the subset of Ψ+(G) consisting of discrete elements. Put Ψ2(G) = Ψ(G)∩Ψ+
+2 (G), Φ2(G) = Φ(G)∩Ψ+
+2 (G),
+and Φ2,bdd(G) = Φbdd(G) ∩ Ψ+
+2 (G).
+We shall write Π(G) for the set of isomorphism classes of irreducible smooth representations of G(F). Let Πtemp(G)
+(resp. Π2(G)) to be the subsets of Π(G) consisting of the tempered (resp. essentially square integrable) representations.
+Put Π2,temp(G) = Π2(G) ∩ Πtemp(G).
+For ψ ∈ Ψ+(G), put Sψ = Cent(Im ψ, “
+G), Sψ = Sψ/Z(“
+G)Γ, Sψ = π0(Sψ) = Sψ/S◦
+ψ, Sψ = π0(Sψ) =
+Sψ/S◦
+ψZ(“
+G)Γ = Sψ/Z(“
+G)Γ, Srad
+ψ
+= Cent(Im ψ, “
+Gder)◦, and S♮
+ψ = Sψ/Srad
+ψ . When we emphasize that the param-
+eter is for G, we write Sψ(G), Sψ(G), Sψ(G), etc. The associated L-parameter φψ of ψ is defined by
+φψ(w) = ψ(w,
+Ç
+|w|
+1
+2
+0
+0
+|w|− 1
+2
+å
+),
+13
+
+for w ∈ LF . Finally, we put
+sψ = ψ(1,
+Å−1
+0
+0
+−1
+ã
+).
+Recall that when G is a general linear group GLN, then the parameters can be regarded as an N-dimensional
+representations of LF or LF × SU(2, R), and the equivalence of parameters is isomorphism of representations. A
+parameter which is irreducible as a representation, is said to be simple. We write Ψ+
+sim(GLN) for the subset of Ψ+(GLN)
+consisting of simple parameters, and put Ψsim(GLN) = Ψ(GLN) ∩ Ψ+
+sim(GLN), Φsim(GLN) = Φ(GLN) ∩ Ψ+
+sim(GLN),
+and Φsim,♥(GLN) = Φ♥(GLN) ∩ Φsim(GLN), for ♥ = 2, bdd. The canonical bijection between Π(GLN) and Φ(GLN),
+which is called the local Langlands correspondence for GLN, is known by Langlands [26] in the archimedean case
+(for any G in fact), by Harris-Taylor [13] and Henniart [14] in the non-archimedean case. As in [7], we shall write
+Π(N) = Π(GLN), Φ(N) = Φ(GLN), and so on. For any φ ∈ Φ(N), we write πφ for the unique corresponding element
+in Π(N). Moreover, for any ψ ∈ Ψ+(N), we define the corresponding (not necessarily irreducible) representation πψ
+as [18, (1.2.4)]. Put Ψ+
+unit(N) = {ψ ∈ Ψ+(N) | πψ is irreducible and unitary.}. Note that Ψ(N) ⊂ Ψ+
+unit(N). When
+ψ ∈ Ψ(N), then πψ = πφψ. A parameter ψ for GLN is said to be self-dual if it is self-dual as a representation. We shall
+write �Φ(N) (resp. �Φsim(N), resp. �Φsim,bdd(N)) for the subset of Φ(N) (resp. Φsim(N), resp. Φsim,bdd(N)) consisting
+of self-dual ones.
+Suppose that G is a simple twisted endoscopy group of (GLN, θN). We say a parameter ψ ∈ Ψ+(G) is simple if
+η ◦ ψ ∈ Ψ+(GLN) is simple, where η : LG → LGLN is the L-homomorphism attached to the endoscopy group G. We
+write Ψ+
+sim(G) for the subset of Ψ+(G) consisting of simple parameters, and put Ψsim(G), Φsim(G), and so on in the
+similar way. Moreover, put ‹Ψ+(G) to be the subset of Ψ+(GLN) consisting of the parameters of the form η◦ψ for some
+ψ ∈ Ψ+(G). Note that ‹Ψ+(G) = Ψ+(G) unless G is an even special orthogonal group, in which case ‹Ψ+(G) coincides
+with Ψ+(G)/ ∼ǫ in the sense of [8]. Put ‹Ψ(G) = ‹Ψ+(G) ∩ Ψ(N), �Φ(G) = ‹Ψ+(G) ∩ Φ(N), �Φsim(G) = �Φ(G) ∩ Φsim(N),
+and �Φsim,bdd(G) = �Φsim(G) ∩ Φbdd(N). Let �Φ2(G) be the subset of �Φ(G) consisting of parameters of multiplicity free
+as representations. Put �Φ2,bdd(G) = �Φ2(G) ∩ Φbdd(N).
+Now let us consider the case when G is an inner form of the split odd special orthogonal group G∗ = SO2n+1, which
+is not necessarily quasi-split. Then the parameters for G∗ are also called parameters for G, since “
+G = ”
+G∗. We omit
+the definition of ”G-relevant”, which is given in [18, §0.4] in a general setting. In this paper G-relevant parameters
+are simply said to be relevant, when there is no danger of confusion. We remark that ”parameter for G” is different
+from ”G-relevant parameter”. Although we use a phrase ”parameter for G”, we never write as Ψ(G). Moreover, we do
+not fix a symbol for the set of G-relevant parameters. Recall from [10, §4] that the parameters can be regarded as an
+2n-dimensional symplectic representations of LF or LF × SU(2, R), and the equivalence of parameters is isomorphism
+of representations. A parameter ψ ∈ Ψ+(G∗) can be written as
+ψ =
+�
+i∈I+
+ψ
+ℓiψi ⊕
+�
+i∈I−
+ψ
+ℓiψi ⊕
+�
+j∈Jψ
+ℓj(ψj ⊕ ψ∨
+j ),
+where ℓi, ℓj are positive integers, I±
+ψ , Jψ indexing sets for mutually inequivalent irreducible representations ψi of
+LF × SU(2, R) such that
+• ψi is symplectic for i ∈ I+
+ψ ;
+• ψi is orthogonal and ℓi is even for i ∈ I−
+ψ ;
+• ψj is not self-dual for j ∈ Jψ.
+Thus we have
+Sψ ∼=
+�
+i∈I+
+ψ
+O(ℓi, C) ×
+�
+i∈I−
+ψ
+Sp(ℓi, C) ×
+�
+j∈Jψ
+GL(ℓj, C),
+and its component group is a free (Z/2Z)-module
+Sψ ∼=
+�
+i∈I+
+ψ
+(Z/2Z)ai,
+14
+
+with a formal basis {ai}, where ai corresponds to the nontrivial coset O(ℓi, C) \ SO(ℓi, C). Note that “
+G = Sp(2n, C) is
+perfect, and thus Srad
+ψ
+= S◦
+ψ and S♮
+ψ = Sψ. It can be easily seen that the parameter ψ is discrete if and only if ℓi = 1
+for all i ∈ I+
+ψ and I−
+ψ = Jψ = ∅. Hence, we have
+Ψ+
+2 (G∗) = { ψ ∈ Ψ+(G∗) | Sψ is finite }
+= { ψ = ψ1 ⊕ · · · ⊕ ψr | ψi are symplectic, irreducible, and mutually distinct, r ≥ 1 } .
+A parameter ψ ∈ Ψ+(G∗) is called to be elliptic if there exists a semisimple element in Sψ whose centralizer in Sψ(G∗)
+is finite. We shall write Ψ+
+ell(G∗) for the set of such parameters. Then we have
+Ψ+
+ell(G∗) = { ψ ∈ Ψ+(G∗) | Cent(s, Sψ) is finite for some s ∈ Sψ,ss }
+=
+ß
+ψ = 2ψ1 ⊕ · · · 2ψq ⊕ ψq+1 ⊕ · · · ⊕ ψr
+����
+r ≥ 1, r ≥ q ≥ 0,
+ψi are symplectic, irreducible, and mutually distinct
+™
+,
+where Sψ,ss denotes the set of semisimple elements in Sψ. We write Ψell(G∗) (resp. Ψell(G∗)) for the subset of Ψ(G∗)
+consisting of elliptic (resp. non-elliptic) ones. We have Ψ2(G∗) ⊂ Ψell(G∗). Put
+Ψ2
+ell(G∗) = Ψell(G∗) \ Ψ2(G∗)
+=
+ß
+ψ = 2ψ1 ⊕ · · · 2ψq ⊕ ψq+1 ⊕ · · · ⊕ ψr
+����
+r ≥ q ≥ 1,
+ψi are symplectic, irreducible, and mutually distinct
+™
+,
+Φell(G∗) = Ψell(G∗) ∩ Φ(G∗), and Φ2
+ell(G∗) = Ψ2
+ell(G∗) ∩ Φ(G∗).
+Let M ∗ be a standard Levi subgroup of G∗ = SO2n+1 isomorphic to
+GLn1 × · · · × GLnk × SO2n0+1,
+(3.4)
+where n1 + · · · nk + n0 = n. Then its dual group �
+M ∗ is isomorphic to
+GL(n1, C) × · · · × GL(nk, C) × Sp(2n0, C),
+and we regard it as a Levi subgroup of ”
+G∗. We have
+Ψ+(M ∗) = { ψ1 ⊕ · · · ⊕ ψk ⊕ ψ0 | ψ1 ∈ Ψ+(n1), . . . , ψk ∈ Ψ+(nk), ψ0 ∈ Ψ+(SO2n0+1) } .
+The canonical injection �
+M ∗ ֒→ ”
+G∗ induces the canonical mapping
+ψ1 ⊕ · · · ⊕ ψk ⊕ ψ0 �→ ψ1 ⊕ · · · ⊕ ψk ⊕ ψ0 ⊕ ψ∨
+1 ⊕ · · · ⊕ ψ∨
+k
+from Ψ+(M ∗) to Ψ+(G∗).
+3.3
+Global parameters
+In this subsection we recall from [7] the notion of global parameters. Let F be a global field, Γ = ΓF its absolute
+Galois group, and WF its absolute Weil group. For a connected reductive group G over F, as in [7, p.19] we put
+G(AF )1 = { g ∈ G(AF ) | |χ(g)| = 1, ∀χ ∈ X∗(G)F } ,
+where X∗(G)F = HomF (G, GL1). The quotient set G(F)\G(AF )1 has finite volume, and there is a sequence
+L2
+cusp(G(F)\G(AF )1) ⊂ L2
+disc(G(F)\G(AF )1) ⊂ L2(G(F)\G(AF )1),
+where L2
+cusp(G(F)\G(AF )1) is the subspace consisting of cuspidal functions, and L2
+disc(G(F)\G(AF )1) the subspace
+that can be decomposed into a direct sum of irreducible representations of G(AF )1. We have
+Acusp(G) ⊂ A2(G) ⊂ A(G),
+15
+
+where Acusp(G), A2(G), and A(G) denote the subsets of irreducible unitary representations of G(AF ) whose restrictions
+to G(AF )1 are constituents of the respective spaces L2
+cusp(G(F)\G(AF )1), L2
+disc(G(F)\G(AF )1), and L2(G(F)\G(AF )1).
+We also write A+
+cusp(G) and A+
+2 (G) for the analogues of Acusp(G) and A2(G) defined without the unitarity.
+For a finite set S of places of F such that v is non-archimedean and Gv is unramified over Fv for any place v /∈ S,
+put CS(G) to be the set consisting of families cS = (cv)v /∈S where each cv is a “
+G-conjugacy class in LGv = “
+G ⋊ WFv
+represented by an element of the form tv ⋊ Frobv with a semisimple element tv ∈ “
+G. For S ⊂ S′ there is a natural
+map cS = (cv)v /∈S �→ (cv)v /∈S′ from CS(G) to CS′(G). Let us define C(G) to be the direct limit of {CS(G)}S.
+Let π = ⊗vπv be an irreducible automorphic representation of G(AF ) and S a finite set of places such that πv is
+unramified for all v /∈ S. The Satake isomorphism associates a “
+G-conjugacy class c(πv) ∈ LGv to π. We can therefore
+associate cS(π) = (c(πv))v /∈S ∈ CS(G) to π. This leads to a mapping π �→ c(π) from a set of equivalence classes of
+irreducible automorphic representations of G to C(G). We shall write Caut(G) for its image.
+Let us next review some fact on automorphic representations of GLN(AF ). Let N be a positive integer. Following
+[7], we write A(N) = A(GLN), and similarly A♥(N) = A♥(GLN) and A+
+♥(N) = A+
+♥(GLN) for ♥ ∈ {2, cusp}. Put
+A+
+iso(N) =
+�
+π = π1 ⊞ · · · ⊞ πr
+�� r ∈ Z≥1, Ni ∈ Z≥1, N1 + · · · + Nr = N, πi ∈ A+
+cusp(Ni),
+∀i = 1, . . . , r
+�
+,
+where ⊞ denotes the isobaric sum. It is known by Mœglin and Waldspurger [29] that any π ∈ A2(N) is isomorphic to
+µ| det |
+n−1
+2
+⊞ µ| det |
+n−3
+2
+⊞ · · · ⊞ µ| det |− n−1
+2 ,
+for some m, n ∈ Z≥1 and µ ∈ Acusp(m) with N = mn. Moreover, (m, n, µ) is uniquely determined by π, and any
+representation of such form is an element of A2(N). In other words, A2(N) is in bijection with {(µ, m) | 1 ≤ m|N, µ ∈
+Acusp(N/m)}. We have
+Acusp(N) ⊂ A2(N) ⊂ A(N).
+(3.5)
+Put Ψcusp(N) = Acusp(N), which is trivially in bijection with Acusp(N). Put Ψsim(N) to be the set of ψ = µ ⊠ ν
+for an irreducible finite dimensional representation ν of SU(2, R) and µ ∈ A2(N/ dim ν). It is in bijection with A2(N)
+by the fact above, for which we shall write ψ �→ πψ. An element in Ψsim(N) is said to be simple. Put Ψ(N) to be
+the set of ψ = ℓ1ψ1 ⊞ · · · ⊞ ℓrψr for some r ∈ Z≥1, ℓi, Ni ∈ Z≥1 with ℓ1N1 + · · · + ℓrNr = N, and mutually distinct
+elements ψi ∈ Ψsim(Ni). To such ψ, we associate πψ = π⊞ℓ1
+ψ1 ⊞ · · · ⊞ π⊞ℓr
+ψr , so that a mapping ψ �→ πψ gives a bijection
+from Ψ(N) to A(N).We have a chain
+Ψcusp(N) ⊂ Ψsim(N) ⊂ Ψ(N),
+which is compatible with the chain (3.5). We say that a parameter ψ = ℓ1(µ1 ⊠ ν1) ⊞ · · · ⊞ ℓr(µr ⊠ νr) is generic if
+ν1, . . . , νr are trivial. Let Φ(N) be the subset of Ψ(N) consisting of generic parameters.
+Let v be a place of F. By LLC for GLN, for any µ ∈ Ψcusp(N), we have an L-parameter φv ∈ Φ(GLN /Fv)
+corresponding to µv ∈ Π(GLN /Fv). By abuse of notation, let us write µv for φv. Then we obtain the localization
+map Ψsim(N) → Ψ+
+v (N), ψ = µ ⊠ ν �→ ψv = µv ⊠ ν, where Ψ+
+v (N) = Ψ+(GLN /Fv). This leads the localization map
+Ψ(N) → Ψ+
+v (N),
+ψ = ℓ1ψ1 ⊞ · · · ⊞ ℓrψr, �→ ψv = ℓ1ψ1,v ⊕ · · · ⊕ ℓrψr,v.
+Let us now recall from [7] the definition of A-parameters for odd special orthogonal groups and other classical
+groups. An irreducible self-dual cuspidal automorphic representation φ of GLN(AF ) is said to be symplectic (resp.
+orthogonal) if the exterior (resp. symmetric) square L-function L(s, φ, ∧2) (resp. L(s, φ, Sym2)) has a pole at s = 1. By
+the theory of Rankin-Selberg L-function, any irreducible self-dual cuspidal automorphic representation φ of GLN(AF )
+is either symplectic or orthogonal, and this is mutually exclusive. Theorems 1.4.1 and 1.5.3 of [7] tell us the following
+proposition.
+Proposition 3.5 (1st seed theorem). For each irreducible self-dual cuspidal automorphic representation φ of GLN(AF ),
+there exists a simple twisted endoscopic triple eφ = (Gφ, sφ, ηφ) of (GLN, θN) and a representation π ∈ A2(Gφ)
+such that c(φ) = ηφ(c(π)). The triple eφ is unique up to isomorphism. If φ is symplectic (hence N is even), then
+Gφ = SON+1 and ηφ is regarded as the unique (conjugacy class of) embedding Sp(N, C) ֒→ GL(N, C). If φ is orthog-
+onal, then LGφ = SO(N, C) ⋊ WF .
+16
+
+Let e = (G∗, s, η) ∈ �Esim(N) be a simple twisted endoscopic triple of (GLN, θN), i.e., G∗ is the split symplectic
+group, the split odd special orthogonal group, or a quasi-split even special orthogonal group. Set �Φsim(G∗) to be the
+set of irreducible self-dual cuspidal automorphic representations φ of GLN(AF ) such that the corresponding endoscopic
+triple eφ is isomorphic to e. Here note that we cannot write Φsim(G∗) because G∗ may be an even special orthogonal
+group.
+An element ψ = µ ⊠ ν in Ψsim(N) is said to be self-dual if ψ = ψ∨, where ψ∨ = µ∨ ⊠ ν. We write ‹Ψsim(N) for the
+subset of Ψsim(N) consisting of such elements, and put �Φsim(N) = ‹Ψsim(N)∩Φ(N). Moreover, an element ψ = ⊞iℓiψi
+in Ψ(N) is said to be self-dual if ψ = ψ∨, where ψ∨ = ⊞iℓiψ∨
+i . Write ‹Ψ(N) for the subset of Ψ(N) consisting of such
+elements, and put �Φ(N) = ‹Ψ(N) ∩ Φ(N).
+For any ψ ∈ ‹Ψ(N), the substitute Lψ for the global Langlands group attached to ψ is defined as follows. We can
+write
+ψ = ⊞
+i∈Iψ
+ℓiψi ⊞ ⊞
+j∈Jψ
+ℓj(ψj ⊞ ψ∨
+j ),
+(3.6)
+where ψi = µi ⊠ νi ∈ ‹Ψsim(N) and ψj = µj ⊠ νj ∈ Ψsim(N) \ ‹Ψsim(N) for any i ∈ Iψ, j ∈ Jψ, and they are mutually
+distinct. Put Kψ = Iψ ⊔ Jψ and let mk be a positive integer such that µk ∈ Ψcusp(mk) for k ∈ Kψ. For i ∈ Iψ, put
+Hi = Gµi of Proposition 3.5 and let ‹
+µi denote for an embedding
+ηµi : LHi = LGµi ֒→ LGLmi,
+of Proposition 3.5. For j ∈ Jψ, put Hj = GLmj and let �
+µj be an embedding LHj ֒→ LGL2mj given by
+GL(mj, C) × WF → GL(2mj, C) × WF ,
+(h, w) �→ (diag(h, �θmj(h)), w),
+where �θmj is the twist given in the introduction. The substitute Lψ is the fiber product of {Hk}k∈Kψ over WF :
+Lψ =
+�
+k∈Kψ
+ÄLHk → WF
+ä
+.
+The associated L-embedding �ψ from Lψ × SU(2, R) to LGLN = GL(N, C) × WF is defined as
+�ψ =
+�
+i∈Iψ
+ℓi(‹
+µi ⊗ νi) ⊕
+�
+j∈Jψ
+ℓj(�
+µj ⊗ νj).
+Now we shall recall the definition of the A-parameters for odd special orthogonal groups. Let G∗ = SO2n+1. An
+A-parameter for G∗ = SO2n+1 is a parameter ψ ∈ ‹Ψ(2n) such that �ψ factors through LG∗ = Sp(2n, C) × WF . We
+shall write Ψ(G∗) for the A-parameters for G∗. Although Arthur[7] defined a set ‹Ψ(G∗) instead of Ψ(G∗), it is not
+needed now since NGL(2n,C)(Sp(2n, C)) = Sp(2n, C), cf. [7, p.31]. Thus for ψ ∈ Ψ(SO2n+1), we have the associated
+L-embedding from Lψ × SU(2, R) to LSO2n+1 = Sp(2n, C) × WF , which we shall also write �ψ. We will write Im ψ for
+the image of �ψ, in order to treat local and global matters uniformly. Let us write Φ(G∗) for the set of equivalence
+classes of generic A-parameters for G, i.e., Φ(G∗) = Ψ(G∗) ∩ �Φ(N).
+We have another characterization of Ψ(G∗) without Lψ. Recall that an irreducible finite dimensional representation
+ν of SU(2, R) is symplectic (resp. orthogonal) if dim ν is even (resp. odd). A self-dual simple parameter µ⊠ν ∈ ‹Ψsim(N)
+is said to be symplectic if one of µ and ν is symplectic and the other is orthogonal, and to be orthogonal otherwise.
+Let ψ be a parameter (3.6) in ‹Ψ(N). Then ψ is in Ψ(G∗) if and only if ℓi is even for i ∈ Iψ with orthogonal ψi.
+For ψ ∈ Ψ(G∗), we put Sψ = Cent(Im �ψ, “
+G), Sψ = Sψ/Z(“
+G)Γ, Sψ = π0(Sψ), and Sψ = π0(Sψ). Note that the
+Galois action is trivial in this case. We also write Sψ(G) and so on if the group G is to be emphasized. Take a partition
+Iψ = I+
+ψ ⊔ I−
+ψ such that I+
+ψ (resp. I−
+ψ ) is the set of i ∈ Iψ such that ψi is symplectic (resp. orthogonal). Then we have
+Sψ ∼=
+�
+i∈I+
+ψ
+O(ℓi, C) ×
+�
+i∈I−
+ψ
+Sp(ℓi, C) ×
+�
+j∈Jψ
+GL(ℓj, C),
+and its component group is a free (Z/2Z)-module
+Sψ ∼=
+�
+i∈I+
+ψ
+(Z/2Z)ai,
+17
+
+with a formal basis {ai}, where ai corresponds to ψi.
+As in [7, §4.1], we put
+Ψsim(G∗) = � ψ ∈ Ψ(G∗)
+�� |Sψ| = 1 � ,
+Ψ2(G∗) = � ψ ∈ Ψ(G∗)
+�� ��Sψ
+�� < ∞ � ,
+Ψell(G∗) =
+�
+ψ ∈ Ψ(G∗)
+�� ��Sψ,s
+�� < ∞, ∃s ∈ Sψ,ss
+�
+,
+and
+Ψdisc(G∗) = � ψ ∈ Ψ(G∗)
+�� ��Z(Sψ)
+�� < ∞ � ,
+where Sψ,s = Cent(s, Sψ). Put also Ψ2
+ell(G∗) = Ψell(G∗) \ Ψ2(G∗). Since we have an explicit description of Sψ, we
+can get a more explicit characterization of these sets. For instance, Ψ2(G∗) is a set of parameters (3.6) in ‹Ψ(N) such
+that ℓi = 1 for i ∈ I+
+ψ and I−
+ψ ⊔ Jψ = ∅. Let us put Φ♥(G∗) = Φ(G∗) ∩ Ψ♥(G∗), for ♥ ∈ {2, sim, ell, disc}. Put also
+Φ2
+ell(G∗) = Φell(G∗) \ Φ2(G∗).
+Consider a parameter ψ ∈ Ψ(G∗) ⊂ ‹Ψ(N) of the form (3.6). Let v be a place of F. The localization of ψ at v is
+defined as
+ψv = ⊞
+i∈Iψ
+ℓiψi,v ⊞ ⊞
+j∈Jψ
+ℓj(ψj,v ⊞ ψ∨
+j,v),
+where ψk,v = µk,v ⊠νk, and µk,v stands for the L-parameter for the local component µk,v of µk at v, for k ∈ Kψ. Since
+µk,v is a map from LFv to GL(mk, C), one can see that ψv is a map from LFv × SU(2, R) to GL(2n, C). By Theorem
+1.4.2 of [7] and our 1st seed theorem (Proposition 3.5), one can see that the image of ψv is contained in Sp(2n, C).
+Thus we obtain our 2nd seed theorem:
+Proposition 3.6 (2nd seed theorem). For an A-parameter ψ ∈ Ψ(G∗) and a place v, the equivalence class of ψv is
+uniquely determined. Hence, we have the localization ψv ∈ Ψ+(G∗
+v).
+Then there is a natural map Sψ → Sψv, which induces natural maps
+Sψ → Sψv,
+Sψ → Sψv,
+and
+Sψ → Sψv.
+If (Ge, se, ηe) ∈ E(G∗) is an endoscopic triple for G∗ = SO2n+1, the endoscopic group Ge is of the form GLn1 × · · ·×
+GLnr × SO2m1+1 × SO2m2+1, where r ∈ Z≥0, n1 + · · · + nr + m1 + m2 = n. Then the set of equivalence classes of
+A-parameters for Ge is defined as
+Ψ(Ge) = Ψ(n1) × · · · × Ψ(nr) × Ψ(SO2m1+1) × Ψ(SO2m2+1),
+or equivalently the set of pairs (ψ, �ψe) of a parameter ψ ∈ Ψ(G∗) and an L-embedding �ψe : Lψ × SU(2, R) → LGe with
+ηe ◦ �ψe = �ψ. We have a canonical mapping (ψ, �ψe) �→ ψ from Ψ(Ge) to Ψ(G∗), for which we will write ψe �→ ηe ◦ ψe.
+The subset of generic parameters is
+Φ(Ge) = Φ(n1) × · · · × Φ(nr) × Φ(SO2m1+1) × Φ(SO2m2+1),
+and we define Ψ♥(Ge) and Φ♥(Ge) (♥ = 2, disc, . . .) similarly. The component groups and localization are defined
+just like the case of GLN or SO2n+1.
+Now we consider the notion of parameters for non-quasi-split odd special orthogonal groups. Let G be an inner
+form of G∗ = SO2n+1. The notion of parameters for G is the same as that for G∗. In addition, a parameter ψ ∈ Ψ(G∗)
+is said to be G-relevant if it is relevant locally everywhere in the sense of [18, §0.4.4].
+Note that an inner twist
+ξ : G∗ → G is uniquely determined by G up to isomorphism, in our case. When there is no danger of confusion, we
+shall simply call relevant. In this paper, as in the local case, we never use a symbol Ψ(G). As in [18, §1.3.7], we have
+the following lemma.
+Lemma 3.7. Let ψ ∈ Ψ(G∗) be a G-relevant parameter and M ∗ ⊂ G∗ a Levi subgroup. If ψ comes from a parameter
+for M ∗, then M ∗ transfers to G.
+18
+
+3.4
+The bijective correspondence
+Here we recall the bijective correspondence (e, ψe) ←→ (ψ, s) from discussions in [7, §1.4] and [18, §1.4] in the case
+of ordinary endoscopy for SO2n+1.
+Let G∗ be a connected reductive split group over a local field F. Then Γ acts trivially on ”
+G∗. Two ordinary
+endoscopic triples e1 = (Ge
+1, se
+1, ηe
+1) and e2 = (Ge
+2, se
+2, ηe
+2) for G∗ are considered strictly equivalent (resp. equivalent) if
+se
+1 = se
+2 (resp. se
+1 = zse
+2 for some z ∈ Z(”
+G∗)) and ηe
+1(LGe
+1) = ηe
+2(LGe
+2). Note that the equivalence here is different from
+the isomorphism.
+Let E(G∗) (resp. E(G∗)) be a complete system of representatives of strict equivalence (resp. equivalence) classes
+of endoscopic data for G∗. We may identify E(G∗) (resp. E(G∗)) with the set of strict equivalence (resp. equivalence)
+classes of endoscopic triples for G∗. (The set E(G) in [7, p.246] does not correspond to E(G∗) here, but to E(G∗).)
+Define F(G∗) to be the set of the parameters ψ : LF × SU(2) → LG∗. (Here we mean the actual L-homomorphisms,
+not the equivalence classes.) For each ψ ∈ F(G∗), we shall write Sψ,ss (resp. Sψ,ss) for the set of semisimple elements
+in Sψ (resp. Sψ). Define
+X(G∗) = { (e, ψe) | e = (Ge, se, ηe) ∈ E(G∗), ψe ∈ F(Ge) } ,
+Y (G∗) = { (ψ, s) | ψ ∈ F(G∗), s ∈ Sψ,ss } .
+They are left ”
+G∗ × Z(”
+G∗)-sets by the following actions: for (g, z) ∈ ”
+G∗ × Z(”
+G∗),
+(g, z) · ((Ge, se, ηe), ψe) = ((Ge
+1, se
+1, ηe
+1), (ηe
+1)−1 ◦ Ad(g) ◦ ηe ◦ ψe),
+(g, z) · (ψ, s) = (Ad(g) ◦ ψ, zgsg−1),
+where (Ge
+1, se
+1, ηe
+1) is an element of E(G∗) that is strictly equivalent to (Ge, zgseg−1, Ad(g) ◦ ηe).
+Put
+X(G∗) = � (e, ψe)
+�� e = (Ge, se, ηe) ∈ E(G∗), ψe ∈ F(Ge) � ,
+Y (G∗) = � (ψ, s)
+�� ψ ∈ F(G∗), s ∈ Sψ,ss
+� ,
+i.e., X(G∗) (resp. Y (G∗)) is the quotient set Z(”
+G∗)\\X(G∗) (resp. Z(”
+G∗)\\Y (G∗)) of X(G∗) (resp. Y (G∗)) by
+Z(”
+G∗) = {1} × Z(”
+G∗). Let us put
+X(G∗) = ”
+G∗\\X(G∗),
+Y(G∗) = ”
+G∗\\Y (G∗),
+standing for the quotient sets of X(G∗) and Y (G∗) by ”
+G∗ = ”
+G∗ × {1}, respectively. Likewise X (G∗) = ”
+G∗\\X(G∗),
+Y(G∗) = ”
+G∗\\Y (G∗). Note that E(G∗) = ”
+G∗\\E(G∗) and E(G∗) = ”
+G∗\\E(G∗) = ”
+G∗ × Z(”
+G∗)\\E(G∗). We can also
+see that
+X(G∗) =
+¶
+(e, ψe)
+��� e = (Ge, se, ηe) ∈ E(G∗), ψe ∈ ‹Ψ(Ge)
+©
+,
+Y(G∗) =
+¶
+(ψ, s)
+��� ψ ∈ Ψ(G∗), s ∈ Sψ,ss/(”
+G∗ − conj.)
+©
+,
+X (G∗) =
+¶
+(e, ψe)
+��� e = (Ge, se, ηe) ∈ E(G∗), ψe ∈ ‹Ψ(Ge)
+©
+,
+Y(G∗) =
+¶
+(ψ, s)
+��� ψ ∈ Ψ(G∗), s ∈ Sψ,ss/(”
+G∗ − conj.)
+©
+,
+where ‹Ψ(Ge) is the quotient set of Ψ(Ge) modulo the action of AutG∗(e). We use the symbol ‹Ψ here, because the idea
+is similar to that of ‹Ψ(G∗). (See [7, p.31].)
+Lemma 3.8. Let G∗ be SO2n+1 defined over a local field F. Then the natural map
+X(G∗) → Y (G∗),
+(e, ψe) �→ (ηe ◦ ψe, se),
+is a ”
+G∗ ×Z(G∗)-equivariant bijection. Here se and ηe are the second and third elements of e respectively, Furthermore,
+this induces a ”
+G∗-equivariant bijection X(G∗) ≃ Y (G∗), a Z(”
+G∗)- equivariant bijection X(G∗) ≃ Y(G∗), and a
+bijection X (G∗) ≃ Y(G∗).
+19
+
+Proof. Recall that E(G∗) is a complete system of representatives, not just a quotient set. The map X(G∗) → Y (G∗)
+is trivially well-defined. The equivariance under the actions of ”
+G∗ × Z(G∗) follows immediately from the definition of
+the actions. The injectivity also follows easily from the definition of strict equivalence of endoscopic triples. Let us
+show the surjectivity. Let (ψ, s) ∈ Y (G∗). Put Ge = Cent(s, ”
+G∗) · ψ(LF × SU(2)). We know that Cent(s, ”
+G∗) is of the
+form (3.2). Let G′ be the group (3.3). Then its dual group �
+G′ is isomorphic to Cent(s, ”
+G∗), with the trivial Galois
+action. Thus we have Ge ≃ LG′. Let us write η′ for an injective map LG′ ≃ Ge ֒→ LG∗, to obtain an endoscopic
+triple (G′, s, η′). Let e = (Ge, se, ηe) ∈ E(G∗) be the unique element that is strictly equivalent to (G′, s, η′). Since
+Im ψ ⊂ Ge = η′(LG′) = ηe(LGe), there exists a parameter ψe ∈ F(Ge) such that ψ = ηe ◦ ψe. Then we obtain
+(e, ψe) ∈ X(G∗), and get a map from Y (G∗) to X(G∗). This shows the surjectivity.
+The latter assertion follows from the former.
+Let F be a number field, and G∗ = SO2n+1. For an endoscopic triple e for G∗, every element in Ψ(Ge) can
+be regarded as an element in Ψ(G∗) in the natural way. For two parameters ψ1, ψ2 ∈ Ψ(Ge), we write ψ1 ∼ ψ2 if
+ηe ◦ ψ1 = ηe ◦ ψ2, i.e., they are same in Ψ(G∗). Let ‹Ψ(Ge) = Ψ(Ge)/ ∼ be the quotient set. The reason why we use
+the symbol ‹Ψ is because the idea is similar to that of ‹Ψ(G∗) (see [7, p.31]). Define
+X(G∗) =
+¶
+(e, ψe)
+��� e = (Ge, se, ηe) ∈ E(G∗), ψe ∈ ‹Ψ(Ge)
+©
+,
+Y(G∗) =
+¶
+(ψ, s)
+��� ψ ∈ Ψ(G∗), s ∈ Sψ,ss/(”
+G∗ − conj.)
+©
+.
+As in the local case, we have the following lemma.
+Lemma 3.9. Let G∗ = SO2n+1 be the global split odd special orthogonal group over a global field F. Then the natural
+map
+X(G∗) → Y(G∗),
+(e, ψe) �→ (ψe, se)
+is bijective.
+Proof. The proof is similar to that of Lemma 3.8, with �ψ(Lψ × SU(2)) in place of ψ(LF × SU(2)).
+3.5
+Main theorems
+In this subsection we recall the statements of the endoscopic classification of representations of the odd special
+orthogonal groups, which will be proved for the generic part in this paper, and was already proven for the quasi-split
+case by Arthur [7].
+Let first F be a local field. Fix a nontrivial additive character ψF : F → C1. Let G∗ = SO2n+1 be the split special
+orthogonal group of size 2n + 1 defined over F, equipped with the fixed pinning. It can be regarded as a twisted
+endoscopic group of (GL2n, θ2n). Arthur proved the existence of the stable linear form satisfying the twisted ECR:
+Proposition 3.10 ([7, Theorem 2.2.1.(a)]). Let ψ ∈ Ψ(G∗). Then there is a unique stable linear form
+H(G∗) ∋ f �→ f G∗(ψ) ∈ C,
+satisfying the twisted endoscopic character relation [7, (2.2.3)].
+We sometimes write f(ψ) for f G∗(ψ) if there is no danger of confusion.
+Let (ξ, z) : G∗ → G be a pure inner twist of G∗. We now state the main local classification theorem.
+Theorem* 3.11.
+1. Let ψ ∈ Ψ(G∗). There is a finite multiset Πψ(G) of irreducible unitary representations of
+G(F), (i.e., a finite set over Πunit(G) in Arthur’s terminology,) satisfying the local and global theorems below.
+We sometimes write Πψ = Πψ(G) if there is no danger of confusion. If ψ = φ ∈ Φbdd(G∗), i.e., ψ is generic,
+then the set Πφ is empty if and only if φ is not G-relevant. In general, Πψ is empty if ψ is not G-relevant. The
+set is called the local A-packet or simply the packet for ψ. When ψ = φ is generic, it is also called the L-packet
+of φ.
+20
+
+2. The local A-packet is equipped with a map
+Πψ(G) → Irr(Sψ, χG),
+π �→ ⟨−, π⟩,
+satisfying the local and global theorems below. The packets Πψ(G) and the pairing ⟨−, π⟩ are independent of the
+choice of a inner twist (ξ, z), and determined only by the isomorphism classes of G.
+3. (ECR) Let e = (Ge, se, ηe) be an endoscopic triple for G with se ∈ Sψ, and ψe ∈ Ψ(Ge) a parameter for Ge such
+that ψ = ηe ◦ ψe is a parameter for G. If f ∈ H(G) and f e ∈ H(Ge) are matching, then we have
+f e(ψe) = e(G)
+�
+π∈Πψ(G)
+⟨sψse, π⟩f(π),
+(3.7)
+where the left hand side is the stable linear form given by Proposition 3.10.
+4. Let ψ = φ ∈ Φbdd(G∗) be a generic parameter.
+Then the packet Πφ(G) is multiplicity free, and Πφ(G) ⊂
+Πtemp(G).
+Moreover, if F is non-archimedean (resp.
+archimedean), then the map Πφ(G) → Irr(Sφ, χG) is
+bijective (resp. injective).
+5. (LLC) As φ runs over Φbdd(G∗) the sets Πφ(G) are disjoint, and we have
+�
+φ∈Φ2,bdd(G∗)
+Πφ(G) = Π2,temp(G),
+�
+φ∈Φbdd(G∗)
+Πφ(G) = Πtemp(G).
+Let ψ ∈ Ψ+(G∗) ∩ Ψ+
+unit(2n). If it is not G-relevant, define Πψ(G) to be empty. Suppose it is G-relevant. Then
+one can choose an inner twist ξ : G∗ → G, standard parabolic subgroups P ∗ ⊂ G∗ and P = ξ(P ∗) ⊂ G over F, the
+Levi subgroups M ∗ ⊂ P ∗ and M = ξ(M ∗) ⊂ P over F, the open chamber U ⊂ a∗
+M associated to P, a point λ ∈ U,
+and a parameter ψM∗ ∈ Ψ(M ∗), such that ψ is the image of the twist ψM∗,λ of ψM∗ by λ. Suppose that the packet
+ΠψM∗(M) and a map ΠψM∗(M) → Irr(SψM∗, χM) is given. Then we can define the packet for ψ and the associated
+map by
+Πψ(G) = { IP (πM,λ) | πM ∈ ΠψM∗(M) } ,
+⟨−, IP (πM,λ)⟩ = ⟨−, πM⟩.
+Note that Sψ = SψM∗ and χG = χM. See [7, pp.44-46] for more details.
+Let next F be a global field. Fix a nontrivial additive character ψF : F\AF → C1. Let G∗ = SO2n+1 be the split
+special orthogonal group of size 2n + 1 defined over F, equipped with the fixed pinning. Let (ξ, z) : G∗ → G be a pure
+inner twist of G∗.
+Theorem 3.12. There exists a decomposition
+L2
+disc(G(F)\G(AF )) =
+�
+ψ∈Ψ(G∗)
+L2
+disc,ψ(G(F)\G(AF )),
+where L2
+disc,ψ(G(F)\G(AF )) is a full near equivalence class of irreducible representations π = ⊗vπv in the discrete
+spectrum such that the L-parameter of πv is φψv for almost all v.
+This theorem will be proved in §5.1.
+For any global A-parameter ψ ∈ Ψ(G∗) and a place v of F, we have the localization ψv ∈ Ψ+(G∗
+v) ∩ Ψ+
+unit(2n)v
+thanks to Proposition 3.6. Then we have the packet Πψv(Gv) and the map Πψv(Gv) → Irr(Sψv, χGv). The global
+packet for ψ is defined as
+Πψ(G) =
+�
+π =
+�
+v
+πv
+����� πv ∈ Πψv(Gv),
+⟨−, πv⟩ = 1 for almost all v
+�
+.
+If a local packet Πψv(Gv) is empty for some v, then the global packet is defined to be empty. For any π = ⊗vπv ∈
+Πψ(G), we define a character ⟨−, π⟩ of Sψ by
+⟨x, π⟩ =
+�
+v
+⟨xv, πv⟩,
+x ∈ Sψ,
+21
+
+where xv denotes the image of x under the natural map Sψ → Sψv. The restriction of this character to Z(“
+G) is
+�
+v χGv, which is trivial by the product formula (2.2). Thus we may regard it as a character of Sψ. We have the
+character εψ : Sψ → {±1} defined by Arthur [7, p.48]. When we emphasize that the parameter ψ is for G∗ (or
+equivalently for G), we shall write εG∗
+ψ
+or εG
+ψ. Note that εψ is trivial if ψ is generic. Put
+Πψ(G, εψ) = { π ∈ Πψ(G) | ⟨−, π⟩ = εψ } .
+We now state the main global classification theorem.
+Theorem* 3.13 (AMF). For ψ ∈ Ψ(G∗), we have
+L2
+disc,ψ(G(F)\G(AF )) =
+
+
+
+
+
+�
+π∈Πψ(G,εψ)
+π,
+if ψ ∈ Ψ2(G∗),
+0,
+if ψ /∈ Ψ2(G∗).
+Arthur [7] proved Theorems 3.11, 3.12, and 3.13 for the case G = G∗. In this paper, following Kaletha-Minguez-
+Shin-White [18], we will give a proof of Theorems 3.11, 3.12, and 3.13 for the case ψ is generic. The main theorem of
+this paper is stated as follows:
+Theorem 3.14 (main theorem). Let F be a local or global field and G an inner form of G∗ = SO2n+1 over F.
+1. Let F be local. Let ψ = φ ∈ Φbdd(G∗) be a generic parameter. Then part 1, 2, 3 of Theorem 3.11 hold true for
+φ. Moreover, part 4 and 5 also hold true.
+2. Let F be global. Then Theorem 3.12 holds. Let moreover ψ = φ ∈ Φ(G∗) be a generic parameter. Then Theorem
+3.13 holds true for φ.
+As stated in Introduction, Arthur [7] also established the endoscopic classification of representations of symplectic
+groups and quasi-split even special orthogonal groups. In the case of symplectic groups, he proved theorems similar
+to Theorems 3.11, 3.12, and 3.13. On the other hand, in the case of quasi-split even special orthogonal groups, he
+proved the similar theorems up to outer automorphisms. For a quasi-split even orthogonal group G over a local field
+F, two representations π and π′ are said to be ǫ-equivalence if there exists ρ ∈ AutF (G) such that π ≃ π′ ◦ ρ, and
+the sets of ǫ-equivalence classes is denoted by �Π instead of Π. Arthur defined a weak local A-packet Πψ(G) for each
+ǫ-equivalence class of A-parameters ψ ∈ ‹Ψ(G), and then he proved a theorem similar to but weaker than Theorem
+3.11 in that Ψ and Π are replaced by ‹Ψ and �Π. For a quasi-split even orthogonal group G over a number field F, two
+representations π = �
+v πv and π′ = �
+v π′
+v are said to be ǫ-equivalence if πv and π′
+v are so for all places v. The weak
+global packet �Πψ(G) for ψ ∈ ‹Ψ(G) is defined as the set of π = �
+v πv such that πv ∈ �Πψv(Gv) for all v and ⟨−, πv⟩
+is trivial for almost all v. As in the local case, he proved theorems similar to but weaker than Theorems 3.12 and
+3.13 in that Ψ and Π are replaced by ‹Ψ and �Π. See also [8] for the weak theorems for even orthogonal groups. These
+classification theorems for symplectic and even special orthogonal groups are needed for the globalization in §6.2
+In the rest of this paper, we shall proof the main theorem 3.14 by a long induction argument following [18]. Let
+n be a positive integer. As an induction hypothesis, we assume that the Theorem 3.14 (or Theorems 3.11, 3.12, and
+3.13) holds for any positive integer n0 < n, and hence for any proper Levi subgroup M ∗ ⊊ G∗ = SO2n+1.
+4
+Local intertwining relation
+In this section, we define the normalized local intertwining operator and state the local intertwining relation, which
+is a key theorem in the theory of the endoscopic classification of representations. After that, we reduce the proof LIR,
+and give a proof of LIR in the special (real and low rank) cases.
+4.1
+The diagram
+We shall describe explicitly the basic commutative diagram ([7, (2.4.3)], [18, (2.1.1)]) for odd special orthogonal
+groups. See [18, §2.1] for the diagram for general connected reductive groups.
+Let F be a local or global field, G∗ = SO2n+1, and M ∗ ⊂ G∗ a standard Levi subgroup. Let ξ : G∗ → G be an
+inner twist over F with a Levi subgroup M = ξ(M ∗) ⊂ G which is an inner form of M ∗ over F. Let ψ ∈ Ψ(M ∗) be a
+22
+
+parameter, and ψG ∈ Ψ(G∗) the image of ψ under the natural map induced by �
+M ֒→ “
+G. Recall that “
+G = Sp(2n, C).
+The parameter ψG ∈ Ψ(G∗) and its centralizer group Sψ(G) = SψG(G) can be written as
+ψG =
+�
+i∈I+
+ψ
+ℓiψi ⊕
+�
+i∈I−
+ψ
+ℓiψi ⊕
+�
+j∈Jψ
+ℓj(ψj ⊕ ψ∨
+j ),
+Sψ(G) = Cent(Im ψ, Sp(2n, C)) ≃
+�
+i∈I+
+ψ
+O(ℓi, C) ×
+�
+i∈I−
+ψ
+Sp(ℓi, C) ×
+�
+j∈Jψ
+GL(ℓj, C).
+Moreover, we can also write as
+ψG =
+r
+�
+t=1
+et
+
+�
+i∈I+
+t
+ℓt
+iψi ⊕
+�
+i∈I−
+t
+ℓt
+iψi ⊕
+�
+j∈Jt
+(ℓt
+jψj ⊕ ℓt∨
+j ψ∨
+j )
+
+
+⊕
+
+�
+i∈I+
+0
+ℓ0
+i ψi ⊕
+�
+i∈I−
+0
+ℓ0
+i ψi ⊕
+�
+j∈J0
+ℓ0
+j(ψj ⊕ ψ∨
+j )
+
+
+⊕
+r
+�
+t=1
+et
+
+�
+i∈I+
+t
+ℓt
+iψi ⊕
+�
+i∈I−
+t
+ℓt
+iψi ⊕
+�
+j∈Jt
+(ℓt∨
+j ψj ⊕ ℓt
+jψ∨
+j )
+
+ ,
+�
+M ≃
+r
+�
+t=1
+GL(kt, C)et × Sp(2n0, C),
+where et, kt are positive integers,
+
+
+
+
+
+I+
+t ⊂ I+
+ψ ,
+I−
+t ⊂ I−
+ψ ,
+Jt ⊂ Jψ,
+for t = 0, 1, . . ., r,
+are subsets of indices, and ℓt
+i, ℓt
+j are positive integers such that
+
+
+
+ℓt
+i ≤ ℓi
+2 ,
+ℓt
+j, ℓt∨
+j
+≤ ℓj,
+ℓt
+j + ℓt∨
+j
+≤ ℓj,
+for t = 0, 1, . . ., r,
+and
+¶
+((ℓt
+i)i∈I+
+t , (ℓt
+i)i∈I−
+t , (ℓt
+j)j∈Jt, (ℓt∨
+j )j∈Jt)
+��� t = 1, . . . , r
+©
+is mutually distinct. Put
+ψt =
+�
+i∈I+
+t
+ℓt
+iψi ⊕
+�
+i∈I−
+t
+ℓt
+iψi ⊕
+�
+j∈Jt
+(ℓt
+jψj ⊕ ℓt∨
+j ψ∨
+j ),
+for t = 1, . . . , r,
+ψ0 =
+�
+i∈I+
+0
+ℓ0
+i ψi ⊕
+�
+i∈I−
+0
+ℓ0
+i ψi ⊕
+�
+j∈J0
+ℓ0
+j(ψj ⊕ ψ∨
+j ).
+Then we have
+ψG =
+r
+�
+t=1
+etψt ⊕ ψ0 ⊕
+r
+�
+t=1
+etψ∨
+t ,
+ψ =
+r
+�
+t=1
+etψt ⊕ ψ0,
+and each ψt corresponds to GL(kt, C) ⊂ �
+M, and ψ0 corresponds to Sp(2n0, C) ⊂ �
+M. Here we note that
+• k1, . . . , kr are not necessarily mutually distinct;
+23
+
+• ψ1, . . . , ψr are mutually distinct;
+• ψ1, . . . , ψr, ψ∨
+1 , . . . , ψ∨
+r are not necessarily mutually distinct.
+Let A �
+M be the maximal central split torus of �
+M. Then �
+M = Cent(A �
+M, “
+G). One has
+A �
+M ≃
+r
+�
+t=1
+(C×)et.
+Put
+Sψ(G) = Cent(Im ψ, “
+G),
+Sψ(M) = Cent(Im ψ, �
+M),
+Nψ(M, G) = Sψ(G) ∩ N(A �
+M, “
+G) = N(A �
+M, Sψ(G)).
+Put moreover
+Wψ(M, G) = N(A �
+M, Sψ(G))
+Z(A �
+M, Sψ(G)) ,
+W ◦
+ψ(M, G) = N(A �
+M, Sψ(G)◦)
+Z(A �
+M, Sψ(G)◦) ,
+Nψ(M, G) = N(A �
+M, Sψ(G))
+Z(A �
+M, Sψ(G)◦),
+Sψ(M, G) = N(A �
+M, Sψ(G))
+N(A �
+M, Sψ(G)◦),
+Rψ(M, G) =
+N(A �
+M, Sψ(G))
+N(A �
+M, Sψ(G)◦) · Z(A �
+M, Sψ(G)),
+and
+Sψ(G) = π0(Sψ(G)) ≃
+�
+i∈I+
+ψ
+O(ℓi, C)/ SO(ℓi, C) ≃
+�
+i∈I+
+ψ
+(Z/2Z)ai,
+Sψ(M) = π0(Sψ(M)) ≃
+�
+i∈I+
+0
+O(ℓ0
+i , C)/ SO(ℓ0
+i , C) ≃
+�
+i∈I+
+0
+(Z/2Z)ai,
+where {ai}i is a formal basis. Direct calculations show that
+Z(A �
+M, Sψ(G)◦) ≃
+r
+�
+t=1
+
+ �
+i∈I+
+t
+GL(ℓt
+i, C) ×
+�
+i∈I−
+t
+GL(ℓt
+i, C) ×
+�
+j∈Jt
+(GL(ℓt
+j, C) × GL(ℓt∨
+j , C))
+
+
+et
+×
+
+ �
+i∈I+
+0
+SO(ℓ0
+i , C) ×
+�
+i∈I−
+0
+Sp(ℓ0
+i , C) ×
+�
+j∈J0
+GL(ℓ0
+j, C)
+
+ ,
+Z(A �
+M, Sψ(G)) ≃
+r
+�
+t=1
+
+ �
+i∈I+
+t
+GL(ℓt
+i, C) ×
+�
+i∈I−
+t
+GL(ℓt
+i, C) ×
+�
+j∈Jt
+(GL(ℓt
+j, C) × GL(ℓt∨
+j , C))
+
+
+et
+×
+
+ �
+i∈I+
+0
+O(ℓ0
+i , C) ×
+�
+i∈I−
+0
+Sp(ℓ0
+i , C) ×
+�
+j∈J0
+GL(ℓ0
+j, C)
+
+ .
+In particular, the connected component of the identity in Z(A �
+M, Sψ(G)) is equal to Z(A �
+M, Sψ(G)◦). Thus, we have
+�
+M ∩ Sψ(G)◦ = Sψ(M)◦,
+(4.1)
+Z(A �
+M, Sψ(G))
+Z(A �
+M, Sψ(G)◦) ≃ Sψ(M) ≃
+�
+i∈I+
+0
+(Z/2Z)ai.
+(4.2)
+24
+
+We now consider the group N(A �
+M, Sψ(G)). Take a partition
+{ 1, . . ., r } = T1 ⊔ T2 ⊔ T3 ⊔ T4
+such that
+T1 = { t | ψt ≃ ψ∨
+t } ,
+T2 = { t | ψt ̸≃ ψ∨
+t′ for any 1 ≤ t′ ≤ r } ,
+T3 = { t | ψt ≃ ψ∨
+t′ for a unique t′ ∈ T4 } ,
+T4 = { t | ψt ≃ ψ∨
+t′ for a unique t′ ∈ T3 } ,
+and for each t ∈ T3, put e∨
+t = et′, where t′ is the element of T4 such that ψt ≃ ψ∨
+t′.
+Before the description of N(A �
+M, Sψ(G)), we must fix notation. For a finite set X, let SX denote its symmetric
+group, and put Sm = S{1,2,...,m} for a positive integer m. For any positive integers e, e∨, let
+W(e, e∨)
+be a subgroup of Se+e∨ ⋉ (Z/2Z)e+e∨ generated by
+S{ 1,...,e }, S{ e+1,...,e+e∨ }, and
+®
+(h, h∨) ⋉ (· · · , 0,
+h
+1, 0, · · · , 0,
+h∨
+1 , 0, · · · )
+���� 1 ≤ h ≤ e, e + 1 ≤ h∨ ≤ e + e∨
+´
+.
+Then we obtain an explicit description of the group N(A �
+M, Sψ(G)). We have
+N(A �
+M, Sψ(G)) ≃ c′ ×
+Ñ
+�
+i∈I+
+0
+O(ℓ0
+i , C), ×
+�
+i∈I−
+0
+Sp(ℓ0
+i , C) ×
+�
+j∈J0
+GL(ℓ0
+j, C)
+é
+,
+where a subgroup c′ of Sp(2(n − n0), C) can be written as
+c′ ≃
+�
+t∈T1
+
+
+�
+wt∈Set⋉(Z/2Z)et
+wt ·
+Ñ
+�
+i∈I+
+t
+GL(ℓt
+i, C) ×
+�
+i∈I−
+t
+GL(ℓt
+i, C) ×
+�
+j∈Jt
+(GL(ℓt
+j, C) × GL(ℓt∨
+j , C))
+éet
+
+×
+�
+t∈T2
+
+
+�
+wt∈Set
+wt ·
+Ñ
+�
+i∈I+
+t
+GL(ℓt
+i, C) ×
+�
+i∈I−
+t
+GL(ℓt
+i, C) ×
+�
+j∈Jt
+(GL(ℓt
+j, C) × GL(ℓt∨
+j , C))
+éet
+
+×
+�
+t∈T3
+
+
+�
+wt∈W(et,e∨
+t )
+wt ·
+Ñ
+�
+i∈I+
+t
+GL(ℓt
+i, C) ×
+�
+i∈I−
+t
+GL(ℓt
+i, C) ×
+�
+j∈Jt
+(GL(ℓt
+j, C) × GL(ℓt∨
+j , C))
+éet+e∨
+t 
+ .
+Therefore we have natural isomorphisms
+Wψ(M, G) ≃
+�
+t∈T1
+(Set ⋉ (Z/2Z)et) ×
+�
+t∈T2
+Set ×
+�
+t∈T3
+W(et, e∨
+t ),
+(4.3)
+Nψ(M, G) ≃ Sψ(M) × Wψ(M, G).
+(4.4)
+We finally consider N(A �
+M, Sψ(G)◦). It is too complicated to write explicitly, but we can describe the natural
+surjection
+Nψ(M, G) = N(A �
+M, Sψ(G))
+Z(A �
+M, Sψ(G)◦) −→ Sψ(M, G) = N(A �
+M, Sψ(G))
+N(A �
+M, Sψ(G)◦),
+to understand the group N(A �
+M, Sψ(G)◦) in terms of its kernel. For e ≥ 1, let us write x for a group homomorphism
+Se ⋉ (Z/2Z)e −→ Z/2Z,
+σ ⋉ (dh)e
+h=1 �→
+e
+�
+h=1
+dh.
+25
+
+For each i ∈ I+
+ψ , we shall define a corresponding function
+xi : Nψ(M, G) −→ Z/2Z
+as follows. By the equations (4.4), (4.3), and (4.2), we may identify Nψ(M, G) with
+�
+i∈I+
+0
+(Z/2Z)ai ×
+�
+t∈T1
+(Set ⋉ (Z/2Z)et) ×
+�
+t∈T2
+Set ×
+�
+t∈T3
+W(et, e∨
+t ).
+(4.5)
+For any element
+u =
+Ñ
+�
+i∈I+
+0
+ciai, (wt)t∈T1, (wt)t∈T2, (wt)t∈T3
+é
+of (4.5), we define
+xi(u) = ci +
+�
+t∈T1⊔T3
+ℓt
+ix(wt),
+where we put ci = 0 for i /∈ I+
+0 , and ℓt
+i = 0 for i /∈ I+
+t . Now, we can describe the natural surjection. Let us define a
+homomorphism xψ : Nψ(M, G) → �
+i∈I+
+ψ (Z/2Z)ai = Sψ(G) by
+xψ(u) =
+�
+i∈I+
+ψ
+xi(u)ai
+for u ∈ Nψ(M, G). Then Sψ(M, G) and W ◦
+ψ(M, G) can be identified with the image and the kernel of xψ, respectively.
+Put O+(l, C) = SO(l, C) and O−(l, C) = O(l, C) \ SO(l, C). The homomorphism xψ gives a description:
+N(A �
+M, Sψ(G)◦) ≃
+�
+u
+{
+�
+t∈T1
+
+wt ·
+Ñ
+�
+i∈I+
+t
+GL(ℓt
+i, C) ×
+�
+i∈I−
+t
+GL(ℓt
+i, C) ×
+�
+j∈Jt
+(GL(ℓt
+j, C) × GL(ℓt∨
+j , C))
+éet
+
+×
+�
+t∈T2
+
+wt ·
+Ñ
+�
+i∈I+
+t
+GL(ℓt
+i, C) ×
+�
+i∈I−
+t
+GL(ℓt
+i, C) ×
+�
+j∈Jt
+(GL(ℓt
+j, C) × GL(ℓt∨
+j , C))
+éet
+
+×
+�
+t∈T3
+
+wt ·
+Ñ
+�
+i∈I+
+t
+GL(ℓt
+i, C) ×
+�
+i∈I−
+t
+GL(ℓt
+i, C) ×
+�
+j∈Jt
+(GL(ℓt
+j, C) × GL(ℓt∨
+j , C))
+éet+e∨
+t 
+
+×
+
+ �
+i∈I+
+0
+Oεi(ℓ0
+i , C), ×
+�
+i∈I−
+0
+Sp(ℓ0
+i , C) ×
+�
+j∈J0
+GL(ℓ0
+j, C)
+
+},
+where u =
+�
+i∈I+
+0 ciai, (wt)t∈T1, (wt)t∈T2, (wt)t∈T3
+ä
+runs over ker(xψ), and εi = (−1)ci.
+We have obtained the explicit description of the following commutative diagram with exact rows and columns.
+1
+1
+�
+�
+W ◦
+ψ(M, G)
+W ◦
+ψ(M, G)
+�
+�
+1 −−−−→ Sψ(M) −−−−→ Nψ(M, G) −−−−→ Wψ(M, G) −−−−→ 1
+���
+xψ
+�
+�
+1 −−−−→ Sψ(M) −−−−→ Sψ(M, G) −−−−→ Rψ(M, G) −−−−→ 1
+�
+�
+1
+1
+(4.6)
+26
+
+Lemma 4.1. If ψ ∈ Ψ2(M ∗), then xψ is surjective. In particular, we have Sψ(M, G) = Sψ(G).
+Proof. Suppose that ψ ∈ Ψ2(M ∗). Then Kt := I+
+t ⊔I−
+t ⊔Jt is a singleton for t = 1, . . . , r, and ℓt
+i = 1 for t = 0, 1, . . ., r.
+Let i ∈ I+
+0 . If we put u = (ai, 1, 1, 1) ∈ Nψ(M, G), then xψ(u) = ai. Let i ∈ I+
+t (t = 1, . . . , r). Then t ∈ T1, as ψt = ψi
+is self-dual. If we put u = (0, (ws)s∈T1, 1, 1) ∈ Nψ(M, G), where wt = 1 ⋉ (1, 0, . . . , 0) and wsis trivial if s ̸= t. Then
+xψ(u) = ai. Since I+
+ψ is the union of I+
+0 , I+
+1 , . . . , T +
+r , this completes the proof.
+Dividing the diagram (4.6) by Z(“
+G)Γ = {±1}, we obtain another diagram
+1
+1
+�
+�
+W ◦
+ψ(M, G)
+W ◦
+ψ(M, G)
+�
+�
+1 −−−−→ Sψ(M) −−−−→ Nψ(M, G) −−−−→ Wψ(M, G) −−−−→ 1
+���
+�
+�
+1 −−−−→ Sψ(M) −−−−→ Sψ(M, G) −−−−→ Rψ(M, G) −−−−→ 1
+�
+�
+1
+1
+where
+Sψ(M) = Sψ(M)
+Z(“
+G)Γ ,
+Nψ(M, G) = Nψ(M, G)
+Z(“
+G)Γ
+,
+Sψ(M, G) = Sψ(M, G)
+Z(“
+G)Γ
+.
+4.2
+The first intertwining operator
+Let F be a local field, ψF : F → C1 a nontrivial additive character, G∗ = SO2n+1, and M ∗ ⊊ G∗ a proper standard
+Levi subgroup. Let ξ : G∗ → G be an inner twist over F which restrict to an inner twist ξ|M∗ : M ∗ → M over F. Let
+ψ ∈ Ψ(M ∗) be a local parameter. Assume that ψ is M-relevant, and the corresponding packet Πψ(M) is not empty.
+Let π ∈ Πψ(M). We shall write Vπ for the representation space of π.
+Let P and P ′ be parabolic subgroups of G defined over F with common Levi factor M. We write (IP (π), HP (π))
+for the representation parabolically representation by (π, Vπ) from P to G.
+A function HM : M(F) → aM is defined by
+exp(⟨HM(m), χ⟩) = |χ(m)|F ,
+for any χ ∈ X∗(M)F , m ∈ M(F). Then each λ ∈ a∗
+M,C gives a character
+M(F) → C×,
+m �→ exp(⟨HM(m), λ⟩).
+We shall write πλ for the tensor product of π and this character.
+Then the unnormalized intertwining operator
+JP ′|P (ξ, ψF ) : HP (πλ) → HP ′(πλ) is given by
+[JP ′|P (ξ, ψF )f](g) =
+�
+N(F )∩N ′(F )\N ′(F )
+f(n′g)dn′,
+where N and N ′ are the unipotent radical of P and P ′, respectively. It is known that the integral converges absolutely
+when the real part of λ lies in a certain open cone. The Haar measure dn′ on N(F) ∩ N ′(F)\N ′(F) that we use here
+is the measure [18, §2.2] defined. Although the case of unitary groups is considered in loc. cit., we apply the same
+definition with the same notation.
+27
+
+We write WF ∋ w �→ |w|λ for the L-parameter of the character M(F) ∋ m �→ exp(⟨HM(m), λ⟩) attached to
+λ ∈ a∗
+M,C. Then ψλ = ψ| − |λ is a parameter of πλ. Let ρP ′|P be the adjoint representation of �
+M on �n′ ∩ �n\�n′. Put
+rP ′|P (ξ, ψλ, ψF ) =
+L(0, ρ∨
+P ′|P ◦ φψλ)
+L(1, ρ∨
+P ′|P ◦ φψλ)
+ǫ( 1
+2, ρ∨
+P ′|P ◦ φψλ, ψF )
+ǫ(0, ρ∨
+P ′|P ◦ φψλ, ψF ) ,
+where the factors are the Artin L- and ǫ- factors. The normalized intertwining operator is given by
+RP ′|P (ξ, ψλ) = rP ′|P (ξ, ψλ, ψF )−1JP ′|P (ξ, ψF ).
+It is known that RP ′|P (ξ, ψλ) is independent of the choice of ψF , and that λ �→ RP ′|P (ξ, ψλ) has meromorphic
+continuation to whole a∗
+M,C.
+Lemma 4.2. Assume that F = R and ψ = φ ∈ Φ(M ∗). Then the function λ �→ RP ′|P (ξ, φλ) has neither a zero nor
+a pole at λ = 0. Moreover, let P ′′ ⊂ G be a parabolic subgroup defined over F with Levi factor M. Then we have
+RP ′′|P (ξ, φ) = RP ′′|P ′(ξ, φ) ◦ RP ′|P (ξ, φ).
+Proof. The idea of the proof is the same as that of [18, Lemma 2.2.1]. So it suffices to show that the measure used in
+the definition of JP ′|P (ξ, ψF ) coincides with the measure introduced by Arthur [5, §3].
+Let us calculate the measure given in loc.
+cit.
+Since F = R, there exist nonnegative integers p and q with
+p + q = 2n + 1 such that the inner twist ξ : G∗ → G is realized by ξp,q : SO2n+1 → SO(p, q) given in §2.3. So,
+we will use the notation introduced there. Let B be a SO(p, q)-invariant bilinear form on gC = so(p, q)C given by
+B(X, Y ) = 1
+2 tr(XY ). Then the quadratic form
+X �→ −B(X, θX)
+is positive definite on gR = so(p, q)R, where θ is the Cartan involution defined by θ(X) = − tX. The straightforward
+calculation shows that the constant αP ′|P defined in [5, §3] is equal to 2
+k
+2 , where k is the number of roots of the form
+χi whose restriction to aM are roots of both (P ′, AM) and (P, AM). Here, AM and P denote the maximal central
+split torus in M and the parabolic subgroup opposite to P containing M, respectively.
+Next, we calculate the Euclidean measure dX defined by the quadratic form X �→ −B(X, θX) ([5, §3]). Put
+T = ξ(T ∗), and write t for its Lie algebra. Then we have a decomposition
+gC = tC ⊕
+�
+a∈R(T,G)/Γ
+ga,C,
+where Γ is the Galois group of F/F = C/R. Note that each ga,C = ⊕α∈agα is defined over R, but gα is not necessarily
+defined over R. Let a ∈ R(T, G)/Γ and α ∈ a. If α is a complex root, then a = {α, σα}, and dim ga = 2, where
+σ ∈ Γ is the complex conjugation. In this case, ga has an R-basis {X1, X2}, where X1 = ξ(Xα) + σ(ξ(Xα)) and
+X2 = −√−1(ξ(Xα) − σ(ξ(Xα))). By a straightforward calculation, we have −B(X1, θX1) = −B(X2, θX2) = 2. If
+gα ⊂ n ∩ n′, the contribution of a to the measure dX is |d(2− 1
+2 X1 ∧ 2− 1
+2 X2)| = |d(ξ(Xα ∧ Xσα))|, which is equal to
+the contribution to our measure. If α is a real root, then a = {α}, and dim ga = 1. In this case, ga has an R-basis
+{X1}, where X1 = ξ(Xα). By a straightforward calculation, we have −B(X1, θX1) = 1 if α is of the form χi ± χj, and
+−B(X1, θX1) = 2 if α is of the form χi. If gα ⊂ n ∩ n′, the contribution of a to the measure dX is |dX1| = |d(ξ(Xα))|
+if α is of the form χi ± χj, and |d(2− 1
+2 X1)| = 2− 1
+2 |d(ξ(X1))| if α is of the form χi. The product of coefficients 2− 1
+2 for
+all α of the form χi and αP ′|P = 2
+k
+2 cancel. If α is an imaginary root, then gα ⊈ n ∩ n′. This completes the proof.
+Lemma 4.3. Assume that F is a p-adic field and ψ = φ ∈ Φ(M ∗) an L-parameter.
+Then the function λ �→
+RP ′|P (ξ, φλ) has neither a zero nor a pole at λ = 0. Moreover, let P ′′ ⊂ G be a parabolic subgroup defined over F with
+Levi factor M. Then we have
+RP ′′|P (ξ, φ) = RP ′′|P ′(ξ, φ) ◦ RP ′|P (ξ, φ).
+Proof. Note that the proofs of Lemmas 5.2, 5.3, and 6.6 are independent from this lemma. In particular, we may use
+them here.
+Let π ∈ Πφ(M) be an arbitrary representation of M corresponding to φ. The operator RP ′|P (ξ, φλ) is an inter-
+twining operator from HP (πλ) to HP ′(πλ). By the reduction steps described in [5, §2] (more precisely, from the last
+28
+
+line in p.29 of loc. cit.), we may assume that π is supercuspidal. Let ˙F, u, v2, ˙G∗, ˙G, ˙ξ, ˙M ∗, ˙M, and ˙π be as in Lemma
+6.6. Furthermore, let ˙P, ˙P ′, and ˙P ′′ be the parabolic subgroups of ˙G such that ˙Pu = P, ˙P ′
+u = P ′, and ˙P ′′
+u = P ′′.
+Since we can choose v2 ̸= u arbitrarily, we may assume that v2 is a real place. By the induction hypothesis, we can
+take the A-parameter of ˙π, write ˙φ for it. Note that ˙φ is a generic parameter because ˙φu = φ.
+The global intertwining operator R ˙P ′| ˙P ( ˙πλ, ˙φλ) has neither a zero nor a pole at λ = 0, and is multiplicative in ˙P ′,
+˙P by Lemma 5.3. For any place v ̸= u, v2, ˙Gv splits over ˙Fv. For u = v2, we have ˙Fv2 = R. Thus for any v ̸= u, the
+local normalized intertwining operator R ˙P ′v| ˙Pv( ˙ξv, ˙φλ,v) has neither a zero nor a pole at λ = 0, and is multiplicative by
+Lemma 4.2 and the assumption from the split case. Now the assertion follows from Lemma 5.2.
+The following lemma is now proven for generic parameters. In general, we expect it can be proven as [18, Lemma
+2.2.4].
+Lemma* 4.4. Let F be a local field. Let ψ ∈ Ψ(M ∗) be a general local A-parameter for M. Then the function
+λ �→ RP ′|P (ξ, ψλ) has neither a zero nor a pole at λ = 0. Moreover, let P ′′ ⊂ G be a parabolic subgroup defined over
+F with Levi factor M. Then we have
+RP ′′|P (ξ, ψ) = RP ′′|P ′(ξ, ψ) ◦ RP ′|P (ξ, ψ).
+4.3
+The second intertwining operator
+We maintain the assumptions of the previous subsection. For every finite separable extension E/F, Langlands [25,
+Theorem 1] defined a complex number λ(E/F, ψF ). By the definition, we have λ(F/F, ψF ) = 1.
+For w ∈ W(T ∗, G∗), Keys-Shahidi [19, (4.1)] defined the constant λ(w, ψF ). In this subsection, we shall construct
+a local intertwining operator denoted by ℓP ′
+P (w, ξ, ψ, ψF ). In the case of unitary groups, its construction involves the
+λ-factor as in [18, §2.3]. However, in the case of odd special orthogonal groups, we do not need the λ-factor since it
+always vanishes:
+Lemma 4.5. For any w ∈ W(T ∗, G∗), we have
+λ(w, ψF ) = 1.
+Proof. By the definition of λ(w, ψF ), it suffices to show that
+λ(Fa/F, ψF ) = 1,
+for any root a ∈ R(T ∗, G∗), where Fa/F is a finite extension such that G∗
+a,sc = ResFa/F (SL2). Here, G∗
+a,sc denotes
+the simply connected cover of the derived group G∗
+a,der of the Levi subgroup G∗
+a of semisimple rank 1 attached to a.
+Since we know that λ(F/F, ψF ) = 1, it suffices to show that Fa = F for all root a ∈ R(T ∗, G∗).
+If a is a long root (i.e., of the form ±χi ± χj), then G∗
+a is isomorphic to a product of GL2 and a finite number of
+GL1. Thus we have G∗
+a,der ∼= SL2 and G∗
+a,sc ∼= SL2, which means that Fa = F. If a is a short root (i.e., of the form
+±χi), then G∗
+a is isomorphic to a product of SO3 and a finite number of GL1. Thus we have G∗
+a,der ∼= SO3 ∼= PGL2
+and G∗
+a,sc ∼= SL2, which means that Fa = F.
+Let ψ ∈ Ψ(M ∗) be a local parameter such that ψ is M-relevant and the corresponding packet Πψ(M) is not empty.
+Let π ∈ Πψ(M). Moreover, let P and P ′ be parabolic subgroups of G defined over F with common Levi factor
+M. For an element w ∈ W(M ∗, G∗) ∼= W(M, G), we write �w ∈ N(M ∗, G∗) for its Langlands-Shelstad lift, and put
+˘w = ξ( �w). By replacing ξ by an equivalent inner twist if necessary, we assume that ˘w ∈ G(F). This gives us another
+representation
+[ ˘wπ](m) = π( ˘w−1m ˘w),
+m ∈ M(F),
+of M(F) on the same vector space as π, which corresponds to a parameter wψ := Ad(w) ◦ ψ.
+First, we define an unnormalized intertwining operator ℓP ′( ˘w) from HP ′w(π) to HP ′( ˘wπ) by
+[ℓP ′( ˘w)f](g) = f( ˘w−1g),
+where P ′w denotes w−1P ′w. Next, our normalizing factor is defined by
+ǫP (w, ψ, ψF ) = ǫ(1
+2, ρ∨
+P w|P ◦ φψ, ψF ),
+and we define
+ℓP ′
+P (w, ξ, ψ, ψF ) = ǫP (w, ψ, ψF )ℓP ′( ˘w).
+29
+
+Lemma 4.6. For any w1, w2 ∈ W(M ∗, G∗) ∼= W(M, G), we have
+ℓP
+P (w2w1, ξ, ψ, ψF ) = ℓP
+P (w2, ξ, w1ψ, ψF ) ◦ ℓP w2
+P
+(w1, ξ, ψ, ψF ).
+Proof. The proof is similar to that of [7, Lemma 2.3.4] or [18, Lemma 2.3.1]. Our case is simpler since the λ-factor is
+trivial and the Galois action on the dual group “
+G is also trivial.
+Lemma 4.7. We have
+ℓP ′
+P (w, ξ, ψ, ψF ) ◦ RP ′w|P w(ξ, ψ) = RP ′|P (ξ, wψ) ◦ ℓP
+P (w, ξ, ψ, ψF ).
+Proof. The assertion follows from two equalities ℓP ′
+P (w, ξ, ψ, ψF )◦RP ′w|P w(ξ, ψ) = RP ′|P (Ad( ˘w)◦ξ, ψ)◦ℓP
+P (w, ξ, ψ, ψF )
+and RP ′|P (Ad( ˘w) ◦ ξ, ψ) = RP ′|P (ξ, wψ). We can show them by a similar way to the second and the third assertions
+in [18, Lemma 2.2.5].
+4.4
+The third intertwining operator
+We maintain the assumptions of the previous subsection. In addition, we assume that we are given z ∈ Z1(F, M ∗) ⊂
+Z1(F, G∗), such that (ξ, z) is a pure inner twist; z commutes with the Langlands-Shelstad lift of every element of
+W(M ∗, G∗); and if we decompose z as z = z+ ×z− according to the decomposition M ∗ = M ∗
++ ×M ∗
+− then z+ takes the
+constant value 1, where M ∗
++ is a product of general linear groups and M ∗
+− is a special orthogonal group. Let u ∈ N( �T, “
+G)
+be such that Ad(u) ◦ ψ = ψ and u preserves the positive roots in �
+M. Let u♮ ∈ Nψ(M, G) and w ∈ W(�
+M, “
+G) be its
+images. We shall regard w as an element of W(M ∗, G∗) or W(M, G) via the natural isomorphisms. Although our
+group G∗ is a special orthogonal group and (ξ, z) is a pure inner twist, by the similar argument as [18, §2.4, §2.4.1],
+we can define
+• the operator π( ˘w)ξ : ( ˘wπ, Vπ) → (π, Vπ);
+• the sophisticated splittings s′ : Wψ(M, G) → Nψ(M, G) and s : Nψ(M, G) → Sψ(M) of the exact sequence
+1 → Sψ(M) → Nψ(M, G) → Wψ(M, G) → 1;
+• the constant ⟨u♮, π⟩ξ,z = ⟨s(u♮)−1, π⟩ξ|M∗,z ∈ C×.
+Then we define the operator π(u♮)ξ,z : ( ˘wπ, Vπ) → (π, Vπ) by π(u♮)ξ,z = ⟨u♮, π⟩ξ,zπ( ˘w)ξ. The assignment u♮ �→ π(u♮)ξ,z
+is multiplicative.
+Next we shall give a more abstract characterization (i.e., another equivalent definition) of the operator π(u♮)ξ,z :
+( ˘wπ, Vπ) → (π, Vπ), following [18, §2.4.3]. Let us put u′ = u−1 ∈ Nψ(M, G), and let ‹
+u′ ∈ N( �T, “
+G) be its Langlands-
+Shelstad lift. Put s = u′‹
+u′−1 ∈ �
+M. Then by the assumption that u preserves the positive roots in �
+M, we have s ∈ �T.
+Put �θ = Ad(‹
+u′), which is an automorphism of �
+M preserving the standard splitting inherited from “
+G. It is the dual of
+an automorphism θ∗ = Ad( �w) of M ∗. Put
+”
+M ′ =
+¶
+x ∈ �
+M
+��� Ad(sψs) ◦ �θ(x) = x
+©
+.
+Let M ′ be the subgroup of M such that ”
+M ′ is the dual of M ′. Note that Im ψ ⊂ ”
+M ′. Then (M ′, sψs, ”
+M ′ ⊂ �
+M) is a
+twisted endoscopic triple of (M ∗, θ∗), and hence of (M, θ), where θ = Ad( ˘w). We shall write eM,ψ for it. The operator
+π(u♮)ξ,z can be characterized as follows:
+Lemma 4.8. For each π ∈ Πψ(M), there exists a unique isomorphism π(u♮)ξ,z : ( ˘wπ, Vπ) → (π, Vπ) such that
+f eM,ψ(ψ) = e(M θ)
+�
+π∈Πψ(M)
+tr(π(u♮)ξ,z ◦ π(f)),
+for all ∆[eM,ψ, ξ|M∗, z]-matching functions f ∈ H(M) and f eM,ψ ∈ H(M ′). Moreover, the operator π(u♮)ξ,z coincides
+with the one constructed above.
+Proof. The proof is similar to that of [18, Lemma 2.4.1] for the case of the quasi-split unitary group UE/F(N).
+30
+
+4.5
+The compound operator
+We shall define the normalized self-intertwining operator RP (u♮, π, ψ, ψF ) ∈ EndG(HP (π)) as a composition
+RP (u♮, π, ψ, ψF ) = IP (π(u♮)ξ,z) ◦ ℓP
+P (w, ξ, ψ, ψF ) ◦ RP w|P (ξ, ψ).
+Recall that the definitions of the Haar measures on the unipotent radicals and the representative ˘w which were used
+to define RP w|P (ξ, ψ) and ℓP
+P (w, ξ, ψ, ψF ), respectively, depend on ξ. Note also that the definition of IP (π(u♮)ξ,z)
+depends on (ξ, z). If the parameter ψ is not generic, we shall assume from now on that Lemma 4.4 holds true.
+Lemma 4.9. The operator RP (u♮, π, ψ, ψF ) does not depend on the choice of the pure inner twist (ξ, z).
+Proof. Suppose that (ξ, z) ≃ (ξ′, z′) as inner twists of G∗ = SO2n+1 over F. Since every automorphism is inner, there
+exists b ∈ G∗(F) such that ξ′ = ξ ◦ Ad(b). The proof is similar to that of [18, Lemma 2.5.1] except that our z is not a
+basic cocycle, but an ordinary 1-cocycle.
+The Kottwitz map (2.1) provides a pairing ⟨−, −⟩ on Z(“
+G) × H1(F, G∗). Note that in our case Z(“
+G) ≃ {±1}.
+Lemma 4.10.
+1. Let x ∈ Z(“
+G). Then RP (xu♮, π, ψ, ψF ) = ⟨x, z⟩−1RP (u♮, π, ψ, ψF ).
+2. Let y ∈ Sψ(M). Then RP (yu♮, π, ψ, ψF ) = ⟨y, π⟩ξ|M∗,zRP (u♮, π, ψ, ψF ).
+Proof. The proof is similar to that of [18, Lemma 2.5.2].
+Lemma 4.11. The operator RP (u♮, π, ψ, ψF ) is multiplicative in u♮.
+Proof. Let u♮
+1 and u♮
+2 be elements in Nψ(M, G), and w1 and w2 their images in Wψ(M, G) respectively. Since the
+assignment u♮ �→ π(u♮)ξ,z is multiplicative, we have
+IP (π(u♮
+2u♮
+1)ξ,z) = IP (π(u♮
+2)ξ,z ◦ π(u♮
+1)ξ,z) = IP (π(u♮
+2)ξ,z) ◦ IP (π(u♮
+1)ξ,z).
+Combining this with Lemmas 4.3, 4.6, and 4.7, we get
+RP (u♮
+2u♮
+1, π, ψ, ψF )
+= IP (π(u♮
+2)ξ,z) ◦ IP (π(u♮
+1)ξ,z) ◦ ℓP
+P (w2, ξ, w1ψ, ψF ) ◦ ℓP w2
+P
+(w1, ξ, ψ, ψF ) ◦ RP w2w1|P w1 (ξ, ψ) ◦ RP w1|P (ξ, ψ)
+= IP (π(u♮
+2)ξ,z) ◦ IP (π(u♮
+1)ξ,z) ◦ ℓP
+P (w2, ξ, ψ, ψF ) ◦ RP w2 |P (ξ, ψ) ◦ ℓP
+P (w1, ξ, ψ, ψF ) ◦ RP w1|P (ξ, ψ).
+We know that the operator IP (π(u♮
+1)ξ,z) commutes with ℓP
+P (w2, ξ, w1ψ, ψF )◦RP w2|P (ξ, ψ), as the operator IP (π(u♮
+1)ξ,z)
+acts on the values of the functions that comprise HP (π), while the operators ℓP
+P (w2, ξ, ψ, ψF ) and RP w2 |P (ξ, ψ) act on
+their variables. Thus we have
+IP (π(u♮
+2)ξ,z) ◦ IP (π(u♮
+1)ξ,z) ◦ ℓP
+P (w2, ξ, ψ, ψF ) ◦ RP w2 |P (ξ, ψ) ◦ ℓP
+P (w1, ξ, ψ, ψF ) ◦ RP w1 |P (ξ, ψ)
+= IP (π(u♮
+2)ξ,z) ◦ ℓP
+P (w2, ξ, ψ, ψF ) ◦ RP w2|P (ξ, ψ) ◦ IP (π(u♮
+1)ξ,z) ◦ ℓP
+P (w1, ξ, ψ, ψF ) ◦ RP w1|P (ξ, ψ)
+= RP (u♮
+2, π, ψ, ψF ) ◦ RP (u♮
+1, π, ψ, ψF ).
+4.6
+The two linear forms, the local intertwining relation, and the construction of the
+non-discrete packets
+Let F be a local field, ψF : F → C1 a nontrivial additive character, G∗ = SO2n+1, and M ∗ ⊊ G∗ a proper standard
+Levi subgroup. Let ξ : G∗ → G be an inner twist over F, and put M = ξ(M ∗). Let ψ ∈ Ψ(M ∗) be a local parameter,
+and u♮ an element of Nψ(M, G). When M ∗ transfers to G, we assume that M ⊊ G is a proper Levi subgroup over F
+and ξ|M∗ : M ∗ → M is an inner twist over F.
+We shall define the first linear form f �→ fG(ψ, u♮) on H(G) as follows:
+fG(ψ, u♮) =
+
+
+
+
+
+�
+π∈Πψ(M)
+tr(RP (u♮, π, ψ, ψF ) ◦ IP (π, f)),
+if ψ is relevant,
+0,
+if ψ is not relevant.
+31
+
+Next we shall give the definition of the second linear form f �→ f ′
+G(ψ, s) on H(G). For f ∈ H(G) and s ∈ Sψ,ss, let
+(e, ψe) ∈ X(G∗) be the element corresponding to (s, ψ) ∈ Y (G∗) under the bijective map given in Lemma 3.8. Then
+we write f ′
+G(ψ, s) for the value f e(ψe) in the endoscopic character relation (3.7). It does not depend on the choice of
+ξ.
+Lemma 4.12. Let s1, s2 ∈ Sψ,ss be two elements such that they have the same image in Sψ(G) and the image belongs
+to Sψ(M, G). Then f ′
+G(ψ, s1) = f ′
+G(ψ, s).
+Proof. The proof is similar to that of [18, Lemma 2.6.1].
+Theorem* 4.13 (LIR). Let u ∈ Nψ(M, G) and f ∈ H(G). Then
+1. If u♮ is in W ◦
+ψ(M, G), then we have RP (u♮, π, ψ, ψF ) = 1 for all π ∈ Πψ(M).
+2. The value fG(ψ, u♮) depends on the image of u♮ in Sψ(M, G).
+3. We have
+f ′
+G(ψ, sψu−1) = e(G)fG(ψ, u♮),
+(4.7)
+where e(G) denotes the Kottwitz sign of G. This equation is called the local intertwining relation.
+Lemma 4.14. For any y ∈ Z(“
+G∗), we have f ′
+G(ψ, sψys) = ⟨y, z⟩f ′
+G(ψ, sψs) and fG(ψ, yu♮) = ⟨y, z⟩−1fG(ψ, u♮).
+Proof. They follow from Lemmas 3.2 and 4.10.
+In this paper, Theorem 4.13 will be completely proved in the case when ψ = φ ∈ Φ(M ∗) is an L-parameter. The
+proof is inductive, and thus we assume that Theorem 4.13 holds for the case the degree is smaller than n, from now
+on. The case of a general parameter ψ ∈ Ψ(M ∗) is expected to be treated as a sequel of [18].
+We will now review the construction of local non-discrete A-packets which is an application of the local intertwining
+relation (Theorem 4.13). Suppose that the theorem is true. Assume that ψM∗ ∈ Ψ2(M ∗) and M ∗ ⊊ G∗, hence ψM∗
+is a discrete parameter for M ∗ and not discrete for G∗. We shall write ψ for the image of ψM∗ under the natural map
+Ψ(M ∗) → Ψ(G∗). We apologize for this change of convention. If ψ is not G-relevant, then we simply set Πψ(G) = ∅.
+Suppose that ψ is G-relevant. Then we may assume that a Levi subgroup M = ξ(M ∗) ⊊ G is defined over F, and
+by the assumption Theorem 4.13 holds true for ψM∗ and M. According to Lemma 4.1, we have SψM∗(M, G) = Sψ(G),
+hence the natural map NψM∗(M, G) → Sψ(G) is surjective. For any πM ∈ ΠψM∗(M), where note that the packet for
+M is given by the induction hypothesis, the map
+NψM∗(M, G) × G(F) → AutG(F )(HP (πM)),
+(u♮, g) �→ RP (u♮, πM, ψM∗, ψF ) ◦ IP (πM, g),
+is a representation of NψM∗(M, G) × G(F). Let ρ(πM) denote this representation. Put
+Π1
+ψ(G) =
+�
+πM∈ΠψM∗ (M)
+ρ(πM).
+Then we have
+tr �Π1
+ψ(G)(u♮, f)� =
+�
+πM∈ΠψM∗ (M)
+tr �ρ(πM)(u♮, f)�
+= fG(ψM∗, u♮),
+for u♮ ∈ NψM∗(M, G) and f ∈ H(G). Thus, Theorem 4.13 (and the character theory of representations of finite groups)
+implies that Π1
+ψ(G) can be regarded as a representation of Sψ(G)×G(F). By Lemma 4.10, we have Π1
+ψ(G)|Z( “
+G)×{1} =
+χ−1
+G . As πM is unitary, Π1
+ψ(G) is also unitary, in particular completely reducible. We define the packet Πψ(G) to be
+the multiset of irreducible representations of G(F) that appear in the direct summand of Π1
+ψ(G). Then we obtain
+Π1
+ψ(G) =
+�
+π∈Πψ(G)
+⟨−, π⟩−1 ⊗ π,
+32
+
+where ⟨−, π⟩ are characters of Sψ(G) whose restriction to Z(“
+G)Γ coincide χG. We equip the packet Πψ(G) with the
+map
+Πψ(G) → Irr(Sψ, χG),
+π �→ ⟨−, π⟩.
+The endoscopic character relation (3.7) follows directly from the local intertwining relation (4.7) and the definition of
+the pairing ⟨−, π⟩ above.
+4.7
+Reduction of LIR to discrete (relative to M) parameters
+In this subsection we review from [18, §2.7] the twisted local intertwining relation and the reduction of the proof of
+the part 1 and 2 of Theorem 4.13 to the case of discrete parameters. Concretely speaking, this section will be devoted
+to the proofs of Lemmas 4.15 and 4.16 below.
+Let us now consider two nested proper standard Levi subgroups
+M ∗
+0 ⊂ M ∗ ⊂ G∗. Let ψ0 ∈ Ψ(M ∗
+0 ) be a local parameter, and ψ ∈ Ψ(M ∗) its image under the natural map from
+Ψ(M ∗
+0 ) to Ψ(M ∗). Note that ψ0 is relevant if and only if ψ is relevant. When ψ and ψ0 are relevant, we assume that
+M = ξ(M ∗) and M0 = ξ(M ∗
+0 ) are proper Levi subgroups of G defined over F.
+Lemma 4.15. For any u ∈ Nψ0(M0, G) ∩ Nψ(M, G), we have
+fG(ψ0, u♮) = fG(ψ, u♮),
+where u♮ in the left (resp. right) hand side is the image of u in Nψ(M0, G) (resp. Nψ(M, G)).
+Note that the lemma is trivial if the parameter ψ0 is not relevant.
+Let P (resp. P0, resp. Q) be the standard parabolic subgroup of G (resp. G, resp. M), with the Levi sub-
+group M (resp. M0, resp. M0). Note that Nψ(M0, M) ⊂ Nψ(M0, G). Note also that the equation (4.1) implies
+Z(A�
+M0, Sψ0(G)◦) = Z(A�
+M0, Sψ0(M)◦), and hence Nψ(M0, M) ⊂ Nψ(M0, G).
+Lemma 4.16. Assume that ψ0 is relevant. For any u ∈ Nψ0(M0, M) and π0 ∈ Πψ0(M0), we have
+RP0(u♮, π0, ψ0, ψF ) = IP (RP0(u♮, π0, ψ0, ψF )).
+Moreover, we have
+fG(ψ0, u♮) = fM(ψ0, u♮).
+Before the proofs of these lemmas, we shall now record a consequence of Lemma 4.15, which is a key ingredient of
+the proof of the local intertwining relation.
+Proposition 4.17. Assume that the second and the third statements of Theorem 4.13 hold for a standard parabolic
+pair (M ∗
+0 , P ∗
+0 ) and a discrete parameter ψ0 ∈ Ψ2(M ∗
+0 ). Then they hold for every standard parabolic pair (M ∗, P ∗) and
+a parameter ψ ∈ Ψ(M ∗) such that M ∗
+0 ⊂ M ∗, where ψ is the image of ψ0 under the canonical map Ψ(M ∗
+0 ) → Ψ(M ∗).
+Proof. The proof is similar to that of [18, Proposition 2.7.3] with Sψ(G)◦ in place of Srad
+ψ . (Precisely speaking, we
+have Srad
+ψ
+= Sψ(G)◦ because G∗ = SO2n+1.)
+The third statement of Theorem 4.13 will also be reduced to the case of discrete parameters, in §6.4 Lemma 6.21.
+As a preparation of the proofs of Lemmas 4.15 and 4.16, we now review the twisted local intertwining relation.
+See [18, §2.7] and [7, pp.115-119] for detail.
+Let N and k be positive integers.
+Put H = GLk
+N.
+The standard
+pinning of GLN gives rise to a pinning of H, which we write (TH, BH, {XαH}αH) and call standard. Let θH be the
+automorphism of H given by θH(h1, . . . , hk) = (θ(hk), h1, . . . , hk−1), where θ is either the identity automorphism of
+GLN, or the automorphism θN. Let (MH, PH) be a standard parabolic pair of H. Recall that any packet for a direct
+product of a finite number of general linear groups, is a singleton. Let ψ ∈ Ψ(MH) and let π be the corresponding
+representation of MH(F). Put Sψ(H, θH) = Cent(Im ψ, “
+H ⋊ ”
+θH), which is not a group. Put also Nψ(MH, H ⋊ θH) =
+N(A ‘
+MH, Sψ(H, θ−1
+H )), W(‘
+MH, “
+H ⋊”
+θH
+−1) = N(A ‘
+MH , “
+H ⋊”
+θH
+−1)/‘
+MH, and W(MH, H ⋊θH) = N(AMH, H ⋊θH)/MH.
+We identify the Weyl sets W(‘
+MH, “
+H ⋊ ”
+θH
+−1) and W(MH, H ⋊ θH) in the standard way. Let u ∈ Nψ(MH, H ⋊ θH)
+and let w ∈ W(‘
+MH, “
+H ⋊ ”
+θH
+−1) ≃ W(MH, H ⋊ θH) be the image of u.
+33
+
+As in §4.2, let us define the first operator
+RP w
+H |PH(πλ) = rP w
+H |PH(πλ, ψF )−1JP w
+H |PH(πλ, ψF ),
+(4.8)
+where JP w
+H |PH(πλ, ψF ) : HPH(πλ) → HP w
+H (πλ) is the unnormalized intertwining operator defined by
+[JP w
+H |PH(πλ, ψF )f](h) =
+�
+NH(F )∩N w
+H(F )\N w
+H(F )
+f(nh)dn,
+and rP w
+H |PH(πλ, ψF ) is the normalizing factor given by
+rP w
+H |PH(πλ, ψF ) =
+L(0, ρ∨
+P w
+H |PH ◦ φψλ)
+L(1, ρ∨
+P w
+H |PH ◦ φψλ)
+ǫ( 1
+2, ρ∨
+P w
+H |PH ◦ φψλ, ψF )
+ǫ(0, ρ∨
+P w
+H |PH ◦ φψλ, ψF ) .
+Then (4.8) is holomorphic at λ = 0, thus we put RP w
+H |PH(π) = RP w
+H |PH(π0).
+In order to define the second operator, we have to fix a representative of w. Let ˙w be the representative of w in
+the Weyl set W(TH, H ⋊ θH) that stabilizes the simple positive roots inside MH. Since both ˙w and θH preserve TH,
+we have ˙w = ˙w0 ⋊ θH for some ˙w0 ∈ W(TH, H). Then we can take the Langlands-Shelstad lift �w0 ∈ N(TH, H) of ˙w0.
+Put �w = �w0 ⋊ θH, which is the representative of w we need. As in §4.3, let us define the second operator
+ℓPH
+PH(w, π, ψF ) = ǫPH(w, ψ, ψF )ℓP H( �w),
+where the normalizing factor is
+ǫPH(w, ψ, ψF ) = ǫ(1
+2, ρ∨
+P w
+H |PH ◦ φψ, ψF ),
+and the operator ℓP H( �w) from (IP w
+H (π), HP w
+H (π)) to (IPH( �wπ) ◦ θH, HPH( �wπ)) is defined by
+[ℓPH( �w)f](h) = f( �w−1 · h ⋊ θH).
+Here the identity component ‹
+H◦ = H ⋊ 1 of the twisted group ‹
+H = H ⋊ ⟨θH⟩ is identified with H.
+Note that �wπ is isomorphic to π. Let us define the third operator
+IPH(π( �w)) : HPH( �wπ) → HPH(π),
+where π( �w) : �wπ → π is the Whittaker normalized isomorphism. Here we use the Whittaker datum corresponding to
+ψF and the standard pinning.
+Now we obtain the normalized intertwining operator
+RPH(w, ψ, ψF ) = IPH(π( �w)) ◦ ℓPH
+PH(w, π, ψF ) ◦ RP w
+H |PH(π),
+from (IPH (π), HPH(π)) to (IPH(π) ◦ θH, HPH(π)), and the twisted first linear form
+H(H) ∋ f �→ fH(ψ, w) := tr (RPH(w, ψ, ψF ) ◦ IPH(π, f)) .
+On the other hand, for any s ∈ Sψ(H, θH), we have a pair (e, ψe) of a twisted endoscopic triple e ∈ E(H ⋊ θH) and
+its parameter ψe ∈ Ψ(He), corresponding to the pair (ψ, s). For any f ∈ H(H), choose f e ∈ H(He) such that f and
+f e are matching. Then we write f ′
+H(ψ, s) for the value f e(ψe). This is the twisted second linear form, which we need.
+We call the following formula the twisted local intertwining relation, or twisted LIR for short.
+Proposition 4.18 (twisted LIR). We have
+fH(ψ, w) = f ′
+H(ψ, sψu−1).
+Proof. The case k = 1 is guaranteed by [7, Corollary 2.5.4]. Then the proof is similar to that of [18, Proposition 2.7.4,
+Lemma 2.7.6].
+34
+
+Now we return to the notation used in Lemmas 4.15 and 4.16.
+If ψ0 and ψ are not relevant, then Lemma
+4.15 is trivial. Assume that ψ0 and ψ are relevant. Hence M0, M, P0, P, and Q are defined over F.
+Let u ∈
+Nψ0(M0, G) ∩ Nψ(M, G), and u♮ ∈ Nψ0(M0, G) be its image. We will also write u♮ for the image in Nψ(M, G), by
+abuse of notation. Let f ∈ H(G). By definition we have
+fG(ψ0, u♮) = tr
+Ñ
+�
+π0∈Πψ0(M0)
+RP0(u♮, π0, ψ0, ψF ) ◦ IP0(π0, f)
+é
+.
+The calculation similar to that in [18, pp.123-126] shows that
+�
+π0∈Πψ0(M0)
+RP0(u♮, π0, ψ0, ψF ) ◦ IP0(π0)(f)
+= ℓP
+P (w, ξ, ψ, ψF ) ◦ RP w|P (ξ, ψ) ◦
+Ñ
+�
+π0∈Πψ0(M0)
+IP (RQ(u♮, π0, ψ0, ψF )) ◦ IP (IQ(π0))(f)
+é
+.
+ECR and LIR for inner form of SO2n0+1 for n0 < n, which is assumed by induction, and Lemma 4.18 imply ECR and
+(twisted) LIR for M. In particular, the operator RQ(u♮, π0, ψ0, ψF ) acts on each irreducible summand of IQ(π0) by
+scalar. Combined with the theory of induced characters ([9]) and Lemma 4.8, we have
+tr
+Ñ
+�
+π0∈Πψ0(M0)
+IP (RQ(u♮, π0, ψ0, ψF )) ◦ IP (IQ(π0))(f)
+é
+= tr
+Ñ
+�
+π0∈Πψ0(M0)
+RQ(u♮, π0, ψ0, ψF ) ◦ IQ(π0)(fM)
+é
+= tr
+Ñ
+�
+π∈Πψ(M)
+π(u♮) ◦ π(fM)
+é
+= tr
+Ñ
+�
+π∈Πψ(M)
+IP (π(u♮)) ◦ IP (π)(f)
+é
+,
+where fM denotes the constant term of f along P. By the linear independence of characters, this means that
+�
+π0∈Πψ0(M0)
+IP (RQ(u♮, π0, ψ0, ψF )) ◦ IP0(π0)(f) =
+�
+π∈Πψ(M)
+IP (π(u♮)) ◦ IP (π)(f).
+Therefore, we have
+ℓP
+P (w, ξ, ψ, ψF ) ◦ RP w|P (ξ, ψ) ◦
+Ñ
+�
+π0∈Πψ0(M0)
+IP (RQ(u♮, π0, ψ0, ψF )) ◦ IP (IQ(π0))(f)
+é
+= ℓP
+P(w, ξ, ψ, ψF ) ◦ RP w|P (ξ, ψ) ◦
+Ñ
+�
+π∈Πψ(M)
+IP (π(u♮)) ◦ IP (π)(f)
+é
+=
+�
+π∈Πψ(M)
+IP (π(u♮)) ◦ ℓP
+P (w, ξ, ψ, ψF ) ◦ RP w|P (ξ, ψ) ◦ IP (π)(f)
+=
+�
+π∈Πψ(M)
+RP (u♮, π, ψ, ψF ) ◦ IP (π, f),
+and hence
+fG(ψ0, u♮) = fG(ψ, u♮).
+35
+
+Thus Lemma 2.7.1 follows.
+Next, let u ∈ Nψ0(M0, M). Then u is trivial in W(�
+M, “
+G), and the calculations similar to those in [18, p.127] prove
+both two equations in Lemma 4.16.
+4.8
+Reduction of LIR to elliptic or exceptional (relative to G) parameters
+In this subsection, we review the reduction of the proof of LIR to the case of elliptic or exceptional parameters.
+Recall from §3.2 that Ψ2
+ell(G∗) is the set of the equivalence classes of parameters ψ of the form
+ψ = 2ψ1 ⊕ · · · 2ψq ⊕ ψq+1 ⊕ · · · ⊕ ψr,
+where ψ1, . . . , ψr are irreducible, symplectic, and mutually distinct, and r ≥ q ≥ 1. Note that then we have
+Sψ ∼= O(2, C)q × O(1, C)r−q.
+Let now Ψexc1(G∗) and Ψexc2(G∗) be the subsets of Ψ(G∗) consisting of the parameters ψ of the form
+(exc1) ψ = 2ψ1 ⊕ ψ2 ⊕ · · · ⊕ ψr, where ψ1 is irreducible and orthogonal, and ψ2, . . . , ψr are irreducible, symplectic, and
+mutually distinct,
+(exc2) ψ = 3ψ1 ⊕ ψ2 ⊕ · · · ⊕ ψr, where ψ1, . . . , ψr are irreducible, symplectic, and mutually distinct,
+respectively. They are disjoint. We then have
+(exc1) Sψ ∼= Sp(2, C) × O(1, C)r−1,
+(exc2) Sψ ∼= O(3, C) × O(1, C)r−1,
+respectively. We put Ψexc(G∗) = Ψexc1(G∗) ⊔ Ψexc2(G∗), and we shall say that ψ is exceptional (resp. of type (exc1),
+resp. of type (exc2)) if ψ ∈ Ψexc(G∗) (resp. Ψexc1(G∗), resp. Ψexc2(G∗)). One can see that Ψexc(G∗) and Ψell(G∗)
+are disjoint. Put Ψell,exc(G∗) = Ψ(G∗) \ (Ψell(G∗) ⊔ Ψexc(G∗)).
+Let M ∗ ⊊ G∗ be a proper standard Levi subgroup, and ξ : G∗ → G an inner twist over F, such that M = ξ(M ∗) ⊊
+G is defined over F if M ∗ transfers to G. In view of §4.7, it is enough to consider discrete parameters for M. Let
+ψM∗ ∈ Ψ2(M ∗) be a discrete local A-parameter, and we shall write ψ for its image in Ψ(G∗) in this subsection. Recall
+from Lemma 4.1 that Sψ(M, G) = Sψ(G). Here we do not assume that M is defined over F, nor do that ψ is relevant.
+Let Tψ be the maximal central torus A �
+M of �
+M. Since a Levi subgroup for which the parameter is discrete is unique,
+the torus Tψ is determined by ψ and is a maximal torus of Sψ = Sψ(G). One can easily see that
+Wψ = Wψ(M, G) = W(Tψ, Sψ),
+W ◦
+ψ = W ◦
+ψ(M, G) = W(Tψ, S◦
+ψ).
+Therefore, if ψ is elliptic then W ◦
+ψ is trivial, as stated in [18, Lemma 2.8.1]. Similarly, let Bψ be the standard Borel
+subgroup of Sψ with a maximal torus Tψ. Then (Tψ, Bψ) is a Borel pair of S◦
+ψ. For any x ∈ Sψ, following [7, p.204],
+put
+Tψ,x = Cent(sx, Tψ)◦,
+where sx ∈ Sψ is a representative of x such that Ad(sx) stabilizes (Tψ, Bψ). Since sx is determined up to a Tψ-translate,
+Tψ,x is uniquely determined by x.
+The following lemma is the key lemma in this subsection.
+Lemma 4.19. If ψ ∈ Ψell,exc(G∗), then
+(1) every simple reflection w ∈ W ◦
+ψ centralizes a torus of positive dimension in Tψ;
+(2) dim Tψ,x ≥ 1, ∀x ∈ Sψ.
+Proof. The proof is similar to that of [18, Lemma 2.8.4].
+36
+
+We now define
+Sψ,ell = { s ∈ Sψ,ss | Z(Cent(s, S◦
+ψ)) is finite } ,
+Sψ,ell = { s ∈ Sψ,ss | Z(Cent(s, S
+◦
+ψ)) is finite } ,
+Sψ,ell = Sψ,ell/(Sψ,ell ∩ S◦
+ψ),
+Sψ,ell = Sψ,ell/(Sψ,ell ∩ S
+◦
+ψ).
+One can see that Sψ,ell (resp. Sψ,ell) is the image of Sψ,ell (resp. Sψ,ell) under the natural surjection Sψ → Sψ (resp.
+Sψ → Sψ).
+Lemma 4.20.
+1. The natural surjection Sψ → Sψ carries Sψ,ell onto Sψ,ell.
+2. The natural surjection Sψ → Sψ carries Sψ,ell onto Sψ,ell.
+3. If ψ ∈ Ψexc(G∗) then Sψ,ell = Sψ and Sψ,ell = Sψ.
+Proof. The first part can be seen easily. The proof of latter two is similar to that of [18, Lemma 2.8.6].
+Lemma 4.21. Assume that either
+(1) ψ is elliptic, or
+(2) every simple reflection w ∈ W ◦
+ψ centralizes a torus of positive dimension in Tψ.
+Then the parts 1 and 2 of Theorem 4.7 hold for ψ (and for all u and f).
+Proof. The proof is similar to that of [18, Lemma 2.8.7]. We appeal to Lemmas 4.11 and 4.16 instead of Lemmas 2.5.3
+and 2.7.2 of loc. cit.
+Lemma 4.22. Let x ∈ Sψ(M, G). Assume that either
+(1) ψ is elliptic and x /∈ Sψ,ell, or
+(2) ψ is not elliptic, every simple reflection w ∈ W ◦
+ψ centralizes a torus of positive dimension in Tψ, and dim Tψ,x ≥ 1.
+Then f ′
+G(ψ, sψs−1) = e(G)fG(ψ, u♮) whenever both u♮ ∈ Nψ(M, G) and s ∈ Sψ,ss map to x.
+Proof. Let sx ∈ Sψ,ss be a representative of x such that Ad(sx) stabilizes (Tψ, Bψ).
+Since Tψ = A �
+M, we have
+sx ∈ Nψ(M, G). Lemmas 4.1, 4.12, and 4.21 tell us that f ′
+G(ψ, sψs−1) = f ′
+G(ψ, sψs−1
+x ) and fG(ψ, u♮) = fG(ψ, s♮
+x).
+We need to show that f ′
+G(ψ, sψs−1
+x ) = e(G)fG(ψ, s♮
+x). Put �
+Mx = Z “
+G(Tψ,x). Let M ∗
+x ⊂ G∗ be the Levi subgroup
+corresponding to �
+Mx. If the assumption (1) holds, then we have |Tψ,x| = ∞ and hence �
+Mx ⊊ “
+G. If the assumption
+(2) holds, then we have dim Tψ,x ≥ 1. This implies |Tψ,x| = ∞ and hence �
+Mx ⊊ “
+G. Therefore, M ∗
+x is a proper Levi
+subgroup of G∗. The rest of the proof is similar to that of [18, Lemma 2.8.8]. We appeal to Lemmas 4.12 and 4.16
+instead of Lemmas 2.6.1 and 2.7.2 of loc. cit.
+Lemmas 4.19, 4.21, and 4.22 imply the following corollary.
+Corollary 4.23. Let x ∈ Sψ(M, G). Then Theorem 4.13 holds for all u♮ ∈ Nψ(M, G) mapping to x unless either
+(1) ψ is elliptic and x ∈ Sψ,ell, or
+(2) ψ is exceptional.
+Let us now define
+Wψ,reg(M, G) = { w ∈ Wψ(M, G) | w has finitely many fixed points on Tψ } .
+We write Nψ,reg(M, G) for the preimage of Wψ,reg(M, G) under Nψ(M, G) → Wψ(M, G).
+Lemma 4.24. Assume that ψ ∈ Ψexc(G∗) and u♮ /∈ Nψ,reg(M, G). Let s be an element of Sψ,ss whose image in
+Sψ(M, G) is the same as that of u♮. Then we have f ′
+G(ψ, sψs−1) = e(G)fG(ψ, u♮).
+37
+
+Proof. The proof is similar to that of [18, Lemma 2.8.10]. We appeal to Lemmas 4.12 and 4.22 instead of Lemmas
+2.6.1 and 2.8.8 of loc. cit.
+Consequently, if ψ ∈ Ψell,exc(G∗) then all the three statements of Theorem 4.13 hold (under the induction hypoth-
+esis). If ψ ∈ Ψ2
+ell(G∗) then the first and second part of Theorem 4.13 hold. In particular, when ψ is not exceptional,
+fG(ψ, x) is well-defined for x ∈ Sψ(M, G). Moreover, the third part of the theorem holds if the image of u in Sψ is
+not contained in Sψ,ell. When ψ is exceptional, for now, the result is Lemma 4.24, i.e., only a part of the third part.
+Let x ∈ Sψ(M, G). Since ψ is exceptional, we have |W ◦
+ψ| = 2, and there is exactly two elements in the fiber of x in
+Nψ(M, G). One is regular in Wψ(M, G), and the other is not. Put ux to be the former one, and define fG(ψ, x) to be
+fG(ψ, ux). Now we have defined fG(ψ, x) for x ∈ Sψ(M, G) for every ψ ∈ Ψ(G∗)
+Put Φ♥(G∗) = Ψ♥(G∗) ∩ Φ(G∗) and Φbdd,♥(G∗) = Ψ♥(G∗) ∩ Φbdd(G∗) for ♥ ∈ {exc, exc1, exc2}.
+4.9
+LIR for special cases
+In this subsection we shall prove the two special cases of Theorem 4.13. Later in §§6.2-6.4, we will reduce the proof
+of Theorem 4.13 for φ ∈ Φell,exc(G∗) to the special cases, which we shall treat in this subsection.
+Before we treat the two cases, we recall LLC for GL1(R) and GL2(R), and fix some notation. We shall realize the
+Weil group WR as C× ⊔ jC×, where j2 = −1 and jz = zj for any z ∈ C×. Recall from [32, §1] that the norm map
+on WR is given by |j| = 1 and |z| = zz for z ∈ C ⊂ WR. We define the (isomorphism classes of) finite dimensional
+representations σ ωt, and τ(l,t) of WR by
+σ : WR → C×,
+®
+C× ∋ z �→ 1,
+j �→ −1,
+ωt : WR → C×,
+®C× ∋ z �→ |z|t = (zz)t,
+j �→ 1,
+τ(l,t) : WR → GL(2, C),
+
+
+
+
+
+
+
+
+
+C× ∋ re
+√−1θ �→ r2t
+Ç
+e
+√−1lθ
+e−√−1lθ
+å
+,
+j �→
+Å
+1
+(−1)l
+ã
+,
+for t ∈ C and l ∈ Z>0, and put τl = τ(l,0). Then the local Langlands correspondence over R says that σεωt (ε = 0, 1)
+is corresponding to the 1-dimensional representation | · |t
+R sgnε, and τ(l,t) = τl ⊗ ωt to the 2-dimensional representation
+Dl ⊗ | det |t
+R, where Dl denotes the discrete series representation of GL2(R) of weight l + 1. Fix the standard additive
+character ψR(x) = exp(2π√−1x) of R. Put ΓR(s) := π− s
+2 Γ( s
+2) and ΓC(s) := 2(2π)−sΓ(s). The L-functions and the
+ǫ-factors are given by the following table. See [20] for more detail, but note that (+, t) and (−, t) in [20, (3.2)] should
+be (+, t
+2) and (−, t
+2) respectively.
+ϕ
+L(s, ϕ)
+ǫ(s, ϕ, ψR)
+ωt
+ΓR(s + t)
+1
+σωt
+ΓR(s + t + 1)
+√−1
+τ(l,t)
+ΓC(s + t + l
+2)
+√−1
+l+1
+For a symmetric matrix Q and an alternative matrix A of degree N, we shall write
+SO(Q) = { g ∈ GLN | tgQg = Q, det g = 1 } ,
+Sp(A) = { g ∈ GLN | tgAg = A } .
+4.9.1
+The first special case
+The first special case we are concerned with is the following. Let F = R, n = 2, and
+G∗ = SO5 = SO
+Ñ
+12
+2
+12
+é
+,
+38
+
+M ∗ =
+
+
+
+
+
+
+
+Ü t
+A
+b
+t−1
+β
+c
+ê ��������
+t ∈ GL1, A ∈ M2×2,
+Å A
+b
+β
+c
+ã
+∈ SO
+Ñ
+1
+2
+1
+é 
+
+
+
+
+
+
+∼= GL1 × SO3 .
+Let φ ∈ Φexc1(G∗) be an L-parameter of type (exc1) of the form
+φ = 2ω0 ⊕ τ1,
+and φM = ω0 ⊕ τ1 ∈ Φ2(M ∗) so that φ is the natural image of φM. More precisely, we write as
+“
+G = SpC
+Ü
+1
+1
+−1
+−1
+ê
+,
+�
+M =
+
+
+
+Ñ a
+X
+a−1
+é
+∈ “
+G
+������
+a ∈ GL(1, C), X ∈ SpC
+Å
+1
+−1
+ã
+= SL(2, C)
+
+
+ ⊂ “
+G,
+φ = φM =
+Ñ
+ω0
+τ1
+ω0
+é
+.
+Up to isomorphism, there exists only one nontrivial inner twist of G∗ for which φ is relevant. Let
+G = SO(1, 4) = SO
+Å 1
+−14
+ã
+.
+Put
+α =
+1
+√
+2
+à 1
+1
+√−1
+√−1
+2√−1
+1
+−1
+1
+−1
+í
+.
+Let z ∈ Z1(R, G∗) be a 1-cocycle such that
+zρ = α−1ρ(α) =
+à 1
+0
+−1
+−1
+1
+−1
+0
+í
+,
+where ρ is the nontrivial element in ΓF = Gal(C/R), i.e., the complex conjugation. Let ξ = Ad(α) : G∗ → G. Put
+M = ξ(M ∗)
+= { m(t, B) | t ∈ GL1, B ∈ SO(−13) = SO(3) } ,
+where
+m(t, B) =
+Ñ c(t)
+s(t)
+B
+s(t)
+c(t)
+é
+,
+c(t) = t + t−1
+2
+,
+s(t) = t − t−1
+2
+.
+39
+
+Then (ξ, z) is a pure inner twist G∗ → G which restricts to a pure inner twist M ∗ → M such that z+ ∈ Z1(R, GL1)
+is trivial. We have Πω0(GL1(R)) = {1}, Πτ1(SO(3)) = {1}, and hence ΠφM (M) = {1}, where 1 denotes the trivial
+representation of each group. Let us write π for the unique element in ΠφM (M), i.e., the trivial representation of
+M(R).
+Proposition 4.25. Theorem 4.13 is valid for G, M, and φ.
+Proof. The centralizer Sφ(G) is
+
+
+
+Ñ a
+b
+ε12
+c
+d
+é ������
+Å a
+b
+c
+d
+ã
+∈ SL(2, C), ε ∈ {±1}
+
+
+
+∼= SL(2, C) × O(1, C).
+Thanks to this explicit structure, one can easily understand the diagram (4.6). The diagram has the form
+1
+1
+�
+�
+{ ±1 } × 1
+{ ±1 } × 1
+�
+�
+1 −−−−→ 1 × { ±1 } −−−−→ { ±1 } × { ±1 } −−−−→ { ±1 } × 1 −−−−→ 1
+���
+�
+�
+1 −−−−→ 1 × { ±1 } −−−−→
+1 × { ±1 }
+−−−−→
+1
+−−−−→ 1
+�
+�
+1
+1
+where (−1, 1) and (1, −1) are represented by
+Ü
+1
+1
+1
+−1
+ê
+∈ N(A �
+M, Sφ(G)◦),
+Ü
+1
+−1
+−1
+1
+ê
+∈ Sφ(M),
+respectively. Note that (−1, 1) vanishes in Sφ(G). Let us write w for the image of (−1, 1) in Wφ(M, G) = Wφ(M, G)◦ ∼=
+W(M ∗, G∗) ∼= W(M, G). One can calculate the Langlands-Shelstad lift in “
+G = Sp(4, C) of w as in [16]. It is
+Ü
+−1
+−1
+−1
+1
+ê
+,
+and thus the first sophisticated splitting s′ : Wψ(M, G) → Nψ(M, G) sends (−1, 1) to (−1, −1). Therefore, the other
+sophisticated splitting s : Nψ(M, G) → Sψ(M) sends (−1, −1) to 1, and (−1, 1) to (1, −1).
+Let P ∗ be the standard parabolic subgroup of G∗ with Levi subgroup M ∗, and P = ξ(P ∗). Now in the same way
+as [18, pp.136-137], we know that the induced representation (IP (π), HP (π)) is irreducible and is the unique element
+of Πφ(G), and that the operator RP ((−1, 1), π, φ, ψR) is a scalar. Moreover, in order to prove the proposition, it is
+sufficient to show
+RP ((−1, 1), π, φ, ψR)f = f,
+40
+
+for a nonzero element f in HP (π). Recall that
+RP ((−1, 1), π, φ, ψR) = IP (π((−1, 1))ξ,z) ◦ ℓP
+P (w, ξ, φ, ψR) ◦ RP w|P (ξ, φ).
+We focus first on IP (π((−1, 1))ξ,z) = IP (⟨(−1, 1), π⟩ξ,zπ( ˘w)ξ).
+Since π is the trivial representation on the 1-
+dimensional vector space C, the operator π( ˘w)ξ is the identity map. On the other hand, we have ⟨(−1, 1), π⟩ξ,z =
+⟨s(−1, 1), π⟩M = ⟨−1, π⟩M = −1. We obtain IP (π((−1, 1))ξ,z) = −1.
+Next we consider ℓP
+P (w, ξ, φ, ψR)◦RP w|P (ξ, φ). Let �w be the Langlands-Shelstad lift of w in G∗(R). The calculation
+of �w has already been done in [16, §8]:
+�w =
+à
+1
+−1
+−1
+1
+−1
+í
+.
+We have ˘w = ξ( �w) =
+Å1
+−14
+ã
+∈ G(R). Let us define a connected compact subgroup K ⊂ G(R) as
+K =
+ß Å 1
+κ
+ã ���� κ ∈ SO(4)
+™
+.
+By abuse of notation, we shall write κ for the element
+Å1
+κ
+ã
+. Although K is not maximal, the Iwasawa decomposition
+tells us that G(R) = P(R)K. Let N ∗ be the unipotent radical of P ∗. Put
+n∗(b) =
+à
+1
+b1
+b2
+−b1b3 − b2
+2
+4
+b3
+1
+−b3
+1
+− b2
+2
+1
+−b1
+1
+í
+,
+for b = (b1, b2, b3) ∈ C3, so that N ∗(R) = {n∗(b) | b ∈ R3}. A direct calculation following §4.2 show that the Haar
+measure on N ∗(R) is d(n∗(b)) =
+1
+2db1db2db3. Put P = ξ(P ∗) and N = ξ(N ∗), which are the standard parabolic
+subgroup with Levi subgroup M and its unipotent radical. Put also
+n(x) =
+Ö
+1 + ∥x∥2
+2
+x
+− ∥x∥2
+2
+tx
+13
+− tx
+∥x∥2
+2
+x
+1 − ∥x∥2
+2
+è
+=
+â
+1 + ∥x∥2
+2
+x1
+x2
+x3
+− ∥x∥2
+2
+x1
+1
+−x1
+x2
+1
+−x2
+x3
+1
+−x3
+∥x∥2
+2
+x1
+x2
+x3
+1 − ∥x∥2
+2
+ì
+,
+for x = (x1, x2, x3) ∈ C3, where ∥x∥2 = x2
+1 + x2
+2 + x2
+3 so that N(R) = {n(x) | x ∈ R3}. A straightforward calculation
+shows that ξ(n∗(b)) = n(x) where
+x1 = −b1 + b3
+2
+√
+−1,
+x2 = −b2
+2
+√
+−1,
+x3 = b1 − b3
+2
+,
+and hence the measure on N(R) is d(n(x)) = 2dx1dx2dx3.
+For λ ∈ C, put φλ = ωλ ⊕ τ1 ⊕ ω−λ and πλ = | · |λ ⊠ 1, so that Πφλ(M) = {πλ} and Πφλ(G) = {IP (πλ)}. Define a
+function ϕ(λ) ∈ HP (πλ) by
+ϕ(λ)(m(t, B)n(x)κ) = |t|λ+ 3
+2 ,
+which is holomorphic in λ ∈ C. Assume that Re(λ) > 0. We have 0 ̸= ϕ(0) ∈ HP (π) and
+î
+ℓP
+P (w, ξ, φ, ψR) ◦ RP w|P (ξ, φ)ϕ(0)ó
+(g)
+41
+
+= lim
+λ→+0
+î
+ℓP
+P (w, ξ, φλ, ψR) ◦ RP w|P (ξ, φλ)ϕ(λ)ó
+(g)
+= lim
+λ→+0 ǫ(0, ρ∨
+P w|P ◦ φλ, ψR)
+L(1, ρ∨
+P w|P ◦ φλ)
+L(0, ρ∨
+P w|P ◦ φλ)
+�
+N(R)
+ϕ(λ)( ˘w−1ng)dn.
+A direct calculation implies that ρ∨
+P w|P ◦ φλ is isomorphic to
+τ(1,λ) ⊕ ω2λ.
+Therefore, we have
+ǫ(0, ρ∨
+P w|P ◦ φλ, ψR)
+L(1, ρ∨
+P w|P ◦ φλ)
+L(0, ρ∨
+P w|P ◦ φλ) = −ΓC(λ + 3
+2)
+ΓC(λ + 1
+2)
+ΓR(2λ + 1)
+ΓR(2λ)
+.
+We now turn to the integral. Put
+[MSO(1,4)ϕ(λ)](g) =
+�
+N(R)
+ϕ(λ)( ˘w−1ng)dn.
+In Lemma 4.26 below, we will show that
+MSO(1,4)ϕ(λ) = 2− 1
+2
+ΓC(λ)
+ΓC(λ + 3
+2)ϕ(−λ).
+This leads to
+ǫ(0, ρ∨
+P w|P ◦ φλ, ψR)
+L(1, ρ∨
+P w|P ◦ φλ)
+L(0, ρ∨
+P w|P ◦ φλ)
+�
+N(R)
+ϕ(λ)( ˘w−1n·)dn
+= −ΓC(λ + 3
+2)
+ΓC(λ + 1
+2)
+ΓR(2λ + 1)
+ΓR(2λ)
+· 2− 1
+2
+ΓC(λ)
+ΓC(λ + 3
+2)ϕ(−λ)
+= −2− 1
+2
+ΓC(λ)
+ΓC(λ + 1
+2)
+ΓR(2λ + 1)
+ΓR(2λ)
+ϕ(−λ),
+whose limit as λ approaching 0 from the right is −ϕ(0). Since ϕ(0) ̸= 0, this completes the proof.
+To finish the proof of Proposition 4.25, it remains to show the following lemma.
+Lemma 4.26. For Re(λ) > 0, we have
+MSO(1,4)ϕ(λ) = 2− 1
+2
+ΓC(λ)
+ΓC(λ + 3
+2)ϕ(−λ).
+Proof. By definition, MSO(1,4)ϕ(λ) is right K-invariant and left N(R)-invariant. Moreover, we have
+[MSO(1,4)ϕ(λ)](m(t, B)g) = |t|−λ+ 3
+2 [MSO(1,4)ϕ(λ)](g).
+Indeed, a direct calculation shows that
+n(x)m(t, B) = m(t, B)n(t−1xB),
+˘w−1m(t, B) = m(t−1, B) ˘w−1.
+Thus we have
+[MSO(1,4)ϕ(λ)](m(t, B)g) =
+�
+x∈R3 ϕ(λ)( ˘w−1n(x)m(t, B)g)d(n(x))
+=
+�
+x∈R3 ϕ(λ)(m(t−1, B) ˘w−1n(t−1xB)g) · 2dx1dx2dx3
+42
+
+= |t|−λ− 3
+2
+�
+y∈R3 ϕ(λ)( ˘w−1n(y)g) · 2|t|3dy1dy2dy3
+= |t|−λ+ 3
+2 [MSO(1,4)ϕ(λ)](g),
+where we have changed the variables y = t−1xB. Combining these properties with ϕ(λ)(1) = 1, we now have
+MSO(1,4)ϕ(λ) = MSO(1,4)ϕ(λ)(1) · ϕ(−λ).
+We must then show that
+MSO(1,4)ϕ(λ)(1) = 2− 1
+2
+ΓC(λ)
+ΓC(λ + 3
+2).
+For x, y ∈ R3, t ∈ R×, B ∈ SO(3), and κ ∈ SO(4) ∼= K, a direct calculation shows that
+˘w−1n(x) =
+Ö
+1 + ∥x∥2
+2
+x
+− ∥x∥2
+2
+− tx
+−13
+tx
+− ∥x∥2
+2
+−x
+−1 + ∥x∥2
+2
+è
+,
+m(t, B)n(y)κ =
+Ö
+c(t) + t∥y∥2
+2
+∗
+∗
+B ty
+∗
+∗
+s(t) + t∥y∥2
+2
+∗
+∗
+è
+,
+so if we write ˘w−1n(x) = m(t, B)n(y)κ then we have
+
+
+
+
+
+
+
+
+
+
+
+1 + ∥x∥2
+2
+= c(t) + t∥y∥2
+2
+,
+− tx
+= B ty,
+−∥x∥2
+2
+= s(t) + t∥y∥2
+2
+,
+which leads t = (1 + ∥x∥2)−1. Now the value MSO(1,4)ϕ(λ)(1) is equal to
+�
+x∈R3 ϕ(λ)( ˘w−1n(x))d(n(x))
+=
+�
+x∈R3
+��1 + ∥x∥2��−λ− 3
+2 · 2dx1dx2dx3
+=
+� ∞
+r=0
+� π
+θ=0
+� 2π
+α=0
+2r2 sin θ
+(1 + r2)λ+ 3
+2 dαdθdr
+= 8π
+� ∞
+0
+r2
+(1 + r2)λ+ 3
+2 dr,
+where we have changed the variables x1 = r cos α sin θ, x2 = r sin α sin θ, and x3 = r cos θ. Since Re(λ) > 0, the
+last integral converges absolutely, and the gamma function Γ(λ + 3
+2) has the integral expression which also converges
+absolutely. Therefore, the product of them is
+8π
+� ∞
+0
+r2
+(1 + r2)λ+ 3
+2 dr × Γ(λ + 3
+2)
+= 8π
+� ∞
+0
+� ∞
+0
+Å
+t
+1 + r2
+ãλ+ 3
+2
+r2e−tdrd×t,
+where d×t = t−1dt. First we fix r and change the variables u = (1 + r2)−1t, and then fix u and change the variables
+s = √ur, to obtain
+8π
+� ∞
+0
+� ∞
+0
+Å
+t
+1 + r2
+ãλ+ 3
+2
+r2e−tdrd×t
+43
+
+= 8π
+� ∞
+0
+uλe−ud×u
+� ∞
+0
+s2e−s2ds
+= 2π
+3
+2 Γ(λ).
+We now have
+MSO(1,4)ϕ(λ)(1) = 8π
+� ∞
+0
+r2
+(1 + r2)λ+ 3
+2 dr
+= 2π
+3
+2
+Γ(λ)
+Γ(λ + 3
+2) = 2− 1
+2
+ΓC(λ)
+ΓC(λ + 3
+2).
+4.9.2
+The second special case
+The second special case we are concerned with is the following. Let F = R, n = 3, and
+G∗ = SO7 = SO
+Ñ
+13
+2
+13
+é
+,
+M ∗ =
+
+
+
+
+
+
+
+Ü A
+X
+b
+tA−1
+β
+c
+ê ��������
+A ∈ GL2, X ∈ M2×2,
+Å X
+b
+β
+c
+ã
+∈ SO
+Ñ
+1
+2
+1
+é 
+
+
+
+
+
+
+∼= GL2 × SO3 .
+Let φ ∈ Φexc2(G∗) be an L-parameter of type (exc2) of the form
+φ = 3τ1,
+and φM = 2τ1 ∈ Φ2(M ∗) (as equivalent classes) so that φ is the natural image of φM. More precisely, we define the
+parameter as
+“
+G = SpC
+Ü
+12
+1
+−1
+−12
+ê
+,
+�
+M =
+
+
+
+Ñ A
+X
+tA−1
+é
+∈ “
+G
+������
+A ∈ GL(2, C), X ∈ SpC
+Å
+1
+−1
+ã
+= SL(2, C)
+
+
+ ⊂ “
+G,
+φ = φM =
+Ñ τ1
+τ1
+tτ1−1
+é
+.
+Up to isomorphism, there exists only one nontrivial inner twist of G∗ for which φ is relevant. Let
+G = SO(2, 5) = SO
+Å 12
+−15
+ã
+.
+Put
+α =
+1
+√
+2
+à 12
+12
+√−1
+√−1
+2√−1
+1
+−1
+12
+−12
+í
+.
+44
+
+Let z ∈ Z1(R, G∗) be a 1-cocycle such that
+zρ = α−1ρ(α) =
+à
+12
+0
+−1
+−1
+12
+−1
+0
+í
+,
+where ρ is the nontrivial element in ΓF = Gal(C/R). Put ξ = Ad(α) : G∗ → G and
+M = ξ(M ∗)
+= { m(A, B) | A ∈ GL2, B ∈ SO(−13) = SO(3) } ,
+where
+m(A, B) =
+Ñ c(A)
+s(A)
+B
+s(A)
+c(A)
+é
+,
+c(A) = A + tA−1
+2
+,
+s(A) = A − tA−1
+2
+.
+Then (ξ, z) is a pure inner twist G∗ → G which restricts to a pure inner twist M ∗ → M such that z+ ∈ Z1(R, GL2)
+is trivial. We have Πτ1(GL2(R)) = {D1}, Πτ1(SO(3)) = {1}, and hence ΠφM (M) = {D1 ⊠ 1}, where 1 denotes the
+trivial representation of SO(3). Let us write π for the unique element in ΠφM (M).
+Proposition 4.27. Theorem 4.13 is valid for G, M, and φ.
+Proof. Put J =
+Å
+1
+−1
+ã
+, and J =
+Å14
+J
+ã
+. Let us define a mapping A �→ A(2) from the set of 3 × 3-matrices to
+those of 6 × 6-matrices by
+A =
+Ñ a1,1
+a1,2
+a1,3
+a2,1
+a2,2
+a2,3
+a3,1
+a3,2
+a3,3
+é
+�→ A(2) :=
+Ñ a1,112
+a1,212
+a1,312
+a2,112
+a2,212
+a2,312
+a3,112
+a3,212
+a3,312
+é
+.
+Since tτ1−1 = Jτ1J−1, the centralizer Sφ(G) is
+
+
+ Jh(2)J−1
+������
+h ∈ OC
+Ñ
+1
+1
+1
+é 
+
+
+∼= O(3, C).
+Thanks to this explicit structure, one can easily understand the diagram (4.6). The diagram has the form
+1
+1
+�
+�
+{ ±1 } × 1
+{ ±1 } × 1
+�
+�
+1 −−−−→ 1 × { ±1 } −−−−→ { ±1 } × { ±1 } −−−−→ { ±1 } × 1 −−−−→ 1
+���
+�
+�
+1 −−−−→ 1 × { ±1 } −−−−→
+1 × { ±1 }
+−−−−→
+1
+−−−−→ 1
+�
+�
+1
+1
+45
+
+where (−1, 1) and (1, −1) are represented by
+Ñ
+J−1
+−12
+J
+é
+∈ N(A �
+M, Sφ(G)◦),
+Ñ 12
+−12
+12
+é
+∈ Sφ(M),
+respectively. Note that (−1, 1) vanishes in Sφ(G). Let us write w for the image of (−1, 1) in Wφ(M, G) = Wφ(M, G)◦ ∼=
+W(M ∗, G∗) ∼= W(M, G). One can calculate the Langlands-Shelstad lift in “
+G of w as in [16]. It is
+Ñ
+J−1
+12
+J
+é
+,
+and thus the first sophisticated splitting s′ : Wψ(M, G) → Nψ(M, G) sends (−1, 1) to (−1, −1). Therefore, the other
+splitting s : Nψ(M, G) → Sψ(M) sends (−1, −1) to 1, and (−1, 1) to (1, −1).
+Let P ∗ be the standard parabolic subgroup of G∗ with Levi subgroup M ∗, and P = ξ(P ∗). Now in the same way
+as [18, pp.136-137], we know that the induced representation (IP (π), HP (π)) is irreducible and is the unique element
+of Πφ(G), and that the operator RP ((−1, 1), π, φ, ψR) is a scalar. Moreover, in order to prove the proposition, it is
+sufficient to show
+RP ((−1, 1), π, φ, ψR)f = f,
+for a nonzero element f in HP (π). Recall that
+RP ((−1, 1), π, φ, ψR) = IP (π((−1, 1))ξ,z) ◦ ℓP
+P (w, ξ, φ, ψR) ◦ RP w|P (ξ, φ).
+We focus first on IP (π((−1, 1))ξ,z) = IP (⟨(−1, 1), π⟩ξ,zπ( ˘w)ξ). Let �w be the Langlands-Shelstad lift of w in G∗(R).
+The calculation of �w has already been done in [16, §8]:
+�w =
+à
+J
+1
+1
+J
+1
+í
+.
+Thus we have
+˘w = ξ( �w) =
+Ñ
+J
+13
+J−1
+é
+∈ G(R).
+Since ˘w−1m(A, B) ˘w = m((det A)−1A, B) and the central character of D1 is trivial, the representations ˘wπ and π
+coincide. Thus π( ˘w)ξ is the identity map. On the other hand, we have ⟨(−1, 1), π⟩ξ,z = ⟨s(−1, 1), π⟩M = ⟨−1, π⟩M =
+−1. We obtain IP (π((−1, 1))ξ,z) = −1.
+Next we consider ℓP
+P (w, ξ, φ, ψR) ◦ RP w|P (ξ, φ). Let us define a connected compact subgroup K ⊂ G(R) as
+K =
+ß
+κ =
+Å κ2
+κ5
+ã ���� κm ∈ SO(m), (m = 2, 5)
+™
+.
+Although K is not maximal, the Iwasawa decomposition tells us that G(R) = P(R)K. Let N ∗ be the unipotent
+radical of P ∗, which is generated by {exp(Xα) | α = χ1 ± χ3, χ1, χ2 ± χ3, χ2, χ1 + χ2}. Here Xα denotes the Chevalley
+basis. The Haar measure on N ∗(R) is also given by the Chevalley basis. Put P = ξ(P ∗) and N = ξ(N ∗), which are
+46
+
+the standard parabolic subgroup with Levi subgroup M and its unipotent radical. Now we describe N(R) explicitly.
+Let ι1 and ι2 be embeddings of SO(1, 4) into G = SO(2, 5) given by
+h =
+Å a
+b
+β
+C
+ã
+�→ ι1(h) =
+Ü a
+b
+1
+β
+C
+1
+ê
+,
+h =
+Å A
+b
+β
+c
+ã
+�→ ι2(h) =
+Ü 1
+A
+b
+1
+β
+c
+ê
+,
+where A and C are 4-by-4 matrices. For x = (x1, x2, x3) ∈ C3 and u ∈ C, put n1(x) = ι1(n(x)) and n2(x) = ι2(n(x)),
+where n(x) is the element defined in the proof of Proposition 4.25, and put
+nc(u) =
+à
+1
+u
+2
+0
+− u
+2
+− u
+2
+1
+u
+2
+0
+13
+0
+u
+2
+1
+− u
+2
+− u
+2
+0
+u
+2
+1
+í
+,
+so that
+n1(x) = ξ(exp(
+√
+−1x1(Xχ1−χ3 + Xχ1+χ3) +
+√
+−1x2Xχ1 + x3(Xχ1−χ3 − Xχ1+χ3))),
+n2(x) = ξ(exp(
+√
+−1x1(Xχ2−χ3 + Xχ2+χ3) +
+√
+−1x2Xχ2 + x3(Xχ2−χ3 − Xχ2+χ3))),
+nc(u) = ξ(exp(uXχ1+χ2)).
+Let N1 (resp. N2, resp. Nc) be the unipotent subgroup of G consisting of n1(x) (resp. n2(x), resp. nc(u)). Then the
+Haar measure on N1(R) (resp. N2(R), resp. Nc(R)) is given by d(n1(x)) = 2dx1dx2dx3 (resp. d(n2(x)) = 2dx1dx2dx3,
+resp. d(nc(u)) = du). We have N(R) = N1(R)N2(R)Nc(R) = {n1(x)nc(u)n2(y) | x, y ∈ R3, u ∈ R} and the Haar
+measure is the product of those on N1(R), N2(R), and Nc(R).
+Let us realize the discrete series representation D1 of GL2(R) as a subrepresentation of a parabolically induced
+representation from a character | − |
+1
+2 ⊠ | − |− 1
+2 on the diagonal maximal torus of GL2(R), and let 0 ̸= v0 ∈ D1 be a
+lowest weight vector such that
+v0
+ÅÅ a
+b
+d
+ã Å
+cos θ
+sin θ
+− sin θ
+cos θ
+ãã
+=
+���a
+d
+��� e2√−1θ.
+For λ ∈ C, put φλ = τ(1,λ) ⊕ τ1 ⊕ τ ∨
+(1,λ) and πλ = (D1 ⊗ | · |λ) ⊠ 1, so that Πφλ(M) = {πλ} and Πφλ(G) = {IP (πλ)}.
+Since D1 is realized as a subrepresentation of a parabolically induced representation, we have a natural injection
+HP (πλ) ֒→ HP0(| − |
+1
+2 +λ ⊠ | − |− 1
+2 +λ ⊠ 1),
+f(·) �→ f(·)(1),
+where P0 is the minimal standard parabolic subgroup of G defined over R.
+Define a function f (λ) ∈ HP (πλ) ⊂
+HP0(| − |
+1
+2 +λ ⊠ | − |− 1
+2 +λ ⊠ 1) by
+f (λ)(m(A, B)nκ) = | det A|λ+2[τ1(A)v0](1)
+= |a|λ+3|d|λ+1e2√−1θ,
+A =
+Å a
+b
+d
+ã Å
+cos θ
+sin θ
+− sin θ
+cos θ
+ã
+,
+which is holomorphic in λ ∈ C. Assume that Re(λ) > 0. We have 0 ̸= f (0) ∈ HP (π) and
+î
+ℓP
+P(w, ξ, φ, ψR) ◦ RP w|P (ξ, φ)f (0)ó
+(g)
+= lim
+λ→+0
+î
+ℓP
+P (w, ξ, φλ, ψR) ◦ RP w|P (ξ, φλ)f (λ)ó
+(g)
+47
+
+= lim
+λ→+0 ǫ(0, ρ∨
+P w|P ◦ φλ, ψR)
+L(1, ρ∨
+P w|P ◦ φλ)
+L(0, ρ∨
+P w|P ◦ φλ)
+�
+N(R)
+f (λ)( ˘w−1ng)dn.
+A direct calculation implies that ρ∨
+P w|P ◦ φλ is isomorphic to
+τ(2,λ) ⊕ σωλ ⊕ ωλ ⊕ τ(2,2λ) ⊕ σω2λ.
+Therefore, we have
+ǫ(0, ρ∨
+P w|P ◦ φλ, ψR)
+L(1, ρ∨
+P w|P ◦ φλ)
+L(0, ρ∨
+P w|P ◦ φλ) = ΓC(λ + 2)
+ΓC(λ + 1)
+ΓR(λ + 2)
+ΓR(λ)
+ΓC(2λ + 2)
+ΓC(2λ + 1)
+ΓR(2λ + 2)
+ΓR(2λ + 1).
+We now turn to the integral. Let ιSL2 be an embedding of SL2 into G = SO(2, 5) given by ιSL2(A) = m(A, 13). Put
+w0 =
+Å1
+−14
+ã
+, which is the Langlands-Shelstad lift in SO(1, 4) given in the proof of proposition 4.25. Equations
+˘w = ι2(w0)ιSL2(J)ι2(w0),
+Ad(ιSL2(J)ι2(w0))−1(n1(x)) = n2(−x),
+Ad(ι2(w0))−1(nc(u)) = ιSL2(
+Å1
+u
+1
+ã
+),
+implies that
+�
+N(R)
+f (λ)( ˘w−1ng)dn = [M2 ◦ Mc ◦ M2(f (λ))](g),
+where we have put
+[M2f](g) =
+�
+N2(R)
+f(ι2(w0)−1n2g)dn2,
+[Mcf](g) =
+�
+Nc(R)
+f(ιSL2(J)−1ncg)dnc,
+for any function f on G(R).
+For α, β ∈ C, we define f (α,β) ∈ HP0(| − |α ⊠ | − |β ⊠ 1) by
+f (α,β)(m(A, B)nκ) = |a|α+ 5
+2 |d|β+ 3
+2 e2√−1θ,
+A =
+Å a
+b
+d
+ã Å
+cos θ
+sin θ
+− sin θ
+cos θ
+ã
+,
+so that f (λ) = f (λ+ 1
+2 ,λ− 1
+2 ). By Lemmas 4.28 and 4.29 below, we have
+M2 ◦ Mc ◦ M2(f (λ))
+= M2 ◦ Mc ◦ M2(f (λ+ 1
+2 ,λ− 1
+2 ))
+= 2− 1
+2 ΓC(λ − 1
+2)
+ΓC(λ + 1) M2 ◦ Mc(f (λ+ 1
+2 ,−λ+ 1
+2 ))
+= −2− 1
+2 ΓC(λ − 1
+2)
+ΓC(λ + 1)
+ΓR(2λ)ΓR(2λ + 1)
+ΓR(2λ + 3)ΓR(2λ − 1)M2(f (−λ+ 1
+2 ,λ+ 1
+2 ))
+= −2−1 ΓC(λ − 1
+2)
+ΓC(λ + 1)
+ΓR(2λ)ΓR(2λ + 1)
+ΓR(2λ + 3)ΓR(2λ − 1)
+ΓC(λ + 1
+2)
+ΓC(λ + 2) f (−λ).
+This leads to
+ǫ(0, ρ∨
+P w|P ◦ φλ, ψR)
+L(1, ρ∨
+P w|P ◦ φλ)
+L(0, ρ∨
+P w|P ◦ φλ)
+�
+N(R)
+f (λ)( ˘w−1ng)dn
+= ΓC(λ + 2)
+ΓC(λ + 1)
+ΓR(λ + 2)
+ΓR(λ)
+ΓC(2λ + 2)
+ΓC(2λ + 1)
+ΓR(2λ + 2)
+ΓR(2λ + 1) · −1
+2
+ΓC(λ − 1
+2)
+ΓC(λ + 1)
+ΓR(2λ)ΓR(2λ + 1)
+ΓR(2λ + 3)ΓR(2λ − 1)
+ΓC(λ + 1
+2)
+ΓC(λ + 2) f (−λ)(g),
+whose limit as λ approaching 0 from the right is −f (λ). Since f (0) ̸= 0, this completes the proof.
+48
+
+In order to finish the proof of Proposition 4.27, it remains to show the following two lemmas.
+Lemma 4.28. Suppose that Re(β) > 0. Then
+M2f (α,β) = 2− 1
+2
+ΓC(β)
+ΓC(β + 3
+2)f (α,−β).
+Proof. As in the proof of Lemma 4.26, one has
+M2f (α,β) = M2f (α,β)(1) · f (α,−β),
+if M2f (α,β)(1) converges absolutely. Since f (α,β) ◦ ι2 is equal to ϕ(β) defined in the previous subsection, we have
+[M2f (α,β)](1) =
+�
+x∈R3 f (α,β)(ι2(w0)−1ι2(n(x)))d(n(x))
+=
+�
+x∈R3 ϕ(β)(w−1
+0 n(x))d(n(x))
+= [MSO(1,4)ϕ(β)](1).
+Hence the assertion follows from Lemma 4.26.
+Lemma 4.29. Suppose that Re(α − β) > 0. Then
+Mcf (α,β) = −
+ΓR(α − β)ΓR(α − β + 1)
+ΓR(α − β + 3)ΓR(α − β − 1)f (β,α).
+Proof. As in the proof of Lemma 4.26, one has
+Mcf (α,β) = Mcf (α,β)(1) · f (β,α),
+if Mcf (α,β)(1) converges absolutely. Following the proof of [15, Lemma 1.4], for s ∈ C we define a function h(s) on
+SL2(R) by
+h(s)
+ÅÅ a
+b
+a−1
+ã Å
+cos θ
+sin θ
+− sin θ
+cos θ
+ãã
+=
+���a
+d
+���
+s+1
+e2√−1θ.
+Then f (α,β) ◦ ιSL2 is equal to h(α−β), and hence we have
+[Mcf (α,β)](1) =
+�
+u∈R
+f (α,β)
+Å
+ιSL2(J)−1ιSL2
+Å 1
+u
+1
+ãã
+du
+=
+�
+u∈R
+h(α−β)
+Å
+J−1
+Å 1
+u
+1
+ãã
+du.
+Hence the assertion follows from the second equation in the proof of [15, Lemma 1.4].
+5
+The decomposition into near equivalence classes and the standard
+model
+In this section we shall roughly recall from [18, 7] the stable multiplicity formula, which implies the decomposition
+of L2
+disc(G(F)\G(AF )) into near equivalence classes, the global intertwining relation, and its weaker identity.
+49
+
+5.1
+Stable multiplicity formula
+Let F be a number field, G∗ the split special orthogonal group SO2n+1 over F, and G an inner form of G∗.
+Although G does not have simply connected derived subgroup, every endoscopic data for G comes from an endoscopic
+triple. Hence the argument in [18, §§3.1-3.2] also holds for our G. Thus we have the decompositions
+L2
+disc(G(F)\G(AF )) =
+�
+c∈C(G)
+t≥0
+L2
+disc,t,c(G(F)\G(AF )),
+tr Rdisc(f) =
+�
+c∈C(G)
+t≥0
+Rdisc,t,c(f),
+for f ∈ H(G), where R♦ denotes the regular representation of G(AF ) on L2
+♦. Moreover, for t ∈ R≥0, c ∈ C(G), and
+e = (Ge, se, ηe) ∈ Eell(G), we have
+• the c-variant IG
+disc,t,c of the discrete part IG
+disc,t of the trace formula on the constituents of which the norm of the
+imaginary part of the infinitesimal character is t, where the central character datum is trivial;
+• the transfer mapping H(G) → S(Ge), f �→ f e = f Ge, where S(Ge) denotes a space of functions on conjugacy
+classes of semisimple elements defined in [7, p.53, p.132];
+• the stable linear form Se
+disc,t,c = SGe
+disc,t,c on H(Ge) and the associated linear form �Se
+disc,t,c on S(Ge) (see [7,
+(2.1.2)] for the notion of associated linear forms),
+and the stabilization
+IG
+disc,t,c(f) =
+�
+e∈Eell(G)
+ι(G, Ge)�Se
+disc,t,c(f e),
+f ∈ H(G).
+Here ι(G, Ge) are the global coefficients introduced by Kottwitz and Shelstad. See [7, (3.2.4)] for an explicit formula
+for them.
+Let ψ ∈ Ψ(G∗) and e = (Ge, se, ηe) ∈ Eell(G). To the former is associated t(ψ) ∈ R≥0 and c(ψ) ∈ C(G). As in [18,
+§3.3], put
+IG
+disc,ψ = IG
+disc,t(ψ),c(ψ),
+SGe
+disc,ψ = SGe
+disc,t(ψ),c(ψ),
+L2
+disc,ψ(G(F)\G(AF )) = L2
+disc,t(ψ),c(ψ)(G(F)\G(AF )),
+Rdisc,ψ = Rdisc,t(ψ),c(ψ).
+We also write RG
+disc,ψ if the group G is to be emphasized. Let us recall from [7, Theorem 4.1.2] the stable trace formula
+for the split odd special orthogonal group G∗ = SO2n+1:
+Proposition 5.1 (Stable trace formula). Let ψ ∈ ‹Ψ(N). Then we have
+SG∗
+disc,ψ(f) =
+�
+|Sψ|−1εG∗
+ψ (sψ)σ(S
+◦
+ψ)f G∗(ψ),
+if ψ ∈ Ψ(G∗),
+0,
+if ψ ∈ ‹Ψ(N) \ Ψ(G∗),
+for f ∈ H(G∗), where εG∗
+ψ
+is the character in the multiplicity formula, σ(S
+◦
+ψ) the constant given in [7, Proposition
+4.1.1], and f G∗(ψ) denotes the linear form defined by [7, (4.1.3)].
+As a consequence we have a decomposition
+L2
+disc(G(F)\G(AF )) =
+�
+ψ∈Ψ(G∗)
+L2
+disc,ψ(G(F)\G(AF )),
+tr Rdisc(f) =
+�
+ψ∈Ψ(G∗)
+Rdisc,ψ(f),
+f ∈ H(G),
+(5.1)
+of the discrete spectrum. Theorem 3.12 is now proven.
+50
+
+5.2
+Global intertwining operator
+Let ξ : G∗ → G be an inner twist. Let P ∗ ⊂ G∗ be a standard parabolic subgroup with a Levi decomposition
+P ∗ = M ∗N ∗ over F, and put P = ξ(P ∗), M = ξ(M ∗), and N = ξ(N ∗). We consider the case that P, M, and N are
+defined over F, and a restriction ξ|M∗ : M ∗ → M is an inner twist.
+Assume that M ̸= G. Let (π, Vπ) be an irreducible component of L2
+disc(A+
+M,∞G(F)\G(AF )), where A+
+M,∞ denotes
+the connected component of 1 in ResF/Q(M)(R). Let ψ ∈ Ψ2(M ∗) be the corresponding A-parameter (given by the
+induction hypothesis). We shall consider the induced representation (IP (π), HP (π)). It is a right regular representation
+on the Hilbert space HP (π) of measurable functions f : G(AF ) → Vπ such that f(nmg) = δ
+1
+2
+P (m)π(m)f(g) for any
+n ∈ N(AF ), m ∈ M(AF ), and g ∈ G(AF ) whose restriction to an open compact subgroup K ⊂ G(AF ) is square-
+integrable.
+Let P ′ ⊂ G be a parabolic subgroup over F with Levi component M. For λ ∈ a∗
+M,C, the intertwining operator
+JP ′|P : IP (πλ) → IP ′(πλ) is defined by
+[JP ′|P (πλ)(f)](g) =
+�
+N(AF )∩N ′(AF )\N ′(AF )
+f(n′g)dn′,
+where N ′ denotes the unipotent radical of P ′, and πλ = π ⊗ λ. As in [18], we take the Haar measure dn′ determined
+by the Haar measure on AF which assigns the quotient AF /F volume 1. It is known that the integral converges
+absolutely when the real part of λ lies in a certain open cone. As a function of λ, it has a meromorphic continuation
+and is nonzero and holomorphic at λ = 0. Then the operator JP ′|P (π) = JP ′|P (πλ)|λ=0 is defined. It is also known
+that JP ′|P (π) is unitary.
+We now consider the case P ′ = P w = w−1Pw for some w ∈ W(M, G)Γ. Let ˘w ∈ N(M, G)(F) be a representative
+of w. Then one has a twist ( ˘wπ, Vπ). We define two intertwining operators ℓ( ˘w) and C ˘
+w by
+ℓ( ˘w) : HP w(π) −→ HP ( ˘wπ),
+[ℓ( ˘w)f](g) = f( ˘w−1g),
+C ˘
+w : ( ˘wπ, Vπ) −→ (π, Vπ),
+[C ˘
+wϕ](m) = ϕ( ˘w−1m ˘w).
+The composition IP (C ˘
+w)◦ℓ( ˘w)◦JP w|P (π) is then a self-intertwining operator of the induced representation (IP (π), HP (π)),
+and independent of the choice of ˘w. We put
+MP (w, π) = IP (C ˘
+w) ◦ ℓ( ˘w) ◦ JP w|P (π).
+Let ρP ′|P be the adjoint representation of �
+M on �n ∩ �n′\�n′ ∼= �n ∩ “n′, where �n, “n′, and �n are the Lie algebras of “
+N,
+�
+N ′, and “
+N, respectively, where N denotes the unipotent radical of the opposite parabolic subgroup P of P containing
+M. Following [18], we define normalizing factors
+rP ′|P (ψ) =
+L(0, ψ, ρ∨
+P ′|P )
+L(1, ψ, ρ∨
+P ′|P )
+ǫ( 1
+2, ψ, ρ∨
+P ′|P )
+ǫ(0, ψ, ρ∨
+P ′|P ) ,
+rP (w, ψ) = rP w|P (ψ)ǫ(1
+2, ψ, ρ∨
+P w|P )−1,
+and normalized intertwining operators
+RP ′|P (π, ψ) = rP ′|P (ψ)−1JP ′|P (π),
+RP (w, π, ψ) = rP (w, ψ)−1MP (w, π),
+where L- and ǫ-factors are the automorphic ones.
+Lemma 5.2. We have
+RP ′|P (πλ, ψλ) =
+�
+v
+RP ′v|Pv(πλ,v, ψλ,v).
+Proof. The proof is similar to that of [18, Lemma 2.2.2].
+Lemma 5.3. Let P ′′ ⊂ G be a parabolic subgroup over F with Levi component M. We have
+RP ′′|P (πλ, ψλ) = RP ′′|P ′(πλ, ψλ) ◦ RP ′|P (πλ, ψλ).
+51
+
+Proof. The unnormalized intertwining operator JP ′|P (πλ) satisfies the multiplicativity property. It is known that at
+each place v the local factors of automorphic L- and ǫ-factors are equal to the Artin L- and ǫ-factors. Thus the
+normalizing factor has a decomposition
+rP ′|P (ψλ) =
+�
+v
+rP ′v|Pv(ξv, ψv, ψF,v).
+Since every local factor rP ′v|Pv(ξv, ψv, ψF,v) has the multiplicativity property, so is rP ′|P (ψλ). This completes the
+proof.
+5.3
+The global intertwining relation
+In this subsection, we define two global linear forms and state the global intertwining relation, which we shall call
+GIR for short. This subsection can be regarded as the global analogue of §4.6.
+Let ψM∗ ∈ Ψ2(M ∗), and ψ ∈ Ψ(G∗) be its image. Note the change of notation from the previous subsection that
+we write ψM∗ for a parameter for M ∗ rather than ψ. Let ψF : AF /F → C1 be a nontrivial additive character. For
+πM = �
+v πM,v ∈ ΠψM∗(M) and u♮ ∈ Nψ(M, G) we define a global intertwining operator
+RP (u♮, πM, ψM∗, ψF ) =
+�
+v
+RPv(u♮
+v, πM,v, ψM∗,v, ψF,v),
+where u♮
+v ∈ Nψv(Mv, Gv), ψM∗,v ∈ Ψ+(M ∗), and ψF,v are the localizations of u♮, ψM∗, and ψF respectively.
+Proposition 5.4. Assume that πM ∈ ΠψM∗(M) is automorphic, i.e., ⟨−, πM⟩ = εψM∗. Then for any u♮ ∈ Nψ(M, G)
+we have an equation
+�
+v
+πM,v(u♮
+v) = εψM∗(u♮)C ˘
+w,
+of isomorphisms ( ˘wπM, VπM ) → (πM, VπM ), where w is the image of u♮ in Wψ(M, G).
+Proof. The proof is similar to that of Proposition 3.5.3 of [18].
+Proposition 5.5. Let u♮ ∈ Nψ(M, G) and πM ∈ ΠψM∗(M). Then
+1. RP (yu♮, πM, ψM∗, ψF ) = ⟨y, πM⟩RP (u♮, πM, ψM∗, ψF ) for any y ∈ Sψ(M), where y is the image of y in Sψ(M).
+2. RP (wu, πM, ψM∗) = εψM∗(u♮)RP (u♮, πM, ψM∗, ψF ) if πM is automorphic.
+Proof. The first assertion follows from Lemma 4.10. The other one follows from Lemma 5.2 and Proposition 5.4.
+The first linear form H(G) ∋ f �→ fG(ψM∗, u♮) is defined by
+fG(ψM∗, u♮) =
+�
+v
+fv,Gv(ψM∗,v, u♮
+v)
+=
+�
+πM∈ΠψM∗ (M)
+tr �RP (u♮, πM, ψM∗, ψF ) ◦ IP (πM, f)� ,
+f =
+�
+v
+fv ∈ H(G).
+Note that the parameter ψM∗ is relevant since it is discrete and M = ξ(M ∗) is defined over F. Put fG(ψM∗, u♮) to be
+0 for M ∗ ⊂ G∗ which does not transfer to G. Proposition 5.5 implies the following lemma.
+Lemma 5.6. The linear form fG(ψM∗, u♮) depends only on the image of u♮ in Nψ(M, G).
+Thus we also write fG(ψM∗, u) for fG(ψM∗, u♮), where u ∈ Nψ(M, G) is the image of u♮.
+Next let us recall the definition of the second linear form. For a parameter ψ ∈ Ψ(G∗) and a semisimple element
+s ∈ Sψ, Lemma 3.9 attaches an endoscopic triple e = (Ge, se, ηe) ∈ E(G) and a parameter ψe ∈ Ψ(Ge), where ψe is
+not unique (determined up to OutG(Ge)-action). The second linear form H(G) ∋ f �→ f ′
+G(ψ, s) is defined by
+f ′
+G(ψ, s) = f e(ψe)
+=
+�
+v
+f e
+v(ψe
+v) =
+�
+v
+f ′
+v,Gv(ψv, sv),
+f =
+�
+v
+fv ∈ H(G),
+where sv ∈ Sψv denotes the localization of s. If ψ is the image of ψM∗ ∈ Ψ2(M ∗), we also write f ′
+G(ψM∗, s) for
+f ′
+G(ψ, s).
+52
+
+Lemma 5.7. The linear form f ′
+G(ψ, s) depends only on the image of s in Sψ, and hence on the image of e in E(G).
+Proof. By Lemma 4.12, we have f ′
+G(ψ, s) = f ′
+G(ψ, ss0) for any s0 ∈ S◦
+ψ. By Lemma 3.2 and the product formula (2.2),
+we have f ′
+G(ψ, s) = f ′
+G(ψ, sx) for any x ∈ Z(“
+G)Γ.
+The following theorem is called the global intertwining relation (GIR), which will follow from LIR.
+Theorem* 5.8 (Global intertwining relation). If u ∈ Nψ(M, G) and s ∈ Sψ,ss have the same image in Sψ(G), then
+we have
+f ′
+G(ψ, sψs−1) = fG(ψM∗, u).
+5.4
+Reduction of GIR to elliptic or exceptional (relative to G) parameters
+This subsection is the global version of §4.8. As in the local case, one can see that Ψ2
+ell(G∗) is the set of the
+equivalence classes of parameters ψ of the form
+ψ = 2ψ1 ⊞ · · · 2ψq ⊞ ψq+1 ⊞ · · · ⊞ ψr,
+(5.2)
+where ψ1, . . . , ψr are simple, symplectic, mutually distinct, and r ≥ q ≥ 1. Then we have
+Sψ ∼= O(2, C)q × O(1, C)r−q.
+Let now Ψexc1(G∗) and Ψexc2(G∗) be the subsets of Ψ(G∗) consisting of the parameters ψ of the form
+(exc1) ψ = 2ψ1 ⊞ψ2 ⊞· · ·⊞ψr, where ψ1 is simple and orthogonal, and ψ2, . . . , ψr are simple, symplectic, and mutually
+distinct,
+(exc2) ψ = 3ψ1 ⊞ ψ2 ⊞ · · · ⊞ ψr, where ψ1, . . . , ψr are simple, symplectic, and mutually distinct,
+respectively. They are disjoint. We then have
+(exc1) Sψ ∼= Sp(2, C) × O(1, C)r−1,
+(exc2) Sψ ∼= O(3, C) × O(1, C)r−1,
+respectively. We put Ψexc(G∗) = Ψexc1(G∗) ⊔ Ψexc2(G∗), and we shall say that ψ is exceptional (resp. of type (exc1),
+resp. of type (exc2)) if ψ ∈ Ψexc(G∗) (resp. ψ ∈ Ψexc1(G∗), resp. ψ ∈ Ψexc2(G∗)). One can see that Ψexc(G∗) and
+Ψell(G∗) are disjoint. Put Ψell,exc(G∗) = Ψ(G∗) \ (Ψell(G∗) ⊔ Ψexc(G∗)). Put also Φ♥(G∗) = Ψ♥(G∗) ∩ Φ(G∗) for
+♥ ∈ {exc, exc1, exc2}, and Φell,exc(G∗) = Ψell,exc(G∗) ∩ Φ(G∗).
+Let ψM∗ ∈ Ψ2(M ∗) and ψ ∈ Ψ(G∗) be parameters such that ψ is the image of ψM∗. For x ∈ Sψ, we define Tψ,
+Bψ, sx ∈ Sψ, and Tψ,x in the same way as in §4.8. Moreover, let T ψ, Bψ, sx, and T ψ,x be their images in Sψ. Then
+it can be seen that sx and T ψ,x are determined by the image of x in Sψ, and hence they are well-defined for x ∈ Sψ.
+Define subsets Sψ,ell ⊂ Sψ and Sψ,ell ⊂ Sψ as in the local case. The following four lemmas can be proved in the
+same way as Lemmas 4.19, 4.20, 4.21, and 4.23 were, respectively.
+Lemma 5.9. Suppose that ψ ∈ Ψell,exc(G∗). Then
+(1) every simple reflection w ∈ W ◦
+ψ(M ∗, G∗) centralizes a torus of positive dimension in T ψ and
+(2) dim T ψ,x ≥ 1 for all x ∈ Sψ.
+Lemma 5.10. If ψ ∈ Ψexc(G∗), then Sψ,ell = Sψ and Sψ,ell = Sψ.
+Let ξ : G∗ → G be an inner twist, and assume that M ∗ ⊊ G∗ is a proper standard Levi subgroup such that
+M = ξ(M ∗) ⊊ G is a proper Levi subgroup defined over F.
+Lemma 5.11. Let x ∈ Sψ(M, G). Assume that either
+(1) ψ is elliptic, or
+53
+
+(2) every simple reflection w ∈ W ◦
+ψ(M ∗, G∗) centralizes a torus of positive dimension in T ψ.
+Then fG(ψM∗, u) is the same for every u ∈ Nψ(M, G) mapping to x.
+Lemma 5.12. Let x ∈ Sψ. We have fG(ψM∗, u) = f ′
+G(ψ, sψs−1) whenever u ∈ Nψ(M, G) and s ∈ Sψ,ss map to x
+unless
+(1) ψ is elliptic and x ∈ Sψ,ell, or
+(2) ψ ∈ Ψexc(G∗).
+Lemmas 5.11 and 5.12 imply that fG(ψM∗, u) depends only on the image of u in Sψ(M, G) = Sψ(G), unless ψ is
+exceptional. Thus fG(ψM∗, x) is well-defined for x ∈ Sψ(M, G). On the other hand, when ψ is exceptional, we can
+define fG(ψM∗, x) for x ∈ Sψ(M, G) similarly to local setting in the end of §4.8. Now we have defined fG(ψM∗, x) for
+x ∈ Sψ(M, G).
+5.5
+The weaker identities
+In the previous subsection, GIR for ψ ∈ Ψell,exc(G∗) was deduced from the induction hypothesis (Lemma 5.12).
+Now we shall see two weaker identities. We omit Arthur’s procedure named the standard model for G, since it is very
+similar to that for other classical groups explained in [7, §4] and [18, §3.6]. We continue to be in the setup of the
+previous subsection, but do not assume that M = ξ(M ∗) is defined over F.
+Lemma 5.13. Suppose that ψ ∈ Ψexc(G∗). Let x ∈ Sψ and f ∈ H(G). If M ∗ does not transfer to G, then we have
+f ′
+G(ψ, sψx−1) = fG(ψ, x) = 0.
+If M ∗ transfers to G, then we have
+�
+x∈Sψ
+εG∗
+ψ (x) �f ′
+G(ψ, sψx−1) − fG(ψ, x)� = 0,
+and RP (w, πM, ψM∗) = 1 for w ∈ Wψ.
+Lemma 5.14. Suppose that ψ ∈ Ψ2
+ell(G∗) is of the form (5.2). Let f ∈ H(G). Then we have
+tr �RG
+disc,ψ(f)� = 2−q|Sψ|−1
+�
+x∈Sψ,ell
+εG∗
+ψ (x) �f ′
+G(ψ, sψx−1) − fG(ψ, x)� .
+Proof of Lemmas 5.13 and 5.14. The proofs of Lemmas 5.13 and 5.14 are similar to those of Lemmas 3.7.1 and 3.8.1
+of [18], respectively.
+6
+Globalizations and the proof of local classification
+In this section, we will finish the proof of LIR and ECR for generic parameters, and of LLC. The argument is
+similar to that of [18, §4]. The primary difference is the globalization of parameters, which is caused by the difference
+between unitary groups and orthogonal groups.
+6.1
+Globalizations of fields, groups, and representations
+First recall from [7, Lemma 6.2.1] the following lemma:
+Lemma 6.1. Let F be a local field other than C, and r0 a positive integer. Then we can find a totally real number
+field ˙F and a place u of ˙F such that ˙Fu is isomorphic to F and ˙F has at least r0 real places.
+As a corollary, we have the following.
+54
+
+Lemma 6.2. Let F be a local field other than C, and r0 a positive integer. Then we can find a totally real number
+field ˙F and places u1 and u2 of ˙F such that ˙Fu is isomorphic to F for u = u1, u2, and that ˙F has at least r0 real
+places.
+Proof. Let ˙F be as in Lemma 6.1. It can be easily seen that there exists an element α ∈ ˙F × that is not square in ˙F,
+totally positive, and square in ˙Fu = F. Then ˙F(√α) is a number field what we want.
+The classification of quadratic forms or the exact sequence (2.2) leads the following lemma of special orthogonal
+groups:
+Lemma 6.3. Let F be a local field, and G an inner form of G∗ = SO2n+1 over F. Let ˙F be a number field with a
+place u such that ˙Fu = F. Assume that ˙F is totally real unless F = C. Then for any place w of ˙F other than u, there
+exists an inner form ˙G of ˙G∗ = SO2n+1 over ˙F with following properties:
+1.
+˙Gu = G;
+2.
+˙Gv is split over ˙Fv for any place v of ˙F except u and w;
+3. if G is non-quasi-split and w is a finite (resp. real) place, then ˙Gw is the unique non-quasi-split inner form
+(resp. isomorphic to SO(n − 1, n + 2)).
+In place of the third property, we can also choose ˙G with the properties 1, 2, and
+3’ if ˙Fw is isomorphic to ˙Fu = F, then ˙Gw is isomorphic to ˙Gu = G.
+We have the following globalization theorems of discrete series representations.
+Lemma 6.4. Let ˙F be a totally real number field, and ˙G a simple twisted endoscopic group of GLN or an inner form
+of ˙G∗ = SO2n+1 over ˙F. Let V be a finite set of places of ˙F such that at least one real place v∞ is not contained
+in V . For all v ∈ V , let πv ∈ Π2,temp( ˙Gv), and let πv∞ be a discrete series representation with a sufficiently regular
+infinitesimal character. Then there exists a cuspidal automorphic representation ˙π of ˙G(A ˙F ) such that ˙πv = πv for all
+v ∈ V and ˙πv∞ = πv∞.
+Proof. The proof is the same as that of [18, Lemma 4.2.1].
+Lemma 6.5. Let ˙F be a totally real number field, and ˙G a simple twisted endoscopic group of GLN. Let V be a finite
+set of places of ˙F such that at least one real place v∞ is not contained in V . For all v ∈ V , let Mv ⊂ ˙Gv be a Levi
+subgroup such that Mv = ˙Gv if v is a real place. For each v ∈ V , let πMv ∈ Π2,temp(Mv). Then there exists a cuspidal
+automorphic representation ˙π of ˙G(A ˙F ) such that for all v ∈ V , if Mv = ˙Gv then ˙πv = πMv and if Mv ̸= ˙Gv then
+˙πv is an irreducible subquotient of the induced representation I
+˙Gv
+Pv (πMv ⊗ χv) for some unramified unitary character
+χv ∈ Ψ(Mv).
+Proof. The proof is the same as that of [18, Lemma 4.2.2].
+The following lemma is used in the proof of Lemma 4.3. Note that the proof of the following lemma is independent
+of §4.
+Lemma 6.6. Let F be a p-adic field, ξ : G∗ → G an inner twist of G∗ = SO2n+1 over F, M ∗ ⊂ G∗ a standard Levi
+subgroup such that M := ξ(M ∗) is defined over F, and π ∈ Πscusp(M) an irreducible supercuspidal representation.
+Let ˙F, u, v2, ˙G∗, and ˙G be as in Lemma 6.3. Then there exist an inner twist ˙ξ : ˙G∗ → ˙G over ˙F, a standard Levi
+subgroup
+˙M ∗ ⊂ ˙G∗, and an irreducible cuspidal automorphic representation ˙π of
+˙M(A ˙F ) with following properties:
+• ˙ξu = ξ;
+•
+˙M := ˙ξ( ˙M ∗) is defined over ˙F, and ˙ξ| ˙
+M∗ : ˙M ∗ → ˙M is an inner twist;
+•
+˙Mu = M, and
+˙Mv is split over ˙Fv for any place v of ˙F except u and v2;
+• ˙πu = π.
+Proof. The former three properties follow from Lemmas 6.1 and 6.3. The last property follows from Lemma 6.4 and
+[31, Proposition 5.1].
+55
+
+6.2
+Globalizations of parameters
+Now we shall globalize generic parameters. Let us first consider globalizations of simple parameters.
+Lemma 6.7. Let ˙F be a totally real number field, and ˙G∗ a simple twisted endoscopic group of GLN, i.e., the split
+symplectic group, the split odd special orthogonal group, or a quasi-split even special orthogonal group over ˙F. Let V
+be a finite set of places of ˙F in which at least one real place is not contained. For each v ∈ V , let φv ∈ �Φ2,bdd( ˙G∗
+v) be
+a square integrable parameter. Assume that for at least one v ∈ V , φv ∈ �Φsim( ˙G∗
+v). Then there exists a simple generic
+parameter ˙φ ∈ �Φsim( ˙G∗) such that ˙φv = φv for all v ∈ V .
+Proof. The proof is similar to that of [18, Lemma 4.3.1], except that we need the following one or two modifications.
+The first one is to use Lemma 6.4 of this paper in place of Lemma 4.2.1 of [18]. Then the same proof carries over word
+by word in the case when ˙G∗ is a symplectic or an odd special orthogonal group. The second one is, in the case of even
+special orthogonal groups, to replace Φsim by �Φsim and the ordinary equivalence classes of the representations by the
+ǫ-equivalence classes of the representations. Here, note that in the case of quasi-split even special orthogonal groups,
+the endoscopic classification (LLC, ECR, and AMF) is known up to ǫ-equivalence, by Arthur [7] and Atobe-Gan
+[8].
+Lemma 6.8. Let N ≥ 2. Let ˙F be a totally real number field, and ˙G∗ a simple twisted endoscopic group of GLN, i.e.,
+the split symplectic group, the split odd special orthogonal group, or a quasi-split even special orthogonal group over ˙F.
+Let V and {φv}v∈V be as in Lemma 6.7. Assume that for at least one v ∈ V , φv ∈ �Φsim( ˙G∗
+v). Let v2 /∈ V be a finite
+place. Then there exists a simple generic parameter ˙φ ∈ �Φsim( ˙G∗) such that ˙φv = φv for all v ∈ V and ˙φv2 is of the
+form
+˙φv2 = φ+ ⊕ φ∨
++ ⊕ φ−,
+where φ+ ∈ Φbdd(GL1) is a non-self-dual parameter and φ− ∈ �Φsim,bdd(N − 2) a simple self-dual parameter.
+Proof. The proof is similar to that of [18, Lemma 4.3.2], except that we need the modifications similar to that of
+Lemma 6.7.
+Next we shall consider globalizations of elliptic or exceptional parameters. Let F be a local field, and G an inner
+form of G∗ = SO2n+1 over F. Let M ∗ ⊂ G∗ be a Levi subgroup. Following [18, §4.4], we shall say that M ∗ is linear
+if M ∗ is isomorphic to a direct product of general linear groups, i.e., n0 = 0 in (3.4). As in the case of even unitary
+groups, any nontrivial inner form G of G∗ does not admit a globalization ˙G which localizes to G at one place and is
+split at all the other places. Hence the same complication is caused.
+Let φ ∈ Φbdd(G∗) be a bounded L-parameter for G, which is the image of a discrete parameter φM∗ ∈ Φ2(M ∗) for
+M ∗. Assume that φ is either elliptic or exceptional. Explicitly M ∗, φ, and φM∗ has the form
+M ∗ ≃ GLe1
+N1 × · · · × GLer
+Nr ×M ∗
+−,
+φ =
+r
+�
+i=1
+ℓiφi =
+r
+�
+i=1
+(ei(φi ⊕ φ∨
+i ) ⊕ δiφi) ,
+φM∗ = e1φ1 ⊕ · · · ⊕ erφr ⊕ φ−,
+where φi ∈ �Φsim,bdd(Ni) are mutually distinct self-dual simple bounded L-parameters for GLNi, φ− = ⊕iδiφi ∈
+Φ2(M ∗
+−), δi ∈ {0, 1}, Ni ∈ Z≥1, ei ∈ Z≥0, ℓi = 2ei + δi, M ∗
+− = SO2n0+1, and 2n0 = �
+i δiNi so that �
+i eiNi + n0 = n
+and ei = ⌊ℓi/2⌋.
+In the lemmas and propositions below, we will start from the following local data with the assumption given above:
+• a local field F;
+• G∗ = SO2n+1 and a Levi subgroup M ∗ ⊂ G∗ over F;
+• an inner form G of G∗;
+• an L-parameter φ ∈ Φbdd(G∗) that is the image of a parameter φM∗ ∈ Φ2(M ∗).
+First we treat the case when M ∗ = G∗.
+56
+
+Lemma 6.9. Assume that M ∗ = G∗, and hence φ ∈ Φ2(G∗). Then there exists a global data ( ˙F, ˙G∗, ˙φ, u, v1, v2),
+where ˙F is a totally real number field, ˙G∗ = SO2n+1 over ˙F, ˙φ ∈ Φ( ˙G∗), and u, v1, v2 are places of ˙F, such that v1 is
+a finite place, v2 is a real place, and
+1.
+˙Fu = F, ˙G∗
+u = G∗, and ˙φu = φ;
+2. ˙φ ∈ Φ2( ˙G∗);
+3. ˙φv ∈ Φ2,bdd( ˙G∗
+v) for v ∈ {v1, v2};
+4. the canonical map S ˙φ → S ˙φv is isomorphic for v ∈ {u, v1}.
+Proof. A totally real number field ˙F and a place u such that ˙Fu = F are given by Lemma 6.1. Here we take ˙F to
+have more than two real places. Let v1 and v2 be finite and real places of ˙F, respectively. Put ˙G∗ = SO2n+1 over ˙F.
+For each i = 1, . . . , r, let G∗
+i ∈ �Esim(Ni) be a classical group over F such that φi ∈ �Φsim,bdd(G∗
+i ), which is given
+in the 1st seed theorem 3.5. Take ˙G∗
+i ∈ �Esim(Ni) to be a simple twisted endoscopic group over ˙F so that ˙G∗
+i,u = G∗
+i .
+Choose a collection φv1,i ∈ �Φsim,bdd( ˙G∗
+i,v1) (i = 1, . . . , r) of pairwise distinct parameters. Choose also a collection
+φv2,i ∈ �Φ2,bdd( ˙G∗
+i,v2) (i = 1, . . . , r) of parameters such that φv2,i and φv2,j do not have a common constituent for all
+i ̸= j. Then Lemma 6.7 gives us a collection of parameters ˙φi ∈ �Φsim( ˙G∗
+i ) such that ˙φi,u = φi, ˙φi,v1 = φv1,i, and
+˙φi,v2 = φv2,i.
+By the assumption, we have ei = 0 and δi = 1 for all i. Put
+˙φ = ˙φ1 ⊞ · · · ⊞ ˙φr ∈ Φ( ˙G∗).
+Then by the construction the canonical map S ˙φ → S ˙φv is isomorphic for v ∈ {u, v1}. Thus the fourth and second
+conditions are satisfied. The first condition is clearly satisfied. The third one follows immediately from the construction.
+Proposition 6.10. Assume that M ∗ = G∗, and φ ∈ Φ2(G∗) is relevant for G. Then there exists a global data
+( ˙F, ˙G∗, ˙G, ˙φ, u, v1, v2),
+where ˙F is a totally real number field, ˙G∗ = SO2n+1 over ˙F, ˙G an inner form of ˙G∗, ˙φ ∈ Φ2( ˙G∗) a parameter relevant
+for ˙G, and u, v1, v2 are places of ˙F, such that v1 is a finite place, v2 is a real place, and
+1.
+˙Fu = F, ˙G∗
+u = G∗, ˙Gu = G, and ˙φu = φ;
+2.
+˙Gv is split unless v ∈ {u, v2};
+3. ˙φv ∈ Φ2,bdd( ˙G∗
+v) for v ∈ {v1, v2};
+4. the canonical map S ˙φ → S ˙φv is isomorphic for v ∈ {u, v1}.
+Proof. By Lemma 6.9, we obtain ˙F, ˙G∗, ˙φ, u, v1, and v2 satisfying the third and fourth conditions. If G is not split
+over F, then by Lemma 6.3, we obtain ˙G that satisfies the second condition for which ˙φ is relevant. The first condition
+is now clear. If G splits over F, then ˙G = SO2n+1 clearly satisfies the conditions.
+Now we treat the case that M ∗ is proper, which implies that φ /∈ Φ2(G∗). At first we consider the case that M ∗
+is not linear excluding the special cases.
+Lemma 6.11. Assume that M ∗ ⊊ G∗ is not linear.
+Assume also that (n, φ) is neither (2, of type (exc1)) nor
+(3, of type (exc2)).
+Then there exists a global data ( ˙F, ˙G∗, ˙M ∗, ˙φ, ˙φ ˙
+M∗, u, v1, v2), where
+˙F is a totally real number
+field, ˙G∗ = SO2n+1 over ˙F,
+˙M ∗ ⊂ ˙G∗ a Levi subgroup over ˙F, ˙φ ∈ Φ( ˙G∗), ˙φ ˙
+M∗ ∈ Φ2( ˙M ∗), and u, v1, v2 are places of
+˙F, such that v1 and v2 are finite places and
+1.
+˙Fu = F, ˙G∗
+u = G∗,
+˙M ∗
+u = M ∗, ˙φu = φ, and ˙φ ˙
+M∗,u = φM∗;
+2. if φ ∈ Φ2
+ell(G∗) (resp. Φexc1(G∗), resp. Φexc2(G∗)), then ˙φ ∈ Φ2
+ell( ˙G∗) (resp. Φexc1( ˙G∗), resp. Φexc2( ˙G∗));
+57
+
+3. ˙φv1 ∈ Φbdd( ˙G∗
+v1) and ˙φ ˙
+M∗,v1 ∈ Φ2,bdd( ˙M ∗
+v1);
+4. ˙φv2 ∈ Φell,exc
+bdd
+( ˙G∗
+v2) and it has a symplectic simple component with odd multiplicity;
+5. the canonical maps S ˙φ → S ˙φv and S ˙φ ˙
+M∗ → S ˙φ ˙
+M∗,v are isomorphic for v ∈ {u, v1}.
+Proof. A totally real number field ˙F and a place u such that ˙Fu = F are given by Lemma 6.1. Let v1 and v2 be finite
+places of ˙F. Put ˙G∗ = SO2n+1 and
+˙M ∗
+− = SO2n0+1 over ˙F. Let
+˙M ∗ ⊂ ˙G∗ be a Levi subgroup over ˙F such that
+˙M ∗ ≃ GLe1
+N1 × · · · × GLer
+Nr × ˙M ∗
+−.
+For each i = 1, . . . , r, let G∗
+i ∈ �Esim(Ni) over F and ˙G∗
+i ∈ �Esim(Ni) over ˙F be as in the proof of Lemma 6.9. Choose
+a collection φv1,i ∈ �Φsim,bdd( ˙G∗
+i,v1), (i = 1, . . . , r) of pairwise distinct parameters.
+Since M ∗ is not linear, we have n0 ≥ 1, which implies that δi = 1 (i.e., ℓi is odd) for some i. For such i, φi must
+be symplectic, in particular Ni ≥ 2.
+Suppose that there exists 1 ≤ s ≤ r such that δs = 1, Ns = 2, and Nt = 1 for all t ̸= s. In this case ℓs is odd and
+ℓt is even for t ̸= s. We may assume s = 1. If φ is elliptic, it can be written as
+φ = φ1 ⊕ 2φq+1 ⊕ · · · ⊕ 2φr,
+where φi are all symplectic, in particular Ni ≥ 2 for all i. This leads to r = 1 and φ = φ1, which contradicts the
+assumption that M ∗ is proper. If φ is of type (exc1), it can be written as
+φ = φ1 ⊕ 2φ2,
+where φ1 is symplectic, and φ2 is orthogonal. Then the dimension of φ equals 2 + 2 × 1 = 4, which contradicts the
+assumption that (n, φ) is not (2, of type (exc1)). If φ is of type (exc2), it can be written as
+φ = 3φ1,
+where φ1 is symplectic. Then the dimension of φ equals 3 × 2 = 6, which contradicts the assumption that (n, φ) is not
+(3, of type (exc2)).
+Therefore, we have that either there exists 1 ≤ s ≤ r such that Ns ≥ 3, or there exists 1 ≤ s ≤ r such that Ns = 2
+and δt = 1 for some t ̸= s. In both cases, by Lemma 6.8, we have ˙φs ∈ �Φsim( ˙G∗
+s) such that ˙φs,u = φs, ˙φs,v1 = φv1,s,
+and ˙φs,v2 is of the form
+˙φs,v2 = φv2+ ⊕ ˙φ∨
+v2+ ⊕ φv2−,
+where φv2+ ∈ Φbdd(GL1) is a non-self-dual parameter, and φv2− ∈ �Φbdd(Ns − 2) a simple self-dual parameter. For
+i ̸= s, choose φv2,i ∈ �Φsim,bdd( ˙G∗
+i ) so that {φv2,i}i̸=s are mutually distinct and φv2,i ̸= φv2− for all i. Lemma 6.7 gives
+us ˙φi ∈ �Φsim( ˙G∗
+i ) such that ˙φi,u = φi, ˙φi,v1 = φv1,i, and ˙φi,v2 = φv2,i.
+Put
+˙φ = ℓ1 ˙φ1 ⊞ · · · ⊞ ℓr ˙φr ∈ Φ( ˙G∗),
+˙φ ˙
+M∗ = e1 ˙φ1 ⊞ · · · ⊞ er ˙φr ⊞ ˙φ− ∈ Φ2( ˙M ∗),
+where ˙φ− = ⊞iδi ˙φi. The first and fifth conditions immediately follow from the construction, and hence so do the
+second and third ones.
+Let us consider the fourth condition. We have
+˙φv2 = ℓsφv2− ⊕ ℓs(φv2+ ⊕ φ∨
+v2+) ⊕
+�
+i̸=s
+ℓiφv2,i,
+which is neither elliptic nor exceptional, because φv2+ is not self-dual. If Ns ≥ 3, then ˙φv2 has a symplectic simple
+component with odd multiplicity because φ has. If Ns = 2 and δt = 1 for t ̸= s, then clearly φv2,t is a symplectic
+simple component with odd multiplicity. This completes the proof.
+58
+
+Proposition 6.12. Assume that M ∗ ⊊ G∗ is not linear. Assume also that (n, φ) is neither (2, of type (exc1)) nor
+(3, of type (exc2)). Then there exists a global data
+( ˙F, ˙G∗, ˙G, ˙φ, ˙M ∗, ˙φ ˙
+M∗, u, v1, v2),
+where ˙F is a totally real number field, ˙G∗ = SO2n+1 over ˙F, ˙G an inner form of ˙G∗, ˙φ ∈ Φ( ˙G∗) a parameter,
+˙M ∗ ⊂ ˙G∗
+a Levi subgroup, ˙φ ˙
+M∗ ∈ Φ2( ˙M ∗) a parameter whose image in Φ( ˙G∗) is ˙φ, and u, v1, v2 are places of ˙F, such that v1
+and v2 are finite place, and
+1.
+˙Fu = F, ˙G∗
+u = G∗
+u, ˙Gu = G, ˙φu = φ,
+˙M ∗
+u = M ∗, and ˙φ ˙
+M∗,u = φM∗;
+2.
+˙Gv is split over ˙Fv unless v ∈ {u, v2};
+3. if φ ∈ Φ2
+ell(G∗) (resp. Φexc1(G∗), resp. Φexc2(G∗)), then ˙φ ∈ Φ2
+ell( ˙G∗) (resp. Φexc1( ˙G∗), resp. Φexc2( ˙G∗));
+4. ˙φv1 ∈ Φbdd( ˙G∗
+v1) and ˙φ ˙
+M∗,v1 ∈ φ2,bdd( ˙M ∗
+v1);
+5. ˙φv2 ∈ Φell,exc
+bdd
+( ˙G∗
+v2) is relevant for ˙Gv2;
+6. the canonical maps S ˙φ → S ˙φv and S ˙φ ˙
+M∗ → S ˙φ ˙
+M∗,v are isomorphic for v ∈ {u, v1}.
+Proof. By Lemma 6.11, we obtain ˙F, ˙G∗, ˙φ, ˙M ∗, ˙φ ˙
+M∗, u, v1, and v2 satisfying the third, fourth, and sixth conditions.
+By Lemma 6.3, we obtain ˙G satisfying the second condition. Since v2 is a finite place, the fifth condition follows from
+the fourth condition of Lemma 6.11, regardless of ˙G. The first condition is now clear.
+Next we shall consider the special cases.
+Lemma 6.13. Assume that M ∗ ⊊ G∗ is not linear. Assume also that n = 2 or 3. Let ♥ = (exc1) (resp. (exc2)) if
+n = 2 (resp. 3), and assume that φ is of type ♥. Then there exists a global data ( ˙F, ˙G∗, ˙M ∗, ˙φ, ˙φ ˙
+M∗, u, v1, v2), where
+˙F is a totally real number field, ˙G∗ = SO2n+1 over ˙F,
+˙M ∗ ⊂ ˙G∗ a Levi subgroup over ˙F, ˙φ ∈ Φ( ˙G∗), ˙φ ˙
+M∗ ∈ Φ2( ˙M ∗),
+and u, v1, v2 are places of ˙F, such that v1 is a finite place, v2 is a real place, and
+1.
+˙Fu = F, ˙G∗
+u = G∗,
+˙M ∗
+u = M ∗, ˙φu = φ, and ˙φ ˙
+M∗,u = φM∗;
+2. ˙φ ∈ Φ♥( ˙G∗);
+3. ˙φv1 ∈ Φbdd( ˙G∗
+v1) and ˙φ ˙
+M∗,v1 ∈ Φ2,bdd( ˙M ∗
+v1);
+4. ˙φv2 = 2ω0 ⊕ τ1 or 3τ1;
+5. the canonical maps S ˙φ → S ˙φv and S ˙φ ˙
+M∗ → S ˙φ ˙
+M∗,v are isomorphic for v ∈ {u, v1}.
+Proof. A totally real number field ˙F and a place u such that ˙Fu = F are given by Lemma 6.1. Let v1 be a finite place
+and v2 a real place of ˙F. Put ˙G∗ = SO2n+1 over ˙F.
+Consider first the case when n = 2 and ♥ = (exc1). By the assumption we have ˙G∗ = SO5, φ = 2φ1 ⊕ φ2,
+Sφ ≃ Sp(2, C) × O(1, C), and M ∗ ≃ GL1 × SO3, where φ1 is 1-dimensional orthogonal and φ2 is irreducible 2-
+dimensional symplectic. We also have N1 = 1 and N2 = 2. For i = 1, 2, choose ( ˙G∗
+i , si, ηi) ∈ �E(Ni) so that φi ∈ Φ( ˙G∗
+i )
+if we regard Φ( ˙G∗
+i ) as a subset of Φ(Ni) via ηi. Concretely, ˙G∗
+1 = Sp0 and ˙G∗
+2 = SO3. Let φv1,i be an element of
+Φsim,bdd( ˙G∗
+i ), for i = 1, 2. Put φv2,1 = ω0 and φv2,2 = τ1. Then Lemma 6.7 gives us a global parameter ˙φi ∈ Φsim( ˙G∗
+i )
+such that ˙φi,u = φi, ˙φi,v1 = φv1,i, and ˙φi,v2 = φv2,i. Put
+˙φ = 2 ˙φ1 ⊞ ˙φ2,
+˙M ∗ = GL1 × SO3,
+˙φ ˙
+M∗ = ˙φ1 ⊞ ˙φ2.
+Consider next the case when n = 3 and ♥ = (exc2). By the assumption we have ˙G∗ = SO7, φ = 3φ1, Sφ ≃ O(3, C),
+and M ∗ ≃ GL2 × SO3, where φ1 is irreducible 2-dimensional symplectic. Hence we regard φ1 ∈ Φ(SO3 /F). Let φv1,1 be
+59
+
+an element of Φsim,bdd(SO3 / ˙Fv1), and put φv2,1 = τ1. Then Lemma 6.7 gives us a global parameter ˙φ1 ∈ Φsim(SO3 / ˙F)
+such that ˙φ1,u = φ1, ˙φ1,v1 = φv1,1, and ˙φ1,v2 = φv2,1. Put
+˙φ = 3 ˙φ1,
+˙M ∗ = GL2 × SO3,
+˙φ ˙
+M∗ = ˙φ1 ⊞ ˙φ1.
+In both cases, the conditions follow from the construction. This completes the proof.
+Proposition 6.14. Assume that M ∗ ⊊ G∗ is not linear. Assume also that n = 2 or 3. Let ♥ = (exc1) (resp. (exc2))
+if n = 2 (resp. 3). Assume that φ ∈ Φbdd,♥(G∗) and that G is not quasi-split. Then there exists a global data
+( ˙F, ˙G∗, ˙G, ˙φ, ˙M ∗, ˙φ ˙
+M∗, u, v1, v2),
+where ˙F is a totally real number field, ˙G∗ = SO2n+1 over ˙F, ˙G an inner form of ˙G∗, ˙φ ∈ Φ( ˙G∗) a parameter,
+˙M ∗ ⊂ ˙G∗
+a Levi subgroup, ˙φ ˙
+M∗ ∈ Φ2( ˙M ∗) a parameter whose image in Φ( ˙G∗) is ˙φ, and u, v1, v2 are places of ˙F, such that v1
+is a finite place, v2 is a real place, and
+1.
+˙Fu = F, ˙G∗
+u = G∗
+u, ˙Gu = G, ˙φu = φ,
+˙M ∗
+u = M ∗, and ˙φ ˙
+M∗,u = φM∗;
+2.
+˙Gv is split over ˙Fv unless v ∈ {u, v2};
+3. ˙φ ∈ Φ♥( ˙G∗);
+4. ˙φv1 ∈ Φbdd( ˙G∗
+v1) and ˙φ ˙
+M∗,v1 ∈ φ2,bdd( ˙M ∗
+v1);
+5. ˙φv2 ∈ Φbdd,♥( ˙G∗
+v2) is relevant for ˙Gv2 and satisfies Theorem 4.13 relative to
+˙M ∗
+v2;
+6. the canonical maps S ˙φ → S ˙φv and S ˙φ ˙
+M∗ → S ˙φ ˙
+M∗,v are isomorphic for v ∈ {u, v1}.
+Proof. By Lemma 6.13, we obtain ˙F, ˙G∗, ˙φ, ˙M ∗, ˙φ ˙
+M∗, u, v1, and v2 satisfying the third, fourth, and sixth conditions.
+By Lemma 6.3, we obtain ˙G such that ˙Gu = G, ˙Gv2 ≃ SO(n − 1, n + 2), and ˙Gv is split if v /∈ {u, v2}. Hence the
+second condition is satisfied. The fifth condition follows from §4.9. The first condition is now clear.
+If M ∗ is linear and G is non-quasi-split, then M ∗ never transfer to G and φ is not relevant. The next proposition
+will be applied in such a case.
+Lemma 6.15. Assume that M ∗ ⊊ G∗ is proper. There exists a global data ( ˙F, ˙G∗, ˙M ∗, ˙φ, ˙φ ˙
+M∗, u1, u2, v1), where ˙F is
+a totally real number field, ˙G∗ = SO2n+1 over ˙F,
+˙M ∗ ⊂ ˙G∗ a Levi subgroup over ˙F, ˙φ ∈ Φ( ˙G∗), ˙φ ˙
+M∗ ∈ Φ2( ˙M ∗), and
+u1, u2, v1 are places of ˙F, such that v1 is a finite place and
+1.
+˙Fu = F, ˙G∗
+u = G∗,
+˙M ∗
+u = M ∗, ˙φu = φ, and ˙φ ˙
+M∗,u = φM∗, for u ∈ {u1, u2};
+2. if φ ∈ Φ2(G∗) (resp. Φ2
+ell(G∗), resp. Φexc1(G∗), resp. Φexc2(G∗)), then ˙φ ∈ Φ2( ˙G∗) (resp. Φ2
+ell( ˙G∗), resp.
+Φexc1( ˙G∗), resp. Φexc2( ˙G∗));
+3. ˙φv1 ∈ Φbdd( ˙G∗
+v1) and ˙φ ˙
+M∗,v1 ∈ Φ2,bdd( ˙M ∗
+v1);
+4. the canonical maps S ˙φ → S ˙φv and S ˙φ ˙
+M∗ → S ˙φ ˙
+M∗,v are isomorphic for v ∈ {u1, u2, v1}.
+Proof. A totally real field ˙F and places u1 and u2 such that ˙Fu1 = ˙Fu2 = F are given by Lemma 6.2. Let v1 be a
+finite place of ˙F. Put ˙G∗ = SO2n+1 and
+˙M ∗
+− = SO2n0+1 over ˙F. Let
+˙M ∗ ⊂ ˙G∗ be a Levi subgroup over ˙F such that
+˙M ∗ ≃ GLe1
+N1 × · · · × GLer
+Nr × ˙M ∗
+−.
+For each i = 1, . . . , r, let G∗
+i ∈ �Esim(Ni) be a classical group over F such that φi ∈ �Φsim,bdd(G∗
+i ), and take
+˙G∗
+i ∈ �Esim(Ni) to be a simple twisted endoscopic group over ˙F so that
+˙G∗
+i,u = G∗
+i for u ∈ {u1, u2}.
+Choose a
+60
+
+collection φv1,i ∈ �Φsim,bdd( ˙G∗
+i,v1) (i = 1, . . . , r) of pairwise distinct parameters. Then Lemma 6.7 gives us a collection
+of parameters ˙φi ∈ �Φsim( ˙G∗
+i ) such that ˙φi,v1 = φv1,i and ˙φi,u = φi for u ∈ {u1, u2}.
+Put
+˙φ = ℓ1 ˙φ1 ⊞ · · · ⊞ ℓr ˙φr ∈ Φ( ˙G∗),
+˙φ ˙
+M∗ = e1 ˙φ1 ⊞ · · · ⊞ er ˙φr ⊞ ˙φ− ∈ Φ2( ˙M ∗),
+where ˙φ− = ⊞iδi ˙φi. Then by the construction, the first and fourth conditions are satisfied. Hence the second and
+third ones follow.
+Proposition 6.16. Assume that M ∗ ⊊ G∗ is proper. Then there exists a global data
+( ˙F, ˙G∗, ˙G, ˙φ, ˙M ∗, ˙φ ˙
+M∗, u1, u2, v1),
+where ˙F is a totally real number field, ˙G∗ = SO2n+1 over ˙F, ˙G an inner form of ˙G∗, ˙φ ∈ Φ( ˙G∗) a parameter,
+˙M ∗ ⊂ ˙G∗
+a Levi subgroup, ˙φ ˙
+M∗ ∈ Φ2( ˙M ∗) a parameter whose image in Φ( ˙G∗) is ˙φ, and u1, u2, v1 are places of ˙F, such that v1
+is a finite place and
+1.
+˙Fu = F, ˙G∗
+u = G∗
+u, ˙Gu = G, ˙φu = φ,
+˙M ∗
+u = M ∗, and ˙φ ˙
+M∗,u = φM∗, for u ∈ {u1, u2};
+2.
+˙Gv is split over ˙Fv unless v ∈ {u1, u2};
+3. if φ ∈ Φ2(G∗) (resp. Φ2
+ell(G∗), resp. Φexc1(G∗), resp. Φexc2(G∗)), then ˙φ ∈ Φ2( ˙G∗) (resp. Φ2
+ell( ˙G∗), resp.
+Φexc1( ˙G∗), resp. Φexc2( ˙G∗));
+4. ˙φv1 ∈ Φbdd( ˙G∗
+v1) and ˙φ ˙
+M∗,v1 ∈ Φ2,bdd( ˙M ∗
+v1);
+5. the canonical maps S ˙φ → S ˙φv and S ˙φ ˙
+M∗ → S ˙φ ˙
+M∗,v are isomorphic for v ∈ {u1, u2, v1}.
+Proof. By Lemma 6.15, we obtain ˙F, ˙G∗, ˙φ,
+˙M ∗, ˙φ ˙
+M∗ u1, u2, and v1 satisfying the first (except the assertion on ˙Gu),
+third, fourth, and fifth conditions. By Lemma 6.3, we obtain ˙G such that ˙Gv = G for v ∈ {u1, u2} and ˙Gv splits over
+˙Fv otherwise. This completes the proof.
+6.3
+On elliptic parameters
+The following lemma is proved in the same way as [18, Lemma 4.5.1].
+Lemma 6.17. Let ˙F be a number field, ˙G∗ = SO2n+1 over ˙F, ˙φ ∈ Φ2
+ell( ˙G∗) a parameter, and ˙G an inner form of ˙G∗.
+Let
+˙M ∗ ⊂ ˙G∗ be a Levi subgroup and ˙φ ˙
+M∗ ∈ Φ2( ˙M ∗) a discrete parameter whose image in Φ( ˙G∗) is ˙φ. Assume that
+there is a place v1 of ˙F such that
+•
+˙Gv1 is split over ˙F;
+• ˙φv1 ∈ Φbdd( ˙G∗
+v1) and ˙φ ˙
+M∗
+v1 ∈ φ2,bdd( ˙M ∗
+v1);
+• the canonical maps S ˙φ → S ˙φv1 and S ˙φ ˙
+M∗ → S ˙φ ˙
+M∗,v1 are isomorphic.
+Then we have
+tr R
+˙G
+disc, ˙φ( ˙f) =
+�
+x∈S ˙φ,ell
+Ä ˙f ′
+˙G( ˙φ, x) − ˙f ˙G( ˙φ, x)
+ä
+= 0,
+for all ˙f ∈ H( ˙G).
+61
+
+6.4
+Proof of LIR for L-parameters
+In this subsection we complete the proof of Theorem 4.13 for generic parameters (i.e., L-parameters). Let F be a
+local field, G∗ = SO2n+1 over F, (M ∗, P ∗) a standard parabolic pair of G∗, and ξ : G∗ → G an inner twist of G∗. If
+M ∗ transfers to G, then we take ξ so that M = ξ(M ∗) is defined over F. Let φM∗ ∈ Φbdd(M ∗) be a generic parameter
+for M, and φ ∈ Φbdd(G∗) its image. If G is split, the theorem is already proven by Arthur [7]. Therefore, we may
+assume that G is non-quasi-split, so in particular F ̸= C.
+Lemma 6.18. Assume that M ∗ ⊊ G∗ is proper, i.e., φ is not discrete. Assume also that φM∗ ∈ Φ2,bdd(M ∗) is
+discrete and that φ is elliptic or exceptional. Then for any x ∈ Sφ,ell, there exists a lift x ∈ Sφ of x such that
+f ′
+G(φ, x−1) = e(G)fG(φ, x),
+for any f ∈ H(G).
+Proof. The proof is similar to that of [18, Case N even of Lemma 4.6.1]. The difference is that we divide the case into
+the following cases:
+• M ∗ is not linear and (n, φ) is neither (2, of type (exc1)) nor (3, of type (exc2));
+• M ∗ is not linear and (n, φ) is either (2, of type (exc1)) or (3, of type (exc2));
+• M ∗ is linear,
+instead of the division into the cases
+• M ∗ is not linear and N ̸= 4;
+• M ∗ is not linear and N = 4;
+• M ∗ is linear,
+in loc. cit., and that we appeal to Lemmas 6.17, 5.13, 5.10, and Propositions 6.12, 6.14 6.16, instead of Lemmas 4.5.1,
+3.7.1, 3.5.10, and Propositions 4.4.4, 4.4.6, 4.4.7 of loc. cit., respectively.
+Lemma 6.19. The assertions 2 and 3 of Theorem 4.13 hold for generic parameters.
+Proof. Note that the assertion 3 implies the assertion 2. Thanks to §4.7 and §4.8, we may assume that φM∗ is discrete
+and that φ ∈ Φ2
+ell(G∗) ⊔ Φexc(G∗). By the consequence of §4.8, if φ is elliptic, then it remains to show that
+f ′
+G(φ, s−1) = e(G)fG(φ, u♮) for any u♮ ∈ Nφ and s ∈ Sφ,ss mapping to the same element in Sφ,ell,
+and if φ is exceptional, then it remains to show that
+f ′
+G(φ, s−1) = e(G)fG(φ, u♮) for any u♮ ∈ Nφ,reg and s ∈ Sφ,ss mapping to the same element in Sφ,ell.
+They can be proved similarly to [18, Lemma 4.6.2]. The difference is that we utilize Lemmas 4.20, 6.18, and 4.14
+instead of Lemmas 2.8.6, 4.6.1, and 2.6.5 of loc. cit. respectively. Note that in our case the equivalence classes of
+inner forms, inner twists, and pure inner twists are in bijection naturally.
+This completes the proof of the second and third assertions of Theorem 4.13 for generic parameters. The next two
+lemmas show the first assertion. The first lemma treats the case of φM∗ ∈ Φ2,bdd(M ∗):
+Lemma 6.20. Assume that φM∗ ∈ Φ2,bdd(M ∗). Then the assertion 1 of Theorem 4.13 holds for φM∗.
+Proof. By Lemmas 4.21 and 4.19, we may assume that φ ∈ Φexc(G∗). The proof is similar to that of [18, Case N
+even of Lemma 4.6.3]. The difference is that we appeal to Lemmas 6.16, 5.13, 6.20, and 6.19 instead of Lemmas 4.4.7,
+3.7.1, 4.6.3, and 4.6.2 of loc. cit. respectively.
+Recall that in §4.7 we reduce only part 2 and 3 of Theorem 4.13 to the case of discrete parameters. The next
+lemma is the reduction of part 1, and hence it completes the proof of part 1 for all generic parameters. As the lemmas
+above, in a similar way to the equation (4.6.3), Lemma 4.6.4, and Lemma 4.6.5 of loc. cit., one can obtain a surjection
+Rφ(M, G) = Wφ(M, G)/W ◦
+φ(M, G) ։ RπM (M, G) := WπM (M, G)/W ◦
+πM (M, G),
+(6.1)
+for πM ∈ ΠφM∗(M), and can show the following lemmas.
+62
+
+Lemma 6.21. Let φM∗ ∈ Φbdd(M ∗). Then the assertion 1 of Theorem 4.13 holds for φM∗.
+Lemma 6.22. The homomorphism (6.1) is bijective if φM∗ is relevant.
+Now Theorem 4.13 holds for any generic parameters. This completes the proof of LIR for generic parameters.
+6.5
+The construction of L-packets
+Theorem 3.11 for generic parameters (i.e., L-parameters) can be proven in the same way as Theorem 1.6.1 of [18].
+Let us roughly review the procedure. See [18, §§4.7-4.9] for detail.
+Let F be a local field, G∗ = SO2n+1 over F, and (ξ, z) : G∗ → G a pure inner twist of G∗ as in §4.4. If G is split,
+the theorem is already proven by Arthur [7]. Therefore, we assume that G is non-quasi-split, so in particular F ̸= C.
+In the archimedean case, the theorem is known by Langlands and Shelstad, so we assume that F is a p-adic field.
+First consider the non-discrete parameters. Since we now have Theorem 4.13 for generic parameters φ, the local
+packets Πφ(G) and the map Πφ(G) → Irr(Sφ, χG) for φ ∈ Φbdd(G∗) \ Φ2(G∗) have already constructed at the
+end of §4.6. The remaining assertions are that the map is bijective and that the packets are disjoint and exhaust
+Πtemp(G) \ Π2(G).
+The general classification [6] (cf. [7, §3.5], [18, §4.7]) of Πtemp(G) by harmonic analysis says that there is a bijective
+correspondence
+(M, σ, µ) �→ πµ ∈ Πtemp(G),
+from the G(F)-orbits of triples consisting of a Levi subgroup M ⊂ G over F, a discrete series representation σ
+of M(F), and an irreducible representation µ of the representation-theoretic R-group R(σ) = Rσ(M, G).
+Here,
+we do not need an extension �R(σ) of the R-group, as explained in [7, p.158] or [18, §4.7].
+The correspondence
+can be described explicitly as follows. For a triple (M, σ, µ), let P be a parabolic subgroup of G with Levi factor
+M. Choose a family {RP (r, σ)}r∈Rσ(M,G) ⊂ AutG(F )(IG
+P (σ)) of self-intertwining operators so that an assignment
+r �→ RP (r, σ) is a homomorphism from Rσ(M, G) to AutG(F )(IG
+P (σ)). Then we have a representation RP (r, σ)◦IG
+P (σ, g)
+of Rσ(M, G)×G(F) on HG
+P (σ), for which we shall write R. The representation πµ is characterized by a decomposition
+R =
+�
+µ∈Irr(Rσ(M,G))
+µ∨ ⊗ πµ.
+Note that Rσ(M, G) is abelian in our case.
+Then the bijectivity of the map Πφ(G) → Irr(Sφ, χG) and the disjointness and exhaustion of L-packets in Πtemp(G)
+follows from the same argument as in [18, §4.7], which we omit.
+Proposition 6.23. Theorem 3.11 holds for generic parameters φ ∈ Πbdd(G∗) \ Π2(G∗).
+Proof. The proof is similar to that of [18, Proposition 4.7.1]. Note that Πφ(G, Ξ), χΞ, S♮
+φ, N ♮
+φ, and S♮♮
+φ (M) in loc. cit.
+should be replaced by Πφ(G), χG, Sφ, Nφ, and Sφ(M) in our case, respectively.
+Next, before consider discrete parameters, we need some preparation. Put Tell(G) to be the set of G(F)-conjugacy
+classes of triples τ = (M, σ, r), where M ⊂ G is a Levi subgroup, σ a unitary discrete series representation of M(F),
+and r ∈ Rσ(M, G) a regular element. Here, r ∈ Rσ(M, G) is said to be regular if ar=1
+M
+:= {λ ∈ aM | rλ = λ} coincides
+with aG. The set Π2,temp(G) is naturally regarded as a subset of Tell(G) via π �→ (G, π, 1). The complement set will
+be denoted by T 2
+ell(G). In general, the trace Paley-Wiener theorem tells that ”orbital integrals” of cuspidal functions
+f ∈ Hcusp(G) are described by the functions on Tell(G) given by some intertwining operators.
+In order to utilize the local intertwining operator defined in §4.5 , we need to consider the set T ♮
+ell(G). Let T 2,♮
+ell (G) be
+the set of G(F)-conjugacy classes of triples τ ♮ = (M, σ, s), where M ⊊ G is a proper Levi subgroup, σ a unitary discrete
+series representation of M(F), and s ∈ Sφσ(G) an element whose image under the surjection Sφσ(G) ։ Rσ(M, G)
+is regular. Here, we write φσ for the L-parameter of σ, and we also write φσ for its image in Φbdd(G∗). Then put
+T ♮
+ell(G) := Π2,temp(G)⊔T 2,♮
+ell (G). The surjection Sφσ(G) ։ Rσ(M, G) induces the natural surjection T ♮
+ell(G) ։ Tell(G).
+For f ∈ H(G), put
+fG(τ ♮) =
+®
+tr �RP (u♮, σ, φσ, ψF ) ◦ IG
+P (σ, f)� ,
+for τ ♮ = (M, σ, s) ∈ T 2,♮
+ell (G),
+tr (π(f)) ,
+for τ ♮ = π ∈ Π2,temp(G),
+63
+
+where u♮ ∈ Nφσ(M, G) is a lift of s. It is independent of the choice of u♮. If τ ♮
+1 and τ ♮
+2 in T 2,♮
+ell (G) maps to a same
+element in Tell(G), then they are of the form τ ♮
+1 = (M, σ, s) and τ ♮
+2 = (M, σ, ys) for some y ∈ Sφσ(M). Then we shall
+write τ ♮
+2 = yτ ♮
+1. In this case, by Lemma 4.10 we have
+fG(τ ♮
+1) = ⟨y, σ⟩fG(τ ♮
+2).
+Let φ ∈ Φ2,bdd(G∗) be a discrete generic parameter, and x ∈ Sφ. Then by Lemma 3.8, one can construct an
+endoscopic triple e = (Ge, se, ηe) of G and a generic parameter φe of Ge, to obtain a linear form H(G) ∋ f �→
+f ′
+G(φ, x) := f e(φe). The trace Paley-Wiener theorem implies that there exists a family {cφ,x(τ ♮)}τ ♮∈T ♮
+ell(G) ⊂ C of
+complex numbers such that for any f ∈ Hcusp(G), we have
+f ′
+G(φ, x) = e(G)
+�
+τ∈Tell(G)
+cφ,x(τ ♮)fG(τ ♮),
+(6.2)
+and
+cφ,x(τ ♮) = ⟨y, τ⟩cφ,x(yτ ♮),
+where τ ♮ is a lift of τ, and ⟨y, τ⟩ = ⟨y, σ⟩ for τ = (M, σ, s) ∈ T 2
+ell(G). Note that the product cφ,x(τ ♮)fG(τ ♮) does not
+depend on the choice of τ ♮. Then one can show an orthogonality relation:
+Proposition 6.24. (a) Let φ ∈ Φ2,bdd(G∗) be a discrete generic parameter, and x ∈ Sφ. Then cφ,x(τ ♮) = 0 for all
+τ ♮ ∈ T 2,♮
+ell (G).
+(b) Let φ1, φ2 ∈ Φ2,bdd(G∗) be discrete generic parameters, and xi ∈ Sφi (i = 1, 2). Then
+�
+π∈Π2,temp(G)
+cφ1,x1(π)cφ2,x2(π) =
+®
+|Sφ1|,
+if φ1 = φ2 and x1 = x2,
+0,
+otherwise.
+Proof. The proof is similar to that of [18, Proposition 4.8.3] or [7, Lemma 6.5.3].
+Finally we shall consider discrete generic parameters, after introducing one lemma.
+Lemma 6.25. Let ξ : ˙G∗ → ˙G be an inner twist of ˙G∗ = SO2n+1 over a number field ˙F, and ˙φ ∈ Φ2( ˙G∗) a discrete
+global parameter. For any ˙f ∈ H( ˙G), we have
+�
+˙π
+n ˙φ( ˙π) ˙f ˙G( ˙π) =
+1
+|S ˙φ|
+�
+x∈S ˙φ
+˙f ′
+˙G( ˙φ, x),
+(6.3)
+where ˙π runs over irreducible representations of ˙G(A ˙F ), and n ˙φ( ˙π) it the multiplicity of ˙π in (R ˙G
+disc, ˙φ, L2
+disc, ˙φ( ˙G( ˙F)\ ˙G(A ˙F ))).
+Proof. The proof is similar to that of [18, Lemma 4.9.1] or [7, (6.6.6)].
+Let φ ∈ Φ2,bdd(G∗). Then Proposition 6.10 and Lemma 6.3 gives us a global data ( ˙F, ˙G∗, ˙G, ˙φ, u, v1, v2) such that
+˙F is totally real, and
+• v1 is a finite place, and v2 is a real place;
+•
+˙Fu = F, ˙G∗
+u = G∗, ˙Gu = G, and ˙φu = φ;
+•
+˙Gv is split for v /∈ {u, v2}, and χG = χ ˙Gv2;
+• ˙φv ∈ Φ2,bdd( ˙G∗
+v) for v ∈ {v1, v2};
+• the canonical map S ˙φ → S ˙φv is isomorphic for v ∈ {u, v1}.
+64
+
+Choose ˙f ∈ H( ˙G) so that ˙f is of the form ˙f = �
+v ˙fv, and that ˙fu ∈ Hcusp(G). Then we have the equation (6.3)
+due to Lemma 6.25. For v /∈ {u, v2}, the group ˙Gv is split and hence we have ECR (3.7). In the case v = v2, since
+˙Fv2 = R, ECR is known by the work of Shelstad. At the place u, we have the equation (6.2) since ˙fu is cuspidal.
+Substituting them, we obtain the following equation from (6.3):
+�
+˙π∈Πunit( ˙G(A ˙
+F ))
+n ˙φ( ˙π) ˙f ˙G( ˙π) =
+1
+|S ˙φ|
+�
+˙πu∈Π ˙φu
+�
+π∈Π2,temp(G)
+�
+x∈S ˙φ
+⟨ ˙xu, ˙πu⟩cφ, ˙xu(π) ˙f ˙G( ˙πu ⊗ π),
+where Πunit( ˙G(A ˙F )) denotes the set of isomorphism classes of irreducible unitary representations of ˙G(A ˙F ), ˙x ∈ S ˙φ
+is a lift of x ∈ S ˙φ, ˙xv ∈ S ˙φv is the image of ˙x, and
+Π ˙φu :=
+
+
+ ˙πu =
+�
+v̸=u
+˙πv
+������
+˙πv ∈ Π ˙φv, ⟨−, ˙πv⟩ = 1 for almost all v
+
+
+ ,
+˙xu := ( ˙xv)v̸=u ∈
+�
+v̸=u
+S ˙φv,
+⟨ ˙xu, ˙πu⟩ :=
+�
+v̸=u
+⟨ ˙xv, ˙πv⟩.
+For each place v other than u, v1 and v2, fix ˙πv ∈ Π ˙φv such that ⟨−, ˙πv⟩ is trivial. For a real place v2, fix ˙πv2 ∈ Π ˙φv2
+arbitrarily. They exist because ˙Gv splits if v /∈ {u, v1, v2} and ˙Fv2 = R.
+For any µ ∈ Irr(Sφ, χG), choose ˙πv1 ∈ Π ˙φv1 so that µ( ˙xu)⟨ ˙xu, ˙πu⟩ = 1 for all ˙x ∈ S ˙φ, where ˙πu = �
+v̸=u ˙πv. Such
+˙πv1 does exist because we have χG = χ ˙Gv2, v1 is a finite place, ˙Gv1 is split, and S ˙φ → S ˙φv is isomorphic for v ∈ {u, v1}.
+Then for π ∈ Π2,temp(G), we set
+nφ(µ, π) := n ˙φ ( ˙πu ⊗ π) .
+Then by the same argument of the proof of [18, Proposition 4.9.2], we have
+nφ(µ, π) =
+1
+|Sφ|
+�
+x∈Sφ
+µ(x)−1cφ,x(π),
+(6.4)
+for any π ∈ Π2,temp(G), where x ∈ Sφ denotes for an arbitrary representative of x. Moreover, the orthogonality
+relation (Proposition 6.24) and the formula (6.4) implies the following formula:
+Proposition 6.26. Let φ, φ′ ∈ Φ2,bdd(G∗) be discrete generic parameters, and µ ∈ Irr(Sφ, χG) and µ′ ∈ Irr(Sφ′, χG).
+Then we have
+�
+π∈Π2,temp(G)
+nφ(µ, π)nφ′(µ′, π) =
+®
+1,
+if (φ, µ) = (φ′, µ′),
+0,
+otherwise.
+Proof. The proof is the similar to that of [18, Proposition 4.9.3].
+Since nφ(µ, π) is a non-negative integer, Proposition 6.26 tells us that for any φ ∈ Φ2,bdd(G∗) and µ ∈ Irr(Sφ, χG),
+there exists a unique π ∈ Π2,temp(G) such that nφ(µ, π) = 1. We shall write π(φ, µ) for this representation π. Then
+an assignment (φ, µ) �→ π(φ, µ) gives an injective map. The packet of φ ∈ Φ2,bdd(G∗) is defined by
+Πφ := { π(φ, µ) | µ ∈ Irr(S, χG) } ,
+and the associated character of the component group is defined by
+⟨−, π(φ, µ)⟩ := µ.
+One can now easily see that the packets are disjoint and the map πφ → Irr(Sφ, χG) is bijective.
+Let φ ∈ Φ2,bdd(G∗) and x ∈ Sφ. Combining the formula (6.4) with Proposition 6.26, we have
+cφ,x(π) =
+®µ(x) = ⟨x, π⟩,
+if π = π(φ, µ),
+0,
+otherwise.
+Then the formula (6.2) is none other than the endoscopic character relation for cuspidal function f ∈ Hcusp(G).
+65
+
+Proposition 6.27. For general f ∈ H(G), we have ECR:
+f ′(φ, x) = e(G)
+�
+π∈Πφ
+⟨x, π⟩fG(π).
+Proof. The proof is similar to that of [7, Corollary 6.7.4].
+Proposition 6.27 characterizes the packet Πφ and the bijection Πφ → Irr(Sφ, χG). Thus they does not depend on
+the choice of ˙φ or ˙πu, but only on φ. By the argument between Theorems 3.11 and 3.12, we now have the packet Πφ
+and the bijection Πφ → Irr(Sφ, χG) for all generic parameters φ ∈ Φ(G∗).
+It remains to show that the packets exhaust Π2,temp(G). Let π ∈ Π2,temp(G). Take a globalization ˙π of π as in
+Lemma 6.4. By the decomposition (5.1), we have the A-parameter ˙φ of ˙π. In particular, we have n ˙φ( ˙π) ̸= 0. Since
+˙πv′ has sufficiently regular infinitesimal character for some real place v′, the parameter ˙φ is generic. Thus Theorem
+4.13 and Hypothesis 7.1 hold for ˙φ, so does Theorem 7.2 for ˙φ. (Note that §7 is independent from the exhaustion.)
+Hence we have π ∈ Π ˙φu. If ˙φu is not bounded (resp. not discrete), then the packet does not contain a tempered
+representation π by definition in §3.5 (resp. the beginning of this subsection). Therefore, we have ˙φu ∈ Φ2,bdd(G∗).
+This completes the proof of the local classification theorem for generic parameters.
+7
+The proof of the global theorem
+Assuming the existence of local A-packets and LIR, Theorem 3.13 can be proven in the same way as Theorem
+1.7.1 of [18]. In particular, since the assumptions hold for generic parameters, generic part of the theorem holds true.
+In this section we record the statements. See [18, §5] for the proof.
+Let F be a number field, G∗ = SO2n+1 over F, and G an inner form of G∗. If G is split, the theorem is already
+proven by Arthur [7]. Therefore, we assume that G is non-quasi-split. Recall from §5.1 (5.1) the decomposition
+L2
+disc(G(F)\G(AF )) =
+�
+ψ∈Ψ(G∗)
+L2
+disc,ψ(G(F)\G(AF )),
+tr Rdisc(f) =
+�
+ψ∈Ψ(G∗)
+Rdisc,ψ(f),
+f ∈ H(G).
+Let us introduce a hypothesis on local A-packets:
+Hypothesis 7.1. Let ψ ∈ Ψ(G∗) be a global parameter for G. We have the local packet Πψv = Πψv(Gv) and the map
+Πψv → Irr(Sψv, χGv) satisfying ECR (3.7) for all places v of F.
+Theorem 7.2. Let ψ ∈ Ψ(G∗) be a global parameter for G. Assume that Theorem 4.13 and Hypothesis 7.1 hold for
+ψ.
+1. If ψ /∈ Ψ2(G∗), then
+L2
+disc,ψ(G(F)\G(AF )) = 0.
+2. If ψ ∈ Ψ2(G∗), then
+L2
+disc,ψ(G(F)\G(AF )) =
+�
+π∈Πψ(G,εψ)
+π.
+Proof. The proof is similar to that of [18, Theorem 5.0.5].
+Since Theorem 4.13 and Hypothesis 7.1 hold true if ψ = φ ∈ Φ(G∗) is generic, the global classification theorem is
+now established for the generic part.
+66
+
+References
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+Congress of Mathematicians, Vol. 1, 2 (Z¨urich, 1994), 721731, Birkh¨auser, Basel, (1995).
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+68
+
diff --git a/29FLT4oBgHgl3EQfrS9R/content/tmp_files/load_file.txt b/29FLT4oBgHgl3EQfrS9R/content/tmp_files/load_file.txt
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@@ -0,0 +1,3320 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf,len=3319
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='12143v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='NT]  28 Jan 2023  The endoscopic classification of representations of non-quasi-split odd  special orthogonal groups  Hiroshi Ishimoto∗  [January 31, 2023]  Abstract  In an earlier book of Arthur, the endoscopic classification of representations of quasi-split orthogonal and  symplectic groups was established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Later Mok gave that of quasi-split unitary groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' After that, Kaletha, Minguez,  Shin, and White gave that of non-quasi-split unitary groups for generic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In this paper we prove the  endoscopic classification of representations of non-quasi-split odd special orthogonal groups for generic parameters,  following Kaletha, Minguez, Shin, and White.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Contents  1  Introduction  1  2  Odd special orthogonal groups  5  3  Notions around endoscopy and parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' and Main theorem  9  4  Local intertwining relation  22  5  The decomposition into near equivalence classes and the standard model  49  6  Globalizations and the proof of local classification  54  7  The proof of the global theorem  66  1  Introduction  Background & Main result  In a 2013 book [7],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Arthur established the endoscopic classification of irreducible representations of quasi-split  special orthogonal and symplectic groups over local fields of characteristic zero and of automorphic representations of  those groups over global fields of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' To local A-parameters he attached local A-packets characterized  by the endoscopic character relation (which we shall call ECR for short), and proved that the local A-packets for  generic parameters give the local Langlands correspondence (which we shall call LLC for short), and that automorphic  representations are classified in terms of automorphic cuspidal representations of general linear groups and local  A-packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let us roughly recall his result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let first F be a local field of characteristic zero, and G a split odd special orthogonal  group over F for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' A local A-parameter for G is a homomorphism  ψ : LF × SU(2, R) → LG,  ∗ishimoto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='hiroshi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='55m@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='com  1  with some conditions, where LF and LG denote the Langlands group of F and the L-group of G, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For every  A-parameter ψ, Arthur first constructed a multiset Πψ(G) over the set Πunit(G) of equivalence classes of irreducible  unitary representations of G(F), and a mapping Πψ(G) → Irr(π0(Sψ)), where Sψ denotes the centralizer of ψ in  the Langlands dual group “  G of G, and then he proved that they satisfy ECR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' An A-parameter is called a bounded  L-parameter if its restriction to SU(2, R) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' He also proved LLC by showing that if ψ is a bounded L-parameter  then the packet Πψ(G) gives the L-packet, which has been expected to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us recall here the basic form of LLC,  which is a conjecture in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1 (LLC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let G be a connected reductive algebraic group over a local field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let Πtemp(G) be the set  of equivalence classes of irreducible tempered representations of G(F), and Φbdd(G) the set of equivalence classes of  bounded L-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' There exists a canonical map  LL : Πtemp(G) −→ Φbdd(G),  such that for each φ ∈ Φbdd(G), the fiber Πφ(G) = LL−1(φ) is a finite set and there is an injective map ιφ : Πφ(G) →  Irr(π0(Sφ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' These two correspondences satisfy some interesting properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The finite set Πφ(G) is called the L-packet  of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We refer the reader to [17] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  On the other hand let next F be a number field, and G a split odd special orthogonal group over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' A global  A-parameter for G is a formal finite unordered sum  ψ =⊞  i  πi ⊠ νi,  with some conditions, where πi is an irreducible cuspidal automorphic representation of a general linear group, and  νi is an irreducible finite dimensional representation of SU(2, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For all place v of F, we have the localization  ψv = �  i φi,v ⊠ νi, where φi,v is the unique L-parameter for a general linear group over Fv corresponding to πi,v  via LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then ψv is a local A-parameter for Gv over Fv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' (Strictly speaking, it may not be an A-parameter, but a  packet for it is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=') Arthur determined an appropriate subset Πψ(εψ) of {�  v πv | πv ∈ Πψv(Gv)} and showed  the decomposition  L2  disc(G(F)\\G(AF )) =  �  ψ  L2  disc,ψ(G(F)\\G(AF )),  L2  disc,ψ(G(F)\\G(AF )) =  �  π∈Πψ(εψ)  π,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1)  of the discrete spectrum into near equivalence classes, and then into irreducible automorphic representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We shall  call a decomposition like (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1) ”Arthur’s multiplicity formula”, and abbreviate it as AMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Later, Mok [30] proved ECR, LLC, and AMF for quasi-split unitary groups by the similar argument, and Kaletha-  Minguez-Shin-White [18] partially proved those for non-quasi-split unitary groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In this paper, following [18], we  shall partially prove the analogous classification (ECR, LLC, and AMF) for non-quasi-split odd special orthogonal  groups by the similar argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In other words, our main theorem (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='14) is the following:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Over a local field of characteristic zero, LLC holds for any non-quasi-split odd special orthogonal  group, and the L-packets equipped with mappings ιφ satisfy ECR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Over a global field of characteristic zero, AMF holds  for any non-quasi-split odd special orthogonal group, except the irreducible decompositions of L2  disc,ψ(G(F)\\G(AF )) for  non-generic parameters ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The most important difference between this paper and [18] is the proof of the local intertwining relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The local  intertwining relation ([18, Theorem* 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2], Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 in this paper), for which we shall write LIR for short, is a  key theorem in the proof of the local main theorems ECR and LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By the similar idea to [18], we can reduce the  proof of LIR to that for two special cases: the real special orthogonal groups SO(1, 4) and SO(2, 5) relative to Levi  subgroups isomorphic to GL1 × SO(0, 3) and GL2 × SO(0, 3) respectively, and the special parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In the case of  SO(1, 4), the special representation of the Levi subgroup under consideration is the trivial representation, and hence  we can prove LIR by the similar argument to §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='9 in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' However, in the case of SO(2, 5), the situation is too  complicated to calculate similarly to loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', since the representation of the Levi subgroup is infinite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For this reason in this paper, we shall prove them by a completely different argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We choose a test function using  2  the Iwasawa decomposition, while the relative Bruhat decomposition was used in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The argument will appear  in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We remark that LLC of non-quasi-split odd special orthogonal groups has already studied by Mœglin-Renard [28],  but their result does not contain LIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Thus our local theorem is differentiated from their work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Application  In [7, Chapter 9], Arthur formulated the classification for non-quasi-split symmetric and orthogonal groups, and he  has the intention of proving it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The results of this paper are included in his project, but we have a different motivation,  the representation theory of the metaplectic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' It is one of the most important application of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us  recall the results on the metaplectic groups by Adams, Barbasch, Gan, Savin, and Ichino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let F be a local field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The metaplectic group, denoted by Mp2n(F), is a unique nonlinear two-fold cover of Sp2n(F)  except F = C, in which case Mp2n(C) = Sp2n(C) × {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We identify Mp2n(F) with Sp2n(F) × {±1} as sets, and  a representation π of Mp2n(F) is said to be genuine if π((1, −1)) is not trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let Πtemp(Mp2n) denote the set of  equivalence classes of genuine tempered irreducible representations of Mp2n(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Adams-Barbasch [2, 3], Adams [1]  (archimedean case), and Gan-Savin [12] (non-archimedean case) showed that the local theta correspondence gives a  canonical bijection  Πtemp(Mp2n) ←→  �  V  Πtemp(SO(V )),  where V runs over all (2n + 1)-dimensional quadratic space of discriminant 1 over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Thanks to their results, LLC  for Mp2n is implied by that for all SO(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In addition, there is an article [16] which proved LIR for Mp2n assuming  that for all SO(V ) over a p-adic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let next F be a number field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The metaplectic group Mp2n(AF ) is a nontrivial two-fold cover of Sp2n(AF ), and there  is a canonical injective homomorphism Sp2n(F) ֒→ Mp2n(AF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Hence the notions of ”automorphic representations”  and ”discrete spectrum” are defined in a canonical way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' AMF for the metaplectic group was studied by Gan-Ichino  [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' They proved the decomposition of the discrete spectrum of Mp2n into near equivalence classes without any  assumption, and proved the decomposition into irreducible automorphic representations of the generic part assuming  that for all non-quasi-split odd special orthogonal groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Those theories will be completed by this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Namely, LLC and LIR for the metaplectic groups will hold true,  and the result of Gan-Ichino [11] on the generic part of AMF will be unconditional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Organization  §2 is the preliminary section, where some notations for odd special orthogonal groups are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  In §3 we first recall the notions of endoscopic triple, transfer factor, local and global parameters, and the canonical  correspondence (e, ψe) ↔ (ψ, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Next we shall state LLC, ECR, and AMF more precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We will recall the result of  Arthur [7] and state our main theorem (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  §4 is the most important section in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We define the local intertwining operator, state LIR, and reduce  its proof to the case when the parameter is discrete for M ∗ and elliptic or exceptional for G∗, following [18, 7, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='9 we give a proof of LIR for the special case explained above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  In §5, we will recall the global theory on the trace formula, and obtain some lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Proofs of some lemmas are  omitted since they are quite similar to those in [18, §3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  In §6, we complete the proof of the local main theorem by the argument involving globalizations and trace Paley-  Wiener theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  In §7, we complete the proof of the global main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Convention & Notation  We marked some theorems, lemmas, and propositions with symbol * to indicate that they are only proved for  generic parameters in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  (We omit * when we refer to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=')  The author expect that their proofs for  non-generic parameters will be completed by an analogous argument to a sequel of [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We do not use ”S” in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Instead, we shall use ”S” to denote the component groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  3  In this paper every field is assumed to be of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In particular, a local field is either R, C, or a finite  extension of Qp for some prime number p ∈ Z, and a global field is a number field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', a finite extension of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For  a field F, we write F for its algebraic closure, and Γ = ΓF for its absolute Galois group Gal(F/F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For a connected  reductive algebraic group G over F, we write e(G) for the Kottwitz sign ([21]) of G, and “  G for the dual group over  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If moreover F is a local or global field, we write WF for the absolute Weil group of F, and the Weil form of the  L-group is defined by LG = “  G ⋊ WF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  If F is a number field, we write AF for the ring of adeles of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We often fix a nontrivial additive character of  F\\AF , which always denoted by ψF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For each place v of F, we write ψF,v for the local component of ψF at v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We  abbreviate ΓFv as Γv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Following [7] and [18], we do not use a symbol �′  v for a restricted tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We simply  write �  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Similarly, we write �  v in place of �′  v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Unless otherwise specified, �  v, �  v, �  v, and �  v denote the certain  operations taken over all places v of F, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  If F is a local field, the Langlands group is defied as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  LF =  ®WF × SU(2, R),  if F is non-archimedean,  WF ,  if F is archimedean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  As in the global case, we often fix a nontrivial additive character of F, which always denoted by ψF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For an algebraic or abstract group G, its center is denoted by Z(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  In addition, when X is a subgroup or  an element of G, we write Cent(X, G), ZX(G), or Z(X, G) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  NG(X) or N(X, G)) for the centralizer (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  normalizer) of X in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For a topological group G, its connected component of the identity element is denoted by G◦, and we put π0(G) =  G/G◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For an algebraic group G over a field F, put X∗(G) = Hom(G, GL1) and X∗(G) = Hom(GL1, G), which are  equipped with the F-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put moreover  aG = Hom(X∗(G)F , R),  a∗  G = X∗(G)F ⊗Z R,  aG,C = Hom(X∗(G)F , C),  a∗  G,C = X∗(G)F ⊗Z C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For any representation π, let π∨ denote its contragredient representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If π is a representation of a topological  group G with a Haar measure dg, for a function f on G, let fG(π) denote its character i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=',  fG(π) = tr(π(f)),  where  π(f) =  �  G  f(g)π(g)dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We also write f(π) if there is no danger of confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  In this paper we shall write Ei,j for the (i, j)-th matrix unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For a positive integer N, we put  J = JN =  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  1  −1  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  (−1)N−2  (−1)N−1  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  ∈ GLN,  and define an automorphism θN of GLN by θN(g) = J tg−1J−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then the standard pinning (TN, BN, {Ei,i+1}N−1  i=1 ),  where TN is the maximal torus consisting of diagonal matrices and BN is the Borel subgroup consisting of upper  triangular matrices, is θN-stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The dual group for GLN is GL(N, C), and the automorphism �θN of GL(N, C),  which is dual to θN, is given by �θN(g) = J tg−1J−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' As GLN is split, the Galois action on GL(N, C) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Acknowledgment  The author would like to thank his doctoral advisor Atsushi Ichino for his helpful advice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' He also thanks the  co-advisor Wen-Wei Li for his helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In addition, he also thanks Masao Oi and Hirotaka Kakuhama for  sincere and useful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' This work was partially supported by JSPS Research Fellowships for Young Scientists  KAKENHI Grant Number 20J10875 and JSPS KAKENHI Grant Number 22K20333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The author also would like to  appreciate Naoki Imai for his great support by JSPS KAKENHI Grant Number 22H00093.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4  2  Odd special orthogonal groups  In this section, we establish some notations for the odd special orthogonal groups, and recall the Kottwitz map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  In the third subsection, we shall describe the real case as a preparation for the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1  Split odd special orthogonal groups SO2n+1  Let F be any field of characteristic zero, and n a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We shall write SO2n+1 for the split odd  special orthogonal group over F of size (2n + 1), which is defined by  SO2n+1 =  \uf8f1  \uf8f2  \uf8f3 g ∈ GL2n+1  ������  tg  Ñ  1n  2  1n  é  g =  Ñ  1n  2  1n  é \uf8fc  \uf8fd  \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We put 2 at the center to make root vectors simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Its Lie algebra is realized as  so2n+1 =  \uf8f1  \uf8f2  \uf8f3 X ∈ M2n+1  ������  tX  Ñ  1n  2  1n  é  +  Ñ  1n  2  1n  é  X = 0  \uf8fc  \uf8fd  \uf8fe ,  over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us fix the standard Borel subgroup and the standard maximal torus  B∗ =  \uf8f1  \uf8f2  \uf8f3  Ñ  a  ∗  ∗  1  ∗  ta−1  é  ∈ SO2n+1  ������  a ∈ GLn, upper triangular  \uf8fc  \uf8fd  \uf8fe ,  T ∗ = � t = diag(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , tn, 1, t−1  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , t−1  n )  �� ti ∈ GL1  � ,  and let χi denote the element of X∗(T ∗) such that χi(t) = ti, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We shall define simple roots αi and  simple root vectors Xαi so that  R(T ∗, SO2n+1) = { ±(χi − χj), ±(χi + χj) }1≤i<j≤n ∪ { ±χi }1≤i≤n ,  ∆(B∗) = { α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , αn }  = { χ1 − χ2, χ2 − χ3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , χn−1 − χn, χn } ,  Xαi = Ei,i+1 − En+1+i+1,n+1+i,  1 ≤ i ≤ n − 1,  Xαn = 2En,n+1 − En+1,2n+1,  where R(T ∗, SO2n+1) and ∆(B∗) denote the root system of T ∗ in SO2n+1 and the set of simple roots with respect to  B∗, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Here we write the group law on X∗(T ∗) additively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put  Hi = Ei,i − Ei+1,i+1 − En+1+i,n+1+i + En+1+i+1,n+1+i+1,  for 1 ≤ i ≤ n − 1,  Hn = 2En,n − 2E2n+1,2n+1,  for i = n,  and  X−αi = Ei+1,i − En+1+i,n+1+i+1,  for 1 ≤ i ≤ n − 1,  X−αn = En+1,n − 2E2n+1,n+1,  for i = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  It can be seen that X−αi is a root vector corresponding to a negative root −αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Also one has dαi(Hi) = 2 and  [Xαi, X−αi] = Hi, so Hi is the coroot for αi and X−αi is the opposite root vector to Xαi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In this paper, by the  standard pinning of SO2n+1, we mean the pinning (T ∗, B∗, {Xα}α∈∆(B∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The dual group of SO2n+1 is the symplectic group Sp(2n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We shall choose a standard pinning of Sp(2n, C)  dual to the standard pinning of SO2n+1, to be the same as in [16, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2  Inner forms of odd special orthogonal groups  Let F be any field of characteristic zero, and G∗ = SO2n+1 the split special orthogonal group over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Since any  automorphism of G∗ is an inner automorphism, the isomorphism classes of inner forms of G∗ and those of inner twists  of G∗ are bijectively corresponding to each other in the natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, since G∗  ad = G∗ and Z(G∗) = {1}, the  isomorphism classes of inner twists, of pure inner twists, and of extended pure inner twists of G∗ are also bijectively  corresponding to each other in the natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We refer the reader to [18, §0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1] for the definitions and basic  properties of inner twists and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Hence there are bijective correspondences between the sets of isomorphism  classes of inner forms, inner twists, and pure inner twists of G∗, and the Galois cohomology set H1(F, G∗):  {inner forms} / ≃  ←→  {inner twists} / ≃  ←→  {pure inner twists} / ≃  ←→  H1(F, G∗),  G  �→  ξ : G∗ → G  �→  (ξ, z) : G∗ → G  �→  z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  By abuse of terminology we often identify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We remark that for any (2n + 1)-dimensional quadratic space V over  F, its special orthogonal group SO(V ) is an inner form of G∗, and every inner form of G∗ is isomorphic to SO(V ) for  some quadratic space V over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let G be an inner form of G∗, and M ∗ ⊂ G∗ a Levi subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We say that M ∗ transfers to G if there exists an  inner twist ξ : G∗ → G such that M := ξ(M ∗) ⊂ G is a Levi subgroup over F and the restriction ξ|M∗ : M ∗ → M is  an inner twist over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' When M ∗ transfers to G, we will always choose an inner twist ξ as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let F be a local field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Kottwitz [22, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2] gave a canonical map of pointed sets  α : H1(F, G∗) → π0(Z(”  G∗)Γ)D,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1)  extending the Tate-Nakayama isomorphism, where D denotes the Pontryagin dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For any pure inner twist (ξ, z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' G∗ →  G, which we regard as an element of H1(F, G∗), put χG = α(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We shall also write ⟨−, z⟩ or ⟨−, ξ⟩ for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that  in our case we have π0(Z(”  G∗)Γ)D = Z(Sp(2n, C))D ≃ Z/2Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  If F is non-archimedean, then there are only two inner forms of G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' One is non-quasi-split, and the other is split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  In this case the map α is the unique isomorphism of pointed sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If F = C, then H1(C, G∗) is singleton, and hence  α is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Suppose that F = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For non-negative integers p and q with p + q = 2n + 1, put  SO(p, q) =  ß  g ∈ GL2n+1(R)  ����  tg  Å1p  −1q  ã  g =  Å1p  −1q  ã ™  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then SO(p, q) is an inner form of G∗, and every inner form is isomorphic to SO(p, q) for some (p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Since SO(p1, q1)  and SO(p2, q2) are isomorphic if and only if (p1, q1) = (p2, q2) or (q2, p2), we can naturally identify H1(R, SO2n+1)  with {(p, q) | p, q ∈ Z≥0, p + q = 2n + 1}/(p, q) ∼ (q, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, It is known that the Kottwitz map is characterized  by  ker α = { SO(p, q) | p − q ≡ 1, 7  mod 8 } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let F be a global field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For each place v of F there is a localization map H1(F, G∗) → H1(Fv, G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let αv denote  the Kottwitz map (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1) from H1(Fv, G∗) to π0(Z(”  G∗)Γv)D = π0(Z(”  G∗)Γ)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By [22, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6] we have an exact  sequence  H1(F, G∗) −→  �  v  H1(Fv, G∗)  �  v αv  −→ π0(Z(”  G∗)Γ)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2)  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Such an exact sequence holds for general G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3  On the real odd special orthogonal groups SO(p, q)  Let us describe the real special orthogonal groups SO(p, q) and their Lie algebras so(p, q) more explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let p > q  be non-negative integers with p + q = 2n + 1, and put r = p − n − 1 = n − q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put  S0 =  Ñ 1n  1n  2  1n  −1n  é  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  6  Then an assignment g �→ S0gS−1  0  determines an isomorphism from SO2n+1 to SO(n + 1, n) over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put next  S′  p,q =  Ñ 1n+1  −i1r  1q  é  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then an assignment g �→ S′  p,qgS′  p,q  −1 determines an isomorphism from SO(n + 1, n) to SO(p, q) over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We thus  obtain an inner twist ξ = ξp,q : G∗ = SO2n+1 → SO(p, q) given by g �→ Sp,qgS−1  p,q, where Sp,q = S′  p,qS0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The standard pinning (T ∗, B∗, {Xα}α) of G∗ gives a Chevalley basis of g∗ = so2n+1:  [Xβ, Xγ] = ±(b + 1)Xβ+γ,  where b is the greatest positive integer such that γ − bβ is a root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Explicitly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Xχi−χj = Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j − En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for 1 ≤ i < j ≤ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Xχi+χj = Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j − Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for 1 ≤ i < j ≤ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Xχi = 2Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1 − En+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for 1 ≤ i ≤ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Hχi−χj = Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i − Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j − En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i + En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for 1 ≤ i < j ≤ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Hχi+χj = Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i + Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j − En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i − En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for 1 ≤ i < j ≤ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Hχi = 2Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i − 2En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for 1 ≤ i ≤ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  X−(χi−χj) = Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i − En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for 1 ≤ i < j ≤ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  X−(χi+χj) = En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i − En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for 1 ≤ i < j ≤ n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  X−χi = En+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i − 2En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  As usual the Lie algebra of SO(p, q) has the real structure  so(p, q)R =  ß  X ∈ M2n+1(R)  ����  tX  Å1p  −1q  ã  +  Å1p  −1q  ã  X = 0  ™  ,  and the inner twist ξ = ξp,q : G∗ = SO2n+1 → SO(p, q) gives an isomorphism ξ : g∗ ∋ X �→ Sp,qXS−1  p,q ∈ so(p, q) of the  complex Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The inner twist ξp,q sends an element  diag(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , 1,  i  t, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , 1,  n+1+i  t−1 , 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=') ∈ T ∗,  to  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  à 1i−1  cos z  − sin z  1n  sin z  cos z  1n−i  í  ,  for i ≤ r,  à 1i−1  t+t−1  2  t−t−1  2  1n  t−t−1  2  t+t−1  2  1n−i  í  ,  for r < i,  where z is a complex number such that t = e  √−1z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' (i ≤ r is non-split part, i > r is split part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=')  The images of the root vectors and the coroots are given as follows:  ξ(Xχi−χj) = 1  2×  7  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  Ä  Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j +  √  −1Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j − Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i +  √  −1Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i  −  √  −1En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j + En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j −  √  −1En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i − En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i  ä  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for i < j ≤ r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Ä  Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j + Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j − Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i +  √  −1Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i  −  √  −1En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j −  √  −1En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j + En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i −  √  −1En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i  ä  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for i ≤ r < j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  (Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j + Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j − Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i + Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i  +En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j + En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j + En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i − En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for r < i < j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ξ(Hχi−χj) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  √  −1 (Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i − Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j − En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i + Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for i < j ≤ r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  √  −1Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i − Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j −  √  −1En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i − Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for i ≤ r < j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i − Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j + En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i − Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for r < i < j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ξ(X−(χi−χj)) = 1  2×  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  Ä  −Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j +  √  −1Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j + Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i +  √  −1Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i  −  √  −1En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j − En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j −  √  −1En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i + En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i  ä  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for i < j ≤ r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Ä  −Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j + Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j + Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i +  √  −1Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i  −  √  −1En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j +  √  −1En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j + En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i +  √  −1En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i  ä  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for i ≤ r < j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  (−Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j + Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j + Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i + Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i  +En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j − En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j + En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i + En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for r < i < j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ξ(Xχi+χj) = 1  2×  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  Ä  Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j −  √  −1Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j − Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i +  √  −1Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i  −  √  −1En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j − En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j +  √  −1En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i + En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i  ä  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for i < j ≤ r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Ä  Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j − Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j − Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i +  √  −1Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i  −  √  −1En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j +  √  −1En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j − En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i +  √  −1En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i  ä  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for i ≤ r < j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  (Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j − Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j − Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i + Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i  +En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j − En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j − En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i + En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for r < i < j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ξ(Hχi+χj) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  √  −1 (Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i + Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j − En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i − Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for i < j ≤ r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  √  −1Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i + Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j −  √  −1En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i + Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for i ≤ r < j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i + Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j + En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i + Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for r < i < j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ξ(X−(χi+χj)) = 1  2×  8  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  Ä  −Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j −  √  −1Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j + Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i +  √  −1Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i  −  √  −1En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j + En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j +  √  −1En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i − En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i  ä  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for i < j ≤ r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Ä  −Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j − Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j + Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i +  √  −1Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i  −  √  −1En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j −  √  −1En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j − En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i −  √  −1En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i  ä  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for i ≤ r < j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  (−Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j − Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j + Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i + Ej,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i  +En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='j + En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+j − En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i − En+1+j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for r < i < j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ξ(Xχi) =  ®  Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1 − En+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i +  √  −1En+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i −  √  −1En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for i ≤ r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1 − En+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i + En+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i + En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for r < i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ξ(Hχi) = 2 ×  ®√  −1 (Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i − En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for i ≤ r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  (Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i + En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for r < i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ξ(X−χi) =  ®  −Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1 + En+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i +  √  −1En+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i −  √  −1En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for i ≤ r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  −Ei,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1 + En+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='i + En+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1+i + En+1+i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='n+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for r < i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  In particular we have  tξ(Xα) = ξ(X−α) for any α ∈ R(T ∗, G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Put T = ξ(T ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The real form T (R) consists of the elements of the form ξ(diag(t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , tr, e  √−1θr+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , e  √−1θn, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' )),  where ti ∈ R× and θi ∈ R/2πZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For an element t ∈ T (R) of such form and a positive root α ∈ R(T, SO(p, q)), which  is equal to R(T ∗, G∗) as sets, we have  Ad(t)(ξ(Xα)) = α(t)ξ(Xα),  where  α(t) =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  e  √−1(θi±θj),  for α = χi ± χj, i < j ≤ r,  e  √−1θit±1  j ,  for α = χi ± χj, i ≤ r < j,  tit±1  j ,  for α = χi ± χj, r < i < j,  e  √−1θi,  for α = χi, i ≤ r,  ti,  for α = χi, r < i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  This means that as roots of T in G, α = ±(χi ± χj) (i < j ≤ r), ±χi (i ≤ r) are imaginary roots (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', α = −α),  α = ±(χi ± χj) (i ≤ r < j) are complex roots (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', α ̸= ±α), and α = ±(χi ± χj) (r < i < j), ±χi (r < i) are real  roots (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', α = α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  3  Notions around endoscopy and parameters, and Main theorem  In this section, we first establish some notations for the endoscopy groups, the transfer factors, and L- or A-  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Afterwards, we shall recall Arthur’s work and state our main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1  Endoscopy  In this subsection, we recall notion of an endoscopic triple and the transfer factor from [18, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1  Endoscopic triples  Let F be a global or local field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Consider a pair (G∗, θ∗) consisting of a connected quasi-split reductive group G∗  defined over F with a fixed F-pinning, and a pinned automorphism θ∗ of G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Here, an automorphism θ∗ is said to  9  be pinned if it preserves the fixed pinning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', the maximal torus, the Borel subgroup, and the set of simple root  vectors are θ∗-stable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we have an automorphism “  θ∗ of ”  G∗, which preserves a Γ-invariant pinning for ”  G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put  Lθ∗ = “  θ∗ ⋊ idWF , which is an L-automorphism of LG∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' An endoscopic triple for (G∗, θ∗) is a triple e = (Ge, se, ηe) of a connected quasi-split reductive group  Ge defined over F, a semisimple element se ∈ ”  G∗, and an L-homomorphism ηe : LGe → LG∗ such that  Ad(se) ◦ Lθ∗ ◦ ηe = ηe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ηe(�  Ge) is the connected component of the subgroup of Ad(se) ◦ “  θ∗-fixed elements in ”  G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  When η(Z(�  Ge)Γ)◦ ⊂ Z(”  G∗), the endoscopic triple e is said to be elliptic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' When θ∗ is trivial (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' not trivial), every  endoscopic triple for (G∗, θ∗) is said to be ordinary (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' twisted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Two endoscopic triples e1 = (Ge  1, se  1, ηe  1) and e2 = (Ge  2, se  2, ηe  2) for G∗ are said to be isomorphic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  strictly isomorphic) if there exists an element g ∈ ”  G∗ such that  gηe  1(LGe  1)g−1 = ηe  2(LGe  2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  zgse  1“  θ∗(g)−1 = se  2 for some z ∈ Z(”  G∗), (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' gse  1“  θ∗(g)−1 = se  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then the element g is called an isomorphism (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' strict isomorphism) from e1 to e2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If g is an isomorphism (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  strict isomorphism) from e to e itself, then it is said to be an automorphism (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' strict automorphism) of e, and we  shall write AutG∗(e) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' AutG∗(e)) for the set of automorphisms (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' strict automorphisms) of e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let ξ : G∗ → G be an inner twist, and put θ = ξ ◦ θ∗ ◦ ξ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then θ is an automorphism of G defined over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By an endoscopic triple for (G, θ), we mean that for (G∗, θ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The notion of elliptic endoscopic triple,  isomorphism and strict isomorphism of endoscopic triples are also the same as those for (G∗, θ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We shall write E(G⋊θ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' E(G⋊θ)) for the set of isomorphism (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' strict isomorphism) classes of endoscopic  triples for (G, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The corresponding subsets of elliptic endoscopic triples will be indicated by the lower right index  ell:  E(G ⋊ θ)  � � E(G ⋊ θ)  Eell(G ⋊ θ)  � �  �  Eell(G ⋊ θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  �  When θ is trivial, then we may write E(G), E(G), Eell(G), and Eell(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Put �E(N) = E(GLN, θN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  If e = (Ge, se, ηe) ∈ �E(N), the group Ge is a direct product of finite numbers of  symplectic or special orthogonal groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We shall say e is simple if the number of factors is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Define �Esim(N) to be  the set of strictly isomorphism classes of simple endoscopic triples of (GLN, θN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2  Normalized transfer factors  Next, we shall recall some properties of transfer factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let G∗ be a quasi-split connected reductive group over  a local or global field F with a fixed F-pinning, and θ∗ an automorphism of G∗ preserving the pinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We assume  that Z(G∗) is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that if G∗ is a direct product of a finite number of SO2m+1 or GLm (m ≥ 1), then  G∗ is quasi-split connected reductive and its center is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let (ξ, z) : G∗ → G be a pure inner twist, and put  θ = ξ ◦θ∗ ◦ξ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then θ is an automorphism of G defined over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Not all twisted groups (G, θ) arise in this way, which  is why an element gθ ∈ G∗ such that θ∗ = Ad(gθ) ◦ ξ−1 ◦ θ ◦ ξ was introduced in [23, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In this paper, however, it  suffices to assume gθ of [23] is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We assume that θ∗(z) = z, which is enough for our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let e = (Ge, se, ηe) be  an endoscopic triple for (G, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let first F be a local field, and ψF : F → C1 a nontrivial additive character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let further δ ∈ Gθ−srss(F) and  γ ∈ Ge(F), where Gθ−srss(F) denotes the set of θ-strongly regular and θ-semisimple elements in G(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we have  the transfer factor:  ∆[e, ξ, z](γ, δ) =  \uf8f1  \uf8f2  \uf8f3  ǫ(1  2, V, ψF )∆new  I  (γ, δ)−1∆II(γ, δ)∆III(γ, δ)−1∆IV (γ, δ),  if γ is a norm of δ,  0,  otherwise,  10  where V , ∆new  I  , ∆II, ∆III, and ∆IV are defined as in [18, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='38-39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Although they treated an extended pure inner  twist in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', the similar arguments also works for a pure inner twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The main difference is that we utilize an  ordinary cocycle, instead of a basic cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' See also [23, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5, Appendices A-C] and [24, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In the case G∗ = SO2n+1 and θ∗ = 1, the transfer factor ∆[e, ξ, z] is determined by the isomorphism class  of G, not by the choice of (ξ, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let γ1, γ2 ∈ Ge(F) be norms of δ1, δ2 ∈ Gθ−srss(F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we have  ∆[e, ξ, z](γ1, δ1)  ∆[e, ξ, z](γ2, δ2) = ∆′(γ1, δ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' γ2, δ2),  where the right hand side is the relative transfer factor of [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let se denote the image of se in (”  G∗/”  G∗der)Γ  �  θ∗,free, which is the set of Γ-fixed points in the torsion free  quotient of the “  θ∗-covariants of ”  G∗/”  G∗der.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For x ∈ Z(”  G∗)Γ and y ∈ Z1(F, (Z(G∗)θ∗)◦), we have  ∆[xe, ξ, z] = ⟨x, z⟩∆[e, ξ, z],  and  ∆[e, ξ, yz] = ⟨se, y⟩∆[e, ξ, z],  where xe denotes the endoscopic triple (Ge, xse, ηe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We shall now recall the notion of matching functions from [23, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For δ ∈ Gθ−srss(F) and f ∈ H(G), the orbital  integral is defined as  Oδθ(f) =  �  Gδθ(F )\\G(F )  f(g−1δθ(g))dg,  where Gδθ = Centθ(δ, G) = {x ∈ G | δθ(x) = xδ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If θ is trivial, it may be simply written Oδ(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For γ ∈ Ge  srss(F)  and f e ∈ H(Ge), the stable orbital integral is defined as  SOγ(f) =  �  γ′  Oγ′(f e),  where the sum is taken over a set of representatives for the conjugacy classes in the stable conjugacy class of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Two functions f e ∈ H(Ge) and f ∈ H(G) are called ∆[e, ξ, z]-matching (or matching when there is  no danger of confusion) if for any strongly G-regular semisimple element γ ∈ Ge(F) we have  SOγ(f e) =  �  δ  ∆[e, ξ, z](γ, δ)Oδθ(f),  where the sum is taken over a set of representatives for the θ-conjugacy classes under G(F) of Gθ−srss(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We also  say that f and f e have ∆[e, ξ, z]-matching orbital integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If we do not assume θ∗(z) = z, then we need θe on Ge as in [23, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Next, let F be a global field, and ψF : AF /F → C1 a nontrivial additive character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For δ ∈ Gθ−srss(AF ) and  γ ∈ Ge(AF ), the global absolute transfer factor is defined by  ∆A[e, ξ](γ, δ) =  \uf8f1  \uf8f2  \uf8f3  �  v  ∆[ev, ξv, zv](γv, δv),  if γ is a norm of δ,  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Note that the product is well-defined because the almost all factors are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2 and the exact sequence  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2) imply that ∆A[e, ξ] is independent of z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  11  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The factor ∆A[e, ξ] coincides with the inverse of the canonical adelic transfer given in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We shall finish this subsubsection stating a lemma regarding local transfer factors and Levi subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let again  F be a local field, and ψF : F → C1 a nontrivial additive character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let M ∗ ⊂ G∗ be a standard Levi subgroup  such that M = ξ(M ∗) ⊂ G is a Levi subgroup defined over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that θ is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let e = (Gese, ηe) and  eM = (M e, se, ηe|LMe) be endoscopic triples for G and M respectively, such that M e ⊂ Ge is a Levi subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For each δ ∈ M(F) and γ ∈ M e(F), we have  ∆[eM, ξ|M∗, z](γ, δ) = ∆[e, ξ, z](γ, δ)  Ç  |DGe  Me(γ)|  |DG  M(δ)|  å 1  2  ,  where DG  M and DGe  Me(γ) are the relative Weyl discriminants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If f ∈ H(G) and f e ∈ H(Ge) are ∆[e, ξ, z]-matching, then their constant terms fM ∈ H(M) and f e  Me ∈ H(M e)  are ∆[eM, ξ|M∗, z]-matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3  Endoscopic triples for odd special orthogonal groups  Here we will explicate the set of representatives of isomorphism or strict isomorphism classes of ordinary endoscopic  triples for odd special orthogonal groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let F be a local or global field, G∗ = SO2n+1 the split special orthogonal  group of size 2n + 1 defined over F, and G an inner form of G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then the Galois group Γ = ΓF acts on “  G = Sp(2n, C)  trivially, and thus we have LG = Sp(2n, C) × WF and Z(“  G)Γ = {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Every semisimple element s ∈ Sp(2n, C) is Sp(2n, C)-conjugate to an element of the form  \uf8eb  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ec  \uf8ed  a11m1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ar1mr  1n1  −1n2  a−1  1 1m1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  a−1  r 1mr  1n1  −1n2  \uf8f6  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f7  \uf8f8  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1)  where r ≥ 0, m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , mr ≥ 1, and n1, n2 ≥ 0 are integers such that m1 + · · · mr + n1 + n2 = n, and a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , ar ∈ C \\  {0, 1, −1} are complex numbers other than 0, 1, −1 that are different to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Thus its centralizer Cent(s, Sp(2n, C))  is isomorphic to  GL(m1, C) × · · · × GL(mr, C) × Sp(2n1, C) × Sp(2n2, C),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2)  which is the dual group of  GLm1 × · · · × GLmr × SO2n1+1 × SO2n2+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3)  Therefore, each endoscopic triple is isomorphic to a triple of a group (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3), a semisimple element (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1), and a natural  inclusion map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Any description of complete systems of representatives of isomorphism or strict isomorphism classes  of them is so complicated that we do not write down it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Elliptic ones are simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For two nonnegative integers n1 and n2 such that n1 + n2 = n, put en1,n2 = (Ge  n1,n2, sn1,n2, ηn1,n2), where  Ge  n1,n2 = SO2n1+1 × SO2n2+1,  12  sn1,n2 =  Ü 1n1  −1n2  1n1  −1n2  ê  ,  ηn1,n2 denotes the direct product of the inclusion map Sp(2n1, C) × Sp(n2, C) ֒→ Sp(2n, C) given by  ÅÅ A1  B1  C1  D1  ã  ,  Å A2  B2  C2  D2  ãã  �→  Ü  A1  B1  A2  B2  C1  D1  C2  D2  ê  ,  and the identity map of WF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then en1,n2 is an elliptic endoscopic triple for G, and four triples ±en1,n2, ±en2,n1 are isomorphic to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Here  −(Ge, se, ηe) denotes (Ge, −se, ηe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Therefore, a set  { en1,n2 | n1 ≥ n2 ≥ 0, n1 + n2 = n }  is a complete system of representatives of isomorphism classes of elliptic endoscopic triples for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The set Eell(G) may  be identified with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In addition, en1,n2 is strictly isomorphic to −en2,n1, but not strictly isomorphic to en2,n1 unless  n1 = n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Therefore, a set  { en1,n2 | n1, n2 ≥ 0, n1 + n2 = n }  is a complete system of representatives of strictly isomorphism classes of elliptic endoscopic triples for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The set  Eell(G) may be identified with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2  Local parameters  In this subsection, we recall the notion of local parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let F be a local field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let G be a quasi-split connected  reductive group over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We say that a continuous homomorphism φ : LF → LG is an L-parameter for G if it commutes  with the canonical projections LF ։ WF and LG ։ WF , and sends semisimple elements to semisimple elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Two L-parameters are said to be equivalent if they are conjugate by an element in “  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We write Φ(G) for the set  of equivalence classes of L-parameters for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' An L-parameter φ is called bounded (or tempered) if the image of the  projection of φ(LF ) to “  G is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We write Φbdd(G) for the subset of Φ(G) consisting of bounded ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We say that a continuous homomorphism ψ : LF × SU(2, R) → LG is a local A-parameter for G if the restriction  ψ|LF to LF is a bounded L-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Two local A-parameters are said to be equivalent if they are conjugate by an  element in “  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We write Ψ(G) for the set of equivalence classes of local A-parameters for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let Ψ+(G) denote the  set of equivalence classes of a continuous homomorphism ψ : LF × SU(2, R) → LG whose restriction ψ|LF to LF is an  L-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' A parameter ψ ∈ Ψ+(G) is called generic if the restriction ψ|SU(2,R) (to the second SU(2, R)) is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The subset of Ψ+(G) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Ψ(G)) consisting of generic elements can be identified with Φ(G) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φbdd(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We say  that ψ ∈ Ψ+(G) is discrete if there is no proper parabolic subgroup of LG which contains the image of ψ, and write  Ψ+  2 (G) for the subset of Ψ+(G) consisting of discrete elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put Ψ2(G) = Ψ(G)∩Ψ+  2 (G), Φ2(G) = Φ(G)∩Ψ+  2 (G),  and Φ2,bdd(G) = Φbdd(G) ∩ Ψ+  2 (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We shall write Π(G) for the set of isomorphism classes of irreducible smooth representations of G(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let Πtemp(G)  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Π2(G)) to be the subsets of Π(G) consisting of the tempered (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' essentially square integrable) representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Put Π2,temp(G) = Π2(G) ∩ Πtemp(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For ψ ∈ Ψ+(G), put Sψ = Cent(Im ψ, “  G), Sψ = Sψ/Z(“  G)Γ, Sψ = π0(Sψ) = Sψ/S◦  ψ, Sψ = π0(Sψ) =  Sψ/S◦  ψZ(“  G)Γ = Sψ/Z(“  G)Γ, Srad  ψ  = Cent(Im ψ, “  Gder)◦, and S♮  ψ = Sψ/Srad  ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' When we emphasize that the param-  eter is for G, we write Sψ(G), Sψ(G), Sψ(G), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The associated L-parameter φψ of ψ is defined by  φψ(w) = ψ(w,  Ç  |w|  1  2  0  0  |w|− 1  2  å  ),  13  for w ∈ LF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Finally, we put  sψ = ψ(1,  Å−1  0  0  −1  ã  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Recall that when G is a general linear group GLN, then the parameters can be regarded as an N-dimensional  representations of LF or LF × SU(2, R), and the equivalence of parameters is isomorphism of representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' A  parameter which is irreducible as a representation, is said to be simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We write Ψ+  sim(GLN) for the subset of Ψ+(GLN)  consisting of simple parameters, and put Ψsim(GLN) = Ψ(GLN) ∩ Ψ+  sim(GLN), Φsim(GLN) = Φ(GLN) ∩ Ψ+  sim(GLN),  and Φsim,♥(GLN) = Φ♥(GLN) ∩ Φsim(GLN), for ♥ = 2, bdd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The canonical bijection between Π(GLN) and Φ(GLN),  which is called the local Langlands correspondence for GLN, is known by Langlands [26] in the archimedean case  (for any G in fact), by Harris-Taylor [13] and Henniart [14] in the non-archimedean case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' As in [7], we shall write  Π(N) = Π(GLN), Φ(N) = Φ(GLN), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For any φ ∈ Φ(N), we write πφ for the unique corresponding element  in Π(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, for any ψ ∈ Ψ+(N), we define the corresponding (not necessarily irreducible) representation πψ  as [18, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put Ψ+  unit(N) = {ψ ∈ Ψ+(N) | πψ is irreducible and unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that Ψ(N) ⊂ Ψ+  unit(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' When  ψ ∈ Ψ(N), then πψ = πφψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' A parameter ψ for GLN is said to be self-dual if it is self-dual as a representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We shall  write �Φ(N) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' �Φsim(N), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' �Φsim,bdd(N)) for the subset of Φ(N) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φsim(N), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φsim,bdd(N)) consisting  of self-dual ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Suppose that G is a simple twisted endoscopy group of (GLN, θN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We say a parameter ψ ∈ Ψ+(G) is simple if  η ◦ ψ ∈ Ψ+(GLN) is simple, where η : LG → LGLN is the L-homomorphism attached to the endoscopy group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We  write Ψ+  sim(G) for the subset of Ψ+(G) consisting of simple parameters, and put Ψsim(G), Φsim(G), and so on in the  similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, put ‹Ψ+(G) to be the subset of Ψ+(GLN) consisting of the parameters of the form η◦ψ for some  ψ ∈ Ψ+(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that ‹Ψ+(G) = Ψ+(G) unless G is an even special orthogonal group, in which case ‹Ψ+(G) coincides  with Ψ+(G)/ ∼ǫ in the sense of [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put ‹Ψ(G) = ‹Ψ+(G) ∩ Ψ(N), �Φ(G) = ‹Ψ+(G) ∩ Φ(N), �Φsim(G) = �Φ(G) ∩ Φsim(N),  and �Φsim,bdd(G) = �Φsim(G) ∩ Φbdd(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let �Φ2(G) be the subset of �Φ(G) consisting of parameters of multiplicity free  as representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put �Φ2,bdd(G) = �Φ2(G) ∩ Φbdd(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Now let us consider the case when G is an inner form of the split odd special orthogonal group G∗ = SO2n+1, which  is not necessarily quasi-split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then the parameters for G∗ are also called parameters for G, since “  G = ”  G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We omit  the definition of ”G-relevant”, which is given in [18, §0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4] in a general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In this paper G-relevant parameters  are simply said to be relevant, when there is no danger of confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We remark that ”parameter for G” is different  from ”G-relevant parameter”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Although we use a phrase ”parameter for G”, we never write as Ψ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, we do  not fix a symbol for the set of G-relevant parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Recall from [10, §4] that the parameters can be regarded as an  2n-dimensional symplectic representations of LF or LF × SU(2, R), and the equivalence of parameters is isomorphism  of representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' A parameter ψ ∈ Ψ+(G∗) can be written as  ψ =  �  i∈I+  ψ  ℓiψi ⊕  �  i∈I−  ψ  ℓiψi ⊕  �  j∈Jψ  ℓj(ψj ⊕ ψ∨  j ),  where ℓi, ℓj are positive integers, I±  ψ , Jψ indexing sets for mutually inequivalent irreducible representations ψi of  LF × SU(2, R) such that  ψi is symplectic for i ∈ I+  ψ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ψi is orthogonal and ℓi is even for i ∈ I−  ψ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ψj is not self-dual for j ∈ Jψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Thus we have  Sψ ∼=  �  i∈I+  ψ  O(ℓi, C) ×  �  i∈I−  ψ  Sp(ℓi, C) ×  �  j∈Jψ  GL(ℓj, C),  and its component group is a free (Z/2Z)-module  Sψ ∼=  �  i∈I+  ψ  (Z/2Z)ai,  14  with a formal basis {ai}, where ai corresponds to the nontrivial coset O(ℓi, C) \\ SO(ℓi, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that “  G = Sp(2n, C) is  perfect, and thus Srad  ψ  = S◦  ψ and S♮  ψ = Sψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' It can be easily seen that the parameter ψ is discrete if and only if ℓi = 1  for all i ∈ I+  ψ and I−  ψ = Jψ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Hence, we have  Ψ+  2 (G∗) = { ψ ∈ Ψ+(G∗) | Sψ is finite }  = { ψ = ψ1 ⊕ · · · ⊕ ψr | ψi are symplectic, irreducible, and mutually distinct, r ≥ 1 } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  A parameter ψ ∈ Ψ+(G∗) is called to be elliptic if there exists a semisimple element in Sψ whose centralizer in Sψ(G∗)  is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We shall write Ψ+  ell(G∗) for the set of such parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we have  Ψ+  ell(G∗) = { ψ ∈ Ψ+(G∗) | Cent(s, Sψ) is finite for some s ∈ Sψ,ss }  =  ß  ψ = 2ψ1 ⊕ · · · 2ψq ⊕ ψq+1 ⊕ · · · ⊕ ψr  ����  r ≥ 1, r ≥ q ≥ 0,  ψi are symplectic, irreducible, and mutually distinct  ™  ,  where Sψ,ss denotes the set of semisimple elements in Sψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We write Ψell(G∗) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Ψell(G∗)) for the subset of Ψ(G∗)  consisting of elliptic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' non-elliptic) ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We have Ψ2(G∗) ⊂ Ψell(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put  Ψ2  ell(G∗) = Ψell(G∗) \\ Ψ2(G∗)  =  ß  ψ = 2ψ1 ⊕ · · · 2ψq ⊕ ψq+1 ⊕ · · · ⊕ ψr  ����  r ≥ q ≥ 1,  ψi are symplectic, irreducible, and mutually distinct  ™  ,  Φell(G∗) = Ψell(G∗) ∩ Φ(G∗), and Φ2  ell(G∗) = Ψ2  ell(G∗) ∩ Φ(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let M ∗ be a standard Levi subgroup of G∗ = SO2n+1 isomorphic to  GLn1 × · · · × GLnk × SO2n0+1,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4)  where n1 + · · · nk + n0 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then its dual group �  M ∗ is isomorphic to  GL(n1, C) × · · · × GL(nk, C) × Sp(2n0, C),  and we regard it as a Levi subgroup of ”  G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We have  Ψ+(M ∗) = { ψ1 ⊕ · · · ⊕ ψk ⊕ ψ0 | ψ1 ∈ Ψ+(n1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , ψk ∈ Ψ+(nk), ψ0 ∈ Ψ+(SO2n0+1) } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The canonical injection �  M ∗ ֒→ ”  G∗ induces the canonical mapping  ψ1 ⊕ · · · ⊕ ψk ⊕ ψ0 �→ ψ1 ⊕ · · · ⊕ ψk ⊕ ψ0 ⊕ ψ∨  1 ⊕ · · · ⊕ ψ∨  k  from Ψ+(M ∗) to Ψ+(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3  Global parameters  In this subsection we recall from [7] the notion of global parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let F be a global field, Γ = ΓF its absolute  Galois group, and WF its absolute Weil group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For a connected reductive group G over F, as in [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='19] we put  G(AF )1 = { g ∈ G(AF ) | |χ(g)| = 1, ∀χ ∈ X∗(G)F } ,  where X∗(G)F = HomF (G, GL1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The quotient set G(F)\\G(AF )1 has finite volume, and there is a sequence  L2  cusp(G(F)\\G(AF )1) ⊂ L2  disc(G(F)\\G(AF )1) ⊂ L2(G(F)\\G(AF )1),  where L2  cusp(G(F)\\G(AF )1) is the subspace consisting of cuspidal functions, and L2  disc(G(F)\\G(AF )1) the subspace  that can be decomposed into a direct sum of irreducible representations of G(AF )1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We have  Acusp(G) ⊂ A2(G) ⊂ A(G),  15  where Acusp(G), A2(G), and A(G) denote the subsets of irreducible unitary representations of G(AF ) whose restrictions  to G(AF )1 are constituents of the respective spaces L2  cusp(G(F)\\G(AF )1), L2  disc(G(F)\\G(AF )1), and L2(G(F)\\G(AF )1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We also write A+  cusp(G) and A+  2 (G) for the analogues of Acusp(G) and A2(G) defined without the unitarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For a finite set S of places of F such that v is non-archimedean and Gv is unramified over Fv for any place v /∈ S,  put CS(G) to be the set consisting of families cS = (cv)v /∈S where each cv is a “  G-conjugacy class in LGv = “  G ⋊ WFv  represented by an element of the form tv ⋊ Frobv with a semisimple element tv ∈ “  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For S ⊂ S′ there is a natural  map cS = (cv)v /∈S �→ (cv)v /∈S′ from CS(G) to CS′(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us define C(G) to be the direct limit of {CS(G)}S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let π = ⊗vπv be an irreducible automorphic representation of G(AF ) and S a finite set of places such that πv is  unramified for all v /∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The Satake isomorphism associates a “  G-conjugacy class c(πv) ∈ LGv to π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We can therefore  associate cS(π) = (c(πv))v /∈S ∈ CS(G) to π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' This leads to a mapping π �→ c(π) from a set of equivalence classes of  irreducible automorphic representations of G to C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We shall write Caut(G) for its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let us next review some fact on automorphic representations of GLN(AF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let N be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Following  [7], we write A(N) = A(GLN), and similarly A♥(N) = A♥(GLN) and A+  ♥(N) = A+  ♥(GLN) for ♥ ∈ {2, cusp}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put  A+  iso(N) =  �  π = π1 ⊞ · · · ⊞ πr  �� r ∈ Z≥1, Ni ∈ Z≥1, N1 + · · · + Nr = N, πi ∈ A+  cusp(Ni),  ∀i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , r  �  ,  where ⊞ denotes the isobaric sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' It is known by Mœglin and Waldspurger [29] that any π ∈ A2(N) is isomorphic to  µ| det |  n−1  2  ⊞ µ| det |  n−3  2  ⊞ · · · ⊞ µ| det |− n−1  2 ,  for some m, n ∈ Z≥1 and µ ∈ Acusp(m) with N = mn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, (m, n, µ) is uniquely determined by π, and any  representation of such form is an element of A2(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In other words, A2(N) is in bijection with {(µ, m) | 1 ≤ m|N, µ ∈  Acusp(N/m)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We have  Acusp(N) ⊂ A2(N) ⊂ A(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5)  Put Ψcusp(N) = Acusp(N), which is trivially in bijection with Acusp(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put Ψsim(N) to be the set of ψ = µ ⊠ ν  for an irreducible finite dimensional representation ν of SU(2, R) and µ ∈ A2(N/ dim ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' It is in bijection with A2(N)  by the fact above, for which we shall write ψ �→ πψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' An element in Ψsim(N) is said to be simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put Ψ(N) to be  the set of ψ = ℓ1ψ1 ⊞ · · · ⊞ ℓrψr for some r ∈ Z≥1, ℓi, Ni ∈ Z≥1 with ℓ1N1 + · · · + ℓrNr = N, and mutually distinct  elements ψi ∈ Ψsim(Ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' To such ψ, we associate πψ = π⊞ℓ1  ψ1 ⊞ · · · ⊞ π⊞ℓr  ψr , so that a mapping ψ �→ πψ gives a bijection  from Ψ(N) to A(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='We have a chain  Ψcusp(N) ⊂ Ψsim(N) ⊂ Ψ(N),  which is compatible with the chain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We say that a parameter ψ = ℓ1(µ1 ⊠ ν1) ⊞ · · · ⊞ ℓr(µr ⊠ νr) is generic if  ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , νr are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let Φ(N) be the subset of Ψ(N) consisting of generic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let v be a place of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By LLC for GLN, for any µ ∈ Ψcusp(N), we have an L-parameter φv ∈ Φ(GLN /Fv)  corresponding to µv ∈ Π(GLN /Fv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By abuse of notation, let us write µv for φv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we obtain the localization  map Ψsim(N) → Ψ+  v (N), ψ = µ ⊠ ν �→ ψv = µv ⊠ ν, where Ψ+  v (N) = Ψ+(GLN /Fv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' This leads the localization map  Ψ(N) → Ψ+  v (N),  ψ = ℓ1ψ1 ⊞ · · · ⊞ ℓrψr, �→ ψv = ℓ1ψ1,v ⊕ · · · ⊕ ℓrψr,v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let us now recall from [7] the definition of A-parameters for odd special orthogonal groups and other classical  groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' An irreducible self-dual cuspidal automorphic representation φ of GLN(AF ) is said to be symplectic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  orthogonal) if the exterior (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' symmetric) square L-function L(s, φ, ∧2) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' L(s, φ, Sym2)) has a pole at s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By  the theory of Rankin-Selberg L-function, any irreducible self-dual cuspidal automorphic representation φ of GLN(AF )  is either symplectic or orthogonal, and this is mutually exclusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3 of [7] tell us the following  proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5 (1st seed theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For each irreducible self-dual cuspidal automorphic representation φ of GLN(AF ),  there exists a simple twisted endoscopic triple eφ = (Gφ, sφ, ηφ) of (GLN, θN) and a representation π ∈ A2(Gφ)  such that c(φ) = ηφ(c(π)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The triple eφ is unique up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If φ is symplectic (hence N is even), then  Gφ = SON+1 and ηφ is regarded as the unique (conjugacy class of) embedding Sp(N, C) ֒→ GL(N, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If φ is orthog-  onal, then LGφ = SO(N, C) ⋊ WF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  16  Let e = (G∗, s, η) ∈ �Esim(N) be a simple twisted endoscopic triple of (GLN, θN), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', G∗ is the split symplectic  group, the split odd special orthogonal group, or a quasi-split even special orthogonal group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Set �Φsim(G∗) to be the  set of irreducible self-dual cuspidal automorphic representations φ of GLN(AF ) such that the corresponding endoscopic  triple eφ is isomorphic to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Here note that we cannot write Φsim(G∗) because G∗ may be an even special orthogonal  group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  An element ψ = µ ⊠ ν in Ψsim(N) is said to be self-dual if ψ = ψ∨, where ψ∨ = µ∨ ⊠ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We write ‹Ψsim(N) for the  subset of Ψsim(N) consisting of such elements, and put �Φsim(N) = ‹Ψsim(N)∩Φ(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, an element ψ = ⊞iℓiψi  in Ψ(N) is said to be self-dual if ψ = ψ∨, where ψ∨ = ⊞iℓiψ∨  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Write ‹Ψ(N) for the subset of Ψ(N) consisting of such  elements, and put �Φ(N) = ‹Ψ(N) ∩ Φ(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For any ψ ∈ ‹Ψ(N), the substitute Lψ for the global Langlands group attached to ψ is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We can  write  ψ = ⊞  i∈Iψ  ℓiψi ⊞ ⊞  j∈Jψ  ℓj(ψj ⊞ ψ∨  j ),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6)  where ψi = µi ⊠ νi ∈ ‹Ψsim(N) and ψj = µj ⊠ νj ∈ Ψsim(N) \\ ‹Ψsim(N) for any i ∈ Iψ, j ∈ Jψ, and they are mutually  distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put Kψ = Iψ ⊔ Jψ and let mk be a positive integer such that µk ∈ Ψcusp(mk) for k ∈ Kψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For i ∈ Iψ, put  Hi = Gµi of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5 and let ‹  µi denote for an embedding  ηµi : LHi = LGµi ֒→ LGLmi,  of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For j ∈ Jψ, put Hj = GLmj and let �  µj be an embedding LHj ֒→ LGL2mj given by  GL(mj, C) × WF → GL(2mj, C) × WF ,  (h, w) �→ (diag(h, �θmj(h)), w),  where �θmj is the twist given in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The substitute Lψ is the fiber product of {Hk}k∈Kψ over WF :  Lψ =  �  k∈Kψ  ÄLHk → WF  ä  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The associated L-embedding �ψ from Lψ × SU(2, R) to LGLN = GL(N, C) × WF is defined as  �ψ =  �  i∈Iψ  ℓi(‹  µi ⊗ νi) ⊕  �  j∈Jψ  ℓj(�  µj ⊗ νj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Now we shall recall the definition of the A-parameters for odd special orthogonal groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let G∗ = SO2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' An  A-parameter for G∗ = SO2n+1 is a parameter ψ ∈ ‹Ψ(2n) such that �ψ factors through LG∗ = Sp(2n, C) × WF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We  shall write Ψ(G∗) for the A-parameters for G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Although Arthur[7] defined a set ‹Ψ(G∗) instead of Ψ(G∗), it is not  needed now since NGL(2n,C)(Sp(2n, C)) = Sp(2n, C), cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Thus for ψ ∈ Ψ(SO2n+1), we have the associated  L-embedding from Lψ × SU(2, R) to LSO2n+1 = Sp(2n, C) × WF , which we shall also write �ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We will write Im ψ for  the image of �ψ, in order to treat local and global matters uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us write Φ(G∗) for the set of equivalence  classes of generic A-parameters for G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', Φ(G∗) = Ψ(G∗) ∩ �Φ(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We have another characterization of Ψ(G∗) without Lψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Recall that an irreducible finite dimensional representation  ν of SU(2, R) is symplectic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' orthogonal) if dim ν is even (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' odd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' A self-dual simple parameter µ⊠ν ∈ ‹Ψsim(N)  is said to be symplectic if one of µ and ν is symplectic and the other is orthogonal, and to be orthogonal otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let ψ be a parameter (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6) in ‹Ψ(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then ψ is in Ψ(G∗) if and only if ℓi is even for i ∈ Iψ with orthogonal ψi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For ψ ∈ Ψ(G∗), we put Sψ = Cent(Im �ψ, “  G), Sψ = Sψ/Z(“  G)Γ, Sψ = π0(Sψ), and Sψ = π0(Sψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that the  Galois action is trivial in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We also write Sψ(G) and so on if the group G is to be emphasized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Take a partition  Iψ = I+  ψ ⊔ I−  ψ such that I+  ψ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' I−  ψ ) is the set of i ∈ Iψ such that ψi is symplectic (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' orthogonal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we have  Sψ ∼=  �  i∈I+  ψ  O(ℓi, C) ×  �  i∈I−  ψ  Sp(ℓi, C) ×  �  j∈Jψ  GL(ℓj, C),  and its component group is a free (Z/2Z)-module  Sψ ∼=  �  i∈I+  ψ  (Z/2Z)ai,  17  with a formal basis {ai}, where ai corresponds to ψi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  As in [7, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1], we put  Ψsim(G∗) = � ψ ∈ Ψ(G∗)  �� |Sψ| = 1 � ,  Ψ2(G∗) = � ψ ∈ Ψ(G∗)  �� ��Sψ  �� < ∞ � ,  Ψell(G∗) =  �  ψ ∈ Ψ(G∗)  �� ��Sψ,s  �� < ∞, ∃s ∈ Sψ,ss  �  ,  and  Ψdisc(G∗) = � ψ ∈ Ψ(G∗)  �� ��Z(Sψ)  �� < ∞ � ,  where Sψ,s = Cent(s, Sψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put also Ψ2  ell(G∗) = Ψell(G∗) \\ Ψ2(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Since we have an explicit description of Sψ, we  can get a more explicit characterization of these sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For instance, Ψ2(G∗) is a set of parameters (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6) in ‹Ψ(N) such  that ℓi = 1 for i ∈ I+  ψ and I−  ψ ⊔ Jψ = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us put Φ♥(G∗) = Φ(G∗) ∩ Ψ♥(G∗), for ♥ ∈ {2, sim, ell, disc}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put also  Φ2  ell(G∗) = Φell(G∗) \\ Φ2(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Consider a parameter ψ ∈ Ψ(G∗) ⊂ ‹Ψ(N) of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let v be a place of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The localization of ψ at v is  defined as  ψv = ⊞  i∈Iψ  ℓiψi,v ⊞ ⊞  j∈Jψ  ℓj(ψj,v ⊞ ψ∨  j,v),  where ψk,v = µk,v ⊠νk, and µk,v stands for the L-parameter for the local component µk,v of µk at v, for k ∈ Kψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Since  µk,v is a map from LFv to GL(mk, C), one can see that ψv is a map from LFv × SU(2, R) to GL(2n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2 of [7] and our 1st seed theorem (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5), one can see that the image of ψv is contained in Sp(2n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Thus we obtain our 2nd seed theorem:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6 (2nd seed theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For an A-parameter ψ ∈ Ψ(G∗) and a place v, the equivalence class of ψv is  uniquely determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Hence, we have the localization ψv ∈ Ψ+(G∗  v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then there is a natural map Sψ → Sψv, which induces natural maps  Sψ → Sψv,  Sψ → Sψv,  and  Sψ → Sψv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  If (Ge, se, ηe) ∈ E(G∗) is an endoscopic triple for G∗ = SO2n+1, the endoscopic group Ge is of the form GLn1 × · · ·×  GLnr × SO2m1+1 × SO2m2+1, where r ∈ Z≥0, n1 + · · · + nr + m1 + m2 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then the set of equivalence classes of  A-parameters for Ge is defined as  Ψ(Ge) = Ψ(n1) × · · · × Ψ(nr) × Ψ(SO2m1+1) × Ψ(SO2m2+1),  or equivalently the set of pairs (ψ, �ψe) of a parameter ψ ∈ Ψ(G∗) and an L-embedding �ψe : Lψ × SU(2, R) → LGe with  ηe ◦ �ψe = �ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We have a canonical mapping (ψ, �ψe) �→ ψ from Ψ(Ge) to Ψ(G∗), for which we will write ψe �→ ηe ◦ ψe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The subset of generic parameters is  Φ(Ge) = Φ(n1) × · · · × Φ(nr) × Φ(SO2m1+1) × Φ(SO2m2+1),  and we define Ψ♥(Ge) and Φ♥(Ge) (♥ = 2, disc, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=') similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The component groups and localization are defined  just like the case of GLN or SO2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Now we consider the notion of parameters for non-quasi-split odd special orthogonal groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let G be an inner  form of G∗ = SO2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The notion of parameters for G is the same as that for G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In addition, a parameter ψ ∈ Ψ(G∗)  is said to be G-relevant if it is relevant locally everywhere in the sense of [18, §0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Note that an inner twist  ξ : G∗ → G is uniquely determined by G up to isomorphism, in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' When there is no danger of confusion, we  shall simply call relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In this paper, as in the local case, we never use a symbol Ψ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' As in [18, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7], we have  the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ψ ∈ Ψ(G∗) be a G-relevant parameter and M ∗ ⊂ G∗ a Levi subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If ψ comes from a parameter  for M ∗, then M ∗ transfers to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  18  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4  The bijective correspondence  Here we recall the bijective correspondence (e, ψe) ←→ (ψ, s) from discussions in [7, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4] and [18, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4] in the case  of ordinary endoscopy for SO2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let G∗ be a connected reductive split group over a local field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then Γ acts trivially on ”  G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Two ordinary  endoscopic triples e1 = (Ge  1, se  1, ηe  1) and e2 = (Ge  2, se  2, ηe  2) for G∗ are considered strictly equivalent (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' equivalent) if  se  1 = se  2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' se  1 = zse  2 for some z ∈ Z(”  G∗)) and ηe  1(LGe  1) = ηe  2(LGe  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that the equivalence here is different from  the isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let E(G∗) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' E(G∗)) be a complete system of representatives of strict equivalence (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' equivalence) classes  of endoscopic data for G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We may identify E(G∗) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' E(G∗)) with the set of strict equivalence (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' equivalence)  classes of endoscopic triples for G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' (The set E(G) in [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='246] does not correspond to E(G∗) here, but to E(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=')  Define F(G∗) to be the set of the parameters ψ : LF × SU(2) → LG∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' (Here we mean the actual L-homomorphisms,  not the equivalence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=') For each ψ ∈ F(G∗), we shall write Sψ,ss (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Sψ,ss) for the set of semisimple elements  in Sψ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Sψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Define  X(G∗) = { (e, ψe) | e = (Ge, se, ηe) ∈ E(G∗), ψe ∈ F(Ge) } ,  Y (G∗) = { (ψ, s) | ψ ∈ F(G∗), s ∈ Sψ,ss } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  They are left ”  G∗ × Z(”  G∗)-sets by the following actions: for (g, z) ∈ ”  G∗ × Z(”  G∗),  (g, z) · ((Ge, se, ηe), ψe) = ((Ge  1, se  1, ηe  1), (ηe  1)−1 ◦ Ad(g) ◦ ηe ◦ ψe),  (g, z) · (ψ, s) = (Ad(g) ◦ ψ, zgsg−1),  where (Ge  1, se  1, ηe  1) is an element of E(G∗) that is strictly equivalent to (Ge, zgseg−1, Ad(g) ◦ ηe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Put  X(G∗) = � (e, ψe)  �� e = (Ge, se, ηe) ∈ E(G∗), ψe ∈ F(Ge) � ,  Y (G∗) = � (ψ, s)  �� ψ ∈ F(G∗), s ∈ Sψ,ss  � ,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', X(G∗) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Y (G∗)) is the quotient set Z(”  G∗)\\\\X(G∗) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Z(”  G∗)\\\\Y (G∗)) of X(G∗) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Y (G∗)) by  Z(”  G∗) = {1} × Z(”  G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us put  X(G∗) = ”  G∗\\\\X(G∗),  Y(G∗) = ”  G∗\\\\Y (G∗),  standing for the quotient sets of X(G∗) and Y (G∗) by ”  G∗ = ”  G∗ × {1}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Likewise X (G∗) = ”  G∗\\\\X(G∗),  Y(G∗) = ”  G∗\\\\Y (G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that E(G∗) = ”  G∗\\\\E(G∗) and E(G∗) = ”  G∗\\\\E(G∗) = ”  G∗ × Z(”  G∗)\\\\E(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We can also  see that  X(G∗) =  ¶  (e, ψe)  ��� e = (Ge, se, ηe) ∈ E(G∗), ψe ∈ ‹Ψ(Ge)  ©  ,  Y(G∗) =  ¶  (ψ, s)  ��� ψ ∈ Ψ(G∗), s ∈ Sψ,ss/(”  G∗ − conj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=')  ©  ,  X (G∗) =  ¶  (e, ψe)  ��� e = (Ge, se, ηe) ∈ E(G∗), ψe ∈ ‹Ψ(Ge)  ©  ,  Y(G∗) =  ¶  (ψ, s)  ��� ψ ∈ Ψ(G∗), s ∈ Sψ,ss/(”  G∗ − conj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=')  ©  ,  where ‹Ψ(Ge) is the quotient set of Ψ(Ge) modulo the action of AutG∗(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We use the symbol ‹Ψ here, because the idea  is similar to that of ‹Ψ(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' (See [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=')  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let G∗ be SO2n+1 defined over a local field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then the natural map  X(G∗) → Y (G∗),  (e, ψe) �→ (ηe ◦ ψe, se),  is a ”  G∗ ×Z(G∗)-equivariant bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Here se and ηe are the second and third elements of e respectively, Furthermore,  this induces a ”  G∗-equivariant bijection X(G∗) ≃ Y (G∗), a Z(”  G∗)- equivariant bijection X(G∗) ≃ Y(G∗), and a  bijection X (G∗) ≃ Y(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  19  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Recall that E(G∗) is a complete system of representatives, not just a quotient set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The map X(G∗) → Y (G∗)  is trivially well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The equivariance under the actions of ”  G∗ × Z(G∗) follows immediately from the definition of  the actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The injectivity also follows easily from the definition of strict equivalence of endoscopic triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us  show the surjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let (ψ, s) ∈ Y (G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put Ge = Cent(s, ”  G∗) · ψ(LF × SU(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We know that Cent(s, ”  G∗) is of the  form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let G′ be the group (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then its dual group �  G′ is isomorphic to Cent(s, ”  G∗), with the trivial Galois  action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Thus we have Ge ≃ LG′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us write η′ for an injective map LG′ ≃ Ge ֒→ LG∗, to obtain an endoscopic  triple (G′, s, η′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let e = (Ge, se, ηe) ∈ E(G∗) be the unique element that is strictly equivalent to (G′, s, η′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Since  Im ψ ⊂ Ge = η′(LG′) = ηe(LGe), there exists a parameter ψe ∈ F(Ge) such that ψ = ηe ◦ ψe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we obtain  (e, ψe) ∈ X(G∗), and get a map from Y (G∗) to X(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' This shows the surjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The latter assertion follows from the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let F be a number field, and G∗ = SO2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For an endoscopic triple e for G∗, every element in Ψ(Ge) can  be regarded as an element in Ψ(G∗) in the natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For two parameters ψ1, ψ2 ∈ Ψ(Ge), we write ψ1 ∼ ψ2 if  ηe ◦ ψ1 = ηe ◦ ψ2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', they are same in Ψ(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ‹Ψ(Ge) = Ψ(Ge)/ ∼ be the quotient set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The reason why we use  the symbol ‹Ψ is because the idea is similar to that of ‹Ψ(G∗) (see [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Define  X(G∗) =  ¶  (e, ψe)  ��� e = (Ge, se, ηe) ∈ E(G∗), ψe ∈ ‹Ψ(Ge)  ©  ,  Y(G∗) =  ¶  (ψ, s)  ��� ψ ∈ Ψ(G∗), s ∈ Sψ,ss/(”  G∗ − conj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=')  ©  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  As in the local case, we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let G∗ = SO2n+1 be the global split odd special orthogonal group over a global field F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then the natural  map  X(G∗) → Y(G∗),  (e, ψe) �→ (ψe, se)  is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8, with �ψ(Lψ × SU(2)) in place of ψ(LF × SU(2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5  Main theorems  In this subsection we recall the statements of the endoscopic classification of representations of the odd special  orthogonal groups, which will be proved for the generic part in this paper, and was already proven for the quasi-split  case by Arthur [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let first F be a local field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Fix a nontrivial additive character ψF : F → C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let G∗ = SO2n+1 be the split special  orthogonal group of size 2n + 1 defined over F, equipped with the fixed pinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' It can be regarded as a twisted  endoscopic group of (GL2n, θ2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Arthur proved the existence of the stable linear form satisfying the twisted ECR:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='10 ([7, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='(a)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ψ ∈ Ψ(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then there is a unique stable linear form  H(G∗) ∋ f �→ f G∗(ψ) ∈ C,  satisfying the twisted endoscopic character relation [7, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We sometimes write f(ψ) for f G∗(ψ) if there is no danger of confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let (ξ, z) : G∗ → G be a pure inner twist of G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We now state the main local classification theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Theorem* 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ψ ∈ Ψ(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' There is a finite multiset Πψ(G) of irreducible unitary representations of  G(F), (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', a finite set over Πunit(G) in Arthur’s terminology,) satisfying the local and global theorems below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We sometimes write Πψ = Πψ(G) if there is no danger of confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If ψ = φ ∈ Φbdd(G∗), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', ψ is generic,  then the set Πφ is empty if and only if φ is not G-relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In general, Πψ is empty if ψ is not G-relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The  set is called the local A-packet or simply the packet for ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' When ψ = φ is generic, it is also called the L-packet  of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  20  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The local A-packet is equipped with a map  Πψ(G) → Irr(Sψ, χG),  π �→ ⟨−, π⟩,  satisfying the local and global theorems below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The packets Πψ(G) and the pairing ⟨−, π⟩ are independent of the  choice of a inner twist (ξ, z), and determined only by the isomorphism classes of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' (ECR) Let e = (Ge, se, ηe) be an endoscopic triple for G with se ∈ Sψ, and ψe ∈ Ψ(Ge) a parameter for Ge such  that ψ = ηe ◦ ψe is a parameter for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If f ∈ H(G) and f e ∈ H(Ge) are matching, then we have  f e(ψe) = e(G)  �  π∈Πψ(G)  ⟨sψse, π⟩f(π),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7)  where the left hand side is the stable linear form given by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ψ = φ ∈ Φbdd(G∗) be a generic parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then the packet Πφ(G) is multiplicity free, and Πφ(G) ⊂  Πtemp(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Moreover, if F is non-archimedean (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  archimedean), then the map Πφ(G) → Irr(Sφ, χG) is  bijective (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' injective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' (LLC) As φ runs over Φbdd(G∗) the sets Πφ(G) are disjoint, and we have  �  φ∈Φ2,bdd(G∗)  Πφ(G) = Π2,temp(G),  �  φ∈Φbdd(G∗)  Πφ(G) = Πtemp(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let ψ ∈ Ψ+(G∗) ∩ Ψ+  unit(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If it is not G-relevant, define Πψ(G) to be empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Suppose it is G-relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then  one can choose an inner twist ξ : G∗ → G, standard parabolic subgroups P ∗ ⊂ G∗ and P = ξ(P ∗) ⊂ G over F, the  Levi subgroups M ∗ ⊂ P ∗ and M = ξ(M ∗) ⊂ P over F, the open chamber U ⊂ a∗  M associated to P, a point λ ∈ U,  and a parameter ψM∗ ∈ Ψ(M ∗), such that ψ is the image of the twist ψM∗,λ of ψM∗ by λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Suppose that the packet  ΠψM∗(M) and a map ΠψM∗(M) → Irr(SψM∗, χM) is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we can define the packet for ψ and the associated  map by  Πψ(G) = { IP (πM,λ) | πM ∈ ΠψM∗(M) } ,  ⟨−, IP (πM,λ)⟩ = ⟨−, πM⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Note that Sψ = SψM∗ and χG = χM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' See [7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='44-46] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let next F be a global field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Fix a nontrivial additive character ψF : F\\AF → C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let G∗ = SO2n+1 be the split  special orthogonal group of size 2n + 1 defined over F, equipped with the fixed pinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let (ξ, z) : G∗ → G be a pure  inner twist of G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' There exists a decomposition  L2  disc(G(F)\\G(AF )) =  �  ψ∈Ψ(G∗)  L2  disc,ψ(G(F)\\G(AF )),  where L2  disc,ψ(G(F)\\G(AF )) is a full near equivalence class of irreducible representations π = ⊗vπv in the discrete  spectrum such that the L-parameter of πv is φψv for almost all v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  This theorem will be proved in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For any global A-parameter ψ ∈ Ψ(G∗) and a place v of F, we have the localization ψv ∈ Ψ+(G∗  v) ∩ Ψ+  unit(2n)v  thanks to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we have the packet Πψv(Gv) and the map Πψv(Gv) → Irr(Sψv, χGv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The global  packet for ψ is defined as  Πψ(G) =  �  π =  �  v  πv  ����� πv ∈ Πψv(Gv),  ⟨−, πv⟩ = 1 for almost all v  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  If a local packet Πψv(Gv) is empty for some v, then the global packet is defined to be empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For any π = ⊗vπv ∈  Πψ(G), we define a character ⟨−, π⟩ of Sψ by  ⟨x, π⟩ =  �  v  ⟨xv, πv⟩,  x ∈ Sψ,  21  where xv denotes the image of x under the natural map Sψ → Sψv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The restriction of this character to Z(“  G) is  �  v χGv, which is trivial by the product formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Thus we may regard it as a character of Sψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We have the  character εψ : Sψ → {±1} defined by Arthur [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' When we emphasize that the parameter ψ is for G∗ (or  equivalently for G), we shall write εG∗  ψ  or εG  ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that εψ is trivial if ψ is generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put  Πψ(G, εψ) = { π ∈ Πψ(G) | ⟨−, π⟩ = εψ } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We now state the main global classification theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Theorem* 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 (AMF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For ψ ∈ Ψ(G∗), we have  L2  disc,ψ(G(F)\\G(AF )) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  �  π∈Πψ(G,εψ)  π,  if ψ ∈ Ψ2(G∗),  0,  if ψ /∈ Ψ2(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Arthur [7] proved Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='11, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='12, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 for the case G = G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In this paper, following Kaletha-Minguez-  Shin-White [18], we will give a proof of Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='11, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='12, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 for the case ψ is generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The main theorem of  this paper is stated as follows:  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='14 (main theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let F be a local or global field and G an inner form of G∗ = SO2n+1 over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let F be local.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ψ = φ ∈ Φbdd(G∗) be a generic parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then part 1, 2, 3 of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='11 hold true for  φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, part 4 and 5 also hold true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let F be global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='12 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let moreover ψ = φ ∈ Φ(G∗) be a generic parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then Theorem  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 holds true for φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  As stated in Introduction, Arthur [7] also established the endoscopic classification of representations of symplectic  groups and quasi-split even special orthogonal groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In the case of symplectic groups, he proved theorems similar  to Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='11, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='12, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' On the other hand, in the case of quasi-split even special orthogonal groups, he  proved the similar theorems up to outer automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For a quasi-split even orthogonal group G over a local field  F, two representations π and π′ are said to be ǫ-equivalence if there exists ρ ∈ AutF (G) such that π ≃ π′ ◦ ρ, and  the sets of ǫ-equivalence classes is denoted by �Π instead of Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Arthur defined a weak local A-packet Πψ(G) for each  ǫ-equivalence class of A-parameters ψ ∈ ‹Ψ(G), and then he proved a theorem similar to but weaker than Theorem  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='11 in that Ψ and Π are replaced by ‹Ψ and �Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For a quasi-split even orthogonal group G over a number field F, two  representations π = �  v πv and π′ = �  v π′  v are said to be ǫ-equivalence if πv and π′  v are so for all places v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The weak  global packet �Πψ(G) for ψ ∈ ‹Ψ(G) is defined as the set of π = �  v πv such that πv ∈ �Πψv(Gv) for all v and ⟨−, πv⟩  is trivial for almost all v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' As in the local case, he proved theorems similar to but weaker than Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='12 and  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 in that Ψ and Π are replaced by ‹Ψ and �Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' See also [8] for the weak theorems for even orthogonal groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' These  classification theorems for symplectic and even special orthogonal groups are needed for the globalization in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2  In the rest of this paper, we shall proof the main theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='14 by a long induction argument following [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let  n be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' As an induction hypothesis, we assume that the Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='14 (or Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='11, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='12, and  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13) holds for any positive integer n0 < n, and hence for any proper Levi subgroup M ∗ ⊊ G∗ = SO2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4  Local intertwining relation  In this section, we define the normalized local intertwining operator and state the local intertwining relation, which  is a key theorem in the theory of the endoscopic classification of representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' After that, we reduce the proof LIR,  and give a proof of LIR in the special (real and low rank) cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1  The diagram  We shall describe explicitly the basic commutative diagram ([7, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3)], [18, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1)]) for odd special orthogonal  groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' See [18, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1] for the diagram for general connected reductive groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let F be a local or global field, G∗ = SO2n+1, and M ∗ ⊂ G∗ a standard Levi subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ξ : G∗ → G be an  inner twist over F with a Levi subgroup M = ξ(M ∗) ⊂ G which is an inner form of M ∗ over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ψ ∈ Ψ(M ∗) be a  22  parameter, and ψG ∈ Ψ(G∗) the image of ψ under the natural map induced by �  M ֒→ “  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Recall that “  G = Sp(2n, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The parameter ψG ∈ Ψ(G∗) and its centralizer group Sψ(G) = SψG(G) can be written as  ψG =  �  i∈I+  ψ  ℓiψi ⊕  �  i∈I−  ψ  ℓiψi ⊕  �  j∈Jψ  ℓj(ψj ⊕ ψ∨  j ),  Sψ(G) = Cent(Im ψ, Sp(2n, C)) ≃  �  i∈I+  ψ  O(ℓi, C) ×  �  i∈I−  ψ  Sp(ℓi, C) ×  �  j∈Jψ  GL(ℓj, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' we can also write as  ψG =  r  �  t=1  et  \uf8ee  \uf8f0�  i∈I+  t  ℓt  iψi ⊕  �  i∈I−  t  ℓt  iψi ⊕  �  j∈Jt  (ℓt  jψj ⊕ ℓt∨  j ψ∨  j )  \uf8f9  \uf8fb  ⊕  \uf8ee  \uf8f0�  i∈I+  0  ℓ0  i ψi ⊕  �  i∈I−  0  ℓ0  i ψi ⊕  �  j∈J0  ℓ0  j(ψj ⊕ ψ∨  j )  \uf8f9  \uf8fb  ⊕  r  �  t=1  et  \uf8ee  \uf8f0�  i∈I+  t  ℓt  iψi ⊕  �  i∈I−  t  ℓt  iψi ⊕  �  j∈Jt  (ℓt∨  j ψj ⊕ ℓt  jψ∨  j )  \uf8f9  \uf8fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  �  M ≃  r  �  t=1  GL(kt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C)et × Sp(2n0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  where et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' kt are positive integers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  I+  t ⊂ I+  ψ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  I−  t ⊂ I−  ψ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Jt ⊂ Jψ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  for t = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', r,  are subsets of indices, and ℓt  i, ℓt  j are positive integers such that  \uf8f1  \uf8f2  \uf8f3  ℓt  i ≤ ℓi  2 ,  ℓt  j, ℓt∨  j  ≤ ℓj,  ℓt  j + ℓt∨  j  ≤ ℓj,  for t = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', r,  and  ¶  ((ℓt  i)i∈I+  t , (ℓt  i)i∈I−  t , (ℓt  j)j∈Jt, (ℓt∨  j )j∈Jt)  ��� t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , r  ©  is mutually distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put  ψt =  �  i∈I+  t  ℓt  iψi ⊕  �  i∈I−  t  ℓt  iψi ⊕  �  j∈Jt  (ℓt  jψj ⊕ ℓt∨  j ψ∨  j ),  for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , r,  ψ0 =  �  i∈I+  0  ℓ0  i ψi ⊕  �  i∈I−  0  ℓ0  i ψi ⊕  �  j∈J0  ℓ0  j(ψj ⊕ ψ∨  j ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then we have  ψG =  r  �  t=1  etψt ⊕ ψ0 ⊕  r  �  t=1  etψ∨  t ,  ψ =  r  �  t=1  etψt ⊕ ψ0,  and each ψt corresponds to GL(kt, C) ⊂ �  M, and ψ0 corresponds to Sp(2n0, C) ⊂ �  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Here we note that  k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , kr are not necessarily mutually distinct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  23  ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , ψr are mutually distinct;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , ψr, ψ∨  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , ψ∨  r are not necessarily mutually distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let A �  M be the maximal central split torus of �  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then �  M = Cent(A �  M, “  G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' One has  A �  M ≃  r  �  t=1  (C×)et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Put  Sψ(G) = Cent(Im ψ, “  G),  Sψ(M) = Cent(Im ψ, �  M),  Nψ(M, G) = Sψ(G) ∩ N(A �  M, “  G) = N(A �  M, Sψ(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Put moreover  Wψ(M, G) = N(A �  M, Sψ(G))  Z(A �  M, Sψ(G)) ,  W ◦  ψ(M, G) = N(A �  M, Sψ(G)◦)  Z(A �  M, Sψ(G)◦) ,  Nψ(M, G) = N(A �  M, Sψ(G))  Z(A �  M, Sψ(G)◦),  Sψ(M, G) = N(A �  M, Sψ(G))  N(A �  M, Sψ(G)◦),  Rψ(M, G) =  N(A �  M, Sψ(G))  N(A �  M, Sψ(G)◦) · Z(A �  M, Sψ(G)),  and  Sψ(G) = π0(Sψ(G)) ≃  �  i∈I+  ψ  O(ℓi, C)/ SO(ℓi, C) ≃  �  i∈I+  ψ  (Z/2Z)ai,  Sψ(M) = π0(Sψ(M)) ≃  �  i∈I+  0  O(ℓ0  i , C)/ SO(ℓ0  i , C) ≃  �  i∈I+  0  (Z/2Z)ai,  where {ai}i is a formal basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Direct calculations show that  Z(A �  M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Sψ(G)◦) ≃  r  �  t=1  \uf8ee  \uf8f0 �  i∈I+  t  GL(ℓt  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  i∈I−  t  GL(ℓt  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  j∈Jt  (GL(ℓt  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) × GL(ℓt∨  j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C))  \uf8f9  \uf8fb  et  ×  \uf8ee  \uf8f0 �  i∈I+  0  SO(ℓ0  i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  i∈I−  0  Sp(ℓ0  i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  j∈J0  GL(ℓ0  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C)  \uf8f9  \uf8fb ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Z(A �  M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Sψ(G)) ≃  r  �  t=1  \uf8ee  \uf8f0 �  i∈I+  t  GL(ℓt  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  i∈I−  t  GL(ℓt  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  j∈Jt  (GL(ℓt  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) × GL(ℓt∨  j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C))  \uf8f9  \uf8fb  et  ×  \uf8ee  \uf8f0 �  i∈I+  0  O(ℓ0  i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  i∈I−  0  Sp(ℓ0  i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  j∈J0  GL(ℓ0  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C)  \uf8f9  \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  In particular, the connected component of the identity in Z(A �  M, Sψ(G)) is equal to Z(A �  M, Sψ(G)◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Thus, we have  �  M ∩ Sψ(G)◦ = Sψ(M)◦,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1)  Z(A �  M, Sψ(G))  Z(A �  M, Sψ(G)◦) ≃ Sψ(M) ≃  �  i∈I+  0  (Z/2Z)ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2)  24  We now consider the group N(A �  M, Sψ(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Take a partition  { 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', r } = T1 ⊔ T2 ⊔ T3 ⊔ T4  such that  T1 = { t | ψt ≃ ψ∨  t } ,  T2 = { t | ψt ̸≃ ψ∨  t′ for any 1 ≤ t′ ≤ r } ,  T3 = { t | ψt ≃ ψ∨  t′ for a unique t′ ∈ T4 } ,  T4 = { t | ψt ≃ ψ∨  t′ for a unique t′ ∈ T3 } ,  and for each t ∈ T3, put e∨  t = et′, where t′ is the element of T4 such that ψt ≃ ψ∨  t′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Before the description of N(A �  M, Sψ(G)), we must fix notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For a finite set X, let SX denote its symmetric  group, and put Sm = S{1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=',m} for a positive integer m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For any positive integers e, e∨, let  W(e, e∨)  be a subgroup of Se+e∨ ⋉ (Z/2Z)e+e∨ generated by  S{ 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=',e }, S{ e+1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=',e+e∨ }, and  ®  (h, h∨) ⋉ (· · · , 0,  h  1, 0, · · · , 0,  h∨  1 , 0, · · · )  ���� 1 ≤ h ≤ e, e + 1 ≤ h∨ ≤ e + e∨  ´  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then we obtain an explicit description of the group N(A �  M, Sψ(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We have  N(A �  M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Sψ(G)) ≃ c′ ×  Ñ  �  i∈I+  0  O(ℓ0  i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ×  �  i∈I−  0  Sp(ℓ0  i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  j∈J0  GL(ℓ0  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C)  é  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  where a subgroup c′ of Sp(2(n − n0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) can be written as  c′ ≃  �  t∈T1  \uf8ee  \uf8f0  �  wt∈Set⋉(Z/2Z)et  wt ·  Ñ  �  i∈I+  t  GL(ℓt  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  i∈I−  t  GL(ℓt  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  j∈Jt  (GL(ℓt  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) × GL(ℓt∨  j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C))  éet\uf8f9  \uf8fb  ×  �  t∈T2  \uf8ee  \uf8f0  �  wt∈Set  wt ·  Ñ  �  i∈I+  t  GL(ℓt  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  i∈I−  t  GL(ℓt  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  j∈Jt  (GL(ℓt  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) × GL(ℓt∨  j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C))  éet\uf8f9  \uf8fb  ×  �  t∈T3  \uf8ee  \uf8ef\uf8f0  �  wt∈W(et,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e∨  t )  wt ·  Ñ  �  i∈I+  t  GL(ℓt  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  i∈I−  t  GL(ℓt  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  j∈Jt  (GL(ℓt  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) × GL(ℓt∨  j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C))  éet+e∨  t \uf8f9  \uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Therefore we have natural isomorphisms  Wψ(M, G) ≃  �  t∈T1  (Set ⋉ (Z/2Z)et) ×  �  t∈T2  Set ×  �  t∈T3  W(et, e∨  t ),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3)  Nψ(M, G) ≃ Sψ(M) × Wψ(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4)  We finally consider N(A �  M, Sψ(G)◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' It is too complicated to write explicitly, but we can describe the natural  surjection  Nψ(M, G) = N(A �  M, Sψ(G))  Z(A �  M, Sψ(G)◦) −→ Sψ(M, G) = N(A �  M, Sψ(G))  N(A �  M, Sψ(G)◦),  to understand the group N(A �  M, Sψ(G)◦) in terms of its kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For e ≥ 1, let us write x for a group homomorphism  Se ⋉ (Z/2Z)e −→ Z/2Z,  σ ⋉ (dh)e  h=1 �→  e  �  h=1  dh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  25  For each i ∈ I+  ψ , we shall define a corresponding function  xi : Nψ(M, G) −→ Z/2Z  as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By the equations (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2), we may identify Nψ(M, G) with  �  i∈I+  0  (Z/2Z)ai ×  �  t∈T1  (Set ⋉ (Z/2Z)et) ×  �  t∈T2  Set ×  �  t∈T3  W(et, e∨  t ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5)  For any element  u =  Ñ  �  i∈I+  0  ciai, (wt)t∈T1, (wt)t∈T2, (wt)t∈T3  é  of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5), we define  xi(u) = ci +  �  t∈T1⊔T3  ℓt  ix(wt),  where we put ci = 0 for i /∈ I+  0 , and ℓt  i = 0 for i /∈ I+  t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Now, we can describe the natural surjection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us define a  homomorphism xψ : Nψ(M, G) → �  i∈I+  ψ (Z/2Z)ai = Sψ(G) by  xψ(u) =  �  i∈I+  ψ  xi(u)ai  for u ∈ Nψ(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then Sψ(M, G) and W ◦  ψ(M, G) can be identified with the image and the kernel of xψ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Put O+(l, C) = SO(l, C) and O−(l, C) = O(l, C) \\ SO(l, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The homomorphism xψ gives a description:  N(A �  M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Sψ(G)◦) ≃  �  u  {  �  t∈T1  \uf8ee  \uf8f0wt ·  Ñ  �  i∈I+  t  GL(ℓt  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  i∈I−  t  GL(ℓt  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  j∈Jt  (GL(ℓt  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) × GL(ℓt∨  j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C))  éet\uf8f9  \uf8fb  ×  �  t∈T2  \uf8ee  \uf8f0wt ·  Ñ  �  i∈I+  t  GL(ℓt  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  i∈I−  t  GL(ℓt  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  j∈Jt  (GL(ℓt  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) × GL(ℓt∨  j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C))  éet\uf8f9  \uf8fb  ×  �  t∈T3  \uf8ee  \uf8ef\uf8f0wt ·  Ñ  �  i∈I+  t  GL(ℓt  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  i∈I−  t  GL(ℓt  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  j∈Jt  (GL(ℓt  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) × GL(ℓt∨  j ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C))  éet+e∨  t \uf8f9  \uf8fa\uf8fb  ×  \uf8ee  \uf8f0 �  i∈I+  0  Oεi(ℓ0  i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ×  �  i∈I−  0  Sp(ℓ0  i ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C) ×  �  j∈J0  GL(ℓ0  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' C)  \uf8f9  \uf8fb},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  where u =  Ä�  i∈I+  0 ciai,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' (wt)t∈T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' (wt)t∈T2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' (wt)t∈T3  ä  runs over ker(xψ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' and εi = (−1)ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We have obtained the explicit description of the following commutative diagram with exact rows and columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  1  1  \uf8e6\uf8e6�  \uf8e6\uf8e6�  W ◦  ψ(M, G)  W ◦  ψ(M, G)  \uf8e6\uf8e6�  \uf8e6\uf8e6�  1 −−−−→ Sψ(M) −−−−→ Nψ(M, G) −−−−→ Wψ(M, G) −−−−→ 1  ���  xψ  \uf8e6\uf8e6�  \uf8e6\uf8e6�  1 −−−−→ Sψ(M) −−−−→ Sψ(M, G) −−−−→ Rψ(M, G) −−−−→ 1  \uf8e6\uf8e6�  \uf8e6\uf8e6�  1  1  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6)  26  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If ψ ∈ Ψ2(M ∗), then xψ is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In particular, we have Sψ(M, G) = Sψ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Suppose that ψ ∈ Ψ2(M ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then Kt := I+  t ⊔I−  t ⊔Jt is a singleton for t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , r, and ℓt  i = 1 for t = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let i ∈ I+  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If we put u = (ai, 1, 1, 1) ∈ Nψ(M, G), then xψ(u) = ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let i ∈ I+  t (t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then t ∈ T1, as ψt = ψi  is self-dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If we put u = (0, (ws)s∈T1, 1, 1) ∈ Nψ(M, G), where wt = 1 ⋉ (1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , 0) and wsis trivial if s ̸= t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then  xψ(u) = ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Since I+  ψ is the union of I+  0 , I+  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , T +  r , this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Dividing the diagram (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6) by Z(“  G)Γ = {±1}, we obtain another diagram  1  1  \uf8e6\uf8e6�  \uf8e6\uf8e6�  W ◦  ψ(M, G)  W ◦  ψ(M, G)  \uf8e6\uf8e6�  \uf8e6\uf8e6�  1 −−−−→ Sψ(M) −−−−→ Nψ(M, G) −−−−→ Wψ(M, G) −−−−→ 1  ���  \uf8e6\uf8e6�  \uf8e6\uf8e6�  1 −−−−→ Sψ(M) −−−−→ Sψ(M, G) −−−−→ Rψ(M, G) −−−−→ 1  \uf8e6\uf8e6�  \uf8e6\uf8e6�  1  1  where  Sψ(M) = Sψ(M)  Z(“  G)Γ ,  Nψ(M, G) = Nψ(M, G)  Z(“  G)Γ  ,  Sψ(M, G) = Sψ(M, G)  Z(“  G)Γ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2  The first intertwining operator  Let F be a local field, ψF : F → C1 a nontrivial additive character, G∗ = SO2n+1, and M ∗ ⊊ G∗ a proper standard  Levi subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ξ : G∗ → G be an inner twist over F which restrict to an inner twist ξ|M∗ : M ∗ → M over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let  ψ ∈ Ψ(M ∗) be a local parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that ψ is M-relevant, and the corresponding packet Πψ(M) is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let π ∈ Πψ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We shall write Vπ for the representation space of π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let P and P ′ be parabolic subgroups of G defined over F with common Levi factor M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We write (IP (π), HP (π))  for the representation parabolically representation by (π, Vπ) from P to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  A function HM : M(F) → aM is defined by  exp(⟨HM(m), χ⟩) = |χ(m)|F ,  for any χ ∈ X∗(M)F , m ∈ M(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then each λ ∈ a∗  M,C gives a character  M(F) → C×,  m �→ exp(⟨HM(m), λ⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We shall write πλ for the tensor product of π and this character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then the unnormalized intertwining operator  JP ′|P (ξ, ψF ) : HP (πλ) → HP ′(πλ) is given by  [JP ′|P (ξ, ψF )f](g) =  �  N(F )∩N ′(F )\\N ′(F )  f(n′g)dn′,  where N and N ′ are the unipotent radical of P and P ′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' It is known that the integral converges absolutely  when the real part of λ lies in a certain open cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The Haar measure dn′ on N(F) ∩ N ′(F)\\N ′(F) that we use here  is the measure [18, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2] defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Although the case of unitary groups is considered in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', we apply the same  definition with the same notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  27  We write WF ∋ w �→ |w|λ for the L-parameter of the character M(F) ∋ m �→ exp(⟨HM(m), λ⟩) attached to  λ ∈ a∗  M,C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then ψλ = ψ| − |λ is a parameter of πλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ρP ′|P be the adjoint representation of �  M on �n′ ∩ �n\\�n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put  rP ′|P (ξ, ψλ, ψF ) =  L(0, ρ∨  P ′|P ◦ φψλ)  L(1, ρ∨  P ′|P ◦ φψλ)  ǫ( 1  2, ρ∨  P ′|P ◦ φψλ, ψF )  ǫ(0, ρ∨  P ′|P ◦ φψλ, ψF ) ,  where the factors are the Artin L- and ǫ- factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The normalized intertwining operator is given by  RP ′|P (ξ, ψλ) = rP ′|P (ξ, ψλ, ψF )−1JP ′|P (ξ, ψF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  It is known that RP ′|P (ξ, ψλ) is independent of the choice of ψF , and that λ �→ RP ′|P (ξ, ψλ) has meromorphic  continuation to whole a∗  M,C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that F = R and ψ = φ ∈ Φ(M ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then the function λ �→ RP ′|P (ξ, φλ) has neither a zero nor  a pole at λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, let P ′′ ⊂ G be a parabolic subgroup defined over F with Levi factor M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we have  RP ′′|P (ξ, φ) = RP ′′|P ′(ξ, φ) ◦ RP ′|P (ξ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The idea of the proof is the same as that of [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' So it suffices to show that the measure used in  the definition of JP ′|P (ξ, ψF ) coincides with the measure introduced by Arthur [5, §3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let us calculate the measure given in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Since F = R, there exist nonnegative integers p and q with  p + q = 2n + 1 such that the inner twist ξ : G∗ → G is realized by ξp,q : SO2n+1 → SO(p, q) given in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' So,  we will use the notation introduced there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let B be a SO(p, q)-invariant bilinear form on gC = so(p, q)C given by  B(X, Y ) = 1  2 tr(XY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then the quadratic form  X �→ −B(X, θX)  is positive definite on gR = so(p, q)R, where θ is the Cartan involution defined by θ(X) = − tX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The straightforward  calculation shows that the constant αP ′|P defined in [5, §3] is equal to 2  k  2 , where k is the number of roots of the form  χi whose restriction to aM are roots of both (P ′, AM) and (P, AM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Here, AM and P denote the maximal central  split torus in M and the parabolic subgroup opposite to P containing M, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Next, we calculate the Euclidean measure dX defined by the quadratic form X �→ −B(X, θX) ([5, §3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put  T = ξ(T ∗), and write t for its Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we have a decomposition  gC = tC ⊕  �  a∈R(T,G)/Γ  ga,C,  where Γ is the Galois group of F/F = C/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that each ga,C = ⊕α∈agα is defined over R, but gα is not necessarily  defined over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let a ∈ R(T, G)/Γ and α ∈ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If α is a complex root, then a = {α, σα}, and dim ga = 2, where  σ ∈ Γ is the complex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In this case, ga has an R-basis {X1, X2}, where X1 = ξ(Xα) + σ(ξ(Xα)) and  X2 = −√−1(ξ(Xα) − σ(ξ(Xα))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By a straightforward calculation, we have −B(X1, θX1) = −B(X2, θX2) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If  gα ⊂ n ∩ n′, the contribution of a to the measure dX is |d(2− 1  2 X1 ∧ 2− 1  2 X2)| = |d(ξ(Xα ∧ Xσα))|, which is equal to  the contribution to our measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If α is a real root, then a = {α}, and dim ga = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In this case, ga has an R-basis  {X1}, where X1 = ξ(Xα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By a straightforward calculation, we have −B(X1, θX1) = 1 if α is of the form χi ± χj, and  −B(X1, θX1) = 2 if α is of the form χi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If gα ⊂ n ∩ n′, the contribution of a to the measure dX is |dX1| = |d(ξ(Xα))|  if α is of the form χi ± χj, and |d(2− 1  2 X1)| = 2− 1  2 |d(ξ(X1))| if α is of the form χi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The product of coefficients 2− 1  2 for  all α of the form χi and αP ′|P = 2  k  2 cancel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If α is an imaginary root, then gα ⊈ n ∩ n′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that F is a p-adic field and ψ = φ ∈ Φ(M ∗) an L-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then the function λ �→  RP ′|P (ξ, φλ) has neither a zero nor a pole at λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, let P ′′ ⊂ G be a parabolic subgroup defined over F with  Levi factor M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we have  RP ′′|P (ξ, φ) = RP ′′|P ′(ξ, φ) ◦ RP ′|P (ξ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that the proofs of Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6 are independent from this lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In particular, we may use  them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let π ∈ Πφ(M) be an arbitrary representation of M corresponding to φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The operator RP ′|P (ξ, φλ) is an inter-  twining operator from HP (πλ) to HP ′(πλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By the reduction steps described in [5, §2] (more precisely, from the last  28  line in p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='29 of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ), we may assume that π is supercuspidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ˙F, u, v2, ˙G∗, ˙G, ˙ξ, ˙M ∗, ˙M, and ˙π be as in Lemma  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Furthermore, let ˙P, ˙P ′, and ˙P ′′ be the parabolic subgroups of ˙G such that ˙Pu = P, ˙P ′  u = P ′, and ˙P ′′  u = P ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Since we can choose v2 ̸= u arbitrarily, we may assume that v2 is a real place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By the induction hypothesis, we can  take the A-parameter of ˙π, write ˙φ for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that ˙φ is a generic parameter because ˙φu = φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The global intertwining operator R ˙P ′| ˙P ( ˙πλ, ˙φλ) has neither a zero nor a pole at λ = 0, and is multiplicative in ˙P ′,  ˙P by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For any place v ̸= u, v2, ˙Gv splits over ˙Fv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For u = v2, we have ˙Fv2 = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Thus for any v ̸= u, the  local normalized intertwining operator R ˙P ′v| ˙Pv( ˙ξv, ˙φλ,v) has neither a zero nor a pole at λ = 0, and is multiplicative by  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2 and the assumption from the split case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Now the assertion follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The following lemma is now proven for generic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In general, we expect it can be proven as [18, Lemma  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma* 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let F be a local field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ψ ∈ Ψ(M ∗) be a general local A-parameter for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then the function  λ �→ RP ′|P (ξ, ψλ) has neither a zero nor a pole at λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, let P ′′ ⊂ G be a parabolic subgroup defined over  F with Levi factor M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we have  RP ′′|P (ξ, ψ) = RP ′′|P ′(ξ, ψ) ◦ RP ′|P (ξ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3  The second intertwining operator  We maintain the assumptions of the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For every finite separable extension E/F, Langlands [25,  Theorem 1] defined a complex number λ(E/F, ψF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By the definition, we have λ(F/F, ψF ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For w ∈ W(T ∗, G∗), Keys-Shahidi [19, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1)] defined the constant λ(w, ψF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In this subsection, we shall construct  a local intertwining operator denoted by ℓP ′  P (w, ξ, ψ, ψF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In the case of unitary groups, its construction involves the  λ-factor as in [18, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' However, in the case of odd special orthogonal groups, we do not need the λ-factor since it  always vanishes:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For any w ∈ W(T ∗, G∗), we have  λ(w, ψF ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By the definition of λ(w, ψF ), it suffices to show that  λ(Fa/F, ψF ) = 1,  for any root a ∈ R(T ∗, G∗), where Fa/F is a finite extension such that G∗  a,sc = ResFa/F (SL2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Here, G∗  a,sc denotes  the simply connected cover of the derived group G∗  a,der of the Levi subgroup G∗  a of semisimple rank 1 attached to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Since we know that λ(F/F, ψF ) = 1, it suffices to show that Fa = F for all root a ∈ R(T ∗, G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  If a is a long root (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', of the form ±χi ± χj), then G∗  a is isomorphic to a product of GL2 and a finite number of  GL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Thus we have G∗  a,der ∼= SL2 and G∗  a,sc ∼= SL2, which means that Fa = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If a is a short root (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', of the form  ±χi), then G∗  a is isomorphic to a product of SO3 and a finite number of GL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Thus we have G∗  a,der ∼= SO3 ∼= PGL2  and G∗  a,sc ∼= SL2, which means that Fa = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let ψ ∈ Ψ(M ∗) be a local parameter such that ψ is M-relevant and the corresponding packet Πψ(M) is not empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let π ∈ Πψ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, let P and P ′ be parabolic subgroups of G defined over F with common Levi factor  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For an element w ∈ W(M ∗, G∗) ∼= W(M, G), we write �w ∈ N(M ∗, G∗) for its Langlands-Shelstad lift, and put  ˘w = ξ( �w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By replacing ξ by an equivalent inner twist if necessary, we assume that ˘w ∈ G(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' This gives us another  representation  [ ˘wπ](m) = π( ˘w−1m ˘w),  m ∈ M(F),  of M(F) on the same vector space as π, which corresponds to a parameter wψ := Ad(w) ◦ ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  First, we define an unnormalized intertwining operator ℓP ′( ˘w) from HP ′w(π) to HP ′( ˘wπ) by  [ℓP ′( ˘w)f](g) = f( ˘w−1g),  where P ′w denotes w−1P ′w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Next, our normalizing factor is defined by  ǫP (w, ψ, ψF ) = ǫ(1  2, ρ∨  P w|P ◦ φψ, ψF ),  and we define  ℓP ′  P (w, ξ, ψ, ψF ) = ǫP (w, ψ, ψF )ℓP ′( ˘w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  29  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For any w1, w2 ∈ W(M ∗, G∗) ∼= W(M, G), we have  ℓP  P (w2w1, ξ, ψ, ψF ) = ℓP  P (w2, ξ, w1ψ, ψF ) ◦ ℓP w2  P  (w1, ξ, ψ, ψF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [7, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4] or [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Our case is simpler since the λ-factor is  trivial and the Galois action on the dual group “  G is also trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We have  ℓP ′  P (w, ξ, ψ, ψF ) ◦ RP ′w|P w(ξ, ψ) = RP ′|P (ξ, wψ) ◦ ℓP  P (w, ξ, ψ, ψF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The assertion follows from two equalities ℓP ′  P (w, ξ, ψ, ψF )◦RP ′w|P w(ξ, ψ) = RP ′|P (Ad( ˘w)◦ξ, ψ)◦ℓP  P (w, ξ, ψ, ψF )  and RP ′|P (Ad( ˘w) ◦ ξ, ψ) = RP ′|P (ξ, wψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We can show them by a similar way to the second and the third assertions  in [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4  The third intertwining operator  We maintain the assumptions of the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In addition, we assume that we are given z ∈ Z1(F, M ∗) ⊂  Z1(F, G∗), such that (ξ, z) is a pure inner twist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' z commutes with the Langlands-Shelstad lift of every element of  W(M ∗, G∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' and if we decompose z as z = z+ ×z− according to the decomposition M ∗ = M ∗  + ×M ∗  − then z+ takes the  constant value 1, where M ∗  + is a product of general linear groups and M ∗  − is a special orthogonal group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let u ∈ N( �T, “  G)  be such that Ad(u) ◦ ψ = ψ and u preserves the positive roots in �  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let u♮ ∈ Nψ(M, G) and w ∈ W(�  M, “  G) be its  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We shall regard w as an element of W(M ∗, G∗) or W(M, G) via the natural isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Although our  group G∗ is a special orthogonal group and (ξ, z) is a pure inner twist, by the similar argument as [18, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1],  we can define  the operator π( ˘w)ξ : ( ˘wπ, Vπ) → (π, Vπ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  the sophisticated splittings s′ : Wψ(M, G) → Nψ(M, G) and s : Nψ(M, G) → Sψ(M) of the exact sequence  1 → Sψ(M) → Nψ(M, G) → Wψ(M, G) → 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  the constant ⟨u♮, π⟩ξ,z = ⟨s(u♮)−1, π⟩ξ|M∗,z ∈ C×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then we define the operator π(u♮)ξ,z : ( ˘wπ, Vπ) → (π, Vπ) by π(u♮)ξ,z = ⟨u♮, π⟩ξ,zπ( ˘w)ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The assignment u♮ �→ π(u♮)ξ,z  is multiplicative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Next we shall give a more abstract characterization (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', another equivalent definition) of the operator π(u♮)ξ,z :  ( ˘wπ, Vπ) → (π, Vπ), following [18, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us put u′ = u−1 ∈ Nψ(M, G), and let ‹  u′ ∈ N( �T, “  G) be its Langlands-  Shelstad lift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put s = u′‹  u′−1 ∈ �  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then by the assumption that u preserves the positive roots in �  M, we have s ∈ �T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Put �θ = Ad(‹  u′), which is an automorphism of �  M preserving the standard splitting inherited from “  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' It is the dual of  an automorphism θ∗ = Ad( �w) of M ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put  ”  M ′ =  ¶  x ∈ �  M  ��� Ad(sψs) ◦ �θ(x) = x  ©  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let M ′ be the subgroup of M such that ”  M ′ is the dual of M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that Im ψ ⊂ ”  M ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then (M ′, sψs, ”  M ′ ⊂ �  M) is a  twisted endoscopic triple of (M ∗, θ∗), and hence of (M, θ), where θ = Ad( ˘w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We shall write eM,ψ for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The operator  π(u♮)ξ,z can be characterized as follows:  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For each π ∈ Πψ(M), there exists a unique isomorphism π(u♮)ξ,z : ( ˘wπ, Vπ) → (π, Vπ) such that  f eM,ψ(ψ) = e(M θ)  �  π∈Πψ(M)  tr(π(u♮)ξ,z ◦ π(f)),  for all ∆[eM,ψ, ξ|M∗, z]-matching functions f ∈ H(M) and f eM,ψ ∈ H(M ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, the operator π(u♮)ξ,z coincides  with the one constructed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1] for the case of the quasi-split unitary group UE/F(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  30  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5  The compound operator  We shall define the normalized self-intertwining operator RP (u♮, π, ψ, ψF ) ∈ EndG(HP (π)) as a composition  RP (u♮, π, ψ, ψF ) = IP (π(u♮)ξ,z) ◦ ℓP  P (w, ξ, ψ, ψF ) ◦ RP w|P (ξ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Recall that the definitions of the Haar measures on the unipotent radicals and the representative ˘w which were used  to define RP w|P (ξ, ψ) and ℓP  P (w, ξ, ψ, ψF ), respectively, depend on ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note also that the definition of IP (π(u♮)ξ,z)  depends on (ξ, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If the parameter ψ is not generic, we shall assume from now on that Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4 holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The operator RP (u♮, π, ψ, ψF ) does not depend on the choice of the pure inner twist (ξ, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Suppose that (ξ, z) ≃ (ξ′, z′) as inner twists of G∗ = SO2n+1 over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Since every automorphism is inner, there  exists b ∈ G∗(F) such that ξ′ = ξ ◦ Ad(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1] except that our z is not a  basic cocycle, but an ordinary 1-cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The Kottwitz map (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1) provides a pairing ⟨−, −⟩ on Z(“  G) × H1(F, G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that in our case Z(“  G) ≃ {±1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let x ∈ Z(“  G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then RP (xu♮, π, ψ, ψF ) = ⟨x, z⟩−1RP (u♮, π, ψ, ψF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let y ∈ Sψ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then RP (yu♮, π, ψ, ψF ) = ⟨y, π⟩ξ|M∗,zRP (u♮, π, ψ, ψF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The operator RP (u♮, π, ψ, ψF ) is multiplicative in u♮.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let u♮  1 and u♮  2 be elements in Nψ(M, G), and w1 and w2 their images in Wψ(M, G) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Since the  assignment u♮ �→ π(u♮)ξ,z is multiplicative, we have  IP (π(u♮  2u♮  1)ξ,z) = IP (π(u♮  2)ξ,z ◦ π(u♮  1)ξ,z) = IP (π(u♮  2)ξ,z) ◦ IP (π(u♮  1)ξ,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Combining this with Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7, we get  RP (u♮  2u♮  1, π, ψ, ψF )  = IP (π(u♮  2)ξ,z) ◦ IP (π(u♮  1)ξ,z) ◦ ℓP  P (w2, ξ, w1ψ, ψF ) ◦ ℓP w2  P  (w1, ξ, ψ, ψF ) ◦ RP w2w1|P w1 (ξ, ψ) ◦ RP w1|P (ξ, ψ)  = IP (π(u♮  2)ξ,z) ◦ IP (π(u♮  1)ξ,z) ◦ ℓP  P (w2, ξ, ψ, ψF ) ◦ RP w2 |P (ξ, ψ) ◦ ℓP  P (w1, ξ, ψ, ψF ) ◦ RP w1|P (ξ, ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We know that the operator IP (π(u♮  1)ξ,z) commutes with ℓP  P (w2, ξ, w1ψ, ψF )◦RP w2|P (ξ, ψ), as the operator IP (π(u♮  1)ξ,z)  acts on the values of the functions that comprise HP (π), while the operators ℓP  P (w2, ξ, ψ, ψF ) and RP w2 |P (ξ, ψ) act on  their variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Thus we have  IP (π(u♮  2)ξ,z) ◦ IP (π(u♮  1)ξ,z) ◦ ℓP  P (w2, ξ, ψ, ψF ) ◦ RP w2 |P (ξ, ψ) ◦ ℓP  P (w1, ξ, ψ, ψF ) ◦ RP w1 |P (ξ, ψ)  = IP (π(u♮  2)ξ,z) ◦ ℓP  P (w2, ξ, ψ, ψF ) ◦ RP w2|P (ξ, ψ) ◦ IP (π(u♮  1)ξ,z) ◦ ℓP  P (w1, ξ, ψ, ψF ) ◦ RP w1|P (ξ, ψ)  = RP (u♮  2, π, ψ, ψF ) ◦ RP (u♮  1, π, ψ, ψF ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6  The two linear forms, the local intertwining relation, and the construction of the  non-discrete packets  Let F be a local field, ψF : F → C1 a nontrivial additive character, G∗ = SO2n+1, and M ∗ ⊊ G∗ a proper standard  Levi subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ξ : G∗ → G be an inner twist over F, and put M = ξ(M ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ψ ∈ Ψ(M ∗) be a local parameter,  and u♮ an element of Nψ(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' When M ∗ transfers to G, we assume that M ⊊ G is a proper Levi subgroup over F  and ξ|M∗ : M ∗ → M is an inner twist over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We shall define the first linear form f �→ fG(ψ, u♮) on H(G) as follows:  fG(ψ, u♮) =  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  �  π∈Πψ(M)  tr(RP (u♮, π, ψ, ψF ) ◦ IP (π, f)),  if ψ is relevant,  0,  if ψ is not relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  31  Next we shall give the definition of the second linear form f �→ f ′  G(ψ, s) on H(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For f ∈ H(G) and s ∈ Sψ,ss, let  (e, ψe) ∈ X(G∗) be the element corresponding to (s, ψ) ∈ Y (G∗) under the bijective map given in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then  we write f ′  G(ψ, s) for the value f e(ψe) in the endoscopic character relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' It does not depend on the choice of  ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let s1, s2 ∈ Sψ,ss be two elements such that they have the same image in Sψ(G) and the image belongs  to Sψ(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then f ′  G(ψ, s1) = f ′  G(ψ, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Theorem* 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 (LIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let u ∈ Nψ(M, G) and f ∈ H(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If u♮ is in W ◦  ψ(M, G), then we have RP (u♮, π, ψ, ψF ) = 1 for all π ∈ Πψ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The value fG(ψ, u♮) depends on the image of u♮ in Sψ(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We have  f ′  G(ψ, sψu−1) = e(G)fG(ψ, u♮),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7)  where e(G) denotes the Kottwitz sign of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' This equation is called the local intertwining relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For any y ∈ Z(“  G∗), we have f ′  G(ψ, sψys) = ⟨y, z⟩f ′  G(ψ, sψs) and fG(ψ, yu♮) = ⟨y, z⟩−1fG(ψ, u♮).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' They follow from Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  In this paper, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 will be completely proved in the case when ψ = φ ∈ Φ(M ∗) is an L-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The  proof is inductive, and thus we assume that Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 holds for the case the degree is smaller than n, from now  on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The case of a general parameter ψ ∈ Ψ(M ∗) is expected to be treated as a sequel of [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We will now review the construction of local non-discrete A-packets which is an application of the local intertwining  relation (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Suppose that the theorem is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that ψM∗ ∈ Ψ2(M ∗) and M ∗ ⊊ G∗, hence ψM∗  is a discrete parameter for M ∗ and not discrete for G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We shall write ψ for the image of ψM∗ under the natural map  Ψ(M ∗) → Ψ(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We apologize for this change of convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If ψ is not G-relevant, then we simply set Πψ(G) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Suppose that ψ is G-relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we may assume that a Levi subgroup M = ξ(M ∗) ⊊ G is defined over F, and  by the assumption Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 holds true for ψM∗ and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' According to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1, we have SψM∗(M, G) = Sψ(G),  hence the natural map NψM∗(M, G) → Sψ(G) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For any πM ∈ ΠψM∗(M), where note that the packet for  M is given by the induction hypothesis, the map  NψM∗(M, G) × G(F) → AutG(F )(HP (πM)),  (u♮, g) �→ RP (u♮, πM, ψM∗, ψF ) ◦ IP (πM, g),  is a representation of NψM∗(M, G) × G(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ρ(πM) denote this representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put  Π1  ψ(G) =  �  πM∈ΠψM∗ (M)  ρ(πM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then we have  tr �Π1  ψ(G)(u♮, f)� =  �  πM∈ΠψM∗ (M)  tr �ρ(πM)(u♮, f)�  = fG(ψM∗, u♮),  for u♮ ∈ NψM∗(M, G) and f ∈ H(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Thus, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 (and the character theory of representations of finite groups)  implies that Π1  ψ(G) can be regarded as a representation of Sψ(G)×G(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='10, we have Π1  ψ(G)|Z( “  G)×{1} =  χ−1  G .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' As πM is unitary, Π1  ψ(G) is also unitary, in particular completely reducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We define the packet Πψ(G) to be  the multiset of irreducible representations of G(F) that appear in the direct summand of Π1  ψ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we obtain  Π1  ψ(G) =  �  π∈Πψ(G)  ⟨−, π⟩−1 ⊗ π,  32  where ⟨−, π⟩ are characters of Sψ(G) whose restriction to Z(“  G)Γ coincide χG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We equip the packet Πψ(G) with the  map  Πψ(G) → Irr(Sψ, χG),  π �→ ⟨−, π⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The endoscopic character relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7) follows directly from the local intertwining relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7) and the definition of  the pairing ⟨−, π⟩ above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7  Reduction of LIR to discrete (relative to M) parameters  In this subsection we review from [18, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7] the twisted local intertwining relation and the reduction of the proof of  the part 1 and 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 to the case of discrete parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Concretely speaking, this section will be devoted  to the proofs of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='15 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='16 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let us now consider two nested proper standard Levi subgroups  M ∗  0 ⊂ M ∗ ⊂ G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ψ0 ∈ Ψ(M ∗  0 ) be a local parameter, and ψ ∈ Ψ(M ∗) its image under the natural map from  Ψ(M ∗  0 ) to Ψ(M ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that ψ0 is relevant if and only if ψ is relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' When ψ and ψ0 are relevant, we assume that  M = ξ(M ∗) and M0 = ξ(M ∗  0 ) are proper Levi subgroups of G defined over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For any u ∈ Nψ0(M0, G) ∩ Nψ(M, G), we have  fG(ψ0, u♮) = fG(ψ, u♮),  where u♮ in the left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' right) hand side is the image of u in Nψ(M0, G) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Nψ(M, G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Note that the lemma is trivial if the parameter ψ0 is not relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let P (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' P0, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Q) be the standard parabolic subgroup of G (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' G, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' M), with the Levi sub-  group M (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' M0, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' M0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that Nψ(M0, M) ⊂ Nψ(M0, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note also that the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1) implies  Z(A�  M0, Sψ0(G)◦) = Z(A�  M0, Sψ0(M)◦), and hence Nψ(M0, M) ⊂ Nψ(M0, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that ψ0 is relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For any u ∈ Nψ0(M0, M) and π0 ∈ Πψ0(M0), we have  RP0(u♮, π0, ψ0, ψF ) = IP (RP0(u♮, π0, ψ0, ψF )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Moreover, we have  fG(ψ0, u♮) = fM(ψ0, u♮).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Before the proofs of these lemmas, we shall now record a consequence of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='15, which is a key ingredient of  the proof of the local intertwining relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that the second and the third statements of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 hold for a standard parabolic  pair (M ∗  0 , P ∗  0 ) and a discrete parameter ψ0 ∈ Ψ2(M ∗  0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then they hold for every standard parabolic pair (M ∗, P ∗) and  a parameter ψ ∈ Ψ(M ∗) such that M ∗  0 ⊂ M ∗, where ψ is the image of ψ0 under the canonical map Ψ(M ∗  0 ) → Ψ(M ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3] with Sψ(G)◦ in place of Srad  ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' (Precisely speaking, we  have Srad  ψ  = Sψ(G)◦ because G∗ = SO2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=')  The third statement of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 will also be reduced to the case of discrete parameters, in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  As a preparation of the proofs of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='15 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='16, we now review the twisted local intertwining relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  See [18, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7] and [7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='115-119] for detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let N and k be positive integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Put H = GLk  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The standard  pinning of GLN gives rise to a pinning of H, which we write (TH, BH, {XαH}αH) and call standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let θH be the  automorphism of H given by θH(h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , hk) = (θ(hk), h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , hk−1), where θ is either the identity automorphism of  GLN, or the automorphism θN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let (MH, PH) be a standard parabolic pair of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Recall that any packet for a direct  product of a finite number of general linear groups, is a singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ψ ∈ Ψ(MH) and let π be the corresponding  representation of MH(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put Sψ(H, θH) = Cent(Im ψ, “  H ⋊ ”  θH), which is not a group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put also Nψ(MH, H ⋊ θH) =  N(A ‘  MH, Sψ(H, θ−1  H )), W(‘  MH, “  H ⋊”  θH  −1) = N(A ‘  MH , “  H ⋊”  θH  −1)/‘  MH, and W(MH, H ⋊θH) = N(AMH, H ⋊θH)/MH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We identify the Weyl sets W(‘  MH, “  H ⋊ ”  θH  −1) and W(MH, H ⋊ θH) in the standard way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let u ∈ Nψ(MH, H ⋊ θH)  and let w ∈ W(‘  MH, “  H ⋊ ”  θH  −1) ≃ W(MH, H ⋊ θH) be the image of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  33  As in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2, let us define the first operator  RP w  H |PH(πλ) = rP w  H |PH(πλ, ψF )−1JP w  H |PH(πλ, ψF ),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8)  where JP w  H |PH(πλ, ψF ) : HPH(πλ) → HP w  H (πλ) is the unnormalized intertwining operator defined by  [JP w  H |PH(πλ, ψF )f](h) =  �  NH(F )∩N w  H(F )\\N w  H(F )  f(nh)dn,  and rP w  H |PH(πλ, ψF ) is the normalizing factor given by  rP w  H |PH(πλ, ψF ) =  L(0, ρ∨  P w  H |PH ◦ φψλ)  L(1, ρ∨  P w  H |PH ◦ φψλ)  ǫ( 1  2, ρ∨  P w  H |PH ◦ φψλ, ψF )  ǫ(0, ρ∨  P w  H |PH ◦ φψλ, ψF ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8) is holomorphic at λ = 0, thus we put RP w  H |PH(π) = RP w  H |PH(π0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  In order to define the second operator, we have to fix a representative of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ˙w be the representative of w in  the Weyl set W(TH, H ⋊ θH) that stabilizes the simple positive roots inside MH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Since both ˙w and θH preserve TH,  we have ˙w = ˙w0 ⋊ θH for some ˙w0 ∈ W(TH, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we can take the Langlands-Shelstad lift �w0 ∈ N(TH, H) of ˙w0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Put �w = �w0 ⋊ θH, which is the representative of w we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' As in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3, let us define the second operator  ℓPH  PH(w, π, ψF ) = ǫPH(w, ψ, ψF )ℓP H( �w),  where the normalizing factor is  ǫPH(w, ψ, ψF ) = ǫ(1  2, ρ∨  P w  H |PH ◦ φψ, ψF ),  and the operator ℓP H( �w) from (IP w  H (π), HP w  H (π)) to (IPH( �wπ) ◦ θH, HPH( �wπ)) is defined by  [ℓPH( �w)f](h) = f( �w−1 · h ⋊ θH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Here the identity component ‹  H◦ = H ⋊ 1 of the twisted group ‹  H = H ⋊ ⟨θH⟩ is identified with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Note that �wπ is isomorphic to π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us define the third operator  IPH(π( �w)) : HPH( �wπ) → HPH(π),  where π( �w) : �wπ → π is the Whittaker normalized isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Here we use the Whittaker datum corresponding to  ψF and the standard pinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Now we obtain the normalized intertwining operator  RPH(w, ψ, ψF ) = IPH(π( �w)) ◦ ℓPH  PH(w, π, ψF ) ◦ RP w  H |PH(π),  from (IPH (π), HPH(π)) to (IPH(π) ◦ θH, HPH(π)), and the twisted first linear form  H(H) ∋ f �→ fH(ψ, w) := tr (RPH(w, ψ, ψF ) ◦ IPH(π, f)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  On the other hand, for any s ∈ Sψ(H, θH), we have a pair (e, ψe) of a twisted endoscopic triple e ∈ E(H ⋊ θH) and  its parameter ψe ∈ Ψ(He), corresponding to the pair (ψ, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For any f ∈ H(H), choose f e ∈ H(He) such that f and  f e are matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we write f ′  H(ψ, s) for the value f e(ψe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' This is the twisted second linear form, which we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We call the following formula the twisted local intertwining relation, or twisted LIR for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='18 (twisted LIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We have  fH(ψ, w) = f ′  H(ψ, sψu−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The case k = 1 is guaranteed by [7, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then the proof is similar to that of [18, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4,  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  34  Now we return to the notation used in Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='15 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  If ψ0 and ψ are not relevant, then Lemma  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='15 is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that ψ0 and ψ are relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Hence M0, M, P0, P, and Q are defined over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let u ∈  Nψ0(M0, G) ∩ Nψ(M, G), and u♮ ∈ Nψ0(M0, G) be its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We will also write u♮ for the image in Nψ(M, G), by  abuse of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let f ∈ H(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By definition we have  fG(ψ0, u♮) = tr  Ñ  �  π0∈Πψ0(M0)  RP0(u♮, π0, ψ0, ψF ) ◦ IP0(π0, f)  é  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The calculation similar to that in [18, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='123-126] shows that  �  π0∈Πψ0(M0)  RP0(u♮, π0, ψ0, ψF ) ◦ IP0(π0)(f)  = ℓP  P (w, ξ, ψ, ψF ) ◦ RP w|P (ξ, ψ) ◦  Ñ  �  π0∈Πψ0(M0)  IP (RQ(u♮, π0, ψ0, ψF )) ◦ IP (IQ(π0))(f)  é  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ECR and LIR for inner form of SO2n0+1 for n0 < n, which is assumed by induction, and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='18 imply ECR and  (twisted) LIR for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In particular, the operator RQ(u♮, π0, ψ0, ψF ) acts on each irreducible summand of IQ(π0) by  scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Combined with the theory of induced characters ([9]) and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8, we have  tr  Ñ  �  π0∈Πψ0(M0)  IP (RQ(u♮, π0, ψ0, ψF )) ◦ IP (IQ(π0))(f)  é  = tr  Ñ  �  π0∈Πψ0(M0)  RQ(u♮, π0, ψ0, ψF ) ◦ IQ(π0)(fM)  é  = tr  Ñ  �  π∈Πψ(M)  π(u♮) ◦ π(fM)  é  = tr  Ñ  �  π∈Πψ(M)  IP (π(u♮)) ◦ IP (π)(f)  é  ,  where fM denotes the constant term of f along P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By the linear independence of characters, this means that  �  π0∈Πψ0(M0)  IP (RQ(u♮, π0, ψ0, ψF )) ◦ IP0(π0)(f) =  �  π∈Πψ(M)  IP (π(u♮)) ◦ IP (π)(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Therefore, we have  ℓP  P (w, ξ, ψ, ψF ) ◦ RP w|P (ξ, ψ) ◦  Ñ  �  π0∈Πψ0(M0)  IP (RQ(u♮, π0, ψ0, ψF )) ◦ IP (IQ(π0))(f)  é  = ℓP  P(w, ξ, ψ, ψF ) ◦ RP w|P (ξ, ψ) ◦  Ñ  �  π∈Πψ(M)  IP (π(u♮)) ◦ IP (π)(f)  é  =  �  π∈Πψ(M)  IP (π(u♮)) ◦ ℓP  P (w, ξ, ψ, ψF ) ◦ RP w|P (ξ, ψ) ◦ IP (π)(f)  =  �  π∈Πψ(M)  RP (u♮, π, ψ, ψF ) ◦ IP (π, f),  and hence  fG(ψ0, u♮) = fG(ψ, u♮).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  35  Thus Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Next, let u ∈ Nψ0(M0, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then u is trivial in W(�  M, “  G), and the calculations similar to those in [18, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='127] prove  both two equations in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8  Reduction of LIR to elliptic or exceptional (relative to G) parameters  In this subsection, we review the reduction of the proof of LIR to the case of elliptic or exceptional parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Recall from §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2 that Ψ2  ell(G∗) is the set of the equivalence classes of parameters ψ of the form  ψ = 2ψ1 ⊕ · · · 2ψq ⊕ ψq+1 ⊕ · · · ⊕ ψr,  where ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , ψr are irreducible, symplectic, and mutually distinct, and r ≥ q ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that then we have  Sψ ∼= O(2, C)q × O(1, C)r−q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let now Ψexc1(G∗) and Ψexc2(G∗) be the subsets of Ψ(G∗) consisting of the parameters ψ of the form  (exc1) ψ = 2ψ1 ⊕ ψ2 ⊕ · · · ⊕ ψr, where ψ1 is irreducible and orthogonal, and ψ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , ψr are irreducible, symplectic, and  mutually distinct,  (exc2) ψ = 3ψ1 ⊕ ψ2 ⊕ · · · ⊕ ψr, where ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , ψr are irreducible, symplectic, and mutually distinct,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' They are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We then have  (exc1) Sψ ∼= Sp(2, C) × O(1, C)r−1,  (exc2) Sψ ∼= O(3, C) × O(1, C)r−1,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We put Ψexc(G∗) = Ψexc1(G∗) ⊔ Ψexc2(G∗), and we shall say that ψ is exceptional (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' of type (exc1),  resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' of type (exc2)) if ψ ∈ Ψexc(G∗) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Ψexc1(G∗), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Ψexc2(G∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' One can see that Ψexc(G∗) and Ψell(G∗)  are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put Ψell,exc(G∗) = Ψ(G∗) \\ (Ψell(G∗) ⊔ Ψexc(G∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let M ∗ ⊊ G∗ be a proper standard Levi subgroup, and ξ : G∗ → G an inner twist over F, such that M = ξ(M ∗) ⊊  G is defined over F if M ∗ transfers to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In view of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7, it is enough to consider discrete parameters for M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let  ψM∗ ∈ Ψ2(M ∗) be a discrete local A-parameter, and we shall write ψ for its image in Ψ(G∗) in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Recall  from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1 that Sψ(M, G) = Sψ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Here we do not assume that M is defined over F, nor do that ψ is relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let Tψ be the maximal central torus A �  M of �  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Since a Levi subgroup for which the parameter is discrete is unique,  the torus Tψ is determined by ψ and is a maximal torus of Sψ = Sψ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' One can easily see that  Wψ = Wψ(M, G) = W(Tψ, Sψ),  W ◦  ψ = W ◦  ψ(M, G) = W(Tψ, S◦  ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Therefore, if ψ is elliptic then W ◦  ψ is trivial, as stated in [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Similarly, let Bψ be the standard Borel  subgroup of Sψ with a maximal torus Tψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then (Tψ, Bψ) is a Borel pair of S◦  ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For any x ∈ Sψ, following [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='204],  put  Tψ,x = Cent(sx, Tψ)◦,  where sx ∈ Sψ is a representative of x such that Ad(sx) stabilizes (Tψ, Bψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Since sx is determined up to a Tψ-translate,  Tψ,x is uniquely determined by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The following lemma is the key lemma in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If ψ ∈ Ψell,exc(G∗), then  (1) every simple reflection w ∈ W ◦  ψ centralizes a torus of positive dimension in Tψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  (2) dim Tψ,x ≥ 1, ∀x ∈ Sψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  36  We now define  Sψ,ell = { s ∈ Sψ,ss | Z(Cent(s, S◦  ψ)) is finite } ,  Sψ,ell = { s ∈ Sψ,ss | Z(Cent(s, S ψ)) is finite } ,  Sψ,ell = Sψ,ell/(Sψ,ell ∩ S◦  ψ),  Sψ,ell = Sψ,ell/(Sψ,ell ∩ S ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  One can see that Sψ,ell (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Sψ,ell) is the image of Sψ,ell (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Sψ,ell) under the natural surjection Sψ → Sψ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Sψ → Sψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The natural surjection Sψ → Sψ carries Sψ,ell onto Sψ,ell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The natural surjection Sψ → Sψ carries Sψ,ell onto Sψ,ell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If ψ ∈ Ψexc(G∗) then Sψ,ell = Sψ and Sψ,ell = Sψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The first part can be seen easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof of latter two is similar to that of [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that either  (1) ψ is elliptic, or  (2) every simple reflection w ∈ W ◦  ψ centralizes a torus of positive dimension in Tψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then the parts 1 and 2 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7 hold for ψ (and for all u and f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We appeal to Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='11 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='16 instead of Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3  and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2 of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let x ∈ Sψ(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that either  (1) ψ is elliptic and x /∈ Sψ,ell, or  (2) ψ is not elliptic, every simple reflection w ∈ W ◦  ψ centralizes a torus of positive dimension in Tψ, and dim Tψ,x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then f ′  G(ψ, sψs−1) = e(G)fG(ψ, u♮) whenever both u♮ ∈ Nψ(M, G) and s ∈ Sψ,ss map to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let sx ∈ Sψ,ss be a representative of x such that Ad(sx) stabilizes (Tψ, Bψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Since Tψ = A �  M, we have  sx ∈ Nψ(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='12, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='21 tell us that f ′  G(ψ, sψs−1) = f ′  G(ψ, sψs−1  x ) and fG(ψ, u♮) = fG(ψ, s♮  x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We need to show that f ′  G(ψ, sψs−1  x ) = e(G)fG(ψ, s♮  x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put �  Mx = Z “  G(Tψ,x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let M ∗  x ⊂ G∗ be the Levi subgroup  corresponding to �  Mx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If the assumption (1) holds, then we have |Tψ,x| = ∞ and hence �  Mx ⊊ “  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If the assumption  (2) holds, then we have dim Tψ,x ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' This implies |Tψ,x| = ∞ and hence �  Mx ⊊ “  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Therefore, M ∗  x is a proper Levi  subgroup of G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The rest of the proof is similar to that of [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We appeal to Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='12 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='16  instead of Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2 of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='19, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='21, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='22 imply the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let x ∈ Sψ(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 holds for all u♮ ∈ Nψ(M, G) mapping to x unless either  (1) ψ is elliptic and x ∈ Sψ,ell, or  (2) ψ is exceptional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let us now define  Wψ,reg(M, G) = { w ∈ Wψ(M, G) | w has finitely many fixed points on Tψ } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We write Nψ,reg(M, G) for the preimage of Wψ,reg(M, G) under Nψ(M, G) → Wψ(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that ψ ∈ Ψexc(G∗) and u♮ /∈ Nψ,reg(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let s be an element of Sψ,ss whose image in  Sψ(M, G) is the same as that of u♮.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we have f ′  G(ψ, sψs−1) = e(G)fG(ψ, u♮).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  37  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We appeal to Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='12 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='22 instead of Lemmas  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8 of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Consequently, if ψ ∈ Ψell,exc(G∗) then all the three statements of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 hold (under the induction hypoth-  esis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If ψ ∈ Ψ2  ell(G∗) then the first and second part of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In particular, when ψ is not exceptional,  fG(ψ, x) is well-defined for x ∈ Sψ(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, the third part of the theorem holds if the image of u in Sψ is  not contained in Sψ,ell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' When ψ is exceptional, for now, the result is Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='24, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', only a part of the third part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let x ∈ Sψ(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Since ψ is exceptional, we have |W ◦  ψ| = 2, and there is exactly two elements in the fiber of x in  Nψ(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' One is regular in Wψ(M, G), and the other is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put ux to be the former one, and define fG(ψ, x) to be  fG(ψ, ux).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Now we have defined fG(ψ, x) for x ∈ Sψ(M, G) for every ψ ∈ Ψ(G∗)  Put Φ♥(G∗) = Ψ♥(G∗) ∩ Φ(G∗) and Φbdd,♥(G∗) = Ψ♥(G∗) ∩ Φbdd(G∗) for ♥ ∈ {exc, exc1, exc2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='9  LIR for special cases  In this subsection we shall prove the two special cases of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Later in §§6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4, we will reduce the proof  of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 for φ ∈ Φell,exc(G∗) to the special cases, which we shall treat in this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Before we treat the two cases, we recall LLC for GL1(R) and GL2(R), and fix some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We shall realize the  Weil group WR as C× ⊔ jC×, where j2 = −1 and jz = zj for any z ∈ C×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Recall from [32, §1] that the norm map  on WR is given by |j| = 1 and |z| = zz for z ∈ C ⊂ WR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We define the (isomorphism classes of) finite dimensional  representations σ ωt, and τ(l,t) of WR by  σ : WR → C×,  ®  C× ∋ z �→ 1,  j �→ −1,  ωt : WR → C×,  ®C× ∋ z �→ |z|t = (zz)t,  j �→ 1,  τ(l,t) : WR → GL(2, C),  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f3  C× ∋ re  √−1θ �→ r2t  Ç  e  √−1lθ  e−√−1lθ  å  ,  j �→  Å  1  (−1)l  ã  ,  for t ∈ C and l ∈ Z>0, and put τl = τ(l,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then the local Langlands correspondence over R says that σεωt (ε = 0, 1)  is corresponding to the 1-dimensional representation | · |t  R sgnε, and τ(l,t) = τl ⊗ ωt to the 2-dimensional representation  Dl ⊗ | det |t  R, where Dl denotes the discrete series representation of GL2(R) of weight l + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Fix the standard additive  character ψR(x) = exp(2π√−1x) of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put ΓR(s) := π− s  2 Γ( s  2) and ΓC(s) := 2(2π)−sΓ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The L-functions and the  ǫ-factors are given by the following table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' See [20] for more detail, but note that (+, t) and (−, t) in [20, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2)] should  be (+, t  2) and (−, t  2) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ϕ  L(s, ϕ)  ǫ(s, ϕ, ψR)  ωt  ΓR(s + t)  1  σωt  ΓR(s + t + 1)  √−1  τ(l,t)  ΓC(s + t + l  2)  √−1  l+1  For a symmetric matrix Q and an alternative matrix A of degree N, we shall write  SO(Q) = { g ∈ GLN | tgQg = Q, det g = 1 } ,  Sp(A) = { g ∈ GLN | tgAg = A } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1  The first special case  The first special case we are concerned with is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let F = R, n = 2, and  G∗ = SO5 = SO  Ñ  12  2  12  é  ,  38  M ∗ =  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  Ü t  A  b  t−1  β  c  ê ��������  t ∈ GL1, A ∈ M2×2,  Å A  b  β  c  ã  ∈ SO  Ñ  1  2  1  é \uf8fc  \uf8f4  \uf8f4  \uf8fd  \uf8f4  \uf8f4  \uf8fe  ∼= GL1 × SO3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let φ ∈ Φexc1(G∗) be an L-parameter of type (exc1) of the form  φ = 2ω0 ⊕ τ1,  and φM = ω0 ⊕ τ1 ∈ Φ2(M ∗) so that φ is the natural image of φM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' More precisely, we write as  “  G = SpC  Ü  1  1  −1  −1  ê  ,  �  M =  \uf8f1  \uf8f2  \uf8f3  Ñ a  X  a−1  é  ∈ “  G  ������  a ∈ GL(1, C), X ∈ SpC  Å  1  −1  ã  = SL(2, C)  \uf8fc  \uf8fd  \uf8fe ⊂ “  G,  φ = φM =  Ñ  ω0  τ1  ω0  é  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Up to isomorphism, there exists only one nontrivial inner twist of G∗ for which φ is relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let  G = SO(1, 4) = SO  Å 1  −14  ã  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Put  α =  1  √  2  à 1  1  √−1  √−1  2√−1  1  −1  1  −1  í  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let z ∈ Z1(R, G∗) be a 1-cocycle such that  zρ = α−1ρ(α) =  à 1  0  −1  −1  1  −1  0  í  ,  where ρ is the nontrivial element in ΓF = Gal(C/R), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', the complex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ξ = Ad(α) : G∗ → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put  M = ξ(M ∗)  = { m(t, B) | t ∈ GL1, B ∈ SO(−13) = SO(3) } ,  where  m(t, B) =  Ñ c(t)  s(t)  B  s(t)  c(t)  é  ,  c(t) = t + t−1  2  ,  s(t) = t − t−1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  39  Then (ξ, z) is a pure inner twist G∗ → G which restricts to a pure inner twist M ∗ → M such that z+ ∈ Z1(R, GL1)  is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We have Πω0(GL1(R)) = {1}, Πτ1(SO(3)) = {1}, and hence ΠφM (M) = {1}, where 1 denotes the trivial  representation of each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us write π for the unique element in ΠφM (M), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', the trivial representation of  M(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 is valid for G, M, and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The centralizer Sφ(G) is  \uf8f1  \uf8f2  \uf8f3  Ñ a  b  ε12  c  d  é ������  Å a  b  c  d  ã  ∈ SL(2, C), ε ∈ {±1}  \uf8fc  \uf8fd  \uf8fe  ∼= SL(2, C) × O(1, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Thanks to this explicit structure, one can easily understand the diagram (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The diagram has the form  1  1  \uf8e6\uf8e6�  \uf8e6\uf8e6�  { ±1 } × 1  { ±1 } × 1  \uf8e6\uf8e6�  \uf8e6\uf8e6�  1 −−−−→ 1 × { ±1 } −−−−→ { ±1 } × { ±1 } −−−−→ { ±1 } × 1 −−−−→ 1  ���  \uf8e6\uf8e6�  \uf8e6\uf8e6�  1 −−−−→ 1 × { ±1 } −−−−→  1 × { ±1 }  −−−−→  1  −−−−→ 1  \uf8e6\uf8e6�  \uf8e6\uf8e6�  1  1  where (−1, 1) and (1, −1) are represented by  Ü  1  1  1  −1  ê  ∈ N(A �  M, Sφ(G)◦),  Ü  1  −1  −1  1  ê  ∈ Sφ(M),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that (−1, 1) vanishes in Sφ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us write w for the image of (−1, 1) in Wφ(M, G) = Wφ(M, G)◦ ∼=  W(M ∗, G∗) ∼= W(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' One can calculate the Langlands-Shelstad lift in “  G = Sp(4, C) of w as in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' It is  Ü  −1  −1  −1  1  ê  ,  and thus the first sophisticated splitting s′ : Wψ(M, G) → Nψ(M, G) sends (−1, 1) to (−1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Therefore, the other  sophisticated splitting s : Nψ(M, G) → Sψ(M) sends (−1, −1) to 1, and (−1, 1) to (1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let P ∗ be the standard parabolic subgroup of G∗ with Levi subgroup M ∗, and P = ξ(P ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Now in the same way  as [18, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='136-137], we know that the induced representation (IP (π), HP (π)) is irreducible and is the unique element  of Πφ(G), and that the operator RP ((−1, 1), π, φ, ψR) is a scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, in order to prove the proposition, it is  sufficient to show  RP ((−1, 1), π, φ, ψR)f = f,  40  for a nonzero element f in HP (π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Recall that  RP ((−1, 1), π, φ, ψR) = IP (π((−1, 1))ξ,z) ◦ ℓP  P (w, ξ, φ, ψR) ◦ RP w|P (ξ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We focus first on IP (π((−1, 1))ξ,z) = IP (⟨(−1, 1), π⟩ξ,zπ( ˘w)ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Since π is the trivial representation on the 1-  dimensional vector space C, the operator π( ˘w)ξ is the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' On the other hand, we have ⟨(−1, 1), π⟩ξ,z =  ⟨s(−1, 1), π⟩M = ⟨−1, π⟩M = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We obtain IP (π((−1, 1))ξ,z) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Next we consider ℓP  P (w, ξ, φ, ψR)◦RP w|P (ξ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let �w be the Langlands-Shelstad lift of w in G∗(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The calculation  of �w has already been done in [16, §8]:  �w =  à  1  −1  −1  1  −1  í  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We have ˘w = ξ( �w) =  Å1  −14  ã  ∈ G(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us define a connected compact subgroup K ⊂ G(R) as  K =  ß Å 1  κ  ã ���� κ ∈ SO(4)  ™  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  By abuse of notation, we shall write κ for the element  Å1  κ  ã  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Although K is not maximal, the Iwasawa decomposition  tells us that G(R) = P(R)K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let N ∗ be the unipotent radical of P ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put  n∗(b) =  à  1  b1  b2  −b1b3 − b2  2  4  b3  1  −b3  1  − b2  2  1  −b1  1  í  ,  for b = (b1, b2, b3) ∈ C3, so that N ∗(R) = {n∗(b) | b ∈ R3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' A direct calculation following §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2 show that the Haar  measure on N ∗(R) is d(n∗(b)) =  1  2db1db2db3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put P = ξ(P ∗) and N = ξ(N ∗), which are the standard parabolic  subgroup with Levi subgroup M and its unipotent radical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put also  n(x) =  Ö  1 + ∥x∥2  2  x  − ∥x∥2  2  tx  13  − tx  ∥x∥2  2  x  1 − ∥x∥2  2  è  =  â  1 + ∥x∥2  2  x1  x2  x3  − ∥x∥2  2  x1  1  −x1  x2  1  −x2  x3  1  −x3  ∥x∥2  2  x1  x2  x3  1 − ∥x∥2  2  ì  ,  for x = (x1, x2, x3) ∈ C3, where ∥x∥2 = x2  1 + x2  2 + x2  3 so that N(R) = {n(x) | x ∈ R3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' A straightforward calculation  shows that ξ(n∗(b)) = n(x) where  x1 = −b1 + b3  2  √  −1,  x2 = −b2  2  √  −1,  x3 = b1 − b3  2  ,  and hence the measure on N(R) is d(n(x)) = 2dx1dx2dx3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For λ ∈ C, put φλ = ωλ ⊕ τ1 ⊕ ω−λ and πλ = | · |λ ⊠ 1, so that Πφλ(M) = {πλ} and Πφλ(G) = {IP (πλ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Define a  function ϕ(λ) ∈ HP (πλ) by  ϕ(λ)(m(t, B)n(x)κ) = |t|λ+ 3  2 ,  which is holomorphic in λ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that Re(λ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We have 0 ̸= ϕ(0) ∈ HP (π) and  î  ℓP  P (w, ξ, φ, ψR) ◦ RP w|P (ξ, φ)ϕ(0)ó  (g)  41  = lim  λ→+0  î  ℓP  P (w, ξ, φλ, ψR) ◦ RP w|P (ξ, φλ)ϕ(λ)ó  (g)  = lim  λ→+0 ǫ(0, ρ∨  P w|P ◦ φλ, ψR)  L(1, ρ∨  P w|P ◦ φλ)  L(0, ρ∨  P w|P ◦ φλ)  �  N(R)  ϕ(λ)( ˘w−1ng)dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  A direct calculation implies that ρ∨  P w|P ◦ φλ is isomorphic to  τ(1,λ) ⊕ ω2λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Therefore, we have  ǫ(0, ρ∨  P w|P ◦ φλ, ψR)  L(1, ρ∨  P w|P ◦ φλ)  L(0, ρ∨  P w|P ◦ φλ) = −ΓC(λ + 3  2)  ΓC(λ + 1  2)  ΓR(2λ + 1)  ΓR(2λ)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We now turn to the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put  [MSO(1,4)ϕ(λ)](g) =  �  N(R)  ϕ(λ)( ˘w−1ng)dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  In Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='26 below, we will show that  MSO(1,4)ϕ(λ) = 2− 1  2  ΓC(λ)  ΓC(λ + 3  2)ϕ(−λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  This leads to  ǫ(0, ρ∨  P w|P ◦ φλ, ψR)  L(1, ρ∨  P w|P ◦ φλ)  L(0, ρ∨  P w|P ◦ φλ)  �  N(R)  ϕ(λ)( ˘w−1n·)dn  = −ΓC(λ + 3  2)  ΓC(λ + 1  2)  ΓR(2λ + 1)  ΓR(2λ)  2− 1  2  ΓC(λ)  ΓC(λ + 3  2)ϕ(−λ)  = −2− 1  2  ΓC(λ)  ΓC(λ + 1  2)  ΓR(2λ + 1)  ΓR(2λ)  ϕ(−λ),  whose limit as λ approaching 0 from the right is −ϕ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Since ϕ(0) ̸= 0, this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  To finish the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='25, it remains to show the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For Re(λ) > 0, we have  MSO(1,4)ϕ(λ) = 2− 1  2  ΓC(λ)  ΓC(λ + 3  2)ϕ(−λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By definition, MSO(1,4)ϕ(λ) is right K-invariant and left N(R)-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, we have  [MSO(1,4)ϕ(λ)](m(t, B)g) = |t|−λ+ 3  2 [MSO(1,4)ϕ(λ)](g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Indeed, a direct calculation shows that  n(x)m(t, B) = m(t, B)n(t−1xB),  ˘w−1m(t, B) = m(t−1, B) ˘w−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Thus we have  [MSO(1,4)ϕ(λ)](m(t, B)g) =  �  x∈R3 ϕ(λ)( ˘w−1n(x)m(t, B)g)d(n(x))  =  �  x∈R3 ϕ(λ)(m(t−1, B) ˘w−1n(t−1xB)g) · 2dx1dx2dx3  42  = |t|−λ− 3  2  �  y∈R3 ϕ(λ)( ˘w−1n(y)g) · 2|t|3dy1dy2dy3  = |t|−λ+ 3  2 [MSO(1,4)ϕ(λ)](g),  where we have changed the variables y = t−1xB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Combining these properties with ϕ(λ)(1) = 1, we now have  MSO(1,4)ϕ(λ) = MSO(1,4)ϕ(λ)(1) · ϕ(−λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We must then show that  MSO(1,4)ϕ(λ)(1) = 2− 1  2  ΓC(λ)  ΓC(λ + 3  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For x, y ∈ R3, t ∈ R×, B ∈ SO(3), and κ ∈ SO(4) ∼= K, a direct calculation shows that  ˘w−1n(x) =  Ö  1 + ∥x∥2  2  x  − ∥x∥2  2  − tx  −13  tx  − ∥x∥2  2  −x  −1 + ∥x∥2  2  è  ,  m(t, B)n(y)κ =  Ö  c(t) + t∥y∥2  2  ∗  ∗  B ty  ∗  ∗  s(t) + t∥y∥2  2  ∗  ∗  è  ,  so if we write ˘w−1n(x) = m(t, B)n(y)κ then we have  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  1 + ∥x∥2  2  = c(t) + t∥y∥2  2  ,  − tx  = B ty,  −∥x∥2  2  = s(t) + t∥y∥2  2  ,  which leads t = (1 + ∥x∥2)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Now the value MSO(1,4)ϕ(λ)(1) is equal to  �  x∈R3 ϕ(λ)( ˘w−1n(x))d(n(x))  =  �  x∈R3  ��1 + ∥x∥2��−λ− 3  2 · 2dx1dx2dx3  =  � ∞  r=0  � π  θ=0  � 2π  α=0  2r2 sin θ  (1 + r2)λ+ 3  2 dαdθdr  = 8π  � ∞  0  r2  (1 + r2)λ+ 3  2 dr,  where we have changed the variables x1 = r cos α sin θ, x2 = r sin α sin θ, and x3 = r cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Since Re(λ) > 0, the  last integral converges absolutely, and the gamma function Γ(λ + 3  2) has the integral expression which also converges  absolutely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Therefore, the product of them is  8π  � ∞  0  r2  (1 + r2)λ+ 3  2 dr × Γ(λ + 3  2)  = 8π  � ∞  0  � ∞  0  Å  t  1 + r2  ãλ+ 3  2  r2e−tdrd×t,  where d×t = t−1dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' First we fix r and change the variables u = (1 + r2)−1t, and then fix u and change the variables  s = √ur, to obtain  8π  � ∞  0  � ∞  0  Å  t  1 + r2  ãλ+ 3  2  r2e−tdrd×t  43  = 8π  � ∞  0  uλe−ud×u  � ∞  0  s2e−s2ds  = 2π  3  2 Γ(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We now have  MSO(1,4)ϕ(λ)(1) = 8π  � ∞  0  r2  (1 + r2)λ+ 3  2 dr  = 2π  3  2  Γ(λ)  Γ(λ + 3  2) = 2− 1  2  ΓC(λ)  ΓC(λ + 3  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2  The second special case  The second special case we are concerned with is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let F = R, n = 3, and  G∗ = SO7 = SO  Ñ  13  2  13  é  ,  M ∗ =  \uf8f1  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f3  Ü A  X  b  tA−1  β  c  ê ��������  A ∈ GL2, X ∈ M2×2,  Å X  b  β  c  ã  ∈ SO  Ñ  1  2  1  é \uf8fc  \uf8f4  \uf8f4  \uf8fd  \uf8f4  \uf8f4  \uf8fe  ∼= GL2 × SO3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let φ ∈ Φexc2(G∗) be an L-parameter of type (exc2) of the form  φ = 3τ1,  and φM = 2τ1 ∈ Φ2(M ∗) (as equivalent classes) so that φ is the natural image of φM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' More precisely, we define the  parameter as  “  G = SpC  Ü  12  1  −1  −12  ê  ,  �  M =  \uf8f1  \uf8f2  \uf8f3  Ñ A  X  tA−1  é  ∈ “  G  ������  A ∈ GL(2, C), X ∈ SpC  Å  1  −1  ã  = SL(2, C)  \uf8fc  \uf8fd  \uf8fe ⊂ “  G,  φ = φM =  Ñ τ1  τ1  tτ1−1  é  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Up to isomorphism, there exists only one nontrivial inner twist of G∗ for which φ is relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let  G = SO(2, 5) = SO  Å 12  −15  ã  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Put  α =  1  √  2  à 12  12  √−1  √−1  2√−1  1  −1  12  −12  í  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  44  Let z ∈ Z1(R, G∗) be a 1-cocycle such that  zρ = α−1ρ(α) =  à  12  0  −1  −1  12  −1  0  í  ,  where ρ is the nontrivial element in ΓF = Gal(C/R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put ξ = Ad(α) : G∗ → G and  M = ξ(M ∗)  = { m(A, B) | A ∈ GL2, B ∈ SO(−13) = SO(3) } ,  where  m(A, B) =  Ñ c(A)  s(A)  B  s(A)  c(A)  é  ,  c(A) = A + tA−1  2  ,  s(A) = A − tA−1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then (ξ, z) is a pure inner twist G∗ → G which restricts to a pure inner twist M ∗ → M such that z+ ∈ Z1(R, GL2)  is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We have Πτ1(GL2(R)) = {D1}, Πτ1(SO(3)) = {1}, and hence ΠφM (M) = {D1 ⊠ 1}, where 1 denotes the  trivial representation of SO(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us write π for the unique element in ΠφM (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 is valid for G, M, and φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put J =  Å  1  −1  ã  , and J =  Å14  J  ã  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us define a mapping A �→ A(2) from the set of 3 × 3-matrices to  those of 6 × 6-matrices by  A =  Ñ a1,1  a1,2  a1,3  a2,1  a2,2  a2,3  a3,1  a3,2  a3,3  é  �→ A(2) :=  Ñ a1,112  a1,212  a1,312  a2,112  a2,212  a2,312  a3,112  a3,212  a3,312  é  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Since tτ1−1 = Jτ1J−1, the centralizer Sφ(G) is  \uf8f1  \uf8f2  \uf8f3 Jh(2)J−1  ������  h ∈ OC  Ñ  1  1  1  é \uf8fc  \uf8fd  \uf8fe  ∼= O(3, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Thanks to this explicit structure, one can easily understand the diagram (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The diagram has the form  1  1  \uf8e6\uf8e6�  \uf8e6\uf8e6�  { ±1 } × 1  { ±1 } × 1  \uf8e6\uf8e6�  \uf8e6\uf8e6�  1 −−−−→ 1 × { ±1 } −−−−→ { ±1 } × { ±1 } −−−−→ { ±1 } × 1 −−−−→ 1  ���  \uf8e6\uf8e6�  \uf8e6\uf8e6�  1 −−−−→ 1 × { ±1 } −−−−→  1 × { ±1 }  −−−−→  1  −−−−→ 1  \uf8e6\uf8e6�  \uf8e6\uf8e6�  1  1  45  where (−1, 1) and (1, −1) are represented by  Ñ  J−1  −12  J  é  ∈ N(A �  M, Sφ(G)◦),  Ñ 12  −12  12  é  ∈ Sφ(M),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that (−1, 1) vanishes in Sφ(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us write w for the image of (−1, 1) in Wφ(M, G) = Wφ(M, G)◦ ∼=  W(M ∗, G∗) ∼= W(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' One can calculate the Langlands-Shelstad lift in “  G of w as in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' It is  Ñ  J−1  12  J  é  ,  and thus the first sophisticated splitting s′ : Wψ(M, G) → Nψ(M, G) sends (−1, 1) to (−1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Therefore, the other  splitting s : Nψ(M, G) → Sψ(M) sends (−1, −1) to 1, and (−1, 1) to (1, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let P ∗ be the standard parabolic subgroup of G∗ with Levi subgroup M ∗, and P = ξ(P ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Now in the same way  as [18, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='136-137], we know that the induced representation (IP (π), HP (π)) is irreducible and is the unique element  of Πφ(G), and that the operator RP ((−1, 1), π, φ, ψR) is a scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, in order to prove the proposition, it is  sufficient to show  RP ((−1, 1), π, φ, ψR)f = f,  for a nonzero element f in HP (π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Recall that  RP ((−1, 1), π, φ, ψR) = IP (π((−1, 1))ξ,z) ◦ ℓP  P (w, ξ, φ, ψR) ◦ RP w|P (ξ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We focus first on IP (π((−1, 1))ξ,z) = IP (⟨(−1, 1), π⟩ξ,zπ( ˘w)ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let �w be the Langlands-Shelstad lift of w in G∗(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The calculation of �w has already been done in [16, §8]:  �w =  à  J  1  1  J  1  í  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Thus we have  ˘w = ξ( �w) =  Ñ  J  13  J−1  é  ∈ G(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Since ˘w−1m(A, B) ˘w = m((det A)−1A, B) and the central character of D1 is trivial, the representations ˘wπ and π  coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Thus π( ˘w)ξ is the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' On the other hand, we have ⟨(−1, 1), π⟩ξ,z = ⟨s(−1, 1), π⟩M = ⟨−1, π⟩M =  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We obtain IP (π((−1, 1))ξ,z) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Next we consider ℓP  P (w, ξ, φ, ψR) ◦ RP w|P (ξ, φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us define a connected compact subgroup K ⊂ G(R) as  K =  ß  κ =  Å κ2  κ5  ã ���� κm ∈ SO(m), (m = 2, 5)  ™  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Although K is not maximal, the Iwasawa decomposition tells us that G(R) = P(R)K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let N ∗ be the unipotent  radical of P ∗, which is generated by {exp(Xα) | α = χ1 ± χ3, χ1, χ2 ± χ3, χ2, χ1 + χ2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Here Xα denotes the Chevalley  basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The Haar measure on N ∗(R) is also given by the Chevalley basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put P = ξ(P ∗) and N = ξ(N ∗), which are  46  the standard parabolic subgroup with Levi subgroup M and its unipotent radical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Now we describe N(R) explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let ι1 and ι2 be embeddings of SO(1, 4) into G = SO(2, 5) given by  h =  Å a  b  β  C  ã  �→ ι1(h) =  Ü a  b  1  β  C  1  ê  ,  h =  Å A  b  β  c  ã  �→ ι2(h) =  Ü 1  A  b  1  β  c  ê  ,  where A and C are 4-by-4 matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For x = (x1, x2, x3) ∈ C3 and u ∈ C, put n1(x) = ι1(n(x)) and n2(x) = ι2(n(x)),  where n(x) is the element defined in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='25, and put  nc(u) =  à  1  u  2  0  − u  2  − u  2  1  u  2  0  13  0  u  2  1  − u  2  − u  2  0  u  2  1  í  ,  so that  n1(x) = ξ(exp(  √  −1x1(Xχ1−χ3 + Xχ1+χ3) +  √  −1x2Xχ1 + x3(Xχ1−χ3 − Xχ1+χ3))),  n2(x) = ξ(exp(  √  −1x1(Xχ2−χ3 + Xχ2+χ3) +  √  −1x2Xχ2 + x3(Xχ2−χ3 − Xχ2+χ3))),  nc(u) = ξ(exp(uXχ1+χ2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let N1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' N2, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Nc) be the unipotent subgroup of G consisting of n1(x) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' n2(x), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' nc(u)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then the  Haar measure on N1(R) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' N2(R), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Nc(R)) is given by d(n1(x)) = 2dx1dx2dx3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' d(n2(x)) = 2dx1dx2dx3,  resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' d(nc(u)) = du).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We have N(R) = N1(R)N2(R)Nc(R) = {n1(x)nc(u)n2(y) | x, y ∈ R3, u ∈ R} and the Haar  measure is the product of those on N1(R), N2(R), and Nc(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let us realize the discrete series representation D1 of GL2(R) as a subrepresentation of a parabolically induced  representation from a character | − |  1  2 ⊠ | − |− 1  2 on the diagonal maximal torus of GL2(R), and let 0 ̸= v0 ∈ D1 be a  lowest weight vector such that  v0  ÅÅ a  b  d  ã Å  cos θ  sin θ  − sin θ  cos θ  ãã  =  ���a  d  ��� e2√−1θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For λ ∈ C, put φλ = τ(1,λ) ⊕ τ1 ⊕ τ ∨  (1,λ) and πλ = (D1 ⊗ | · |λ) ⊠ 1, so that Πφλ(M) = {πλ} and Πφλ(G) = {IP (πλ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Since D1 is realized as a subrepresentation of a parabolically induced representation, we have a natural injection  HP (πλ) ֒→ HP0(| − |  1  2 +λ ⊠ | − |− 1  2 +λ ⊠ 1),  f(·) �→ f(·)(1),  where P0 is the minimal standard parabolic subgroup of G defined over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Define a function f (λ) ∈ HP (πλ) ⊂  HP0(| − |  1  2 +λ ⊠ | − |− 1  2 +λ ⊠ 1) by  f (λ)(m(A, B)nκ) = | det A|λ+2[τ1(A)v0](1)  = |a|λ+3|d|λ+1e2√−1θ,  A =  Å a  b  d  ã Å  cos θ  sin θ  − sin θ  cos θ  ã  ,  which is holomorphic in λ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that Re(λ) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We have 0 ̸= f (0) ∈ HP (π) and  î  ℓP  P(w, ξ, φ, ψR) ◦ RP w|P (ξ, φ)f (0)ó  (g)  = lim  λ→+0  î  ℓP  P (w, ξ, φλ, ψR) ◦ RP w|P (ξ, φλ)f (λ)ó  (g)  47  = lim  λ→+0 ǫ(0, ρ∨  P w|P ◦ φλ, ψR)  L(1, ρ∨  P w|P ◦ φλ)  L(0, ρ∨  P w|P ◦ φλ)  �  N(R)  f (λ)( ˘w−1ng)dn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  A direct calculation implies that ρ∨  P w|P ◦ φλ is isomorphic to  τ(2,λ) ⊕ σωλ ⊕ ωλ ⊕ τ(2,2λ) ⊕ σω2λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Therefore, we have  ǫ(0, ρ∨  P w|P ◦ φλ, ψR)  L(1, ρ∨  P w|P ◦ φλ)  L(0, ρ∨  P w|P ◦ φλ) = ΓC(λ + 2)  ΓC(λ + 1)  ΓR(λ + 2)  ΓR(λ)  ΓC(2λ + 2)  ΓC(2λ + 1)  ΓR(2λ + 2)  ΓR(2λ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We now turn to the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ιSL2 be an embedding of SL2 into G = SO(2, 5) given by ιSL2(A) = m(A, 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put  w0 =  Å1  −14  ã  , which is the Langlands-Shelstad lift in SO(1, 4) given in the proof of proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Equations  ˘w = ι2(w0)ιSL2(J)ι2(w0),  Ad(ιSL2(J)ι2(w0))−1(n1(x)) = n2(−x),  Ad(ι2(w0))−1(nc(u)) = ιSL2(  Å1  u  1  ã  ),  implies that  �  N(R)  f (λ)( ˘w−1ng)dn = [M2 ◦ Mc ◦ M2(f (λ))](g),  where we have put  [M2f](g) =  �  N2(R)  f(ι2(w0)−1n2g)dn2,  [Mcf](g) =  �  Nc(R)  f(ιSL2(J)−1ncg)dnc,  for any function f on G(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For α, β ∈ C, we define f (α,β) ∈ HP0(| − |α ⊠ | − |β ⊠ 1) by  f (α,β)(m(A, B)nκ) = |a|α+ 5  2 |d|β+ 3  2 e2√−1θ,  A =  Å a  b  d  ã Å  cos θ  sin θ  − sin θ  cos θ  ã  ,  so that f (λ) = f (λ+ 1  2 ,λ− 1  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='28 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='29 below, we have  M2 ◦ Mc ◦ M2(f (λ))  = M2 ◦ Mc ◦ M2(f (λ+ 1  2 ,λ− 1  2 ))  = 2− 1  2 ΓC(λ − 1  2)  ΓC(λ + 1) M2 ◦ Mc(f (λ+ 1  2 ,−λ+ 1  2 ))  = −2− 1  2 ΓC(λ − 1  2)  ΓC(λ + 1)  ΓR(2λ)ΓR(2λ + 1)  ΓR(2λ + 3)ΓR(2λ − 1)M2(f (−λ+ 1  2 ,λ+ 1  2 ))  = −2−1 ΓC(λ − 1  2)  ΓC(λ + 1)  ΓR(2λ)ΓR(2λ + 1)  ΓR(2λ + 3)ΓR(2λ − 1)  ΓC(λ + 1  2)  ΓC(λ + 2) f (−λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  This leads to  ǫ(0, ρ∨  P w|P ◦ φλ, ψR)  L(1, ρ∨  P w|P ◦ φλ)  L(0, ρ∨  P w|P ◦ φλ)  �  N(R)  f (λ)( ˘w−1ng)dn  = ΓC(λ + 2)  ΓC(λ + 1)  ΓR(λ + 2)  ΓR(λ)  ΓC(2λ + 2)  ΓC(2λ + 1)  ΓR(2λ + 2)  ΓR(2λ + 1) · −1  2  ΓC(λ − 1  2)  ΓC(λ + 1)  ΓR(2λ)ΓR(2λ + 1)  ΓR(2λ + 3)ΓR(2λ − 1)  ΓC(λ + 1  2)  ΓC(λ + 2) f (−λ)(g),  whose limit as λ approaching 0 from the right is −f (λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Since f (0) ̸= 0, this completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  48  In order to finish the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='27, it remains to show the following two lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Suppose that Re(β) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then  M2f (α,β) = 2− 1  2  ΓC(β)  ΓC(β + 3  2)f (α,−β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' As in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='26, one has  M2f (α,β) = M2f (α,β)(1) · f (α,−β),  if M2f (α,β)(1) converges absolutely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Since f (α,β) ◦ ι2 is equal to ϕ(β) defined in the previous subsection, we have  [M2f (α,β)](1) =  �  x∈R3 f (α,β)(ι2(w0)−1ι2(n(x)))d(n(x))  =  �  x∈R3 ϕ(β)(w−1  0 n(x))d(n(x))  = [MSO(1,4)ϕ(β)](1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Hence the assertion follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Suppose that Re(α − β) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then  Mcf (α,β) = −  ΓR(α − β)ΓR(α − β + 1)  ΓR(α − β + 3)ΓR(α − β − 1)f (β,α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' As in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='26, one has  Mcf (α,β) = Mcf (α,β)(1) · f (β,α),  if Mcf (α,β)(1) converges absolutely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Following the proof of [15, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4], for s ∈ C we define a function h(s) on  SL2(R) by  h(s)  ÅÅ a  b  a−1  ã Å  cos θ  sin θ  − sin θ  cos θ  ãã  =  ���a  d  ���  s+1  e2√−1θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then f (α,β) ◦ ιSL2 is equal to h(α−β), and hence we have  [Mcf (α,β)](1) =  �  u∈R  f (α,β)  Å  ιSL2(J)−1ιSL2  Å 1  u  1  ãã  du  =  �  u∈R  h(α−β)  Å  J−1  Å 1  u  1  ãã  du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Hence the assertion follows from the second equation in the proof of [15, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  5  The decomposition into near equivalence classes and the standard  model  In this section we shall roughly recall from [18, 7] the stable multiplicity formula, which implies the decomposition  of L2  disc(G(F)\\G(AF )) into near equivalence classes, the global intertwining relation, and its weaker identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  49  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1  Stable multiplicity formula  Let F be a number field, G∗ the split special orthogonal group SO2n+1 over F, and G an inner form of G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Although G does not have simply connected derived subgroup, every endoscopic data for G comes from an endoscopic  triple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Hence the argument in [18, §§3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2] also holds for our G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Thus we have the decompositions  L2  disc(G(F)\\G(AF )) =  �  c∈C(G)  t≥0  L2  disc,t,c(G(F)\\G(AF )),  tr Rdisc(f) =  �  c∈C(G)  t≥0  Rdisc,t,c(f),  for f ∈ H(G), where R♦ denotes the regular representation of G(AF ) on L2  ♦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, for t ∈ R≥0, c ∈ C(G), and  e = (Ge, se, ηe) ∈ Eell(G), we have  the c-variant IG  disc,t,c of the discrete part IG  disc,t of the trace formula on the constituents of which the norm of the  imaginary part of the infinitesimal character is t, where the central character datum is trivial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  the transfer mapping H(G) → S(Ge), f �→ f e = f Ge, where S(Ge) denotes a space of functions on conjugacy  classes of semisimple elements defined in [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='53, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='132];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  the stable linear form Se  disc,t,c = SGe  disc,t,c on H(Ge) and the associated linear form �Se  disc,t,c on S(Ge) (see [7,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2)] for the notion of associated linear forms),  and the stabilization  IG  disc,t,c(f) =  �  e∈Eell(G)  ι(G, Ge)�Se  disc,t,c(f e),  f ∈ H(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Here ι(G, Ge) are the global coefficients introduced by Kottwitz and Shelstad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' See [7, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4)] for an explicit formula  for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let ψ ∈ Ψ(G∗) and e = (Ge, se, ηe) ∈ Eell(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' To the former is associated t(ψ) ∈ R≥0 and c(ψ) ∈ C(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' As in [18,  §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3], put  IG  disc,ψ = IG  disc,t(ψ),c(ψ),  SGe  disc,ψ = SGe  disc,t(ψ),c(ψ),  L2  disc,ψ(G(F)\\G(AF )) = L2  disc,t(ψ),c(ψ)(G(F)\\G(AF )),  Rdisc,ψ = Rdisc,t(ψ),c(ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We also write RG  disc,ψ if the group G is to be emphasized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us recall from [7, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2] the stable trace formula  for the split odd special orthogonal group G∗ = SO2n+1:  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1 (Stable trace formula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ψ ∈ ‹Ψ(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we have  SG∗  disc,ψ(f) =  �  |Sψ|−1εG∗  ψ (sψ)σ(S ψ)f G∗(ψ),  if ψ ∈ Ψ(G∗),  0,  if ψ ∈ ‹Ψ(N) \\ Ψ(G∗),  for f ∈ H(G∗), where εG∗  ψ  is the character in the multiplicity formula, σ(S ψ) the constant given in [7, Proposition  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1], and f G∗(ψ) denotes the linear form defined by [7, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  As a consequence we have a decomposition  L2  disc(G(F)\\G(AF )) =  �  ψ∈Ψ(G∗)  L2  disc,ψ(G(F)\\G(AF )),  tr Rdisc(f) =  �  ψ∈Ψ(G∗)  Rdisc,ψ(f),  f ∈ H(G),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1)  of the discrete spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='12 is now proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  50  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2  Global intertwining operator  Let ξ : G∗ → G be an inner twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let P ∗ ⊂ G∗ be a standard parabolic subgroup with a Levi decomposition  P ∗ = M ∗N ∗ over F, and put P = ξ(P ∗), M = ξ(M ∗), and N = ξ(N ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We consider the case that P, M, and N are  defined over F, and a restriction ξ|M∗ : M ∗ → M is an inner twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Assume that M ̸= G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let (π, Vπ) be an irreducible component of L2  disc(A+  M,∞G(F)\\G(AF )), where A+  M,∞ denotes  the connected component of 1 in ResF/Q(M)(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ψ ∈ Ψ2(M ∗) be the corresponding A-parameter (given by the  induction hypothesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We shall consider the induced representation (IP (π), HP (π)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' It is a right regular representation  on the Hilbert space HP (π) of measurable functions f : G(AF ) → Vπ such that f(nmg) = δ  1  2  P (m)π(m)f(g) for any  n ∈ N(AF ), m ∈ M(AF ), and g ∈ G(AF ) whose restriction to an open compact subgroup K ⊂ G(AF ) is square-  integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let P ′ ⊂ G be a parabolic subgroup over F with Levi component M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For λ ∈ a∗  M,C, the intertwining operator  JP ′|P : IP (πλ) → IP ′(πλ) is defined by  [JP ′|P (πλ)(f)](g) =  �  N(AF )∩N ′(AF )\\N ′(AF )  f(n′g)dn′,  where N ′ denotes the unipotent radical of P ′, and πλ = π ⊗ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' As in [18], we take the Haar measure dn′ determined  by the Haar measure on AF which assigns the quotient AF /F volume 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' It is known that the integral converges  absolutely when the real part of λ lies in a certain open cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' As a function of λ, it has a meromorphic continuation  and is nonzero and holomorphic at λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then the operator JP ′|P (π) = JP ′|P (πλ)|λ=0 is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' It is also known  that JP ′|P (π) is unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We now consider the case P ′ = P w = w−1Pw for some w ∈ W(M, G)Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ˘w ∈ N(M, G)(F) be a representative  of w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then one has a twist ( ˘wπ, Vπ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We define two intertwining operators ℓ( ˘w) and C ˘  w by  ℓ( ˘w) : HP w(π) −→ HP ( ˘wπ),  [ℓ( ˘w)f](g) = f( ˘w−1g),  C ˘  w : ( ˘wπ, Vπ) −→ (π, Vπ),  [C ˘  wϕ](m) = ϕ( ˘w−1m ˘w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The composition IP (C ˘  w)◦ℓ( ˘w)◦JP w|P (π) is then a self-intertwining operator of the induced representation (IP (π), HP (π)),  and independent of the choice of ˘w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We put  MP (w, π) = IP (C ˘  w) ◦ ℓ( ˘w) ◦ JP w|P (π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let ρP ′|P be the adjoint representation of �  M on �n ∩ �n′\\�n′ ∼= �n ∩ “n′, where �n, “n′, and �n are the Lie algebras of “  N,  �  N ′, and “  N, respectively, where N denotes the unipotent radical of the opposite parabolic subgroup P of P containing  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Following [18], we define normalizing factors  rP ′|P (ψ) =  L(0, ψ, ρ∨  P ′|P )  L(1, ψ, ρ∨  P ′|P )  ǫ( 1  2, ψ, ρ∨  P ′|P )  ǫ(0, ψ, ρ∨  P ′|P ) ,  rP (w, ψ) = rP w|P (ψ)ǫ(1  2, ψ, ρ∨  P w|P )−1,  and normalized intertwining operators  RP ′|P (π, ψ) = rP ′|P (ψ)−1JP ′|P (π),  RP (w, π, ψ) = rP (w, ψ)−1MP (w, π),  where L- and ǫ-factors are the automorphic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We have  RP ′|P (πλ, ψλ) =  �  v  RP ′v|Pv(πλ,v, ψλ,v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let P ′′ ⊂ G be a parabolic subgroup over F with Levi component M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We have  RP ′′|P (πλ, ψλ) = RP ′′|P ′(πλ, ψλ) ◦ RP ′|P (πλ, ψλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  51  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The unnormalized intertwining operator JP ′|P (πλ) satisfies the multiplicativity property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' It is known that at  each place v the local factors of automorphic L- and ǫ-factors are equal to the Artin L- and ǫ-factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Thus the  normalizing factor has a decomposition  rP ′|P (ψλ) =  �  v  rP ′v|Pv(ξv, ψv, ψF,v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Since every local factor rP ′v|Pv(ξv, ψv, ψF,v) has the multiplicativity property, so is rP ′|P (ψλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' This completes the  proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3  The global intertwining relation  In this subsection, we define two global linear forms and state the global intertwining relation, which we shall call  GIR for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' This subsection can be regarded as the global analogue of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let ψM∗ ∈ Ψ2(M ∗), and ψ ∈ Ψ(G∗) be its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note the change of notation from the previous subsection that  we write ψM∗ for a parameter for M ∗ rather than ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ψF : AF /F → C1 be a nontrivial additive character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For  πM = �  v πM,v ∈ ΠψM∗(M) and u♮ ∈ Nψ(M, G) we define a global intertwining operator  RP (u♮, πM, ψM∗, ψF ) =  �  v  RPv(u♮  v, πM,v, ψM∗,v, ψF,v),  where u♮  v ∈ Nψv(Mv, Gv), ψM∗,v ∈ Ψ+(M ∗), and ψF,v are the localizations of u♮, ψM∗, and ψF respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that πM ∈ ΠψM∗(M) is automorphic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', ⟨−, πM⟩ = εψM∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then for any u♮ ∈ Nψ(M, G)  we have an equation  �  v  πM,v(u♮  v) = εψM∗(u♮)C ˘  w,  of isomorphisms ( ˘wπM, VπM ) → (πM, VπM ), where w is the image of u♮ in Wψ(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3 of [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let u♮ ∈ Nψ(M, G) and πM ∈ ΠψM∗(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' RP (yu♮, πM, ψM∗, ψF ) = ⟨y, πM⟩RP (u♮, πM, ψM∗, ψF ) for any y ∈ Sψ(M), where y is the image of y in Sψ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' RP (wu, πM, ψM∗) = εψM∗(u♮)RP (u♮, πM, ψM∗, ψF ) if πM is automorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The first assertion follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The other one follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2 and Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The first linear form H(G) ∋ f �→ fG(ψM∗, u♮) is defined by  fG(ψM∗, u♮) =  �  v  fv,Gv(ψM∗,v, u♮  v)  =  �  πM∈ΠψM∗ (M)  tr �RP (u♮, πM, ψM∗, ψF ) ◦ IP (πM, f)� ,  f =  �  v  fv ∈ H(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Note that the parameter ψM∗ is relevant since it is discrete and M = ξ(M ∗) is defined over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put fG(ψM∗, u♮) to be  0 for M ∗ ⊂ G∗ which does not transfer to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5 implies the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The linear form fG(ψM∗, u♮) depends only on the image of u♮ in Nψ(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Thus we also write fG(ψM∗, u) for fG(ψM∗, u♮), where u ∈ Nψ(M, G) is the image of u♮.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Next let us recall the definition of the second linear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For a parameter ψ ∈ Ψ(G∗) and a semisimple element  s ∈ Sψ, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='9 attaches an endoscopic triple e = (Ge, se, ηe) ∈ E(G) and a parameter ψe ∈ Ψ(Ge), where ψe is  not unique (determined up to OutG(Ge)-action).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The second linear form H(G) ∋ f �→ f ′  G(ψ, s) is defined by  f ′  G(ψ, s) = f e(ψe)  =  �  v  f e  v(ψe  v) =  �  v  f ′  v,Gv(ψv, sv),  f =  �  v  fv ∈ H(G),  where sv ∈ Sψv denotes the localization of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If ψ is the image of ψM∗ ∈ Ψ2(M ∗), we also write f ′  G(ψM∗, s) for  f ′  G(ψ, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  52  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The linear form f ′  G(ψ, s) depends only on the image of s in Sψ, and hence on the image of e in E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='12, we have f ′  G(ψ, s) = f ′  G(ψ, ss0) for any s0 ∈ S◦  ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2 and the product formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2),  we have f ′  G(ψ, s) = f ′  G(ψ, sx) for any x ∈ Z(“  G)Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The following theorem is called the global intertwining relation (GIR), which will follow from LIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Theorem* 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8 (Global intertwining relation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If u ∈ Nψ(M, G) and s ∈ Sψ,ss have the same image in Sψ(G), then  we have  f ′  G(ψ, sψs−1) = fG(ψM∗, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4  Reduction of GIR to elliptic or exceptional (relative to G) parameters  This subsection is the global version of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' As in the local case, one can see that Ψ2  ell(G∗) is the set of the  equivalence classes of parameters ψ of the form  ψ = 2ψ1 ⊞ · · · 2ψq ⊞ ψq+1 ⊞ · · · ⊞ ψr,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2)  where ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , ψr are simple, symplectic, mutually distinct, and r ≥ q ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we have  Sψ ∼= O(2, C)q × O(1, C)r−q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let now Ψexc1(G∗) and Ψexc2(G∗) be the subsets of Ψ(G∗) consisting of the parameters ψ of the form  (exc1) ψ = 2ψ1 ⊞ψ2 ⊞· · ·⊞ψr, where ψ1 is simple and orthogonal, and ψ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , ψr are simple, symplectic, and mutually  distinct,  (exc2) ψ = 3ψ1 ⊞ ψ2 ⊞ · · · ⊞ ψr, where ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , ψr are simple, symplectic, and mutually distinct,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' They are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We then have  (exc1) Sψ ∼= Sp(2, C) × O(1, C)r−1,  (exc2) Sψ ∼= O(3, C) × O(1, C)r−1,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We put Ψexc(G∗) = Ψexc1(G∗) ⊔ Ψexc2(G∗), and we shall say that ψ is exceptional (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' of type (exc1),  resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' of type (exc2)) if ψ ∈ Ψexc(G∗) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ψ ∈ Ψexc1(G∗), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ψ ∈ Ψexc2(G∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' One can see that Ψexc(G∗) and  Ψell(G∗) are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put Ψell,exc(G∗) = Ψ(G∗) \\ (Ψell(G∗) ⊔ Ψexc(G∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put also Φ♥(G∗) = Ψ♥(G∗) ∩ Φ(G∗) for  ♥ ∈ {exc, exc1, exc2}, and Φell,exc(G∗) = Ψell,exc(G∗) ∩ Φ(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let ψM∗ ∈ Ψ2(M ∗) and ψ ∈ Ψ(G∗) be parameters such that ψ is the image of ψM∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For x ∈ Sψ, we define Tψ,  Bψ, sx ∈ Sψ, and Tψ,x in the same way as in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, let T ψ, Bψ, sx, and T ψ,x be their images in Sψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then  it can be seen that sx and T ψ,x are determined by the image of x in Sψ, and hence they are well-defined for x ∈ Sψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Define subsets Sψ,ell ⊂ Sψ and Sψ,ell ⊂ Sψ as in the local case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The following four lemmas can be proved in the  same way as Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='19, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='20, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='21, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='23 were, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Suppose that ψ ∈ Ψell,exc(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then  (1) every simple reflection w ∈ W ◦  ψ(M ∗, G∗) centralizes a torus of positive dimension in T ψ and  (2) dim T ψ,x ≥ 1 for all x ∈ Sψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If ψ ∈ Ψexc(G∗), then Sψ,ell = Sψ and Sψ,ell = Sψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let ξ : G∗ → G be an inner twist, and assume that M ∗ ⊊ G∗ is a proper standard Levi subgroup such that  M = ξ(M ∗) ⊊ G is a proper Levi subgroup defined over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let x ∈ Sψ(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that either  (1) ψ is elliptic, or  53  (2) every simple reflection w ∈ W ◦  ψ(M ∗, G∗) centralizes a torus of positive dimension in T ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then fG(ψM∗, u) is the same for every u ∈ Nψ(M, G) mapping to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let x ∈ Sψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We have fG(ψM∗, u) = f ′  G(ψ, sψs−1) whenever u ∈ Nψ(M, G) and s ∈ Sψ,ss map to x  unless  (1) ψ is elliptic and x ∈ Sψ,ell, or  (2) ψ ∈ Ψexc(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='11 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='12 imply that fG(ψM∗, u) depends only on the image of u in Sψ(M, G) = Sψ(G), unless ψ is  exceptional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Thus fG(ψM∗, x) is well-defined for x ∈ Sψ(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' On the other hand, when ψ is exceptional, we can  define fG(ψM∗, x) for x ∈ Sψ(M, G) similarly to local setting in the end of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Now we have defined fG(ψM∗, x) for  x ∈ Sψ(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5  The weaker identities  In the previous subsection, GIR for ψ ∈ Ψell,exc(G∗) was deduced from the induction hypothesis (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Now we shall see two weaker identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We omit Arthur’s procedure named the standard model for G, since it is very  similar to that for other classical groups explained in [7, §4] and [18, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We continue to be in the setup of the  previous subsection, but do not assume that M = ξ(M ∗) is defined over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Suppose that ψ ∈ Ψexc(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let x ∈ Sψ and f ∈ H(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If M ∗ does not transfer to G, then we have  f ′  G(ψ, sψx−1) = fG(ψ, x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  If M ∗ transfers to G, then we have  �  x∈Sψ  εG∗  ψ (x) �f ′  G(ψ, sψx−1) − fG(ψ, x)� = 0,  and RP (w, πM, ψM∗) = 1 for w ∈ Wψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Suppose that ψ ∈ Ψ2  ell(G∗) is of the form (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let f ∈ H(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we have  tr �RG  disc,ψ(f)� = 2−q|Sψ|−1  �  x∈Sψ,ell  εG∗  ψ (x) �f ′  G(ψ, sψx−1) − fG(ψ, x)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof of Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proofs of Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='14 are similar to those of Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1  of [18], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  6  Globalizations and the proof of local classification  In this section, we will finish the proof of LIR and ECR for generic parameters, and of LLC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The argument is  similar to that of [18, §4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The primary difference is the globalization of parameters, which is caused by the difference  between unitary groups and orthogonal groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1  Globalizations of fields, groups, and representations  First recall from [7, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1] the following lemma:  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let F be a local field other than C, and r0 a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we can find a totally real number  field ˙F and a place u of ˙F such that ˙Fu is isomorphic to F and ˙F has at least r0 real places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  As a corollary, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  54  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let F be a local field other than C, and r0 a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we can find a totally real number  field ˙F and places u1 and u2 of ˙F such that ˙Fu is isomorphic to F for u = u1, u2, and that ˙F has at least r0 real  places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ˙F be as in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' It can be easily seen that there exists an element α ∈ ˙F × that is not square in ˙F,  totally positive, and square in ˙Fu = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then ˙F(√α) is a number field what we want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The classification of quadratic forms or the exact sequence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2) leads the following lemma of special orthogonal  groups:  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let F be a local field, and G an inner form of G∗ = SO2n+1 over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ˙F be a number field with a  place u such that ˙Fu = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that ˙F is totally real unless F = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then for any place w of ˙F other than u, there  exists an inner form ˙G of ˙G∗ = SO2n+1 over ˙F with following properties:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ˙Gu = G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ˙Gv is split over ˙Fv for any place v of ˙F except u and w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' if G is non-quasi-split and w is a finite (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' real) place, then ˙Gw is the unique non-quasi-split inner form  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' isomorphic to SO(n − 1, n + 2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  In place of the third property, we can also choose ˙G with the properties 1, 2, and  3’ if ˙Fw is isomorphic to ˙Fu = F, then ˙Gw is isomorphic to ˙Gu = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  We have the following globalization theorems of discrete series representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ˙F be a totally real number field, and ˙G a simple twisted endoscopic group of GLN or an inner form  of ˙G∗ = SO2n+1 over ˙F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let V be a finite set of places of ˙F such that at least one real place v∞ is not contained  in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For all v ∈ V , let πv ∈ Π2,temp( ˙Gv), and let πv∞ be a discrete series representation with a sufficiently regular  infinitesimal character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then there exists a cuspidal automorphic representation ˙π of ˙G(A ˙F ) such that ˙πv = πv for all  v ∈ V and ˙πv∞ = πv∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is the same as that of [18, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ˙F be a totally real number field, and ˙G a simple twisted endoscopic group of GLN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let V be a finite  set of places of ˙F such that at least one real place v∞ is not contained in V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For all v ∈ V , let Mv ⊂ ˙Gv be a Levi  subgroup such that Mv = ˙Gv if v is a real place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For each v ∈ V , let πMv ∈ Π2,temp(Mv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then there exists a cuspidal  automorphic representation ˙π of ˙G(A ˙F ) such that for all v ∈ V , if Mv = ˙Gv then ˙πv = πMv and if Mv ̸= ˙Gv then  ˙πv is an irreducible subquotient of the induced representation I  ˙Gv  Pv (πMv ⊗ χv) for some unramified unitary character  χv ∈ Ψ(Mv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is the same as that of [18, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The following lemma is used in the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that the proof of the following lemma is independent  of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let F be a p-adic field, ξ : G∗ → G an inner twist of G∗ = SO2n+1 over F, M ∗ ⊂ G∗ a standard Levi  subgroup such that M := ξ(M ∗) is defined over F, and π ∈ Πscusp(M) an irreducible supercuspidal representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let ˙F, u, v2, ˙G∗, and ˙G be as in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then there exist an inner twist ˙ξ : ˙G∗ → ˙G over ˙F, a standard Levi  subgroup  ˙M ∗ ⊂ ˙G∗, and an irreducible cuspidal automorphic representation ˙π of  ˙M(A ˙F ) with following properties:  ˙ξu = ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙M := ˙ξ( ˙M ∗) is defined over ˙F, and ˙ξ| ˙  M∗ : ˙M ∗ → ˙M is an inner twist;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙Mu = M, and  ˙Mv is split over ˙Fv for any place v of ˙F except u and v2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ˙πu = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The former three properties follow from Lemmas 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The last property follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4 and  [31, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  55  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2  Globalizations of parameters  Now we shall globalize generic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let us first consider globalizations of simple parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ˙F be a totally real number field, and ˙G∗ a simple twisted endoscopic group of GLN, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', the split  symplectic group, the split odd special orthogonal group, or a quasi-split even special orthogonal group over ˙F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let V  be a finite set of places of ˙F in which at least one real place is not contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For each v ∈ V , let φv ∈ �Φ2,bdd( ˙G∗  v) be  a square integrable parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that for at least one v ∈ V , φv ∈ �Φsim( ˙G∗  v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then there exists a simple generic  parameter ˙φ ∈ �Φsim( ˙G∗) such that ˙φv = φv for all v ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1], except that we need the following one or two modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The first one is to use Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4 of this paper in place of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1 of [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then the same proof carries over word  by word in the case when ˙G∗ is a symplectic or an odd special orthogonal group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The second one is, in the case of even  special orthogonal groups, to replace Φsim by �Φsim and the ordinary equivalence classes of the representations by the  ǫ-equivalence classes of the representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Here, note that in the case of quasi-split even special orthogonal groups,  the endoscopic classification (LLC, ECR, and AMF) is known up to ǫ-equivalence, by Arthur [7] and Atobe-Gan  [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let N ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ˙F be a totally real number field, and ˙G∗ a simple twisted endoscopic group of GLN, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=',  the split symplectic group, the split odd special orthogonal group, or a quasi-split even special orthogonal group over ˙F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let V and {φv}v∈V be as in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that for at least one v ∈ V , φv ∈ �Φsim( ˙G∗  v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let v2 /∈ V be a finite  place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then there exists a simple generic parameter ˙φ ∈ �Φsim( ˙G∗) such that ˙φv = φv for all v ∈ V and ˙φv2 is of the  form  ˙φv2 = φ+ ⊕ φ∨  + ⊕ φ−,  where φ+ ∈ Φbdd(GL1) is a non-self-dual parameter and φ− ∈ �Φsim,bdd(N − 2) a simple self-dual parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2], except that we need the modifications similar to that of  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Next we shall consider globalizations of elliptic or exceptional parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let F be a local field, and G an inner  form of G∗ = SO2n+1 over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let M ∗ ⊂ G∗ be a Levi subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Following [18, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4], we shall say that M ∗ is linear  if M ∗ is isomorphic to a direct product of general linear groups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', n0 = 0 in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' As in the case of even unitary  groups, any nontrivial inner form G of G∗ does not admit a globalization ˙G which localizes to G at one place and is  split at all the other places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Hence the same complication is caused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let φ ∈ Φbdd(G∗) be a bounded L-parameter for G, which is the image of a discrete parameter φM∗ ∈ Φ2(M ∗) for  M ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that φ is either elliptic or exceptional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Explicitly M ∗, φ, and φM∗ has the form  M ∗ ≃ GLe1  N1 × · · · × GLer  Nr ×M ∗  −,  φ =  r  �  i=1  ℓiφi =  r  �  i=1  (ei(φi ⊕ φ∨  i ) ⊕ δiφi) ,  φM∗ = e1φ1 ⊕ · · · ⊕ erφr ⊕ φ−,  where φi ∈ �Φsim,bdd(Ni) are mutually distinct self-dual simple bounded L-parameters for GLNi, φ− = ⊕iδiφi ∈  Φ2(M ∗  −), δi ∈ {0, 1}, Ni ∈ Z≥1, ei ∈ Z≥0, ℓi = 2ei + δi, M ∗  − = SO2n0+1, and 2n0 = �  i δiNi so that �  i eiNi + n0 = n  and ei = ⌊ℓi/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  In the lemmas and propositions below, we will start from the following local data with the assumption given above:  a local field F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  G∗ = SO2n+1 and a Levi subgroup M ∗ ⊂ G∗ over F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  an inner form G of G∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  an L-parameter φ ∈ Φbdd(G∗) that is the image of a parameter φM∗ ∈ Φ2(M ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  First we treat the case when M ∗ = G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  56  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that M ∗ = G∗, and hence φ ∈ Φ2(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then there exists a global data ( ˙F, ˙G∗, ˙φ, u, v1, v2),  where ˙F is a totally real number field, ˙G∗ = SO2n+1 over ˙F, ˙φ ∈ Φ( ˙G∗), and u, v1, v2 are places of ˙F, such that v1 is  a finite place, v2 is a real place, and  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ˙Fu = F, ˙G∗  u = G∗, and ˙φu = φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙φ ∈ Φ2( ˙G∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙φv ∈ Φ2,bdd( ˙G∗  v) for v ∈ {v1, v2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' the canonical map S ˙φ → S ˙φv is isomorphic for v ∈ {u, v1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' A totally real number field ˙F and a place u such that ˙Fu = F are given by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Here we take ˙F to  have more than two real places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let v1 and v2 be finite and real places of ˙F, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put ˙G∗ = SO2n+1 over ˙F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , r, let G∗  i ∈ �Esim(Ni) be a classical group over F such that φi ∈ �Φsim,bdd(G∗  i ), which is given  in the 1st seed theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Take ˙G∗  i ∈ �Esim(Ni) to be a simple twisted endoscopic group over ˙F so that ˙G∗  i,u = G∗  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Choose a collection φv1,i ∈ �Φsim,bdd( ˙G∗  i,v1) (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , r) of pairwise distinct parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Choose also a collection  φv2,i ∈ �Φ2,bdd( ˙G∗  i,v2) (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , r) of parameters such that φv2,i and φv2,j do not have a common constituent for all  i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7 gives us a collection of parameters ˙φi ∈ �Φsim( ˙G∗  i ) such that ˙φi,u = φi, ˙φi,v1 = φv1,i, and  ˙φi,v2 = φv2,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  By the assumption, we have ei = 0 and δi = 1 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put  ˙φ = ˙φ1 ⊞ · · · ⊞ ˙φr ∈ Φ( ˙G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then by the construction the canonical map S ˙φ → S ˙φv is isomorphic for v ∈ {u, v1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Thus the fourth and second  conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The first condition is clearly satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The third one follows immediately from the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that M ∗ = G∗, and φ ∈ Φ2(G∗) is relevant for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then there exists a global data  ( ˙F, ˙G∗, ˙G, ˙φ, u, v1, v2),  where ˙F is a totally real number field, ˙G∗ = SO2n+1 over ˙F, ˙G an inner form of ˙G∗, ˙φ ∈ Φ2( ˙G∗) a parameter relevant  for ˙G, and u, v1, v2 are places of ˙F, such that v1 is a finite place, v2 is a real place, and  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ˙Fu = F, ˙G∗  u = G∗, ˙Gu = G, and ˙φu = φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ˙Gv is split unless v ∈ {u, v2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙φv ∈ Φ2,bdd( ˙G∗  v) for v ∈ {v1, v2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' the canonical map S ˙φ → S ˙φv is isomorphic for v ∈ {u, v1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='9, we obtain ˙F, ˙G∗, ˙φ, u, v1, and v2 satisfying the third and fourth conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If G is not split  over F, then by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3, we obtain ˙G that satisfies the second condition for which ˙φ is relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The first condition  is now clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If G splits over F, then ˙G = SO2n+1 clearly satisfies the conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Now we treat the case that M ∗ is proper, which implies that φ /∈ Φ2(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' At first we consider the case that M ∗  is not linear excluding the special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that M ∗ ⊊ G∗ is not linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Assume also that (n, φ) is neither (2, of type (exc1)) nor  (3, of type (exc2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then there exists a global data ( ˙F, ˙G∗, ˙M ∗, ˙φ, ˙φ ˙  M∗, u, v1, v2), where  ˙F is a totally real number  field, ˙G∗ = SO2n+1 over ˙F,  ˙M ∗ ⊂ ˙G∗ a Levi subgroup over ˙F, ˙φ ∈ Φ( ˙G∗), ˙φ ˙  M∗ ∈ Φ2( ˙M ∗), and u, v1, v2 are places of  ˙F, such that v1 and v2 are finite places and  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ˙Fu = F, ˙G∗  u = G∗,  ˙M ∗  u = M ∗, ˙φu = φ, and ˙φ ˙  M∗,u = φM∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' if φ ∈ Φ2  ell(G∗) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φexc1(G∗), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φexc2(G∗)), then ˙φ ∈ Φ2  ell( ˙G∗) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φexc1( ˙G∗), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φexc2( ˙G∗));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  57  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙φv1 ∈ Φbdd( ˙G∗  v1) and ˙φ ˙  M∗,v1 ∈ Φ2,bdd( ˙M ∗  v1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙φv2 ∈ Φell,exc  bdd  ( ˙G∗  v2) and it has a symplectic simple component with odd multiplicity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' the canonical maps S ˙φ → S ˙φv and S ˙φ ˙  M∗ → S ˙φ ˙  M∗,v are isomorphic for v ∈ {u, v1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' A totally real number field ˙F and a place u such that ˙Fu = F are given by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let v1 and v2 be finite  places of ˙F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put ˙G∗ = SO2n+1 and  ˙M ∗  − = SO2n0+1 over ˙F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let  ˙M ∗ ⊂ ˙G∗ be a Levi subgroup over ˙F such that  ˙M ∗ ≃ GLe1  N1 × · · · × GLer  Nr × ˙M ∗  −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , r, let G∗  i ∈ �Esim(Ni) over F and ˙G∗  i ∈ �Esim(Ni) over ˙F be as in the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Choose  a collection φv1,i ∈ �Φsim,bdd( ˙G∗  i,v1), (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , r) of pairwise distinct parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Since M ∗ is not linear, we have n0 ≥ 1, which implies that δi = 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', ℓi is odd) for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For such i, φi must  be symplectic, in particular Ni ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Suppose that there exists 1 ≤ s ≤ r such that δs = 1, Ns = 2, and Nt = 1 for all t ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In this case ℓs is odd and  ℓt is even for t ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We may assume s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If φ is elliptic, it can be written as  φ = φ1 ⊕ 2φq+1 ⊕ · · · ⊕ 2φr,  where φi are all symplectic, in particular Ni ≥ 2 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' This leads to r = 1 and φ = φ1, which contradicts the  assumption that M ∗ is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If φ is of type (exc1), it can be written as  φ = φ1 ⊕ 2φ2,  where φ1 is symplectic, and φ2 is orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then the dimension of φ equals 2 + 2 × 1 = 4, which contradicts the  assumption that (n, φ) is not (2, of type (exc1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If φ is of type (exc2), it can be written as  φ = 3φ1,  where φ1 is symplectic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then the dimension of φ equals 3 × 2 = 6, which contradicts the assumption that (n, φ) is not  (3, of type (exc2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Therefore, we have that either there exists 1 ≤ s ≤ r such that Ns ≥ 3, or there exists 1 ≤ s ≤ r such that Ns = 2  and δt = 1 for some t ̸= s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In both cases, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8, we have ˙φs ∈ �Φsim( ˙G∗  s) such that ˙φs,u = φs, ˙φs,v1 = φv1,s,  and ˙φs,v2 is of the form  ˙φs,v2 = φv2+ ⊕ ˙φ∨  v2+ ⊕ φv2−,  where φv2+ ∈ Φbdd(GL1) is a non-self-dual parameter, and φv2− ∈ �Φbdd(Ns − 2) a simple self-dual parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For  i ̸= s, choose φv2,i ∈ �Φsim,bdd( ˙G∗  i ) so that {φv2,i}i̸=s are mutually distinct and φv2,i ̸= φv2− for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7 gives  us ˙φi ∈ �Φsim( ˙G∗  i ) such that ˙φi,u = φi, ˙φi,v1 = φv1,i, and ˙φi,v2 = φv2,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Put  ˙φ = ℓ1 ˙φ1 ⊞ · · · ⊞ ℓr ˙φr ∈ Φ( ˙G∗),  ˙φ ˙  M∗ = e1 ˙φ1 ⊞ · · · ⊞ er ˙φr ⊞ ˙φ− ∈ Φ2( ˙M ∗),  where ˙φ− = ⊞iδi ˙φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The first and fifth conditions immediately follow from the construction, and hence so do the  second and third ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let us consider the fourth condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We have  ˙φv2 = ℓsφv2− ⊕ ℓs(φv2+ ⊕ φ∨  v2+) ⊕  �  i̸=s  ℓiφv2,i,  which is neither elliptic nor exceptional, because φv2+ is not self-dual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If Ns ≥ 3, then ˙φv2 has a symplectic simple  component with odd multiplicity because φ has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If Ns = 2 and δt = 1 for t ̸= s, then clearly φv2,t is a symplectic  simple component with odd multiplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  58  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that M ∗ ⊊ G∗ is not linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume also that (n, φ) is neither (2, of type (exc1)) nor  (3, of type (exc2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then there exists a global data  ( ˙F, ˙G∗, ˙G, ˙φ, ˙M ∗, ˙φ ˙  M∗, u, v1, v2),  where ˙F is a totally real number field, ˙G∗ = SO2n+1 over ˙F, ˙G an inner form of ˙G∗, ˙φ ∈ Φ( ˙G∗) a parameter,  ˙M ∗ ⊂ ˙G∗  a Levi subgroup, ˙φ ˙  M∗ ∈ Φ2( ˙M ∗) a parameter whose image in Φ( ˙G∗) is ˙φ, and u, v1, v2 are places of ˙F, such that v1  and v2 are finite place, and  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ˙Fu = F, ˙G∗  u = G∗  u, ˙Gu = G, ˙φu = φ,  ˙M ∗  u = M ∗, and ˙φ ˙  M∗,u = φM∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ˙Gv is split over ˙Fv unless v ∈ {u, v2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' if φ ∈ Φ2  ell(G∗) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φexc1(G∗), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φexc2(G∗)), then ˙φ ∈ Φ2  ell( ˙G∗) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φexc1( ˙G∗), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φexc2( ˙G∗));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙φv1 ∈ Φbdd( ˙G∗  v1) and ˙φ ˙  M∗,v1 ∈ φ2,bdd( ˙M ∗  v1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙φv2 ∈ Φell,exc  bdd  ( ˙G∗  v2) is relevant for ˙Gv2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' the canonical maps S ˙φ → S ˙φv and S ˙φ ˙  M∗ → S ˙φ ˙  M∗,v are isomorphic for v ∈ {u, v1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='11, we obtain ˙F, ˙G∗, ˙φ, ˙M ∗, ˙φ ˙  M∗, u, v1, and v2 satisfying the third, fourth, and sixth conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3, we obtain ˙G satisfying the second condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Since v2 is a finite place, the fifth condition follows from  the fourth condition of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='11, regardless of ˙G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The first condition is now clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Next we shall consider the special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that M ∗ ⊊ G∗ is not linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume also that n = 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ♥ = (exc1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' (exc2)) if  n = 2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' 3), and assume that φ is of type ♥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then there exists a global data ( ˙F, ˙G∗, ˙M ∗, ˙φ, ˙φ ˙  M∗, u, v1, v2), where  ˙F is a totally real number field, ˙G∗ = SO2n+1 over ˙F,  ˙M ∗ ⊂ ˙G∗ a Levi subgroup over ˙F, ˙φ ∈ Φ( ˙G∗), ˙φ ˙  M∗ ∈ Φ2( ˙M ∗),  and u, v1, v2 are places of ˙F, such that v1 is a finite place, v2 is a real place, and  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ˙Fu = F, ˙G∗  u = G∗,  ˙M ∗  u = M ∗, ˙φu = φ, and ˙φ ˙  M∗,u = φM∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙φ ∈ Φ♥( ˙G∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙φv1 ∈ Φbdd( ˙G∗  v1) and ˙φ ˙  M∗,v1 ∈ Φ2,bdd( ˙M ∗  v1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙φv2 = 2ω0 ⊕ τ1 or 3τ1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' the canonical maps S ˙φ → S ˙φv and S ˙φ ˙  M∗ → S ˙φ ˙  M∗,v are isomorphic for v ∈ {u, v1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' A totally real number field ˙F and a place u such that ˙Fu = F are given by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let v1 be a finite place  and v2 a real place of ˙F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put ˙G∗ = SO2n+1 over ˙F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Consider first the case when n = 2 and ♥ = (exc1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By the assumption we have ˙G∗ = SO5, φ = 2φ1 ⊕ φ2,  Sφ ≃ Sp(2, C) × O(1, C), and M ∗ ≃ GL1 × SO3, where φ1 is 1-dimensional orthogonal and φ2 is irreducible 2-  dimensional symplectic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We also have N1 = 1 and N2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For i = 1, 2, choose ( ˙G∗  i , si, ηi) ∈ �E(Ni) so that φi ∈ Φ( ˙G∗  i )  if we regard Φ( ˙G∗  i ) as a subset of Φ(Ni) via ηi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Concretely, ˙G∗  1 = Sp0 and ˙G∗  2 = SO3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let φv1,i be an element of  Φsim,bdd( ˙G∗  i ), for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put φv2,1 = ω0 and φv2,2 = τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7 gives us a global parameter ˙φi ∈ Φsim( ˙G∗  i )  such that ˙φi,u = φi, ˙φi,v1 = φv1,i, and ˙φi,v2 = φv2,i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put  ˙φ = 2 ˙φ1 ⊞ ˙φ2,  ˙M ∗ = GL1 × SO3,  ˙φ ˙  M∗ = ˙φ1 ⊞ ˙φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Consider next the case when n = 3 and ♥ = (exc2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By the assumption we have ˙G∗ = SO7, φ = 3φ1, Sφ ≃ O(3, C),  and M ∗ ≃ GL2 × SO3, where φ1 is irreducible 2-dimensional symplectic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Hence we regard φ1 ∈ Φ(SO3 /F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let φv1,1 be  59  an element of Φsim,bdd(SO3 / ˙Fv1), and put φv2,1 = τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7 gives us a global parameter ˙φ1 ∈ Φsim(SO3 / ˙F)  such that ˙φ1,u = φ1, ˙φ1,v1 = φv1,1, and ˙φ1,v2 = φv2,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put  ˙φ = 3 ˙φ1,  ˙M ∗ = GL2 × SO3,  ˙φ ˙  M∗ = ˙φ1 ⊞ ˙φ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  In both cases, the conditions follow from the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that M ∗ ⊊ G∗ is not linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume also that n = 2 or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ♥ = (exc1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' (exc2))  if n = 2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that φ ∈ Φbdd,♥(G∗) and that G is not quasi-split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then there exists a global data  ( ˙F, ˙G∗, ˙G, ˙φ, ˙M ∗, ˙φ ˙  M∗, u, v1, v2),  where ˙F is a totally real number field, ˙G∗ = SO2n+1 over ˙F, ˙G an inner form of ˙G∗, ˙φ ∈ Φ( ˙G∗) a parameter,  ˙M ∗ ⊂ ˙G∗  a Levi subgroup, ˙φ ˙  M∗ ∈ Φ2( ˙M ∗) a parameter whose image in Φ( ˙G∗) is ˙φ, and u, v1, v2 are places of ˙F, such that v1  is a finite place, v2 is a real place, and  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ˙Fu = F, ˙G∗  u = G∗  u, ˙Gu = G, ˙φu = φ,  ˙M ∗  u = M ∗, and ˙φ ˙  M∗,u = φM∗;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ˙Gv is split over ˙Fv unless v ∈ {u, v2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙φ ∈ Φ♥( ˙G∗);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙φv1 ∈ Φbdd( ˙G∗  v1) and ˙φ ˙  M∗,v1 ∈ φ2,bdd( ˙M ∗  v1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙φv2 ∈ Φbdd,♥( ˙G∗  v2) is relevant for ˙Gv2 and satisfies Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 relative to  ˙M ∗  v2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' the canonical maps S ˙φ → S ˙φv and S ˙φ ˙  M∗ → S ˙φ ˙  M∗,v are isomorphic for v ∈ {u, v1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13, we obtain ˙F, ˙G∗, ˙φ, ˙M ∗, ˙φ ˙  M∗, u, v1, and v2 satisfying the third, fourth, and sixth conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3, we obtain ˙G such that ˙Gu = G, ˙Gv2 ≃ SO(n − 1, n + 2), and ˙Gv is split if v /∈ {u, v2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Hence the  second condition is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The fifth condition follows from §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The first condition is now clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  If M ∗ is linear and G is non-quasi-split, then M ∗ never transfer to G and φ is not relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The next proposition  will be applied in such a case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that M ∗ ⊊ G∗ is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' There exists a global data ( ˙F, ˙G∗, ˙M ∗, ˙φ, ˙φ ˙  M∗, u1, u2, v1), where ˙F is  a totally real number field, ˙G∗ = SO2n+1 over ˙F,  ˙M ∗ ⊂ ˙G∗ a Levi subgroup over ˙F, ˙φ ∈ Φ( ˙G∗), ˙φ ˙  M∗ ∈ Φ2( ˙M ∗), and  u1, u2, v1 are places of ˙F, such that v1 is a finite place and  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ˙Fu = F, ˙G∗  u = G∗,  ˙M ∗  u = M ∗, ˙φu = φ, and ˙φ ˙  M∗,u = φM∗, for u ∈ {u1, u2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' if φ ∈ Φ2(G∗) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φ2  ell(G∗), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φexc1(G∗), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φexc2(G∗)), then ˙φ ∈ Φ2( ˙G∗) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φ2  ell( ˙G∗), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Φexc1( ˙G∗), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φexc2( ˙G∗));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙φv1 ∈ Φbdd( ˙G∗  v1) and ˙φ ˙  M∗,v1 ∈ Φ2,bdd( ˙M ∗  v1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' the canonical maps S ˙φ → S ˙φv and S ˙φ ˙  M∗ → S ˙φ ˙  M∗,v are isomorphic for v ∈ {u1, u2, v1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' A totally real field ˙F and places u1 and u2 such that ˙Fu1 = ˙Fu2 = F are given by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let v1 be a  finite place of ˙F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put ˙G∗ = SO2n+1 and  ˙M ∗  − = SO2n0+1 over ˙F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let  ˙M ∗ ⊂ ˙G∗ be a Levi subgroup over ˙F such that  ˙M ∗ ≃ GLe1  N1 × · · · × GLer  Nr × ˙M ∗  −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , r, let G∗  i ∈ �Esim(Ni) be a classical group over F such that φi ∈ �Φsim,bdd(G∗  i ), and take  ˙G∗  i ∈ �Esim(Ni) to be a simple twisted endoscopic group over ˙F so that  ˙G∗  i,u = G∗  i for u ∈ {u1, u2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Choose a  60  collection φv1,i ∈ �Φsim,bdd( ˙G∗  i,v1) (i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' , r) of pairwise distinct parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7 gives us a collection  of parameters ˙φi ∈ �Φsim( ˙G∗  i ) such that ˙φi,v1 = φv1,i and ˙φi,u = φi for u ∈ {u1, u2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Put  ˙φ = ℓ1 ˙φ1 ⊞ · · · ⊞ ℓr ˙φr ∈ Φ( ˙G∗),  ˙φ ˙  M∗ = e1 ˙φ1 ⊞ · · · ⊞ er ˙φr ⊞ ˙φ− ∈ Φ2( ˙M ∗),  where ˙φ− = ⊞iδi ˙φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then by the construction, the first and fourth conditions are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Hence the second and  third ones follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that M ∗ ⊊ G∗ is proper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then there exists a global data  ( ˙F, ˙G∗, ˙G, ˙φ, ˙M ∗, ˙φ ˙  M∗, u1, u2, v1),  where ˙F is a totally real number field, ˙G∗ = SO2n+1 over ˙F, ˙G an inner form of ˙G∗, ˙φ ∈ Φ( ˙G∗) a parameter,  ˙M ∗ ⊂ ˙G∗  a Levi subgroup, ˙φ ˙  M∗ ∈ Φ2( ˙M ∗) a parameter whose image in Φ( ˙G∗) is ˙φ, and u1, u2, v1 are places of ˙F, such that v1  is a finite place and  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ˙Fu = F, ˙G∗  u = G∗  u, ˙Gu = G, ˙φu = φ,  ˙M ∗  u = M ∗, and ˙φ ˙  M∗,u = φM∗, for u ∈ {u1, u2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ˙Gv is split over ˙Fv unless v ∈ {u1, u2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' if φ ∈ Φ2(G∗) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φ2  ell(G∗), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φexc1(G∗), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φexc2(G∗)), then ˙φ ∈ Φ2( ˙G∗) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φ2  ell( ˙G∗), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Φexc1( ˙G∗), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Φexc2( ˙G∗));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙φv1 ∈ Φbdd( ˙G∗  v1) and ˙φ ˙  M∗,v1 ∈ Φ2,bdd( ˙M ∗  v1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' the canonical maps S ˙φ → S ˙φv and S ˙φ ˙  M∗ → S ˙φ ˙  M∗,v are isomorphic for v ∈ {u1, u2, v1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='15, we obtain ˙F, ˙G∗, ˙φ,  ˙M ∗, ˙φ ˙  M∗ u1, u2, and v1 satisfying the first (except the assertion on ˙Gu),  third, fourth, and fifth conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3, we obtain ˙G such that ˙Gv = G for v ∈ {u1, u2} and ˙Gv splits over  ˙Fv otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3  On elliptic parameters  The following lemma is proved in the same way as [18, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ˙F be a number field, ˙G∗ = SO2n+1 over ˙F, ˙φ ∈ Φ2  ell( ˙G∗) a parameter, and ˙G an inner form of ˙G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let  ˙M ∗ ⊂ ˙G∗ be a Levi subgroup and ˙φ ˙  M∗ ∈ Φ2( ˙M ∗) a discrete parameter whose image in Φ( ˙G∗) is ˙φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that  there is a place v1 of ˙F such that ˙Gv1 is split over ˙F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ˙φv1 ∈ Φbdd( ˙G∗  v1) and ˙φ ˙  M∗  v1 ∈ φ2,bdd( ˙M ∗  v1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  the canonical maps S ˙φ → S ˙φv1 and S ˙φ ˙  M∗ → S ˙φ ˙  M∗,v1 are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then we have  tr R  ˙G  disc, ˙φ( ˙f) =  �  x∈S ˙φ,ell  Ä ˙f ′  ˙G( ˙φ, x) − ˙f ˙G( ˙φ, x)  ä  = 0,  for all ˙f ∈ H( ˙G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  61  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4  Proof of LIR for L-parameters  In this subsection we complete the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 for generic parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', L-parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let F be a  local field, G∗ = SO2n+1 over F, (M ∗, P ∗) a standard parabolic pair of G∗, and ξ : G∗ → G an inner twist of G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If  M ∗ transfers to G, then we take ξ so that M = ξ(M ∗) is defined over F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let φM∗ ∈ Φbdd(M ∗) be a generic parameter  for M, and φ ∈ Φbdd(G∗) its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If G is split, the theorem is already proven by Arthur [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Therefore, we may  assume that G is non-quasi-split, so in particular F ̸= C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that M ∗ ⊊ G∗ is proper, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', φ is not discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume also that φM∗ ∈ Φ2,bdd(M ∗) is  discrete and that φ is elliptic or exceptional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then for any x ∈ Sφ,ell, there exists a lift x ∈ Sφ of x such that  f ′  G(φ, x−1) = e(G)fG(φ, x),  for any f ∈ H(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Case N even of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The difference is that we divide the case into  the following cases:  M ∗ is not linear and (n, φ) is neither (2, of type (exc1)) nor (3, of type (exc2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  M ∗ is not linear and (n, φ) is either (2, of type (exc1)) or (3, of type (exc2));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  M ∗ is linear,  instead of the division into the cases  M ∗ is not linear and N ̸= 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  M ∗ is not linear and N = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  M ∗ is linear,  in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', and that we appeal to Lemmas 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='17, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='10, and Propositions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='12, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='14 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='16, instead of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='10, and Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7 of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The assertions 2 and 3 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 hold for generic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that the assertion 3 implies the assertion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Thanks to §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7 and §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8, we may assume that φM∗ is discrete  and that φ ∈ Φ2  ell(G∗) ⊔ Φexc(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By the consequence of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8, if φ is elliptic, then it remains to show that  f ′  G(φ, s−1) = e(G)fG(φ, u♮) for any u♮ ∈ Nφ and s ∈ Sφ,ss mapping to the same element in Sφ,ell,  and if φ is exceptional, then it remains to show that  f ′  G(φ, s−1) = e(G)fG(φ, u♮) for any u♮ ∈ Nφ,reg and s ∈ Sφ,ss mapping to the same element in Sφ,ell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  They can be proved similarly to [18, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The difference is that we utilize Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='20, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='18, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='14  instead of Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5 of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that in our case the equivalence classes of  inner forms, inner twists, and pure inner twists are in bijection naturally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  This completes the proof of the second and third assertions of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 for generic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The next two  lemmas show the first assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The first lemma treats the case of φM∗ ∈ Φ2,bdd(M ∗):  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that φM∗ ∈ Φ2,bdd(M ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then the assertion 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 holds for φM∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='21 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='19, we may assume that φ ∈ Φexc(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Case N  even of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The difference is that we appeal to Lemmas 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='16, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='20, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='19 instead of Lemmas 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7,  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2 of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Recall that in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7 we reduce only part 2 and 3 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 to the case of discrete parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The next  lemma is the reduction of part 1, and hence it completes the proof of part 1 for all generic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' As the lemmas  above, in a similar way to the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3), Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4, and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5 of loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', one can obtain a surjection  Rφ(M, G) = Wφ(M, G)/W ◦  φ(M, G) ։ RπM (M, G) := WπM (M, G)/W ◦  πM (M, G),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1)  for πM ∈ ΠφM∗(M), and can show the following lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  62  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let φM∗ ∈ Φbdd(M ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then the assertion 1 of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 holds for φM∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The homomorphism (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1) is bijective if φM∗ is relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Now Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 holds for any generic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' This completes the proof of LIR for generic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5  The construction of L-packets  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='11 for generic parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=', L-parameters) can be proven in the same way as Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1 of [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let us roughly review the procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' See [18, §§4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='9] for detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let F be a local field, G∗ = SO2n+1 over F, and (ξ, z) : G∗ → G a pure inner twist of G∗ as in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If G is split,  the theorem is already proven by Arthur [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Therefore, we assume that G is non-quasi-split, so in particular F ̸= C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  In the archimedean case, the theorem is known by Langlands and Shelstad, so we assume that F is a p-adic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  First consider the non-discrete parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Since we now have Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 for generic parameters φ, the local  packets Πφ(G) and the map Πφ(G) → Irr(Sφ, χG) for φ ∈ Φbdd(G∗) \\ Φ2(G∗) have already constructed at the  end of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The remaining assertions are that the map is bijective and that the packets are disjoint and exhaust  Πtemp(G) \\ Π2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The general classification [6] (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' [7, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5], [18, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7]) of Πtemp(G) by harmonic analysis says that there is a bijective  correspondence  (M, σ, µ) �→ πµ ∈ Πtemp(G),  from the G(F)-orbits of triples consisting of a Levi subgroup M ⊂ G over F, a discrete series representation σ  of M(F), and an irreducible representation µ of the representation-theoretic R-group R(σ) = Rσ(M, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Here,  we do not need an extension �R(σ) of the R-group, as explained in [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='158] or [18, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  The correspondence  can be described explicitly as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For a triple (M, σ, µ), let P be a parabolic subgroup of G with Levi factor  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Choose a family {RP (r, σ)}r∈Rσ(M,G) ⊂ AutG(F )(IG  P (σ)) of self-intertwining operators so that an assignment  r �→ RP (r, σ) is a homomorphism from Rσ(M, G) to AutG(F )(IG  P (σ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we have a representation RP (r, σ)◦IG  P (σ, g)  of Rσ(M, G)×G(F) on HG  P (σ), for which we shall write R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The representation πµ is characterized by a decomposition  R =  �  µ∈Irr(Rσ(M,G))  µ∨ ⊗ πµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Note that Rσ(M, G) is abelian in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then the bijectivity of the map Πφ(G) → Irr(Sφ, χG) and the disjointness and exhaustion of L-packets in Πtemp(G)  follows from the same argument as in [18, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7], which we omit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='11 holds for generic parameters φ ∈ Πbdd(G∗) \\ Π2(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that Πφ(G, Ξ), χΞ, S♮  φ, N ♮  φ, and S♮♮  φ (M) in loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  should be replaced by Πφ(G), χG, Sφ, Nφ, and Sφ(M) in our case, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Next, before consider discrete parameters, we need some preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Put Tell(G) to be the set of G(F)-conjugacy  classes of triples τ = (M, σ, r), where M ⊂ G is a Levi subgroup, σ a unitary discrete series representation of M(F),  and r ∈ Rσ(M, G) a regular element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Here, r ∈ Rσ(M, G) is said to be regular if ar=1  M  := {λ ∈ aM | rλ = λ} coincides  with aG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The set Π2,temp(G) is naturally regarded as a subset of Tell(G) via π �→ (G, π, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The complement set will  be denoted by T 2  ell(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In general, the trace Paley-Wiener theorem tells that ”orbital integrals” of cuspidal functions  f ∈ Hcusp(G) are described by the functions on Tell(G) given by some intertwining operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  In order to utilize the local intertwining operator defined in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5 , we need to consider the set T ♮  ell(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let T 2,♮  ell (G) be  the set of G(F)-conjugacy classes of triples τ ♮ = (M, σ, s), where M ⊊ G is a proper Levi subgroup, σ a unitary discrete  series representation of M(F), and s ∈ Sφσ(G) an element whose image under the surjection Sφσ(G) ։ Rσ(M, G)  is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Here, we write φσ for the L-parameter of σ, and we also write φσ for its image in Φbdd(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then put  T ♮  ell(G) := Π2,temp(G)⊔T 2,♮  ell (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The surjection Sφσ(G) ։ Rσ(M, G) induces the natural surjection T ♮  ell(G) ։ Tell(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For f ∈ H(G), put  fG(τ ♮) =  ®  tr �RP (u♮, σ, φσ, ψF ) ◦ IG  P (σ, f)� ,  for τ ♮ = (M, σ, s) ∈ T 2,♮  ell (G),  tr (π(f)) ,  for τ ♮ = π ∈ Π2,temp(G),  63  where u♮ ∈ Nφσ(M, G) is a lift of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' It is independent of the choice of u♮.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If τ ♮  1 and τ ♮  2 in T 2,♮  ell (G) maps to a same  element in Tell(G), then they are of the form τ ♮  1 = (M, σ, s) and τ ♮  2 = (M, σ, ys) for some y ∈ Sφσ(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we shall  write τ ♮  2 = yτ ♮  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In this case, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='10 we have  fG(τ ♮  1) = ⟨y, σ⟩fG(τ ♮  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let φ ∈ Φ2,bdd(G∗) be a discrete generic parameter, and x ∈ Sφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8, one can construct an  endoscopic triple e = (Ge, se, ηe) of G and a generic parameter φe of Ge, to obtain a linear form H(G) ∋ f �→  f ′  G(φ, x) := f e(φe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The trace Paley-Wiener theorem implies that there exists a family {cφ,x(τ ♮)}τ ♮∈T ♮  ell(G) ⊂ C of  complex numbers such that for any f ∈ Hcusp(G), we have  f ′  G(φ, x) = e(G)  �  τ∈Tell(G)  cφ,x(τ ♮)fG(τ ♮),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2)  and  cφ,x(τ ♮) = ⟨y, τ⟩cφ,x(yτ ♮),  where τ ♮ is a lift of τ, and ⟨y, τ⟩ = ⟨y, σ⟩ for τ = (M, σ, s) ∈ T 2  ell(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Note that the product cφ,x(τ ♮)fG(τ ♮) does not  depend on the choice of τ ♮.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then one can show an orthogonality relation:  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' (a) Let φ ∈ Φ2,bdd(G∗) be a discrete generic parameter, and x ∈ Sφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then cφ,x(τ ♮) = 0 for all  τ ♮ ∈ T 2,♮  ell (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  (b) Let φ1, φ2 ∈ Φ2,bdd(G∗) be discrete generic parameters, and xi ∈ Sφi (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then  �  π∈Π2,temp(G)  cφ1,x1(π)cφ2,x2(π) =  ®  |Sφ1|,  if φ1 = φ2 and x1 = x2,  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3] or [7, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Finally we shall consider discrete generic parameters, after introducing one lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ξ : ˙G∗ → ˙G be an inner twist of ˙G∗ = SO2n+1 over a number field ˙F, and ˙φ ∈ Φ2( ˙G∗) a discrete  global parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For any ˙f ∈ H( ˙G), we have  �  ˙π  n ˙φ( ˙π) ˙f ˙G( ˙π) =  1  |S ˙φ|  �  x∈S ˙φ  ˙f ′  ˙G( ˙φ, x),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3)  where ˙π runs over irreducible representations of ˙G(A ˙F ), and n ˙φ( ˙π) it the multiplicity of ˙π in (R ˙G  disc, ˙φ, L2  disc, ˙φ( ˙G( ˙F)\\ ˙G(A ˙F ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1] or [7, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='6)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let φ ∈ Φ2,bdd(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='10 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3 gives us a global data ( ˙F, ˙G∗, ˙G, ˙φ, u, v1, v2) such that  ˙F is totally real, and  v1 is a finite place, and v2 is a real place;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙Fu = F, ˙G∗  u = G∗, ˙Gu = G, and ˙φu = φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙Gv is split for v /∈ {u, v2}, and χG = χ ˙Gv2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ˙φv ∈ Φ2,bdd( ˙G∗  v) for v ∈ {v1, v2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  the canonical map S ˙φ → S ˙φv is isomorphic for v ∈ {u, v1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  64  Choose ˙f ∈ H( ˙G) so that ˙f is of the form ˙f = �  v ˙fv, and that ˙fu ∈ Hcusp(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then we have the equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3)  due to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For v /∈ {u, v2}, the group ˙Gv is split and hence we have ECR (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In the case v = v2, since  ˙Fv2 = R, ECR is known by the work of Shelstad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' At the place u, we have the equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2) since ˙fu is cuspidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Substituting them, we obtain the following equation from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3):  �  ˙π∈Πunit( ˙G(A ˙  F ))  n ˙φ( ˙π) ˙f ˙G( ˙π) =  1  |S ˙φ|  �  ˙πu∈Π ˙φu  �  π∈Π2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='temp(G)  �  x∈S ˙φ  ⟨ ˙xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙πu⟩cφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙xu(π) ˙f ˙G( ˙πu ⊗ π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  where Πunit( ˙G(A ˙F )) denotes the set of isomorphism classes of irreducible unitary representations of ˙G(A ˙F ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙x ∈ S ˙φ  is a lift of x ∈ S ˙φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙xv ∈ S ˙φv is the image of ˙x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' and  Π ˙φu :=  \uf8f1  \uf8f2  \uf8f3 ˙πu =  �  v̸=u  ˙πv  ������  ˙πv ∈ Π ˙φv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ⟨−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙πv⟩ = 1 for almost all v  \uf8fc  \uf8fd  \uf8fe ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ˙xu := ( ˙xv)v̸=u ∈  �  v̸=u  S ˙φv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  ⟨ ˙xu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙πu⟩ :=  �  v̸=u  ⟨ ˙xv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' ˙πv⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For each place v other than u, v1 and v2, fix ˙πv ∈ Π ˙φv such that ⟨−, ˙πv⟩ is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For a real place v2, fix ˙πv2 ∈ Π ˙φv2  arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' They exist because ˙Gv splits if v /∈ {u, v1, v2} and ˙Fv2 = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  For any µ ∈ Irr(Sφ, χG), choose ˙πv1 ∈ Π ˙φv1 so that µ( ˙xu)⟨ ˙xu, ˙πu⟩ = 1 for all ˙x ∈ S ˙φ, where ˙πu = �  v̸=u ˙πv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Such  ˙πv1 does exist because we have χG = χ ˙Gv2, v1 is a finite place, ˙Gv1 is split, and S ˙φ → S ˙φv is isomorphic for v ∈ {u, v1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then for π ∈ Π2,temp(G), we set  nφ(µ, π) := n ˙φ ( ˙πu ⊗ π) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then by the same argument of the proof of [18, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2], we have  nφ(µ, π) =  1  |Sφ|  �  x∈Sφ  µ(x)−1cφ,x(π),  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4)  for any π ∈ Π2,temp(G), where x ∈ Sφ denotes for an arbitrary representative of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Moreover, the orthogonality  relation (Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='24) and the formula (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4) implies the following formula:  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let φ, φ′ ∈ Φ2,bdd(G∗) be discrete generic parameters, and µ ∈ Irr(Sφ, χG) and µ′ ∈ Irr(Sφ′, χG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then we have  �  π∈Π2,temp(G)  nφ(µ, π)nφ′(µ′, π) =  ®  1,  if (φ, µ) = (φ′, µ′),  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is the similar to that of [18, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Since nφ(µ, π) is a non-negative integer, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='26 tells us that for any φ ∈ Φ2,bdd(G∗) and µ ∈ Irr(Sφ, χG),  there exists a unique π ∈ Π2,temp(G) such that nφ(µ, π) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We shall write π(φ, µ) for this representation π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Then  an assignment (φ, µ) �→ π(φ, µ) gives an injective map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The packet of φ ∈ Φ2,bdd(G∗) is defined by  Πφ := { π(φ, µ) | µ ∈ Irr(S, χG) } ,  and the associated character of the component group is defined by  ⟨−, π(φ, µ)⟩ := µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  One can now easily see that the packets are disjoint and the map πφ → Irr(Sφ, χG) is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let φ ∈ Φ2,bdd(G∗) and x ∈ Sφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Combining the formula (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4) with Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='26, we have  cφ,x(π) =  ®µ(x) = ⟨x, π⟩,  if π = π(φ, µ),  0,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Then the formula (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2) is none other than the endoscopic character relation for cuspidal function f ∈ Hcusp(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  65  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' For general f ∈ H(G), we have ECR:  f ′(φ, x) = e(G)  �  π∈Πφ  ⟨x, π⟩fG(π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [7, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='27 characterizes the packet Πφ and the bijection Πφ → Irr(Sφ, χG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Thus they does not depend on  the choice of ˙φ or ˙πu, but only on φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By the argument between Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='11 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='12, we now have the packet Πφ  and the bijection Πφ → Irr(Sφ, χG) for all generic parameters φ ∈ Φ(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  It remains to show that the packets exhaust Π2,temp(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let π ∈ Π2,temp(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Take a globalization ˙π of π as in  Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' By the decomposition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1), we have the A-parameter ˙φ of ˙π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In particular, we have n ˙φ( ˙π) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Since  ˙πv′ has sufficiently regular infinitesimal character for some real place v′, the parameter ˙φ is generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Thus Theorem  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 and Hypothesis 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1 hold for ˙φ, so does Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2 for ˙φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' (Note that §7 is independent from the exhaustion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=')  Hence we have π ∈ Π ˙φu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If ˙φu is not bounded (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' not discrete), then the packet does not contain a tempered  representation π by definition in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' the beginning of this subsection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Therefore, we have ˙φu ∈ Φ2,bdd(G∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  This completes the proof of the local classification theorem for generic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  7  The proof of the global theorem  Assuming the existence of local A-packets and LIR, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 can be proven in the same way as Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1 of [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' In particular, since the assumptions hold for generic parameters, generic part of the theorem holds true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  In this section we record the statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' See [18, §5] for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let F be a number field, G∗ = SO2n+1 over F, and G an inner form of G∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If G is split, the theorem is already  proven by Arthur [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Therefore, we assume that G is non-quasi-split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Recall from §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1) the decomposition  L2  disc(G(F)\\G(AF )) =  �  ψ∈Ψ(G∗)  L2  disc,ψ(G(F)\\G(AF )),  tr Rdisc(f) =  �  ψ∈Ψ(G∗)  Rdisc,ψ(f),  f ∈ H(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Let us introduce a hypothesis on local A-packets:  Hypothesis 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ψ ∈ Ψ(G∗) be a global parameter for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' We have the local packet Πψv = Πψv(Gv) and the map  Πψv → Irr(Sψv, χGv) satisfying ECR (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='7) for all places v of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Let ψ ∈ Ψ(G∗) be a global parameter for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Assume that Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 and Hypothesis 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1 hold for  ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If ψ /∈ Ψ2(G∗), then  L2  disc,ψ(G(F)\\G(AF )) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' If ψ ∈ Ψ2(G∗), then  L2  disc,ψ(G(F)\\G(AF )) =  �  π∈Πψ(G,εψ)  π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' The proof is similar to that of [18, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  Since Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='13 and Hypothesis 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='1 hold true if ψ = φ ∈ Φ(G∗) is generic, the global classification theorem is  now established for the generic part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content='  66  References  [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
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+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
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+page_content=' Barbasch, Genuine representations of the metaplectic group, Compositio Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' 113, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
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+page_content=' Barbasch, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
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+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' 84, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
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+page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FLT4oBgHgl3EQfrS9R/content/2301.12143v1.pdf'}
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+arXiv:2301.00743v1  [cs.SC]  2 Jan 2023
+1
+Computing square roots in quaternion algebras
+Przemysław Koprowski
+Institute of Mathematics
+University of Silesia in Katowice
+ul. Bankowa 14, 40-007 Katowice, Poland
+przemyslaw.koprowski@us.edu.pl
+Abstract. We present an explicit algorithmic method for computing square roots in quaternion
+algebras over global fields of characteristic different from 2.
+Keywords:
+square root computation, quaternion algebra, number fields, global fields
+1.
+Introduction
+The computation of square roots is one of the most basic operations in mathematics. Effective methods
+for computing square roots are among the oldest algorithms in the realm of computational mathemat-
+ics. In fact, Heron’s method for a numerical approximation of a square root of a real number is two
+thousand years old and preceded by the Euclidean algorithm (wildly believed to be the oldest mathe-
+matical algorithm) by only about three to four centuries (for an in-depth discussion on the chronology
+see [1]). Although numerous methods for computing square roots in various algebraic structures are
+known nowadays, some important omissions prevail. Among them are general quaternion algebras.
+Computation of square roots in the algebra of Hamilton quaternions H =
+� −1,−1
+R
+�
+is well-known (see
+[2]) and very simple as for every quaternion
+∈ H there is a subfield K ∼= C of H containing , and
+so the computation the square root in H can be reduced to the computation of the square root in C. It
+is no longer so in a general quaternion algebra Q =
+� α,β
+K
+�
+for an arbitrary field K and two elements
+α, β ∈ K×. To the best of our knowledge, no algorithm for computing quaternionic square roots ex-
+ists in the literature. One possible explanation for this (quite surprising) fact is that in the commutative
+Address for correspondence: Institute of Mathematics, University of Silesia, ul. Bankowa 14, 40-007 Katowice, Poland
+
+2
+P. Koprowski / Computing square roots in quaternion algebras
+case when one considers a field extension L/K, a typical way to compute a square root of an element
+a ∈ L is to factor the polynomial x2 − a in L[x]. However, for quaternion algebras, there are no
+known polynomial factorization algorithms.
+The sole purpose of this paper is to correct this evident omission and present an explicit algorithm
+for computing square roots in quaternion algebras over arbitrary global fields of characteristic different
+from 2.
+2.
+Notation
+Throughout this paper, K will denote an arbitrary global field of characteristic char K ̸= 2. Hence, K
+is either a number field, i.e. a finite extension of Q (then its characteristic is just 0) or a global function
+field, that is, a finite extension of a rational function field over a finite field Fq, where q is a power of
+an odd prime. The set of nonzero elements of K is denoted K×.
+Recall that a quaternion algebra Q =
+� α,β
+K
+�
+over K is a 4-dimensional K-algebra with a basis
+{1, i, j, k} and a multiplication gathered by the rules:
+i2 = α,
+j2 = β,
+ij = k = −ji.
+As usual, we shall identify the field K with the subfield K · 1 of Q, which is known to coincide with
+the center Z(Q) of Q. We refer the reader to [3, 4] for a comprehensive presentation of the theory of
+quaternion algebras.
+A quaternion
+is called pure (see e.g., [4, Definition 5.2.1]) if
+∈ spanK{i, j, k}. Every quater-
+nion
+∈ Q can be uniquely expressed as a sum
+= a +
+0 of a scalar a ∈ K and a pure quaternion
+0. We write
+:= a −
+0 for the conjugate of . The map that sends a quaternion to its conjugate is an
+involution.
+If x is an element of either a quadratic field extension L = K
+�√α
+�
+of K or a quaternion algebra
+Q =
+� α,β
+K
+�
+over K, we write N (x) := xx and call it the norm of x. If the domain is not clear from
+the context, we write NL/K or NQ/K.
+Remark 2.1. When Q is a quaternion algebra, the norm of in the above sense should not be confused
+with the determinant of the endomorphism of Q defined by the multiplication by , which is often also
+called the norm. For this reason, in [3, 4] the map
+�→
+is called the reduced norm and denoted nrd.
+In that manner, our terminology in the present paper agrees with the one used by Lam in [5] but not
+with the one used by Vigneras in [3] and Voight in [4].
+Equivalency classes of valuations on K are called places. Throughout this paper, places are de-
+noted using fraktur letters p, q, r. Every place of a global field is either archimedean, when it extends
+the standard absolute value on Q (then the field K is necessarily a number field) or non-archimedean.
+Over a global function field, every place is non-archimedean. To avoid monotonous repetitions, non-
+archimedean places will also be called primes (or finite primes when we want to emphasize the fact
+that they are non-archimedean). The completion of K with respect to a place p is denoted Kp. If p is
+a finite prime, we write ordp : K → Z to denote the corresponding (normalized) discrete valuation
+
+P. Koprowski / Computing square roots in quaternion algebras
+3
+on K. The prime p is called dyadic if ordp 2 ̸= 0. The map ordp induces a natural map K×/K×2 → Z/2Z
+on the group of square classes of K that is again denoted ordp.
+If p is an archimedean place, then the completion Kp is isomorphic either to C or to R. The places
+of the second kind are called real. We write sgnr a for the sign of a ∈ K with respect to for the unique
+ordering of K induced by a real place r.
+Given some nonzero elements a1, . . . , an ∈ K we denote by ⟨a1, . . . , an⟩ the quadratic form
+a1x2
+1 + · · · + anx2
+n. Further, if p is a place and a, b ∈ K× we write (a, b)p for the Hilber symbol of a
+and b at p, that is
+(a, b)p :=
+�
+1
+if
+�a,b
+K
+� ∼= M2Kp,
+−1
+otherwise.
+For a quadratic form ξ = ⟨a1, . . . , an⟩ we define its Hasse invariant spξ at p by the formula (see e.g.,
+[5, Definition V.3.17]):
+spξ :=
+�
+i<j
+(ai, aj)p.
+Finally, abusing the notation harmlessly, by log-1 we will denote the (unique) isomorphism from the
+multiplicative group {±1} to the additive group {0, 1} with addition modulo 2.
+3.
+Square roots of non-central elements
+Let us begin by writing down the explicit formula for a square in quaternion algebra so that we can
+easily reference it in the discussion that follows.
+Observation 3.1. If
+= q0 + q1i + q2j + q3k ∈ Q is a quaternion, then
+2 = (q2
+0 + q2
+1α + q2
+2β − q2
+3αβ) + 2q0q1i + 2q0q2j + 2q0q3k
+= (2q2
+0 − N( )) + 2q0 · (q1i + q2j + q3k).
+(1)
+An immediate consequence of the previous observation is the following rather well-known fact.
+Corollary 3.2. If
+∈ Q is a pure quaternion, then
+2 ∈ Z(Q) = K.
+Another direct consequence of Eq. (1) is the following observation that may be treated as a partial
+converse of Corollary 3.2.
+Observation 3.3. Let
+∈ Q be a square root of some element a ∈ K. Then
+is either pure or
+∈ K.
+Proof:
+Let
+= q0 + q1i + q2j + q3k. If
+2 = a ∈ K then by Eq. (1) we have
+2q0q1 = 2q0q2 = 2q0q3 = 0.
+Therefore, if
+is not pure, that is if q0 ̸= 0, then q1 = q2 = q3 = 0, hence
+∈ K.
+⊓⊔
+
+4
+P. Koprowski / Computing square roots in quaternion algebras
+Combining Corollary 3.2 with Observation 3.3 we see that for computing the square roots in
+quaternion algebras it is crucial to distinguish between the case when one computes a quaternionic
+square root of an element in K (i.e., in the center of Q) and the case when the argument comes from
+Q \Z(Q). It turns out that the latter case is, in fact, trivial and requires nothing more than high-school
+mathematics.
+Algorithm 1. Let Q =
+� α,β
+K
+�
+be a quaternion algebra over a field K of characteristic char K ̸= 2.
+Given a quaternion
+= q0 + q1i + q2j + q3k ∈ Q \ Z(Q), this algorithm outputs its square root or
+reports a failure when
+is not a square.
+1. Check if the norm N( ) of
+is a square in K.
+(a) If it is not, then report a failure and quit.
+(b) If it is, let d be an element of K such that d2 = N( ).
+2. Check if any of the following two elements is a square in K:
+a+ := q0 + d
+2
+,
+a− := q0 − d
+2
+.
+3. If neither of them is a square, then report a failure and quit.
+4. Otherwise, fix r0 such that either r2
+0 = a+ or r2
+0 = a−.
+5. Set
+r1 := q1
+2r0
+,
+r2 := q2
+2r0
+,
+r3 := q3
+2r0
+.
+6. Output r = r0 + r1i + r2j + r3k.
+Proof of correctness:
+Since the norm N : Q → K is multiplicative, it is obvious that if N( ) /∈ K2, then
+cannot be a
+square in Q. This fact justifies the early exit in step (1a) of the algorithm. Assume that N( ) = d2 and
+let r = r0 + r1i + r2j + r3k be the sought square root of , if it exists. By Eq. (1) we have
+q1 = 2r0r1,
+q2 = 2r0r2,
+q3 = 2r0r3.
+It is, thus, clear that it suffices to find r0. Again by Eq. (1) we may write
+q0 = r2
+0 + r2
+1α + r2
+2β − r2
+3αβ = r2
+0 +
+� q1
+2r0
+�2
+α +
+� q2
+2r0
+�2
+β −
+� q3
+2r0
+�2
+αβ.
+The above formula can be rewritten in the form of a bi-quadratic equation:
+4r4
+0 − 4q0r2
+0 +
+�
+q2
+1α + q2
+2β − q2
+3αβ
+�
+= 0.
+
+P. Koprowski / Computing square roots in quaternion algebras
+5
+If we treat the left-hand-side as a quadratic equation in r2
+0, then its discriminant equals 16 · N( ) =
+(4d)2, hence
+r2
+0 = q0 ± d
+2
+= a±.
+It follows that the sought quaternion r exists if and only if either a+ or a− is a square in K. This
+proves the correctness of the algorithm.
+⊓⊔
+Remark 3.4. In the above proof, we constructed the square root r of a quaternion
+∈ Q \ Z(Q) by
+solving a bi-quadratic equation. Such equations in general, may have four roots. Hence, one may
+suspect that there are four distinct quaternions r such that r2 = . It is not the case. It is clear from
+the above proof that
+∈ Q \ Z(Q) has only finitely moan square roots in Q. Now, if r2 = Q, then r
+is a root of a quaternionic polynomial x2 − . But [6, Theorem 5] asserts that a quadratic polynomial
+over Q which has more than two zeros must have infinitely many of them. This way, we conclude
+that
+has just two square roots. Notice that for hamiltonian quaternions this fact has been observed
+already 80 years ago by Niven in [2].
+4.
+Square roots of central elements. Split case
+It is evident from the preceding section that the only non-trivial case that must be considered is how
+to compute a quaternionic square root of an element of the base field K, which is not a square in K.
+In contrast to the previous case (cf. Remark 3.4), in general, an element a ∈ K = Z(Q) may have
+infinitely many square roots in Q. Once again, for hamiltonian quaternions it has been observed
+already by Niven.
+First, we need, however, to introduce an auxiliary algorithm that is not specific to quaternions, as
+it deals with an arbitrary quadratic form. Recall that a quadratic form is called isotropic (see e.g., [5,
+Definition I.3.1]) if it represents zero non-trivially. It is well known (see, e.g., [5, Theorem I.3.4]) that
+every isotropic form represents all elements of K.
+Algorithm 2. Let ξ be an isotropic quadratic form of dimension n over a field K of characteristic
+char K ̸= 2. Given an element a ∈ K and a vector V ∈ Kn such that ξ(V ) = 0, this algorithm
+outputs a vector W ∈ Kn satisfying the condition ξ(W) = a.
+1. Find a vector U ∈ Kn such that U and V are linearly independent.
+2. Set b := ξ(U) and c := 1/2 ·
+�
+ξ(U + V ) − ξ(U)
+�
+.
+3. Output
+W := U + a − b
+2c
+· V.
+Proof of correctness:
+
+6
+P. Koprowski / Computing square roots in quaternion algebras
+Just compute:
+ξ(W) = ξ
+�
+U + a − b
+2c
+· V
+�
+= ξ(U) + a − b
+2c
+·
+�
+ξ(U + V ) − ξ(U) − ξ(V )
+�
++ (a − b)2
+4c2
+ξ(V )
+= b + a − b
+2c
+· 2c + 0 = a
+⊓⊔
+Recall that a quaternion algebra Q =
+� α,β
+K
+�
+is said to split (see e.g., [4, Definition 5.4.5]) if Q
+is isomorphic to the matrix ring M2K. It is well known (see e.g., [4, Theorem 5.4.4] or [5, Theo-
+rem III.2.7]) that Q is split if and only if the quadratic form ⟨−α, −β, αβ⟩ is isotropic. If it is the case,
+the preceding algorithm combined with Eq. (1) lets us compute the quaternionic square root of any
+element of the base field. In particular, when K is a global field, char K ̸= 2, then the computation
+of the square root of a ∈ K in a split quaternion algebra boils down to solving a norm equation in
+a quadratic extension of K. Algorithms for the latter task are well known. They can be found in
+[7, 8, 9, 10, 11].
+Algorithm 3. Let Q =
+�α,β
+K
+�
+be a split quaternion algebra over a global field K of characteristic
+char K ̸= 2. Given a nonzero element a ∈ K, this algorithm outputs a pure quaternion
+∈ Q such
+that
+2 = a.
+1. Check if α is a square in K. If there is c ∈ K× such that c2 = α, then set V := (0, c, 1).
+2. Otherwise, if α /∈ K×2, then:
+(a) Construct a quadratic field extension L = K
+�√α
+�
+of K.
+(b) Solve the norm equation
+NL/K (x) = −α
+β
+and denote the solution by λ = b + c√α.
+(c) Set V := (1, b, c).
+3. Let ξ := ⟨−α, −β, αβ⟩ be the pure subform of the norm form of Q. Execute Algorithm 2 with
+the input (−a, V, ξ) to construct a vector W = (w1, w2, w3) such that ξ(W) = −a.
+4. Output
+= 0 + w1i + w2j + w3k.
+Proof of correctness:
+We claim that the vector V constructed either in step (1) or in step (2) of the algorithm is an isotropic
+vector for ξ. First, suppose that α is a square in K. Say α = c2 for some c ∈ K×. Then
+−α · 02 − β · c2 + αβ · 12 = 0.
+
+P. Koprowski / Computing square roots in quaternion algebras
+7
+Conversely, assume that α /∈ K×2 and so L = K
+�√α
+�
+is a proper extension of K. Let λ = b + c√α
+be an element of L such that N (λ) = −α/β. Then
+−α
+β = λλ = b2 − αc2.
+It follows that
+−α · 12 − β · b2 + αβ · c2 = 0.
+Hence, in both cases V is an isotropic vector of ξ, as claimed. Consequently, executing Algorithm 2
+in step (3) we obtain a vector W satisfying the condition ξ(W) = −a. Now, by Eq. (1) the square of
+the quaternion
+outputted by the algorithm equals
+2 = −N( ) = −ξ(W) = a.
+Thus, to conclude the proof, we only need to show that the norm equation in step (2b) is solvable. But
+this follows immediately from the fact that Q is split. Hence ξ is isotropic. Indeed, if V = (v1, v2, v3)
+is an isotropic vector of ξ, then
+−α · v2
+1 − β · v2
+2 + αβ · v2
+3 = 0.
+Observe that v1 must be nonzero since otherwise, α would be a square. It follows that
+−α
+β =
+�v2
+v1
+�2
+− α
+�v3
+v1
+�2
+= NL/K
+�v2
+v1
++ v3
+v1
+√α
+�
+.
+Therefore, the norm equation is solvable, as claimed.
+⊓⊔
+Remark 4.1. The construction of the isotropic vector V in steps (1–2) of Algorithm 3 is equivalent
+to establishing an explicit isomorphism Q ∼= M2K. For details, see [5, Chapter III]. Of course, if
+the quaternion algebra Q is fixed, the vector V should be computed only once and cached between
+successive computations of square roots.
+Remark 4.2. If the isomorphism Q ∼= M2K is a priori known explicitly, then the computation of the
+quaternionic square root of any a ∈ K× trivializes, as we have the identity
+�
+0
+a
+1
+0
+�2
+=
+�
+a
+0
+0
+a
+�
+.
+5.
+Square roots of central elements. Non-split case
+Now the only case left to be dealt with is when a ∈ K× but Q is not split. Here we have to solve
+not one but two norm equations (see Algorithm 5 below). First, however, we need to introduce the
+following auxiliary algorithm that constructs an element simultaneously represented by two binary
+
+8
+P. Koprowski / Computing square roots in quaternion algebras
+forms. Recall (see e.g., [5, Definition I.2.1]) that for a given quadratic form ξ of dimension d, we
+denote the set of nonzero elements of K represented by ξ by the symbol
+DK(ξ) :=
+�
+ξ(V ) | V ∈ Kd and ξ(V ) ̸= 0
+�
+.
+Let P be any finite set of primes of K. Recall that an element a ∈ K× is called P-singular if
+ordp a ≡ 0 (mod 2) for all finite primes p /∈ P. The set of all P-singular elements forms a subgroup
+of the group K× containing K×2. Thus, the notion of P-singularity generalizes naturally to the square
+classes. Define the set
+EP :=
+�
+aK×2 | a is P-singular
+�
+of P-singular square classes. It is a subgroup of the group K×/K×2 of square classes of K, hence a
+vector space over F2. It is known that the dimension of this vector space is finite. In fact it equals (see
+e.g., [12, p. 607])
+dimF2 EP = |P| + dimF2 CP/C2
+P,
+where CP is the P-class group of K. There is a number of know algorithms to construct a basis of
+this vector space. For details see e.g., [13, 14, 15].
+Algorithm 4. Let K be a global field of characteristic char K ̸= 2. Given two binary quadratic
+forms ξ = ⟨x0, x1⟩ and ζ = ⟨z0, z1⟩ over K with x0, x1, z0, z1 ̸= 0, this algorithm outputs a nonzero
+element d ∈ K× such that d ∈ DK(ξ) ∩ DK(ζ) or reports a failure if there is no such d.
+1. If −x0x1 is a square in K, then output z0 and quit.
+2. Likewise, if −z0z1 is a square in K, then output x0 and quit.
+3. Check (using e.g., [16, Algorithm 5]) whether the form
+ξ ⊥ (−ζ) = ⟨x0, x1, −z0, −z1⟩
+is isotropic. If it is not, then report a failure and quit.
+4. Construct a set P consisting of all dyadic places of K (if there are any) and of all these non-
+dyadic primes of K where at least one of the elements x0, x1, z0, z1 has an odd valuation.
+5. If K is a formally real number field, then:
+(a) Construct the set R of all the real places of K, where either ξ or ζ is definite and denote
+its cardinality by r, i.e.
+R =
+�
+r | sgnr x0x1 = 1 or sgnr z0z1 = 1
+�
+,
+r = |R|.
+(b) [Notation only] Let r1, . . . , rr be all the elements of R.
+(c) Construct a vector W = (w1, . . . , wr) ∈ {0, 1}r setting
+wi =
+�
+log-1 sgnri x0
+if sgnri x0x1 = 1,
+log-1 sgnri z0
+if sgnri x0x1 = −1.
+
+P. Koprowski / Computing square roots in quaternion algebras
+9
+Otherwise, if the field K is non-real, set R := ∅, r = 0 and W := ().
+6. Repeat the following steps until the sought element d is found:
+(a) [Notation only] Let p1, . . . , ps be all the elements of P.
+(b) Construct a basis B = {β1, . . . , βk} of the group EP of P-singular square classes.
+(c) Construct vectors U = (u1, . . . , us) and V = (v1, . . . , vs) setting
+ui = log-1(x0, x1)pi
+and
+vi = log-1(z0, z1)pi.
+(d) Construct matrices A = (aij) and B = (bij), with k = |B| columns and s = |P| rows,
+setting
+aij = log-1(−x0x1, βj)pi
+and
+bij = log-1(−z0z1, βj)pi.
+(e) If R ̸= ∅ construct a matrix C = (cij) with k columns and r = |R| rows, setting
+cij = log-1 sgnri βj.
+Otherwise, when R = ∅, set C = ().
+(f) Check if the following system of F2-linear equations has a solution
+
+
+
+A
+B
+C
+
+
+ ·
+
+
+
+
+x1
+...
+xk
+
+
+
+ =
+
+
+
+U
+V
+W
+
+
+
+(Ϙ)
+(g) If it does, denote the solution by (ε1, . . . , εk) ∈ {0, 1}k. Output d = βε1
+1 · · · βεk
+k and quit.
+(h) If the system (Ϙ) has no solution, then append a new prime p to P (see Remark 5.1 below)
+and reiterate the loop.
+Proof of correctness:
+First, suppose that −x0x1 is a square in K. This means that the form ξ is isotropic (see, e.g., [5,
+Theorem I.3.2]). Hence, by [5, Theorem I.3.4] it represents every element of K. In particular, it
+represents z0. Since ζ also represents z0 (trivially), step (1) of the algorithm outputs the correct result.
+The same argument also applies to step (2), when it is the form ζ that is isotropic. It is also clear that
+the sets DK(ξ) and DK(ζ) of elements represented by ξ and ζ, intersect if and only if ξ ⊥ (−ζ) is
+isotropic. This justifies the test in step (3). Therefore, without loss of generality, for the remainder of
+the proof, we may assume that ξ ⊥ (−ζ) is isotropic while both forms ξ and ζ are anisotropic.
+We will first show that the algorithm terminates. Let W = (w0, w1, w2, w3) ∈ K4 be an isotropic
+vector of ξ ⊥ (−ζ). Denote e := ξ(w0, w1) = ζ(w2, w3). Further, let R and P be the sets of
+places (real and non-archimedean, respectively) constructed in steps (4–5) of the algorithm. Now, [17,
+Lemma 2.1] asserts that there exists a finite prime p0 of K and an element d ∈ K× such that:
+i. ordp d = 0 for every finite prime p /∈ P ∪ {p0};
+
+10
+P. Koprowski / Computing square roots in quaternion algebras
+ii. d ≡ e (mod p1+ordp 4) for every p ∈ P;
+iii. ordp0 d = 1;
+iv. sgnr d = sgnr e for every real places r of K.
+Let B = {β1, . . . , βk} be a basis of the group EP∪{p0} of
+�
+P ∪ {p0}
+�
+-singular square classes.
+The element d is
+�
+P ∪ {p0}
+�
+-singular, hence it can be expressed in the form
+d = βε1
+1 · · · βεk
+k ,
+where ε1, . . . , εk ∈ F2 are the coordinates of d with respect to B.
+Fix a real place ri ∈ R. First, suppose that sgnri x0x1 = 1, so the form ξ ⊗ Kp is definite. Then
+sgnri(−d) = sgnri(−e) = sgnri x0 since ⟨−e, x0, x1⟩ is isotropic. But this implies that
+k
+�
+j=1
+(−1)cijεj =
+k
+�
+j=1
+sgnri βεj
+j = sgnri d = sgnri x0 = (−1)wi.
+Consequently
+ci1ε1 + · · · + cikεk = wi.
+(2)
+Conversely, assume that ξ ⊗ Kri is indefinete, hence ζ ⊗ Kri must be definete. Applying the same
+arguments to the form ζ instead of ξ, we show that Eq. (2) also holds in this case.
+Now fix a finite prime pi ∈ P. Observe that by the local square theorem (see, e.g., [5, The-
+orem VI.2.19]) condition (ii) implies that the local squares classes dK×2
+pi
+and eK×2
+pi
+coincide. It
+follows that the form
+⟨−d, x0, x1⟩ ⊗ Kpi ∼= ⟨−e, x0, x1⟩ ⊗ Kpi
+is isotropic. Now, [5, Proposition V.3.22] asserts that the Hasse invariant of ⟨−d, x0, x1⟩ ⊗ Kpi equals
+spi⟨−d, x0, x1⟩ = (−1, x0x1 · d)pi.
+This can be rewritten as
+(−x0x1, d)pi = (x0, x1)pi.
+Substituting βε1
+1 · · · βεk
+k for d we obtain
+k
+�
+j=1
+(−x0x1, βj)εj
+pi = (x0, x1)pi.
+Now, (x0, x1)pi = (−1)ui and (−x0x1, βj)pi = (−1)aij, where ui, aij ∈ {0, 1} are the elements
+constructed in steps (6c–6d). Therefore, the last condition can be expressed as a linear equation over
+F2:
+ai1ε1 + · · · + aikεk = ui.
+(3)
+
+P. Koprowski / Computing square roots in quaternion algebras
+11
+Finally, we will show that the above equation also holds for the index i = 0, that is for the prime p0
+appended to P. This fact follows from Hilbert reciprocity law (see, e.g., [5, Theorem VI.5.5]). We
+already know that for every i ∈ {1, . . . , s} we have
+(−x0x1, d)pi = (x0, x1)pi.
+The same also holds for primes not in P. Indeed, if q /∈ P ∪ {p0} then q is non-dyadin and all three
+elements x0, x1 and d have even valuations at q. Consequently, by [5, Corollary VI.2.5] one obtains
+(−x0x1, d)q = (x0, x1)q = 1.
+Now, by Hilbert reciprocity law, we can write
+1 =
+�
+p
+(−x0x1, d)p ·
+�
+p
+(x0, x1)p
+= (−x0x1, d)p0(x0, x1)p0 ·
+�
+p∈P
+�
+(−x0x1, d)p(x0, x1)p
+�
+·
+�
+q/∈P∪{p0}
+�
+(−x0x1, d)p(x0, x1)p
+�
+= (−x0x1, d)p0(x0, x1)p0.
+Hence, in the same way as above, we show that Eq. (3) also holds for i = 0. Applying the same
+arguments to the form ζ, we obtain
+bi1ε1 + · · · + bikεk = vi,
+(4)
+for all i ∈ {0, 1, . . . , s}.
+All in all, we have proved that Eq. (Ϙ) has a solution in EP∪{p0}. Now, for every P′ ⊇ P ∪ {p0}
+we have EP∪{p0} ⊆ EP′, hence once the prime p0 is appended to P the algorithm terminates (see also
+Remark 5.1 w below).
+Now, when we have proved that the algorithm stops, we must show that it outputs a correct result.
+To this end, we will show that the forms ⟨−d, x0, x1⟩ and ⟨−d, z0, z1⟩ are locally isotropic in every
+completion of K. The assumptions are symmetric with respect to both forms, except in real places.
+Hence it generally suffices to prove the isotropy of one of them.
+Both forms are trivially isotropic in all complex completions of K (provided that there are any)
+and in all real completions Kr for r /∈ R. Fix now a real place ri ∈ R. First, assume that the form
+⟨x0, x1⟩ ⊗ Kri is definite. From the preceding part we know that the element d = βε1
+1 · · · βεk
+k , con-
+structed by the algorithm, satisfies the condition sgnri d = sgnri x0. Therefore the form ⟨−d, x0, x1⟩⊗
+Kri is isotropic. Now, the form ξ ⊥ (−ζ) is isotropic because otherwise, the execution of the algo-
+rithm would have been interrupted already in step (3). Thus, either sgnri z0 = sgnri x0 = sgnri d or
+sgnri z1 = sgnri x0 = sgnri d. In both cases, we have that the form ⟨−d, z0, z1⟩ ⊗ Kri is isotropic, as
+well. Conversely, assume that ξ ⊗ Kri is indefinite, and so it is ζ ⊗ Kri that must be definite. Then,
+⟨−d, x0, x1⟩⊗Kri is trivially isotropic and to the form ⟨−d, z0, z1⟩⊗Kri we apply the some argument
+as to the form ⟨−d, x0, x1⟩ ⊗ Kri in the previous case.
+We may now concentrate on finite primes. Fix a prime p. Suppose p is not among the primes
+constituting P (here, we allow P to have been already enlarged during the execution of the algorithm).
+
+12
+P. Koprowski / Computing square roots in quaternion algebras
+In that case, p is certainly non-dyadic, and all three elements x0, x1, and d have even valuations at p.
+Hence, [5, Corollary VI.2.5] asserts that ⟨−d, x0, x1⟩ ⊗ Kp is isotropic. On the other hand, we know
+from the first part of the proof that if p = pi ∈ P, then d satisfies the condition (−x0x1, d)p =
+(x0, x1)p, which is equivalent to sp⟨−d, x0, x1⟩ = (−1, x0x1 · d)p. The later condition implies that
+⟨−d, x0, x1⟩ ⊗ Kp is isotropic, again by [5, Proposition V.3.22]. The very same arguments may be
+applied to the form ⟨−d, z0, z1⟩ ⊗ Kp.
+All in all, we have shown that the forms ⟨−d, x0, x1⟩ and ⟨−d, z0, z1⟩ are locally isotropic in every
+completion of K. Thus, they are isotropic over K by the Hasse–Minkowski principle (see e.g., [5,
+Theorem VI.3.1]). This means that the forms ξ and ζ represent d over K by [5, Corollary I.3.5].
+⊓⊔
+Remark 5.1. To rigorously prove that Algorithm 4 terminates, we must show that the new primes
+may be appended to the set P in such a way that after finitely many iterations the set will contain the
+prime p0 specified in the proof of correctness. If K is a number field, let p1, p2, p3, . . . = 2, 3, 5, . . .
+be the (strictly increasing) sequence of all prime numbers. On the other hand, if K is a global function
+field, i.e., a finite extension of a rational function field Fq(x), let p1, p2, p3, . . . be a sequence of all
+the irreducible polynomials from Fq[x] ordered in such a way that deg pj ≤ deg pj+1 for every j.
+Now, appending new primes to P, we can first use the places of K that extend p1, then the ones that
+extend p2, then p3, and so on. It is clear that this is an exhaustive method, so eventually, we will
+append p0 and consequently terminate the algorithm. This proves that the algorithm stops but does
+not present a complete picture. The existence of the prime p0 follows from [17, Lemma 2.1], which in
+turn uses Chebotarev’s density theorem. In particular, if P0 denotes the set of primes of K such that
+appending any of them makes the algorithm stop, then the set P0 has positive density. This means that
+in practice, one can just append primes to P at random with exponentially diminishing probability
+that the system (Ϙ) fails to be solvable.
+We are now in a position to present an algorithm that computes a square root of a scalar in a
+non-split quaternion algebra.
+Algorithm 5. Let Q =
+� α,β
+K
+�
+be a non-split quaternion algebra over a global field of characteristic
+char K ̸= 2. Given a nonzero element a ∈ K this algorithm outputs a quaternion
+∈ Q such that
+2 = a or reports a failure if a is not a square in Q.
+1. Check if a is a square in K. If there is c ∈ K× such that a = c2, then output
+= c+0i+0j+0k
+and quit.
+2. Check if aα is a square in K. If there is c ∈ K× such that aα = c2, then output
+= 0+(c/α)i+
+0j + 0k and quit.
+3. Check if aβ is a square in K. If there is c ∈ K× such that aβ = c2, then output
+= 0 + 0i +
+(c/β)j + 0k and quit.
+4. Execute Algorithm 4 with input ξ = ⟨a, −α⟩ and ζ = ⟨β, −αβ⟩. If it fails, then report a failure
+and quit. Otherwise, let d ∈ K× denote the outputted element represented by these two binary
+forms.
+
+P. Koprowski / Computing square roots in quaternion algebras
+13
+5. Construct two quadratic extensions of K:
+L := K
+�√α
+�
+and
+M := K
+�√aα
+�
+.
+6. Solve the following two norm equations:
+d
+β = NL/K (x)
+and
+d
+a = NM/K (y) .
+Denote the solutions by
+λ = l0 + l1
+√α
+and
+µ = m0 + m1
+√aα,
+respectively.
+7. Output
+= 0 + a · m1
+m0 i + l0
+m0 j + l1
+m0 k.
+Proof of correctness:
+The correctness of the results outputted in step (1) is obvious as is the correctness of output of steps
+(2–3). Indeed, if aα = c2 for some c ∈ K× and
+= (c/α)i, then
+2 = α · c2/α2 = a. In the remainder
+of the proof, we can, thus, assume that neither a nor aα is a square in K. Likewise, α is not a square,
+either, since otherwise, the quaternion algebra Q would split. Therefore, L and M are proper quadratic
+extensions of K. It follows from Eq. (1) that a is a square of some pure quaternion
+= q1i+q2j+q3k
+if and only if
+a · 12 − α · q2
+1 = β · q2
+2 − αβ · q2
+3.
+This equality is equivalent to the condition that the sets of elements of K represented by the binary
+forms ξ = ⟨a, −α⟩ and ζ = ⟨β, −αβ⟩ have a non-empty intersection. This proves the correctness of
+step (4). Now, assume that Algorithm 4 returned some element d ∈ DK(ξ) ∩ DK(ζ). Then there are
+l0, l1, m0, m1 ∈ K such that
+�
+d = am2
+0 − α(am1)2 = a · NM/K (m0 + m1
+√aα)
+d = βl2
+0 − αβl2
+1 = β · NL/K (l0 + l1
+√α) .
+Rearranging the terms we have
+a = α
+�am1
+m0
+�2
++ β
+� l0
+m0
+�2
+− αβ
+� l1
+m0
+�2
+.
+Now, the right-hand-side is nothing else but the square of the quaternion
+constructed in step (7). This
+proves that the algorithm is correct.
+⊓⊔
+
+14
+P. Koprowski / Computing square roots in quaternion algebras
+References
+[1] Heath T. A history of Greek mathematics. Vol. I. Dover Publications, Inc., New York, 1981. ISBN
+0-486-24073-8. From Thales to Euclid, Corrected reprint of the 1921 original.
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+https://doi.org/10.2307/2303134.
+[3] Vign´eras MF. Arithm´etique des alg`ebres de quaternions, volume 800 of Lecture Notes in Mathematics.
+Springer, Berlin, 1980. ISBN 3-540-09983-2.
+[4] Voight J.
+Quaternion algebras, volume 288 of Graduate Texts in Mathematics.
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+[5] Lam TY. Introduction to quadratic forms over fields, volume 67 of Graduate Studies in Mathematics.
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+[12] Czogała A.
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+J. Algebra, 2001. 244(2):604–630. doi:
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+[13] Cannon J, Bosma W, Fieker C, (eds) AS. Handbook of Magma Functions, 2.26-4 edition, 2021.
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+P. Koprowski / Computing square roots in quaternion algebras
+15
+[15] Koprowski
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+
diff --git a/3tAyT4oBgHgl3EQf1_ko/content/tmp_files/load_file.txt b/3tAyT4oBgHgl3EQf1_ko/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf,len=702
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='00743v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='SC]  2 Jan 2023  1  Computing square roots in quaternion algebras  Przemysław Koprowski  Institute of Mathematics  University of Silesia in Katowice  ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Bankowa 14, 40-007 Katowice, Poland  przemyslaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='koprowski@us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='pl  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' We present an explicit algorithmic method for computing square roots in quaternion  algebras over global fields of characteristic different from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Keywords:  square root computation, quaternion algebra, number fields, global fields  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Introduction  The computation of square roots is one of the most basic operations in mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Effective methods  for computing square roots are among the oldest algorithms in the realm of computational mathemat-  ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' In fact, Heron’s method for a numerical approximation of a square root of a real number is two  thousand years old and preceded by the Euclidean algorithm (wildly believed to be the oldest mathe-  matical algorithm) by only about three to four centuries (for an in-depth discussion on the chronology  see [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Although numerous methods for computing square roots in various algebraic structures are  known nowadays, some important omissions prevail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Among them are general quaternion algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Computation of square roots in the algebra of Hamilton quaternions H =  � −1,−1  R  �  is well-known (see  [2]) and very simple as for every quaternion  ∈ H there is a subfield K ∼= C of H containing , and  so the computation the square root in H can be reduced to the computation of the square root in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' It  is no longer so in a general quaternion algebra Q =  � α,β  K  �  for an arbitrary field K and two elements  α, β ∈ K×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' To the best of our knowledge, no algorithm for computing quaternionic square roots ex-  ists in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' One possible explanation for this (quite surprising) fact is that in the commutative  Address for correspondence: Institute of Mathematics, University of Silesia, ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Bankowa 14, 40-007 Katowice, Poland  2  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Koprowski / Computing square roots in quaternion algebras  case when one considers a field extension L/K, a typical way to compute a square root of an element  a ∈ L is to factor the polynomial x2 − a in L[x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' However, for quaternion algebras, there are no  known polynomial factorization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  The sole purpose of this paper is to correct this evident omission and present an explicit algorithm  for computing square roots in quaternion algebras over arbitrary global fields of characteristic different  from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Notation  Throughout this paper, K will denote an arbitrary global field of characteristic char K ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Hence, K  is either a number field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' a finite extension of Q (then its characteristic is just 0) or a global function  field, that is, a finite extension of a rational function field over a finite field Fq, where q is a power of  an odd prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' The set of nonzero elements of K is denoted K×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Recall that a quaternion algebra Q =  � α,β  K  �  over K is a 4-dimensional K-algebra with a basis  {1, i, j, k} and a multiplication gathered by the rules:  i2 = α,  j2 = β,  ij = k = −ji.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  As usual, we shall identify the field K with the subfield K · 1 of Q, which is known to coincide with  the center Z(Q) of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' We refer the reader to [3, 4] for a comprehensive presentation of the theory of  quaternion algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  A quaternion  is called pure (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=', [4, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='1]) if  ∈ spanK{i, j, k}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Every quater-  nion  ∈ Q can be uniquely expressed as a sum  = a +  0 of a scalar a ∈ K and a pure quaternion  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' We write  := a −  0 for the conjugate of .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' The map that sends a quaternion to its conjugate is an  involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  If x is an element of either a quadratic field extension L = K  �√α  �  of K or a quaternion algebra  Q =  � α,β  K  �  over K, we write N (x) := xx and call it the norm of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' If the domain is not clear from  the context, we write NL/K or NQ/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' When Q is a quaternion algebra, the norm of in the above sense should not be confused  with the determinant of the endomorphism of Q defined by the multiplication by , which is often also  called the norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' For this reason, in [3, 4] the map  �→  is called the reduced norm and denoted nrd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  In that manner, our terminology in the present paper agrees with the one used by Lam in [5] but not  with the one used by Vigneras in [3] and Voight in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Equivalency classes of valuations on K are called places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Throughout this paper, places are de-  noted using fraktur letters p, q, r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Every place of a global field is either archimedean, when it extends  the standard absolute value on Q (then the field K is necessarily a number field) or non-archimedean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Over a global function field, every place is non-archimedean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' To avoid monotonous repetitions, non-  archimedean places will also be called primes (or finite primes when we want to emphasize the fact  that they are non-archimedean).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' The completion of K with respect to a place p is denoted Kp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' If p is  a finite prime, we write ordp : K → Z to denote the corresponding (normalized) discrete valuation  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Koprowski / Computing square roots in quaternion algebras  3  on K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' The prime p is called dyadic if ordp 2 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' The map ordp induces a natural map K×/K×2 → Z/2Z  on the group of square classes of K that is again denoted ordp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  If p is an archimedean place, then the completion Kp is isomorphic either to C or to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' The places  of the second kind are called real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' We write sgnr a for the sign of a ∈ K with respect to for the unique  ordering of K induced by a real place r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Given some nonzero elements a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' , an ∈ K we denote by ⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' , an⟩ the quadratic form  a1x2  1 + · · · + anx2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Further, if p is a place and a, b ∈ K× we write (a, b)p for the Hilber symbol of a  and b at p, that is  (a, b)p :=  �  1  if  �a,b  K  � ∼= M2Kp,  −1  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  For a quadratic form ξ = ⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' , an⟩ we define its Hasse invariant spξ at p by the formula (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=',  [5, Definition V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='17]):  spξ :=  �  i<j  (ai, aj)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Finally, abusing the notation harmlessly, by log-1 we will denote the (unique) isomorphism from the  multiplicative group {±1} to the additive group {0, 1} with addition modulo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Square roots of non-central elements  Let us begin by writing down the explicit formula for a square in quaternion algebra so that we can  easily reference it in the discussion that follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' If  = q0 + q1i + q2j + q3k ∈ Q is a quaternion, then  2 = (q2  0 + q2  1α + q2  2β − q2  3αβ) + 2q0q1i + 2q0q2j + 2q0q3k  = (2q2  0 − N( )) + 2q0 · (q1i + q2j + q3k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  (1)  An immediate consequence of the previous observation is the following rather well-known fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' If  ∈ Q is a pure quaternion, then  2 ∈ Z(Q) = K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Another direct consequence of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' (1) is the following observation that may be treated as a partial  converse of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Let  ∈ Q be a square root of some element a ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Then  is either pure or  ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Proof:  Let  = q0 + q1i + q2j + q3k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' If  2 = a ∈ K then by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' (1) we have  2q0q1 = 2q0q2 = 2q0q3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Therefore, if  is not pure, that is if q0 ̸= 0, then q1 = q2 = q3 = 0, hence  ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  ⊓⊔  4  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Koprowski / Computing square roots in quaternion algebras  Combining Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='2 with Observation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='3 we see that for computing the square roots in  quaternion algebras it is crucial to distinguish between the case when one computes a quaternionic  square root of an element in K (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=', in the center of Q) and the case when the argument comes from  Q \\Z(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' It turns out that the latter case is, in fact, trivial and requires nothing more than high-school  mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Let Q =  � α,β  K  �  be a quaternion algebra over a field K of characteristic char K ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Given a quaternion  = q0 + q1i + q2j + q3k ∈ Q \\ Z(Q), this algorithm outputs its square root or  reports a failure when  is not a square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Check if the norm N( ) of  is a square in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  (a) If it is not, then report a failure and quit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  (b) If it is, let d be an element of K such that d2 = N( ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Check if any of the following two elements is a square in K:  a+ := q0 + d  2  ,  a− := q0 − d  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' If neither of them is a square, then report a failure and quit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Otherwise, fix r0 such that either r2  0 = a+ or r2  0 = a−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Set  r1 := q1  2r0  ,  r2 := q2  2r0  ,  r3 := q3  2r0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Output r = r0 + r1i + r2j + r3k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Proof of correctness:  Since the norm N : Q → K is multiplicative, it is obvious that if N( ) /∈ K2, then  cannot be a  square in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' This fact justifies the early exit in step (1a) of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Assume that N( ) = d2 and  let r = r0 + r1i + r2j + r3k be the sought square root of , if it exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' By Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' (1) we have  q1 = 2r0r1,  q2 = 2r0r2,  q3 = 2r0r3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  It is, thus, clear that it suffices to find r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Again by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' (1) we may write  q0 = r2  0 + r2  1α + r2  2β − r2  3αβ = r2  0 +  � q1  2r0  �2  α +  � q2  2r0  �2  β −  � q3  2r0  �2  αβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  The above formula can be rewritten in the form of a bi-quadratic equation:  4r4  0 − 4q0r2  0 +  �  q2  1α + q2  2β − q2  3αβ  �  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Koprowski / Computing square roots in quaternion algebras  5  If we treat the left-hand-side as a quadratic equation in r2  0, then its discriminant equals 16 · N( ) =  (4d)2, hence  r2  0 = q0 ± d  2  = a±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  It follows that the sought quaternion r exists if and only if either a+ or a− is a square in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' This  proves the correctness of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  ⊓⊔  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' In the above proof, we constructed the square root r of a quaternion  ∈ Q \\ Z(Q) by  solving a bi-quadratic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Such equations in general, may have four roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Hence, one may  suspect that there are four distinct quaternions r such that r2 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' It is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' It is clear from  the above proof that  ∈ Q \\ Z(Q) has only finitely moan square roots in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Now, if r2 = Q, then r  is a root of a quaternionic polynomial x2 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' But [6, Theorem 5] asserts that a quadratic polynomial  over Q which has more than two zeros must have infinitely many of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' This way, we conclude  that  has just two square roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Notice that for hamiltonian quaternions this fact has been observed  already 80 years ago by Niven in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Square roots of central elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Split case  It is evident from the preceding section that the only non-trivial case that must be considered is how  to compute a quaternionic square root of an element of the base field K, which is not a square in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  In contrast to the previous case (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='4), in general, an element a ∈ K = Z(Q) may have  infinitely many square roots in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Once again, for hamiltonian quaternions it has been observed  already by Niven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  First, we need, however, to introduce an auxiliary algorithm that is not specific to quaternions, as  it deals with an arbitrary quadratic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Recall that a quadratic form is called isotropic (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=', [5,  Definition I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='1]) if it represents zero non-trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' It is well known (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=', [5, Theorem I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='4]) that  every isotropic form represents all elements of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Let ξ be an isotropic quadratic form of dimension n over a field K of characteristic  char K ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Given an element a ∈ K and a vector V ∈ Kn such that ξ(V ) = 0, this algorithm  outputs a vector W ∈ Kn satisfying the condition ξ(W) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Find a vector U ∈ Kn such that U and V are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Set b := ξ(U) and c := 1/2 ·  �  ξ(U + V ) − ξ(U)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Output  W := U + a − b  2c  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Proof of correctness:  6  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Koprowski / Computing square roots in quaternion algebras  Just compute:  ξ(W) = ξ  �  U + a − b  2c  V  �  = ξ(U) + a − b  2c �  ξ(U + V ) − ξ(U) − ξ(V )  �  + (a − b)2  4c2  ξ(V )  = b + a − b  2c  2c + 0 = a  ⊓⊔  Recall that a quaternion algebra Q =  � α,β  K  �  is said to split (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=', [4, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='5]) if Q  is isomorphic to the matrix ring M2K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' It is well known (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=', [4, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='4] or [5, Theo-  rem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='7]) that Q is split if and only if the quadratic form ⟨−α, −β, αβ⟩ is isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' If it is the case,  the preceding algorithm combined with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' (1) lets us compute the quaternionic square root of any  element of the base field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' In particular, when K is a global field, char K ̸= 2, then the computation  of the square root of a ∈ K in a split quaternion algebra boils down to solving a norm equation in  a quadratic extension of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Algorithms for the latter task are well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' They can be found in  [7, 8, 9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Let Q =  �α,β  K  �  be a split quaternion algebra over a global field K of characteristic  char K ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Given a nonzero element a ∈ K, this algorithm outputs a pure quaternion  ∈ Q such  that  2 = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Check if α is a square in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' If there is c ∈ K× such that c2 = α, then set V := (0, c, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Otherwise, if α /∈ K×2, then:  (a) Construct a quadratic field extension L = K  �√α  �  of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  (b) Solve the norm equation  NL/K (x) = −α  β  and denote the solution by λ = b + c√α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  (c) Set V := (1, b, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Let ξ := ⟨−α, −β, αβ⟩ be the pure subform of the norm form of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Execute Algorithm 2 with  the input (−a, V, ξ) to construct a vector W = (w1, w2, w3) such that ξ(W) = −a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Output  = 0 + w1i + w2j + w3k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Proof of correctness:  We claim that the vector V constructed either in step (1) or in step (2) of the algorithm is an isotropic  vector for ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' First, suppose that α is a square in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Say α = c2 for some c ∈ K×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Then  −α · 02 − β · c2 + αβ · 12 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Koprowski / Computing square roots in quaternion algebras  7  Conversely, assume that α /∈ K×2 and so L = K  �√α  �  is a proper extension of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Let λ = b + c√α  be an element of L such that N (λ) = −α/β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Then  −α  β = λλ = b2 − αc2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  It follows that  −α · 12 − β · b2 + αβ · c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Hence, in both cases V is an isotropic vector of ξ, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Consequently, executing Algorithm 2  in step (3) we obtain a vector W satisfying the condition ξ(W) = −a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Now, by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' (1) the square of  the quaternion  outputted by the algorithm equals  2 = −N( ) = −ξ(W) = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Thus, to conclude the proof, we only need to show that the norm equation in step (2b) is solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' But  this follows immediately from the fact that Q is split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Hence ξ is isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Indeed, if V = (v1, v2, v3)  is an isotropic vector of ξ, then  −α · v2  1 − β · v2  2 + αβ · v2  3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Observe that v1 must be nonzero since otherwise, α would be a square.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' It follows that  −α  β =  �v2  v1  �2  − α  �v3  v1  �2  = NL/K  �v2  v1  + v3  v1  √α  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Therefore, the norm equation is solvable, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  ⊓⊔  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' The construction of the isotropic vector V in steps (1–2) of Algorithm 3 is equivalent  to establishing an explicit isomorphism Q ∼= M2K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' For details, see [5, Chapter III].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Of course, if  the quaternion algebra Q is fixed, the vector V should be computed only once and cached between  successive computations of square roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' If the isomorphism Q ∼= M2K is a priori known explicitly, then the computation of the  quaternionic square root of any a ∈ K× trivializes, as we have the identity  �  0  a  1  0  �2  =  �  a  0  0  a  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Square roots of central elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Non-split case  Now the only case left to be dealt with is when a ∈ K× but Q is not split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Here we have to solve  not one but two norm equations (see Algorithm 5 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' First, however, we need to introduce the  following auxiliary algorithm that constructs an element simultaneously represented by two binary  8  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Koprowski / Computing square roots in quaternion algebras  forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Recall (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=', [5, Definition I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='1]) that for a given quadratic form ξ of dimension d, we  denote the set of nonzero elements of K represented by ξ by the symbol  DK(ξ) :=  �  ξ(V ) | V ∈ Kd and ξ(V ) ̸= 0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Let P be any finite set of primes of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Recall that an element a ∈ K× is called P-singular if  ordp a ≡ 0 (mod 2) for all finite primes p /∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' The set of all P-singular elements forms a subgroup  of the group K× containing K×2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Thus, the notion of P-singularity generalizes naturally to the square  classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Define the set  EP :=  �  aK×2 | a is P-singular  �  of P-singular square classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' It is a subgroup of the group K×/K×2 of square classes of K, hence a  vector space over F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' It is known that the dimension of this vector space is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' In fact it equals (see  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=', [12, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' 607])  dimF2 EP = |P| + dimF2 CP/C2  P,  where CP is the P-class group of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' There is a number of know algorithms to construct a basis of  this vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' For details see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=', [13, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Let K be a global field of characteristic char K ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Given two binary quadratic  forms ξ = ⟨x0, x1⟩ and ζ = ⟨z0, z1⟩ over K with x0, x1, z0, z1 ̸= 0, this algorithm outputs a nonzero  element d ∈ K× such that d ∈ DK(ξ) ∩ DK(ζ) or reports a failure if there is no such d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' If −x0x1 is a square in K, then output z0 and quit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Likewise, if −z0z1 is a square in K, then output x0 and quit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Check (using e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=', [16, Algorithm 5]) whether the form  ξ ⊥ (−ζ) = ⟨x0, x1, −z0, −z1⟩  is isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' If it is not, then report a failure and quit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Construct a set P consisting of all dyadic places of K (if there are any) and of all these non-  dyadic primes of K where at least one of the elements x0, x1, z0, z1 has an odd valuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' If K is a formally real number field, then:  (a) Construct the set R of all the real places of K, where either ξ or ζ is definite and denote  its cardinality by r, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  R =  �  r | sgnr x0x1 = 1 or sgnr z0z1 = 1  �  ,  r = |R|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  (b) [Notation only] Let r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' , rr be all the elements of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  (c) Construct a vector W = (w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' , wr) ∈ {0, 1}r setting  wi =  �  log-1 sgnri x0  if sgnri x0x1 = 1,  log-1 sgnri z0  if sgnri x0x1 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Koprowski / Computing square roots in quaternion algebras  9  Otherwise, if the field K is non-real, set R := ∅, r = 0 and W := ().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Repeat the following steps until the sought element d is found:  (a) [Notation only] Let p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' , ps be all the elements of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  (b) Construct a basis B = {β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' , βk} of the group EP of P-singular square classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  (c) Construct vectors U = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' , us) and V = (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' , vs) setting  ui = log-1(x0, x1)pi  and  vi = log-1(z0, z1)pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  (d) Construct matrices A = (aij) and B = (bij), with k = |B| columns and s = |P| rows,  setting  aij = log-1(−x0x1, βj)pi  and  bij = log-1(−z0z1, βj)pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  (e) If R ̸= ∅ construct a matrix C = (cij) with k columns and r = |R| rows, setting  cij = log-1 sgnri βj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Otherwise, when R = ∅, set C = ().' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  (f) Check if the following system of F2-linear equations has a solution  \uf8eb  \uf8ec  \uf8ed  A  B  C  \uf8f6  \uf8f7  \uf8f8 ·  \uf8eb  \uf8ec  \uf8ec  \uf8ed  x1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  xk  \uf8f6  \uf8f7  \uf8f7  \uf8f8 =  \uf8eb  \uf8ec  \uf8ed  U  V  W  \uf8f6  \uf8f7  \uf8f8  (Ϙ)  (g) If it does, denote the solution by (ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' , εk) ∈ {0, 1}k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Output d = βε1  1 · · · βεk  k and quit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  (h) If the system (Ϙ) has no solution, then append a new prime p to P (see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='1 below)  and reiterate the loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Proof of correctness:  First, suppose that −x0x1 is a square in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' This means that the form ξ is isotropic (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=', [5,  Theorem I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Hence, by [5, Theorem I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='4] it represents every element of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' In particular, it  represents z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Since ζ also represents z0 (trivially), step (1) of the algorithm outputs the correct result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  The same argument also applies to step (2), when it is the form ζ that is isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' It is also clear that  the sets DK(ξ) and DK(ζ) of elements represented by ξ and ζ, intersect if and only if ξ ⊥ (−ζ) is  isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' This justifies the test in step (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Therefore, without loss of generality, for the remainder of  the proof, we may assume that ξ ⊥ (−ζ) is isotropic while both forms ξ and ζ are anisotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  We will first show that the algorithm terminates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Let W = (w0, w1, w2, w3) ∈ K4 be an isotropic  vector of ξ ⊥ (−ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Denote e := ξ(w0, w1) = ζ(w2, w3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Further, let R and P be the sets of  places (real and non-archimedean, respectively) constructed in steps (4–5) of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Now, [17,  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='1] asserts that there exists a finite prime p0 of K and an element d ∈ K× such that:  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' ordp d = 0 for every finite prime p /∈ P ∪ {p0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  10  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Koprowski / Computing square roots in quaternion algebras  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' d ≡ e (mod p1+ordp 4) for every p ∈ P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' ordp0 d = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' sgnr d = sgnr e for every real places r of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Let B = {β1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' , βk} be a basis of the group EP∪{p0} of  �  P ∪ {p0}  �  singular square classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  The element d is  �  P ∪ {p0}  �  singular, hence it can be expressed in the form  d = βε1  1 · · · βεk  k ,  where ε1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' , εk ∈ F2 are the coordinates of d with respect to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Fix a real place ri ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' First, suppose that sgnri x0x1 = 1, so the form ξ ⊗ Kp is definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Then  sgnri(−d) = sgnri(−e) = sgnri x0 since ⟨−e, x0, x1⟩ is isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' But this implies that  k  �  j=1  (−1)cijεj =  k  �  j=1  sgnri βεj  j = sgnri d = sgnri x0 = (−1)wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Consequently  ci1ε1 + · · · + cikεk = wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  (2)  Conversely, assume that ξ ⊗ Kri is indefinete, hence ζ ⊗ Kri must be definete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Applying the same  arguments to the form ζ instead of ξ, we show that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' (2) also holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Now fix a finite prime pi ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Observe that by the local square theorem (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=', [5, The-  orem VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='19]) condition (ii) implies that the local squares classes dK×2  pi  and eK×2  pi  coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' It  follows that the form  ⟨−d, x0, x1⟩ ⊗ Kpi ∼= ⟨−e, x0, x1⟩ ⊗ Kpi  is isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Now, [5, Proposition V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='22] asserts that the Hasse invariant of ⟨−d, x0, x1⟩ ⊗ Kpi equals  spi⟨−d, x0, x1⟩ = (−1, x0x1 · d)pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  This can be rewritten as  (−x0x1, d)pi = (x0, x1)pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Substituting βε1  1 · · · βεk  k for d we obtain  k  �  j=1  (−x0x1, βj)εj  pi = (x0, x1)pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Now, (x0, x1)pi = (−1)ui and (−x0x1, βj)pi = (−1)aij, where ui, aij ∈ {0, 1} are the elements  constructed in steps (6c–6d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Therefore, the last condition can be expressed as a linear equation over  F2:  ai1ε1 + · · · + aikεk = ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  (3)  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Koprowski / Computing square roots in quaternion algebras  11  Finally, we will show that the above equation also holds for the index i = 0, that is for the prime p0  appended to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' This fact follows from Hilbert reciprocity law (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=', [5, Theorem VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' We  already know that for every i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' , s} we have  (−x0x1, d)pi = (x0, x1)pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  The same also holds for primes not in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Indeed, if q /∈ P ∪ {p0} then q is non-dyadin and all three  elements x0, x1 and d have even valuations at q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Consequently, by [5, Corollary VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='5] one obtains  (−x0x1, d)q = (x0, x1)q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Now, by Hilbert reciprocity law, we can write  1 =  �  p  (−x0x1, d)p ·  �  p  (x0, x1)p  = (−x0x1, d)p0(x0, x1)p0 ·  �  p∈P  �  (−x0x1, d)p(x0, x1)p  � �  q/∈P∪{p0}  �  (−x0x1, d)p(x0, x1)p  �  = (−x0x1, d)p0(x0, x1)p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Hence, in the same way as above, we show that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' (3) also holds for i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Applying the same  arguments to the form ζ, we obtain  bi1ε1 + · · · + bikεk = vi,  (4)  for all i ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' , s}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  All in all, we have proved that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' (Ϙ) has a solution in EP∪{p0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Now, for every P′ ⊇ P ∪ {p0}  we have EP∪{p0} ⊆ EP′, hence once the prime p0 is appended to P the algorithm terminates (see also  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='1 w below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Now, when we have proved that the algorithm stops, we must show that it outputs a correct result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  To this end, we will show that the forms ⟨−d, x0, x1⟩ and ⟨−d, z0, z1⟩ are locally isotropic in every  completion of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' The assumptions are symmetric with respect to both forms, except in real places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Hence it generally suffices to prove the isotropy of one of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Both forms are trivially isotropic in all complex completions of K (provided that there are any)  and in all real completions Kr for r /∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Fix now a real place ri ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' First, assume that the form  ⟨x0, x1⟩ ⊗ Kri is definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' From the preceding part we know that the element d = βε1  1 · · · βεk  k , con-  structed by the algorithm, satisfies the condition sgnri d = sgnri x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Therefore the form ⟨−d, x0, x1⟩⊗  Kri is isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Now, the form ξ ⊥ (−ζ) is isotropic because otherwise, the execution of the algo-  rithm would have been interrupted already in step (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Thus, either sgnri z0 = sgnri x0 = sgnri d or  sgnri z1 = sgnri x0 = sgnri d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' In both cases, we have that the form ⟨−d, z0, z1⟩ ⊗ Kri is isotropic, as  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Conversely, assume that ξ ⊗ Kri is indefinite, and so it is ζ ⊗ Kri that must be definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Then,  ⟨−d, x0, x1⟩⊗Kri is trivially isotropic and to the form ⟨−d, z0, z1⟩⊗Kri we apply the some argument  as to the form ⟨−d, x0, x1⟩ ⊗ Kri in the previous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  We may now concentrate on finite primes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Fix a prime p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Suppose p is not among the primes  constituting P (here, we allow P to have been already enlarged during the execution of the algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  12  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Koprowski / Computing square roots in quaternion algebras  In that case, p is certainly non-dyadic, and all three elements x0, x1, and d have even valuations at p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Hence, [5, Corollary VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='5] asserts that ⟨−d, x0, x1⟩ ⊗ Kp is isotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' On the other hand, we know  from the first part of the proof that if p = pi ∈ P, then d satisfies the condition (−x0x1, d)p =  (x0, x1)p, which is equivalent to sp⟨−d, x0, x1⟩ = (−1, x0x1 · d)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' The later condition implies that  ⟨−d, x0, x1⟩ ⊗ Kp is isotropic, again by [5, Proposition V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' The very same arguments may be  applied to the form ⟨−d, z0, z1⟩ ⊗ Kp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  All in all, we have shown that the forms ⟨−d, x0, x1⟩ and ⟨−d, z0, z1⟩ are locally isotropic in every  completion of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Thus, they are isotropic over K by the Hasse–Minkowski principle (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=', [5,  Theorem VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' This means that the forms ξ and ζ represent d over K by [5, Corollary I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  ⊓⊔  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' To rigorously prove that Algorithm 4 terminates, we must show that the new primes  may be appended to the set P in such a way that after finitely many iterations the set will contain the  prime p0 specified in the proof of correctness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' If K is a number field, let p1, p2, p3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' = 2, 3, 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  be the (strictly increasing) sequence of all prime numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' On the other hand, if K is a global function  field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=', a finite extension of a rational function field Fq(x), let p1, p2, p3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' be a sequence of all  the irreducible polynomials from Fq[x] ordered in such a way that deg pj ≤ deg pj+1 for every j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Now, appending new primes to P, we can first use the places of K that extend p1, then the ones that  extend p2, then p3, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' It is clear that this is an exhaustive method, so eventually, we will  append p0 and consequently terminate the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' This proves that the algorithm stops but does  not present a complete picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' The existence of the prime p0 follows from [17, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='1], which in  turn uses Chebotarev’s density theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' In particular, if P0 denotes the set of primes of K such that  appending any of them makes the algorithm stop, then the set P0 has positive density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' This means that  in practice, one can just append primes to P at random with exponentially diminishing probability  that the system (Ϙ) fails to be solvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  We are now in a position to present an algorithm that computes a square root of a scalar in a  non-split quaternion algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Let Q =  � α,β  K  �  be a non-split quaternion algebra over a global field of characteristic  char K ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Given a nonzero element a ∈ K this algorithm outputs a quaternion  ∈ Q such that  2 = a or reports a failure if a is not a square in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Check if a is a square in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' If there is c ∈ K× such that a = c2, then output  = c+0i+0j+0k  and quit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Check if aα is a square in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' If there is c ∈ K× such that aα = c2, then output  = 0+(c/α)i+  0j + 0k and quit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Check if aβ is a square in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' If there is c ∈ K× such that aβ = c2, then output  = 0 + 0i +  (c/β)j + 0k and quit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Execute Algorithm 4 with input ξ = ⟨a, −α⟩ and ζ = ⟨β, −αβ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' If it fails, then report a failure  and quit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Otherwise, let d ∈ K× denote the outputted element represented by these two binary  forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Koprowski / Computing square roots in quaternion algebras  13  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Construct two quadratic extensions of K:  L := K  �√α  �  and  M := K  �√aα  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Solve the following two norm equations:  d  β = NL/K (x)  and  d  a = NM/K (y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Denote the solutions by  λ = l0 + l1  √α  and  µ = m0 + m1  √aα,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Output  = 0 + a · m1  m0 i + l0  m0 j + l1  m0 k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Proof of correctness:  The correctness of the results outputted in step (1) is obvious as is the correctness of output of steps  (2–3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Indeed, if aα = c2 for some c ∈ K× and  = (c/α)i, then  2 = α · c2/α2 = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' In the remainder  of the proof, we can, thus, assume that neither a nor aα is a square in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Likewise, α is not a square,  either, since otherwise, the quaternion algebra Q would split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Therefore, L and M are proper quadratic  extensions of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' It follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' (1) that a is a square of some pure quaternion  = q1i+q2j+q3k  if and only if  a · 12 − α · q2  1 = β · q2  2 − αβ · q2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  This equality is equivalent to the condition that the sets of elements of K represented by the binary  forms ξ = ⟨a, −α⟩ and ζ = ⟨β, −αβ⟩ have a non-empty intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' This proves the correctness of  step (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Now, assume that Algorithm 4 returned some element d ∈ DK(ξ) ∩ DK(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Then there are  l0, l1, m0, m1 ∈ K such that  �  d = am2  0 − α(am1)2 = a · NM/K (m0 + m1  √aα)  d = βl2  0 − αβl2  1 = β · NL/K (l0 + l1  √α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Rearranging the terms we have  a = α  �am1  m0  �2  + β  � l0  m0  �2  − αβ  � l1  m0  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Now, the right-hand-side is nothing else but the square of the quaternion  constructed in step (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' This  proves that the algorithm is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  ⊓⊔  14  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Koprowski / Computing square roots in quaternion algebras  References  [1] Heath T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' A history of Greek mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Dover Publications, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=', New York, 1981.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' ISBN  0-486-24073-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' From Thales to Euclid, Corrected reprint of the 1921 original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  [2] Niven I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' The roots of a quaternion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Monthly, 1942.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' 49:386–388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
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+page_content='2307/2303134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' URL  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='2307/2303134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  [3] Vign´eras MF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Arithm´etique des alg`ebres de quaternions, volume 800 of Lecture Notes in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Springer, Berlin, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' ISBN 3-540-09983-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  [4] Voight J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Quaternion algebras, volume 288 of Graduate Texts in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Springer, Cham,  [2021] ©2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' ISBN 978-3-030-56692-0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' 978-3-030-56694-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='1007/978-3-030-56694-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' URL  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='1007/978-3-030-56694-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  [5] Lam TY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Introduction to quadratic forms over fields, volume 67 of Graduate Studies in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  American Mathematical Society, Providence, RI, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' ISBN 0-8218-1095-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  [6] Gordon B, Motzkin TS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' On the zeros of polynomials over division rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=', 1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  116:218–226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='2307/1994114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='2307/1994114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  [7] Cohen  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  Advanced  topics  in  computational  number  theory,  volume  193  of  Graduate  Texts in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
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+page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  [16] Koprowski P, Czogała A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Computing with quadratic forms over number fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Symbolic Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=', 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  89:129–145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='jsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='jsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  [17] Leep D, Wadsworth A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' The Hasse norm theorem mod squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' Number Theory, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' 42(3):337–348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='1016/0022-314X(92)90098-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
+page_content='1016/0022-314X(92)90098-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tAyT4oBgHgl3EQf1_ko/content/2301.00743v1.pdf'}
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+arXiv:2301.03299v1  [math.NA]  9 Jan 2023
+ASYMPTOTIC ERROR ANALYSIS FOR THE DISCRETE ITERATED
+GALERKIN SOLUTION OF URYSOHN INTEGRAL EQUATIONS WITH
+GREEN’S KERNELS
+GOBINDA RAKSHIT
+ABSTRACT. Consider a Urysohn integral equation x − K(x) = f, where f and the integral
+operator K with kernel of the type of Green’s function are given. In the computation of
+approximate solutions of the given integral equation by Galerkin method, all the integrals
+are needed to be evaluated by some numerical integration formula. This gives rise to the
+discrete version of the Galerkin method. For r ≥ 1, a space of piecewise polynomials
+of degree ≤ r − 1 with respect to a uniform partition is chosen to be the approximating
+space. For the appropriate choice of a numerical integration formula, an asymptotic series
+expansion of the discrete iterated Galerkin solution is obtained at the above partition points.
+Richardson extrapolation is used to improve the order of convergence. Using this method
+we can restore the rate of convergence when the error is measured in the continuous case. A
+numerical example is given to illustrate this theory.
+1 Introduction
+Let X = L∞[0, 1]. Consider the problem of solving Urysohn integral equation
+x(s) −
+� 1
+0
+κ(s, t, x(t)) dt = f(s), s ∈ [0, 1],
+(1.1)
+where f ∈ X and κ ∈ C ([0, 1] × [0, 1] × R) are given. Let the Urysohn integral operator
+K : L∞[0, 1] → C[0, 1] be defined by
+(1.2)
+K(x)(s) =
+� 1
+0
+κ(s, t, x(t)) dt,
+x ∈ X , s ∈ [0, 1].
+Since the kernel κ is continuous, K is compact operator on X . Denoting the equation (1.1)
+by
+(1.3)
+x − Kx = f.
+We assume that the above equation has a solution, say ϕ. We also assume that K is twice
+Frechét differentiable and 1 is not an eigenvalue of the compact linear operator K′(ϕ). This
+gives us that ϕ is an isolated solution of (1.3). See [12], [14]. We are looking for Galerkin
+approximations of ϕ.
+For r ≥ 1, consider the approximating space Xn as a space of piecewise polynomials of
+degree ≤ r − 1 with respect to a uniform partition, say ∆(n), of [0, 1] with n subintervals
+Date: January 10, 2023.
+2020 Mathematics Subject Classification. 45G10, 65B05, 65J15, 65R20.
+Key words and phrases. Urysohn integral operator, Green’s kernel, Galerkin method, Nyström approxima-
+tion, Richardson extrapolation.
+1
+
+2
+G. RAKSHIT
+each of length h = 1
+n. Let πn be the restriction to L∞[0, 1] of the orthogonal projection from
+L2[0, 1] to Xn. Then the Galerkin solution ϕG
+n satisfies the following integral equation
+ϕG
+n − πnK(ϕG
+n ) = πnf.
+Galerkin method for Urysohn integral equation has been studied extensively in research
+literature. See [5], [12], [13], [14]. The iterated Galerkin solution is defined by
+ϕS
+n = K(ϕG
+n ) + f.
+In [5], the following orders of convergence are also obtained.
+∥ϕG
+n − ϕ∥∞ = O (h) ,
+∥ϕS
+n − ϕ∥∞ = O
+�
+h2�
+,
+if r = 1,
+and
+∥ϕG
+n − ϕ∥∞ = O (hr) ,
+∥ϕS
+n − ϕ∥∞ = O
+�
+hr+2�
+,
+if r ≥ 2.
+It is also shown that the order of convergence of ϕS
+n at the points of partition ∆(n), is h2r.
+If an asymptotic expansion for the error exists, one can apply a well-known techniques
+to obtain more accurate approximations. Richardson extrapolation one such method for
+application. In [24], an asymptotic expansion for the iterated Galerkin solution of Urysohn
+integral equation with Green’s function type of kernel, is obtained at the above mentioned
+partition points. Then, by [11] and using Richardson extrapolation, an approximate solution
+with order of convergence h2r+2 can be obtained.
+In the computation of of above approximations, various integrals are involved. There is
+an integral in the definition of the Urysohn integral operator K. In the definition of the
+orthogonal projection πn, the standard inner product on L2[0, 1] comes into picture. In
+practice, it is necessary to replace all these integrals by a numerical quadrature formula.
+This gives rise to the discrete versions of the projection methods. The discrete versions of
+the Galerkin methods for Urysohn integral with Green’s kernel, are investigated in [6], [4].
+Whereas, in [17], a different version of discrete projection method is discussed.
+In this article, we consider the Urysohn integral equation with Green’s kernel, and discrete
+Galerkin method is applied for approximations. Then, an asymptotic expansion for the
+discrete iterated Galerkin solution is obtained.
+We choose a fine partition of [0, 1] with m subintervals each of length ˜h = 1
+m and define a
+composite numerical quadrature formula. Replacing the integrals in the definition of K and
+πn, we define the Nyström operator Km and the discrete orthogonal projection Pn. Then the
+discrete Galerkin and the discrete iterated Galerkin equations are given by
+zG
+n − PnKm(zG
+n ) = Pnf and zS
+n − Km(PnzS
+n) = f
+respectively. If ϕ ∈ Cr+2[0, 1], then from [6] and [17], we have
+(1.4)
+��zG
+n − ϕ
+��
+∞ = O
+�
+max
+�
+hr, ˜h2��
+(1.5)
+��zS
+n − ϕ
+��
+∞ =
+
+
+
+O
+�
+max
+�
+h2, ˜h2��
+,
+r = 1,
+O
+�
+max
+�
+hr+2, ˜h2��
+, r ≥ 2.
+
+Section 2. Preliminaries
+3
+In this article, first we find an asymptotic error expansion due to the discrete orthogonal
+projection. Then using this, the following asymptotic expansion is obtained:
+(1.6)
+zS
+n(ti) = ϕ(ti) + γ(ti)h2r + O
+�
+max
+�
+h2r+2, ˜h2��
+,
+where the function γ is independent of h. If we choose m such that ˜h ≤ h2r+2, then using
+the Richardson extrapolation, an approximation of ϕ of the order of h2r+2 could be obtained.
+See [11].
+This article is organized as follows. Definitions, notations and some preliminary results
+are given in section 2. In Section 3, a quadrature rule is defined, and using it the discrete
+orthogonal projection and the Nyström approximations of the integral operators are defined.
+Section 4 contains the asymptotic error analysis for the approximations. Numerical example
+is given in Section 5.
+2 Preliminaries
+For an integer α ≥ 0, let Cα[0, 1] denotes the space of all real valued α-times continuously
+differentiable functions on [0, 1] with the norm
+∥x∥α,∞ = max
+0≤j≤α
+��x(j)��
+∞ ,
+where x(j) is the jth derivative of the function x, and
+���x(j)���
+∞ = sup
+0≤t≤1
+|x(j)(t)|. Define
+∥κ∥α,∞ =
+max
+0≤i+j+k≤α
+���D(i,j,k)κ(s, t, u)
+���
+∞ ,
+where
+D(i,j,k)κ(s, t, u) =
+∂i+j+kκ
+∂si∂tj∂uk (s, t, u).
+2.0 Green’s function type kernel
+Let r ≥ 1 be an integer and assume that the kernel κ has the following properties.
+(1) For i = 1, 2, 3, 4, the functions κ, ∂iκ
+∂ui ∈ C(Ω), where C(Ω) denotes the space of all
+real valued continuous function on Ω = [0, 1] × [0, 1] × R.
+(2) Let Ω1 = {(s, t, u) : 0 ≤ t ≤ s ≤ 1, u ∈ R} and Ω2 = {(s, t, u) : 0 ≤ s ≤ t ≤
+1, u ∈ R}. There are two functions κj ∈ Cr(Ωj), j = 1, 2, such that
+κ(s, t, u) =
+�
+κ1(s, t, u),
+(s, t, u) ∈ Ω1,
+κ2(s, t, u),
+(s, t, u) ∈ Ω2.
+(3) Denote ℓ(s, t, u) = ∂κ
+∂u(s, t, u) and λ(s, t, u) = ∂2κ
+∂u2(s, t, u), (s, t, u) ∈ Ω. The partial
+derivatives of ℓ(s, t, u) and λ(s, t, u) with respect to s and t have jump discontinu-
+ities on s = t.
+(4) There are functions ℓj, λj ∈ Cr(Ωj), j = 1, 2, with
+ℓ(s, t, u) =
+�
+ℓ1(s, t, u),
+(s, t, u) ∈ Ω1,
+ℓ2(s, t, u),
+(s, t, u) ∈ Ω2,
+λ(s, t, u) =
+�
+λ1(s, t, u),
+(s, t, u) ∈ Ω1,
+λ2(s, t, u),
+(s, t, u) ∈ Ω2.
+
+4
+G. RAKSHIT
+Under the above assumptions, the operator K is four times Fréchet differentiable, and its
+Fréchet derivatives at x ∈ X are given by
+K′(x)v1(s) =
+� 1
+0
+∂κ
+∂u (s, t, x(t)) v1(t) dt,
+K(i)(x)(v1, . . . , vi)(s) =
+� 1
+0
+∂iκ
+∂ui (s, t, x(t)) v1(t) · · ·vi(t) dt,
+i = 2, 3, 4,
+where
+∂iκ
+∂ui (s, t, x(t)) = ∂iκ
+∂ui (s, t, u)|u=x(t),
+i = 1, 2, 3, 4
+and v1, v2, v3, v4 ∈ X . Note that K′(x) : X → X is linear and K(i)(x) : X i → X are multi-
+linear operators, where X i is the cartesian product of i copies of X . See [25]. The norms of
+these operators are defined by
+��K(i)(x)
+�� = sup
+���K(i)(x)(v1, . . . , vi)
+��
+∞ : ∥vj∥∞ ≤ 1, j = 1, . . . , i
+�
+for i = 1, 2, 3, 4. It follows that
+��K(i)(x)
+��
+≤
+sup
+0≤s,t≤1
+����
+∂iκ
+∂ui (s, t, x(t))
+���� ,
+i = 1, 2, 3, 4.
+Note that, if f ∈ Cα[0, 1] for any positive integer α, then ϕ ∈ Cα[0, 1]. See [5, Corollary
+3.2], [6, Corollary 4.2].
+3 Discretization of Integrals by numerical quadrature
+Rule
+In this section, first we consider a numerical integration formula. We replace the integral
+in the standard inner product of L2[0, 1] (i.e. ⟨x , y⟩ =
+� 1
+0 x(t)y(t)dt) by the quadrature rule
+and define a discrete inner product. Subsequently, the corresponding discrete orthogonal
+projection is defined. After that, an asymptotic error expansion for the discrete orthogonal
+projection is obtained. Next we define the Nyström approximations of the integral operator
+K and its Fréchet derivatives.
+Consider a basic numerical integration formula by
+(3.1)
+� 1
+0
+x(t)dt ≈
+ρ
+�
+q=1
+wq x(µq),
+which is exact at least for polynomials of degree ≤ 3r. If r = 0, then it is assumed that the
+quadrature rule is exact atleast for linear polynomials. It follows that �ρ
+q=1 wq = 1.
+Let n ∈ N and consider the following uniform partition of [0, 1] :
+(3.2)
+∆(n) :
+0 < 1
+n < · · · < n − 1
+n
+< 1.
+Define tj = j
+n, ∆j = [tj−1, tj] and h = tj − tj−1 = 1
+n, j = 1, . . . , n. Define the subspace
+Cα
+∆(n)[0, 1] = {x ∈ X : x ∈ Cα[tj−1, tj], j = 1, 2, 3, . . ., n} . For r ≥ 1, the approximating
+space
+Xn =
+�
+x ∈ X : x|∆j is a polynomial of degree ≤ r − 1
+�
+.
+
+Section 3. Discretization of Integrals by numerical quadrature Rule
+5
+Let p be a positive integer and m = pn. Consider the following uniform partition of [0, 1] :
+(3.3)
+∆(m) :
+0 < 1
+m < · · · < m − 1
+m
+< 1.
+Let ˜h = 1
+m
+and
+si = i
+m,
+i = 0, . . . , m.
+Note : As our goal to find the equation (1.6), where the higher order term is max
+�
+h2r+2, ˜h2�
+,
+we choose the partition ∆(m) such that ˜h2 ≤ hr.
+A composite integration rule with respect to the partition (3.3) is then defined as
+� 1
+0
+x(t) dt
+=
+m
+�
+i=1
+� si
+si−1
+x(t) dt ≈ ˜h
+m
+�
+i=1
+ρ
+�
+q=1
+wq x(si−1 + µq˜h).
+Thus,
+� tj
+tj−1
+x(t) dt =
+� jh
+(j−1)h
+x(t) dt =
+p
+�
+ν=1
+� (j−1)h+ν˜h
+(j−1)h+(ν−1)˜h
+x(t) dt.
+Since h = p˜h,
+� tj
+tj−1
+x(t)dt
+=
+p
+�
+ν=1
+�
+(j−1)p+ν
+p
+h
+(j−1)p+ν−1
+p
+h
+x(t) dt.
+Substituting t =
+(j−1)p+ν−1
+p
+h + ˜hσ =
+(j−1)p+ν−1+σ
+p
+h in the above equation, we obtain
+� tj
+tj−1
+x(t) dt
+=
+h
+p
+p
+�
+ν=1
+� 1
+0
+x
+�(j − 1)p + ν − 1 + σ
+p
+h
+�
+dσ
+=
+h
+p
+p
+�
+ν=1
+� 1
+0
+x
+�
+tj−1 + ν − 1 + σ
+p
+h
+�
+dσ.
+Note that ν−1+σ
+p
+∈ [0, 1]. Now using the numerical quadrature formula (3.1), we obtain
+� tj
+tj−1
+x(t) dt
+≈
+h
+p
+p
+�
+ν=1
+ρ
+�
+q=1
+wq x
+�
+tj−1 + ν − 1 + µq
+p
+h
+�
+.
+Let
+µqν = ν − 1 + µq
+p
+,
+q = 1, 2, . . . , ρ; ν = 1, 2, . . . , p.
+Then,
+� tj
+tj−1
+x(t) dt
+≈
+h
+p
+p
+�
+ν=1
+ρ
+�
+q=1
+wq x(tj−1 + µqνh).
+(3.4)
+We prove the following lemma which will be used to find an asymptotic error expansion for
+the discrete orthogonal projection.
+Lemma 3.1. Let Lη be the Legendre polynomial of degree η ∈ {0, 1, . . . , r − 1} defined on
+[0, 1]. Then for any k = 1, 2, . . . , 2r − 1,
+1
+p
+r−1
+�
+η=0
+p
+�
+ν=1
+ρ
+�
+q=1
+wq Lη(µqν)Lη(τ)(µqν − τ)k
+k!
+=
+� 1
+0
+Λr(τ, s)(s − τ)k
+k!
+ds,
+
+6
+G. RAKSHIT
+where
+r−1
+�
+η=0
+Lη(τ)Lη(s) = Λr(τ, s), for τ, s ∈ [0, 1].
+Proof. Since Lη is a polynomial of degree 0 ≤ η ≤ r − 1,
+1
+p
+p
+�
+ν=1
+ρ
+�
+q=1
+wq Lη
+�ν − 1 + µq
+p
+�
+= 1
+p
+p
+�
+ν=1
+� 1
+0
+Lη
+�ν − 1 + t
+p
+�
+dt
+=
+p
+�
+ν=1
+�
+ν
+p
+ν−1
+p
+Lη(s) ds
+=
+� 1
+0
+Lη(s) ds.
+Since the basic quadrature formula (3.1) is exact for polynomials of degree ≤ 3r,
+1
+p
+p
+�
+ν=1
+ρ
+�
+q=1
+wq Lη
+�ν − 1 + µq
+p
+� �ν − 1 + µq
+p
+− τ
+�k
+=
+� 1
+0
+Lη(s) (s − τ)k ds.
+It follows that
+1
+p
+p
+�
+ν=1
+ρ
+�
+q=1
+wq Lη(µqν) (µqν − τ)k
+k!
+=
+� 1
+0
+Lη(s) (s − τ)k
+k!
+ds,
+where µqν = ν − 1 + µq
+p
+. This gives
+1
+p
+r−1
+�
+η=0
+p
+�
+ν=1
+ρ
+�
+q=1
+wq Lη(τ)Lη(µqν) (µqν − τ)k
+k!
+=
+r−1
+�
+η=0
+Lη(τ)
+� 1
+0
+Lη(s) (s − τ)k
+k!
+ds.
+Let
+r−1
+�
+η=0
+Lη(τ)Lη(s) = Λr(τ, s),
+τ, s ∈ [0, 1].
+Then,
+1
+p
+r−1
+�
+η=0
+p
+�
+ν=1
+ρ
+�
+q=1
+wq Lη(τ)Lη(µqν) (µqν − τ)k
+k!
+=
+� 1
+0
+Λr(τ, s) (s − τ)k
+k!
+ds.
+Hence the required result follows.
+□
+3.1 Discrete Orthogonal Projection
+Let j ∈ {1, 2, . . . , n} and x, y ∈ C(∆j). Define a discrete inner product on ∆j by
+(3.5)
+⟨x , y⟩∆j,m = ˜h
+p
+�
+ν=1
+ρ
+�
+q=1
+wq x(tj−1 + µqνh) y(tj−1 + µqνh).
+Note that, this is an indefinite inner product. For more details on indefinite inner product
+spaces, see [8]. However, the properties which we need to define a discrete orthogonal
+projection, hold true for (3.5). For η = 0, 1, . . . , r−1, let Lη denote the Legendre polynomial
+of degree η on [0, 1]. For j = 2, . . . , n, and for η = 0, 1, . . . , r − 1, define
+ϕj,η(t)
+=
+� �
+1
+hLη
+�
+t−tj−1
+h
+�
+,
+t ∈ (tj−1, tj],
+0,
+otherwise
+
+3.1
+Discrete Orthogonal Projection
+7
+and, ϕ1,η(t) =
+�
+1
+hLη
+� t−t0
+h
+�
+if t ∈ [t0, t1] and 0 otherwise. Note that
+(3.6)
+ϕj,η (tj−1 + µqνh) = h− 1
+2Lη(µqν)
+for all j = 1, 2, . . . , n.
+Note that {ϕj,η : j = 1, . . . , n, η = 0, 1, . . . , r − 1} be a set of orthonormal basis for Xn,
+where ϕj,η is the Legendre polynomial of degree η defined on [tj−1, tj]. Since the basic
+numerical integration (3.1) has degree of precision 3r, the set {ϕj,η} is also orthonormal
+with respect to the discrete inner product (3.5). Let Pr,∆j be the space of polynomials of
+degree ≤ r − 1 on ∆j. Define the discrete orthogonal projection Pn,j : C[tj−1, tj] → Pr,∆j
+as follows:
+(3.7)
+Pn,jx =
+r−1
+�
+η=0
+⟨x , ϕj,η⟩∆j ϕj,η.
+See [4], [6] for more details. A discrete orthogonal projection Pn : C[0, 1] → Xn is defined
+by
+Pnx =
+n
+�
+j=1
+Pn,jx.
+(3.8)
+It follows that Pnx(t) = Pn,jx(t), for all t ∈ [tj−1, tj]. We also have the following error
+bound:
+∥Pn∥ < ∞ and also, if x ∈ Cr[tj−1, tj], then
+(3.9)
+∥x − Pn,jx∥∆j,∞ ≤ C1
+��x(r)��
+∆j,∞ hr,
+if x ∈ Cr[0, 1], then
+(3.10)
+∥x − Pnx∥∆j,∞ ≤ C1
+��x(r)��
+∞ hr,
+where ∥x∥∆j,∞ =
+sup
+t∈[tj−1,tj]
+|x(t)| and, C1 is a constant independent of h. For details see
+[17].
+In (3.10) we have a error bound for the discrete orthogonal projection. But, by the follow-
+ing lemma we obtain an asymptotic error expansion for the discrete orthogonal projection,
+which is more stronger result than (3.10).
+Lemma 3.2. Let Pn be the discrete orthogonal projection defined by (3.7) - (3.8). Let
+x ∈ C2r+2
+∆(n) [0, 1] and t = tj−1 + τh with τ ∈ [0, 1]. Then
+Pnx(t) − x(t) =
+2r+1
+�
+k=1
+Jk(τ) x(k)(tj−1 + τh) hk + O
+�
+h2r+2�
+,
+where Jk(τ) =
+� 1
+0
+Λr(τ, s)(s − τ)k
+k!
+ds,
+k = 1, 2, . . . , 2r + 1.
+Proof. Define a function vj : [tj−1, tj] → R by
+vj(t) = 1,
+t ∈ [tj−1, tj].
+For τ ∈ [0, 1], let t = tj−1 + hτ ∈ [tj−1, tj]. From (3.7) it is easy to see that
+Pn,jvj = vj.
+
+8
+G. RAKSHIT
+It follows that
+r−1
+�
+η=0
+⟨vj , ϕj,η⟩∆j ϕj,η(t) = 1.
+Since ϕj,η is a polynomial of degree 0 ≤ η ≤ r − 1 on [tj−1, tj],
+⟨vj , ϕj,η⟩∆j =
+� tj
+tj−1
+ϕj,η(s) ds
+=
+h
+p
+p
+�
+ν=1
+ρ
+�
+q=1
+wq ϕj,η(tj−1 + µqνh).
+Thus for any function x : [tj−1, tj] → R, we have
+x(t) = x(t) h
+p
+r−1
+�
+η=0
+p
+�
+ν=1
+ρ
+�
+q=1
+wq ϕj,η(tj−1 + µqνh) ϕj,η(t).
+It follows that
+Pn,jx(t) − x(t)
+= h
+p
+r−1
+�
+η=0
+p
+�
+ν=1
+ρ
+�
+q=1
+wq ϕj,η(tj−1 + µqνh) [x(tj−1 + µqνh) − x(t)] ϕj,η(t),
+where t = tj−1 + hτ ∈ [tj−1, tj] and τ ∈ [0, 1]. From (3.6), we have
+(3.11)
+Pn,jx(t) − x(t) = 1
+p
+r−1
+�
+η=0
+p
+�
+ν=1
+ρ
+�
+q=1
+wq Lη(µqν) [x(tj−1 + µqνh) − x(t)] Lη(τ).
+Since x ∈ C2r+2[tj−1, tj], using Taylor series expansion we obtain
+x(tj−1 + µqνh) − x(tj−1 + hτ) =
+2r+1
+�
+k=1
+x(k)(tj−1 + τh) (µqν − τ)k
+k!
+hk + O
+�
+h2r+2�
+.
+Thus
+Pnx(t) − x(t) = Pn,jx(t) − x(t)
+=
+2r+1
+�
+k=1
+x(k)(tj−1 + τh) hk
+�
+1
+p
+r−1
+�
+η=0
+p
+�
+ν=1
+ρ
+�
+q=1
+wqLη(µqν)Lη(τ)(µqν − τ)k
+k!
+�
++ O
+�
+h2r+2�
+.
+Let
+Jk(τ) = 1
+p
+r−1
+�
+η=0
+p
+�
+ν=1
+ρ
+�
+q=1
+wq Lη(µqν) Lη(τ) (µqν − τ)k
+k!
+,
+τ ∈ [0, 1].
+By Lemma 3.1, we can write Jk(τ) =
+� 1
+0 Λr(τ, s) (s−τ)k
+k!
+ds. Hence
+Pnx(t) − x(t) =
+2r+1
+�
+k=1
+Jk(τ) x(k)(tj−1 + τh) hk + O
+�
+h2r+2�
+.
+The result follows.
+□
+
+3.2
+Approximation of the Integral Operator
+9
+Let
+L = (I − K′(ϕ))−1 K′(ϕ).
+Then L is a compact linear integral operator with kernel ˜ℓ. Note that the smoothness of ˜ℓ is
+same as the kernel ℓ. See [26], [5, Lemma 5.1] for details. It follows that
+LPnx(s) − Lx(s) =
+n
+�
+j=1
+� tj
+tj−1
+˜ℓ(s, t) (Pnx(t) − x(t)) dt
+∀x ∈ X .
+Then using Lemma 3.2, and following the proofs of [22, Theorem 5.1] and [23, Theorem
+3.2], it can be shown that
+(3.12)
+L(I − Pn)ϕ(ti) = E2r(ϕ)(ti)h2r + O
+�
+h2r+2�
+,
+i = 0, 1, . . . , n,
+where
+E2r(ϕ)(ti) = ¯b2r,2r
+� 1
+0
+˜ℓ(ti, t)(t) ϕ(2r)(t) dt
++
+2r−1
+�
+p=1
+¯b2r,p
+� �� ∂
+∂t
+�2r−p−1 �
+˜ℓ(ti, t)ϕ(p)(t)
+��t=1
+t=0
+−
+�� ∂
+∂t
+�2r−p−1 �
+˜ℓ(ti, t)ϕ(p)(t)
+��t=ti+
+t=ti−
+�
+with
+¯b2r,p =
+� 1
+0
+� 1
+0
+Λr(τ, s)(τ − s)p
+p!
+B2r−p(s)
+(2r − p)! dτ ds
+and Bk is the Bernoulli polynomial of degree k ≥ 0.
+3.2 Approximation of the Integral Operator
+Let x ∈ X . Recall that
+K(x)(s) =
+� 1
+0
+κ(s, t, x(t)) dt,
+s ∈ [0, 1].
+Replacing the above integral by the numerical quadrature rule (3.4), we define the Nyström
+approximation of K by
+Km(x)(s) = h
+p
+n
+�
+j=1
+ρ
+�
+q=1
+p
+�
+ν=1
+wq κ(s, tj−1 + µqνh, x(tj−1 + µqνh)),
+s ∈ [0, 1].
+Let {µj
+qν = tj−1 + µqνh : j = 1, 2, . . . , n; q = 1, 2, . . . , ρ; ν = 1, 2, . . . , p} denotes the set
+of all quadrature nodes in [0, 1]. Then
+Km(x)(s) = h
+p
+n
+�
+j=1
+ρ
+�
+q=1
+p
+�
+ν=1
+wq κ
+�
+s, µj
+qν, x
+�
+µj
+qν
+��
+,
+s ∈ [0, 1].
+The Nyström method for solving (1.1) is to find the element xm for which
+xm − Km(xm) = f.
+For sufficiently large m, the above equation has a unique solution ϕm in a neighborhood
+B(ϕ, ǫ) of ϕ, and
+∥ϕ − ϕm∥∞
+≤
+C2 ∥K(ϕ) − Km(ϕ)∥∞ = O
+�
+˜h2�
+,
+(3.13)
+
+10
+G. RAKSHIT
+where C2 is a constant independent of m. See [2, Theorem 4]. We write
+(I − Pn) ϕm = (I − Pn) (ϕm − ϕ) + (I − Pn) ϕ.
+Then from (3.10), (3.13), we have
+(3.14)
+(I − Pn) ϕm = O
+�
+max
+�
+hr, ˜h2��
+.
+Let v1, v2 ∈ X and x ∈ B(ϕ, ǫ). Then the Fréchet derivatives of Km at x are given by
+K′
+m(x)v1(s) = h
+p
+n
+�
+j=1
+ρ
+�
+q=1
+p
+�
+ν=1
+wq D(0,0,1)κ
+�
+s, µj
+qν, x
+�
+µj
+qν
+��
+v1
+�
+µj
+qν
+�
+,
+s ∈ [0, 1],
+K′′
+m(x) (v1, v2) (s) = h
+p
+n
+�
+j=1
+ρ
+�
+q=1
+p
+�
+ν=1
+wq
+∂2κ
+∂u2
+�
+s, µj
+qν, x
+�
+µj
+qν
+��
+v1
+�
+µj
+qν
+�
+v2
+�
+µj
+qν
+�
+.
+It follows that
+∥K′′
+m(x) (v1, v2)∥∞ ≤
+
+
+
+sup
+s,t∈[0,1]
+|u|≤∥ϕ∥∞+ǫ
+����
+∂2κ
+∂u2(s, t, u)
+����
+
+
+ ∥v1∥∞ ∥v2∥∞ .
+This implies
+∥K′′
+m(x)∥ < ∞
+Similarly, it can be shown that
+���K(3)
+m (x)
+��� ≤
+
+
+
+sup
+s,t∈[0,1]
+|u|≤∥ϕ∥∞+ǫ
+����
+∂3κ
+∂u3 (s, t, u)
+����
+
+
+ = C3 < ∞
+Lemma 3.3. Let x1, x2 ∈ B(ϕ, ǫ). If D(0,0,3)κ ∈ C(Ω) then
+∥K′′
+m(x1) − K′′
+m(x2)∥ ≤ C3 ∥x1 − x2∥∞ ,
+where C3 is constant independent of n.
+Proof. For v1, v2 ∈ X , we have
+[K′′
+m(x1) − K′′
+m(x2)] (v1, v2)(s)
+= h
+p
+n
+�
+j=1
+ρ
+�
+q=1
+p
+�
+ν=1
+wq
+�∂2κ
+∂u2
+�
+s, µj
+qν, x1
+�
+µj
+qν
+��
+− ∂2κ
+∂u2
+�
+s, µj
+qν, x2
+�
+µj
+qν
+���
+v1
+�
+µj
+qν
+�
+v2
+�
+µj
+qν
+�
+for all s ∈ [0, 1]. Since D(0,0,3)κ ∈ C(Ω), applying mean value theorem on ∂2κ
+∂u2 with respect
+to its third variable u, we obtain
+∂2κ
+∂u2
+�
+s, µj
+qν, x1
+�
+µj
+qν
+��
+−∂2κ
+∂u2
+�
+s, µj
+qν, x2
+�
+µj
+qν
+��
+=
+�
+x1
+�
+µj
+qν
+�
+− x2
+�
+µj
+qν
+�� ∂3κ
+∂u3
+�
+s, µj
+qν, ζj
+qν
+�
+,
+where ζj
+qν lies in the line segment joining the points x1
+�
+µj
+qν
+�
+and x2
+�
+µj
+qν
+�
+. Then
+����
+∂2κ
+∂u2
+�
+s, µj
+qν, x1
+�
+µj
+qν
+��
+− ∂2κ
+∂u2
+�
+s, µj
+qν, x2
+�
+µj
+qν
+������ ≤ C3 ∥x1 − x2∥∞ .
+Hence
+∥[K′′
+m(x1) − K′′
+m(x2)] (v1, v2)∥∞ ≤ C3 ∥x1 − x2∥∞ ∥v1∥∞ ∥v2∥∞ ,
+which follows the result.
+□
+
+Section 4. Asymptotic Error Analysis
+11
+We will now quote some error estimates for the Nyström approximations.
+For α ≥ 0, if v1, v2 ∈ Cα
+∆(m)[0, 1], then from [27] or [4, Corollary 1], we obtain the
+following errors for numerical integration.
+∥[K′
+m(ϕ) − K′(ϕ)] v1∥∞ = O
+�
+˜h2�
+,
+(3.15)
+∥[K′′
+m(ϕ) − K′′(ϕ)] (v1, v2)∥∞ = O
+�
+˜h2�
+.
+Also from [18, Proposition 3.3], we have
+∥K′
+m(ϕm) − K′
+m(ϕ)∥ ≤ C4 ∥ϕm − ϕ∥∞ = O
+�
+˜h2�
+.
+(3.16)
+Therefore combining (3.15) and the above equation, we obtain
+∥[K′
+m(ϕm) − K′(ϕ)] v∥∞ = O
+�
+˜h2�
+,
+for all v ∈ Cν
+∆m[0, 1].
+(3.17)
+Similarly,
+∥[K′′
+m(ϕm) − K′′(ϕ)] (v1, v2)∥∞ = O
+�
+˜h2�
+,
+∀v1, v2 ∈ Cν
+∆m[0, 1],
+(3.18)
+for all v1, v2, v3 ∈ Cν
+∆m[0, 1] implies
+���
+K(3)
+m (ϕm) − K(3)(ϕ)
+�
+(v1, v2, v3)
+��
+∞ = O
+�
+˜h2�
+.
+(3.19)
+4 Asymptotic Error Analysis
+Replacing K by Km and πn by Pn in the Galerkin equation x − πnK(x) = πnf, the
+discrete Galerkin equation is defined by zG
+n − PnKm(zG
+n ) = Pnf, where zG
+n is the discrete
+Galerkin solution. Then the discrete iterated Galerkin solution is defined by
+zS
+n = Km(zG
+n ) + f.
+Note that PnzS
+n = zG
+n . From the equations ϕm − Km(ϕm) = f and zS
+n − Km(PnzS
+n) = f,
+we obtain the following error term.
+zS
+n − ϕm = [I − K′
+m(ϕm)]−1 �
+Km(zG
+n ) − Km(ϕm) − K′
+m(ϕm)(zG
+n − ϕm)
+�
+− Lm(I − Pn)
+�
+Km(zG
+n ) − Km(ϕm) − K′
+m(ϕm)(zG
+n − ϕm)
+�
+− Lm(I − Pn)K′
+m(ϕm)(zG
+n − ϕm)
+− Lm(I − Pn)ϕm,
+(4.1)
+where
+Lm = [I − K′
+m(ϕm)]−1 K′
+m(ϕm).
+Using the Resolvent Identity, we get
+(I − K′
+m(ϕm))−1 − (I − K′(ϕ))−1
+= (I − K′(ϕ))−1 [K′
+m(ϕm) − K′(ϕ)] (I − K′
+m(ϕm))−1
+
+12
+G. RAKSHIT
+Therefore
+(I − K′
+m(ϕm))−1 = (I − K′(ϕ))−1
++ (I − K′(ϕ))−1 [K′
+m(ϕm) − K′
+m(ϕ)] (I − K′
+m(ϕm))−1
++ (I − K′(ϕ))−1 [K′
+m(ϕ) − K′(ϕ)] (I − K′
+m(ϕm))−1
+(4.2)
+Now, we will analyze each of the terms appearing in the RHS of the equation (4.1). Error
+estimates for each of the said terms will be obtained by the following propositions.
+Proposition 4.1. Let {ti : i = 0, 1, . . . , n} be the set of partition points of [0, 1] defined by
+(3.2), then
+Lm(I − Pn)ϕm(ti) = E2r(ϕ)(ti)h2r + O
+�
+max
+�
+h2r+2, ˜h2��
+,
+where E2r is defined by (3.12).
+Proof. It can be easily verified that (using (4.2))
+Lm(I − Pn)ϕm =
+�
+I − K′
+m(ϕm)
+�−1 K′
+m(ϕm)(I − Pn)ϕm
+=
+�
+I − K′(ϕ)
+�−1 K′
+m(ϕm)(I − Pn)ϕm
++
+�
+I − K′(ϕ)
+�−1 �
+K′
+m(ϕm) − K′
+m(ϕ)
+� �
+I − K′
+m(ϕm)
+�−1 K′
+m(ϕm)(I − Pn)ϕm
++
+�
+I − K′(ϕ)
+�−1 �
+K′
+m(ϕ) − K′(ϕ)
+� �
+I − K′
+m(ϕm)
+�−1 K′
+m(ϕm)(I − Pn)ϕm.
+(4.3)
+Consider the first term of the above equation, we have
+(I − K′(ϕ))−1K′
+m(ϕm)(I − Pn)ϕm
+= (I − K′(ϕ))−1 K′(ϕ)(I − Pn)ϕ
++ (I − K′(ϕ))−1 K′(ϕ)(I − Pn)(ϕm − ϕ)
++ (I − K′(ϕ))−1 [K′
+m(ϕm) − K′(ϕ)] (I − Pn)ϕm.
+Using (3.12), (3.13) and (3.17), we obtain
+(4.4)
+(I − K′(ϕ))−1 K′
+m(ϕm)(I −Pn)ϕm(ti) = E2r(ϕ)(ti)h2r +O
+�
+max
+�
+h2r+2, ˜h2��
+.
+Note that
+∥K′
+m(ϕm)∥ ≤
+sup
+s,t∈[0,1]
+|u|≤∥ϕ∥∞+ǫ
+|κu(s, t, u)|
+and from [18, Proposition 4.2], we have
+��(I − K′
+m(ϕm))−1�� < ∞. Thus, from (3.15) and
+(3.16), we have the followings
+���
+�
+I − K′(ϕ)
+�−1 �
+K′
+m(ϕm) − K′
+m(ϕ)
+� �
+I − K′
+m(ϕm)
+�−1 K′
+m(ϕm)(I − Pn)ϕm
+���
+∞ = O
+�
+˜h2�
+,
+���
+�
+I − K′(ϕ)
+�−1 �
+K′
+m(ϕm) − K′
+m(ϕ)
+� �
+I − K′
+m(ϕm)
+�−1 K′
+m(ϕm)(I − Pn)ϕm
+���
+∞ = O
+�
+˜h2�
+.
+Hence the required result follows from (4.3), (4.4) and the above two estimates.
+□
+Before each of the following propositions, we prove lemmas and its corollaries which are
+used to prove next propositions.
+
+Section 4. Asymptotic Error Analysis
+13
+Lemma 4.1. Let Pn be the discrete orthogonal projection defined by (3.8). If D(0,0,3)κ ∈
+C(Ω) and v ∈ Cr+2([0, 1]), then for r ≥ 1
+K′′(ϕ)(Pnv − v)2 = T(v)h2r + O(h2r+2),
+(4.5)
+where
+T(v) =
+�� 1
+0
+Jr(τ)2 dτ
+�
+K′′(ϕ)
+�
+v(r)�2 .
+Furthermore, when r = 1, then
+(4.6)
+K(3)(ϕ)(Pnv − v)3 = O
+�
+h4�
+.
+Proof. The proofs of (4.5) and (4.6) follows from Lemma 3.2, [16, Lemma 2.4] and [16,
+Remark 2.4] respectively.
+□
+Given that (I − K′(ϕ))−1 is a bounded linear operator. Let
+M = (I − K′(ϕ))−1 K′′(ϕ).
+Note that M is a compact bi-linear integral operator. Also the smoothness of the kernel of
+M, is same as the kernels of K′′(ϕ). See [4], [26].
+As a consequence of the above lemma, we get the following result.
+Corollary 4.1. For r ≥ 1,
+M(Pnv − v)2 = T (v)h2r + O(h2r+2),
+(4.7)
+where
+T (v) =
+�� 1
+0
+Jr(τ)2 dτ
+�
+M
+�
+v(r)�2
+.
+Lemma 4.2. If ϕ ∈ Cr+2([0, 1]), then for r ≥ 1,
+K′′
+m(ϕm)(zG
+n − ϕm)2 = T(ϕ)h2r + O
+�
+max
+�
+h2r+2, ˜h2��
+,
+where T is defined in Lemma 4.1.
+Proof. Note that
+zG
+n − ϕm = PnzS
+n − Pnϕm − ϕm + Pnϕm + ϕ
+= Pn
+�
+zS
+n − ϕm
+�
+− (I − Pn) ϕm
+(4.8)
+Thus,
+K′′
+m(ϕm)(zG
+n − ϕm)2
+= K′′
+m(ϕm)
+�
+Pn
+�
+zS
+n − ϕm
+��2 − 2K′′
+m(ϕm)
+�
+Pn
+�
+zS
+n − ϕm
+�
+, (I − Pn) ϕm
+�
++ K′′
+m(ϕm) ((I − Pn) ϕm)2
+(4.9)
+Since ∥K′′
+m(ϕm)∥ < ∞ and ∥Pn∥ < ∞, from (1.4) it is easy to see that
+(4.10)
+���K′′
+m(ϕm)
+�
+Pn
+�
+zS
+n − ϕm
+��2���
+∞ = O
+�
+max
+�
+h2r+4, ˜h4��
+,
+(4.11)
+��K′′
+m(ϕm)
+�
+Pn
+�
+zS
+n − ϕm
+�
+, (I − Pn) ϕm
+���
+∞ = O
+�
+˜h2 max
+�
+hr+2, ˜h2��
+.
+
+14
+G. RAKSHIT
+Now we write
+K′′
+m(ϕm) ((I − Pn) ϕm)2 =
+�
+K′′
+m(ϕm) − K′′(ϕ)
+�
+((I − Pn) ϕm)2
++ K′′(ϕ) ((I − Pn) ϕm)2
+= K′′(ϕ) ((I − Pn) ϕ)2 + K′′(ϕ) ((I − Pn) (ϕ − ϕm))2
++
+�
+K′′
+m(ϕm) − K′′(ϕ)
+�
+((I − Pn) ϕm)2 .
+Since ∥K′′
+m(ϕm)∥ < ∞ and ∥Pn∥ < ∞, from (3.13), (3.18) and the above estimate, we
+obtain
+K′′
+m(ϕm) ((I − Pn) ϕm)2 = K′′(ϕ) ((I − Pn) ϕ)2 + O
+�
+˜h2�
+.
+Therefore, from (4.5) we obtain
+K′′
+m(ϕm) ((I − Pn) ϕm)2 = T(ϕ)h2r + O
+�
+max
+�
+h2r+2, ˜h2��
+.
+Hence, the required result follows from (4.9), (4.10), (4.11) and the above equation.
+□
+From the above lemma, we obtain
+(4.12)
+(I − K′(ϕ))−1 K′′
+m(ϕm)(zG
+n − ϕm)2 = T (ϕ)h2r + O
+�
+max
+�
+h2r+2, ˜h2��
+,
+where T is defined by (4.7).
+Lemma 4.3. If ϕ ∈ Cr+2([0, 1]), then
+[I − K′
+m(ϕm)]−1 K(3)
+m (ϕm)(zG
+n − ϕm)3 =
+
+
+
+O
+�
+max
+�
+h4, ˜h2��
+,
+r = 1,
+O
+�
+max
+�
+h3r, ˜h6��
+, r ≥ 2.
+Proof. First, we consider the case when r ≥ 2.
+Since
+��[I − K′
+m(ϕm)]−1�� < ∞ and
+���K(3)
+m (ϕm)
+��� < ∞, from (1.4) we obtain
+���[I − K′
+m(ϕm)]−1 K(3)
+m (ϕm)(zG
+n − ϕm)3���
+∞ = O
+�
+max
+�
+h3r, ˜h6��
+.
+Now consider the case, when r = 1. We rewrite (4.8) as
+(zG
+n − ϕm)3 =
+�
+Pn
+�
+zS
+n − ϕm
+�
+− (I − Pn) ϕm
+�3 .
+Thus
+K(3)
+m (ϕm)(zG
+n − ϕm)3 = K(3)
+m (ϕm)
+�
+Pn
+�
+zS
+n − ϕm
+��3 − K(3)
+m (ϕm) ((I − Pn) ϕm)3
+− K(3)
+m (ϕm)
+��
+Pn
+�
+zS
+n − ϕm
+��2 , (I − Pn) ϕm
+�
++ K(3)
+m (ϕm)
+�
+Pn
+�
+zS
+n − ϕm
+�
+, ((I − Pn) ϕm)2�
+.
+(4.13)
+Since
+���K(3)
+m (ϕm)
+��� < ∞ and ∥Pn∥ < ∞, from (1.5) and (3.14) we obtain
+K(3)
+m (ϕm)
+�
+Pn
+�
+zS
+n − ϕm
+��3 = O
+�
+h6�
+,
+K(3)
+m (ϕm)
+��
+Pn
+�
+zS
+n − ϕm
+��2 , (I − Pn) ϕm
+�
+= O
+�
+h5�
+,
+K(3)
+m (ϕm)
+�
+Pn
+�
+zS
+n − ϕm
+�
+, ((I − Pn) ϕm)2�
+= O
+�
+h4�
+.
+
+Section 4. Asymptotic Error Analysis
+15
+Note that, we have used the fact ˜h ≤ h in the above three expressions. On the other hand,
+from (3.13), (3.19) we have
+K(3)
+m (ϕm) ((I − Pn) ϕm)3 = K(3)(ϕ) ((I − Pn) ϕ)3 + O
+�
+˜h2�
+.
+From (4.6), it follows that
+(4.14)
+K(3)
+m (ϕm) ((I − Pn) ϕm)3 = O
+�
+max
+�
+h4, ˜h2��
+.
+Now, combining the results (4.13) - (4.14), we obtain
+K(3)
+m (ϕm)(zG
+n − ϕm)3 = O
+�
+max
+�
+h4, ˜h2��
+.
+Therefore
+[I − K′
+m(ϕm)]−1 K(3)
+m (ϕm)(zG
+n − ϕm)3 = O
+�
+max
+�
+h4, ˜h2��
+.
+Hence follows the result.
+□
+Proposition 4.2. Let ϕ ∈ Cr+2([0, 1]). Then for r ≥ 1,
+[I − K′
+m(ϕm)]−1 �
+Km(zG
+n ) − Km(ϕm) − K′
+m(ϕm)(zG
+n − ϕm)
+�
+(s)
+= 1
+2T (ϕ)(s)h2r + O
+�
+max
+�
+h2r+2, ˜h2��
+,
+for all s ∈ [0, 1].
+Proof. Applying the generalized Taylor’s series expansion of Km about ϕm in the neigh-
+bourhood B(ϕ, ǫ), we obtain
+(4.15)
+Km(zG
+n ) − Km(ϕm) − K′
+m(ϕm)(zG
+n − ϕm)
+= 1
+2K′′
+m(ϕm)(zG
+n − ϕm)2 + 1
+6K(3)
+m (ϕm)(zG
+n − ϕm)3 + R4,m
+�
+zG
+n − ϕm
+�
+,
+where
+R4,m
+�
+zG
+n − ϕm
+�
+=
+� 1
+0
+(1 − θ)3
+3!
+K(4)
+m
+�
+ϕm + θ(zG
+n − ϕm)
+�
+(zG
+n − ϕm)4 dθ.
+Note that for any x ∈ B(ϕ, ǫ), v ∈ X ,
+K(4)
+m (x)v4(s) = h
+p
+n
+�
+j=1
+ρ
+�
+q=1
+p
+�
+ν=1
+wq
+∂4κ
+∂u4
+�
+s, µj
+qν, x
+�
+µj
+qν
+��
+v4�
+µj
+qν
+�
+,
+s ∈ [0, 1].
+It follows that
+��K(4)
+m (x)v4��
+∞ ≤
+
+
+
+sup
+s,t∈[0,1]
+|u|≤∥ϕ∥∞+ǫ
+����
+∂4κ
+∂u4 (s, t, u)
+����
+
+
+ ∥v∥4
+∞ = C5 ∥v∥4
+∞ .
+Since ϕm and zG
+n ∈ B(ϕ, ǫ), ϕm + θ(zG
+n − ϕm) ∈ B(ϕ, ǫ) and therefore
+��K(4)
+m
+�
+ϕm + θ(zG
+n − ϕm)
+�
+(zG
+n − ϕm)4��
+∞ ≤ C5
+��zG
+n − ϕm
+��4
+∞ = O
+�
+h4r�
+.
+It follows that
+R4,m
+�
+zG
+n − ϕm
+�
+= O
+�
+h4r�
+.
+
+16
+G. RAKSHIT
+Using the resolvent identity (3.17) and (4.2), we obtain
+[I − K′
+m(ϕm)]−1 K′′
+m(ϕm)(zG
+n −ϕm)2 = [I − K′(ϕ)]−1 K′′
+m(ϕm)(zG
+n −ϕm)2 +O
+�
+˜h2�
+.
+By (4.12), it follows that
+[I − K′
+m(ϕm)]−1 K′′
+m(ϕm)(zG
+n − ϕm)2 = T (ϕ)h2r + O
+�
+max
+�
+h2r+2, ˜h2��
+.
+From the Lemma 4.3, we have
+(4.16)
+[I − K′
+m(ϕm)]−1 K(3)
+m (ϕm)(zG
+n − ϕm)3 =
+
+
+
+O
+�
+max
+�
+h4, ˜h2��
+,
+r = 1,
+O
+�
+max
+�
+h3r, ˜h6��
+, r ≥ 2.
+Combining the results from (4.15) to (4.16), we obtain for r ≥ 1,
+[I − K′
+m(ϕm)]−1 �
+Km(zG
+n ) − Km(ϕm) − K′
+m(ϕm)(zG
+n − ϕm)
+�
+= 1
+2T (ϕ)h2r + O
+�
+max
+�
+h2r+2, ˜h2��
+,
+which completes the proof.
+□
+Lemma 4.4. If v ∈ X , then for r ≥ 1,
+max
+0≤i≤n |K′
+m(ϕm)(I − Pn)v(ti)|
+≤
+C7 ∥(I − Pn)v∥∞ hr,
+where C7 is a constant independent of h.
+Proof. We write
+(4.17)
+K′
+m(ϕm)(I − Pn)v = K′
+m(ϕ)(I − Pn)v + [K′
+m(ϕm) − K′
+m(ϕ)](I − Pn)v.
+For fixed s ∈ [0, 1], let
+ℓ∗,s(t) = ℓ∗(s, t) = ℓ(s, t, ϕ(t)) = ∂κ
+∂u(s, t, ϕ(t)),
+t ∈ [0, 1].
+From the definition of K′
+m(ϕ) and the discrete inner product, we have
+K′
+m(ϕ)(I − Pn)v(s)
+=
+n
+�
+j=1
+⟨ℓ∗,s , (I − Pn,j)v⟩∆j,m .
+Since Pn,j is self-adjoint on C(∆j), so as I − Pn,j. Therefore
+K′
+m(ϕ)(I − Pn)v(s)
+=
+n
+�
+j=1
+⟨(I − Pn,j)ℓ∗,s , (I − Pn,j)v⟩∆j,m .
+Note that, if s = ti for some i ∈ {0, 1, . . . , n}, then ℓ∗,s ∈ Cr[tj−1, tj] for all j = 1, . . . , n.
+Hence from (3.9),
+∥(I − Pn,j)ℓ∗,ti∥∆j,∞
+≤
+C1
+�
+sup
+t∈[tj−1,tj]
+|D(0,r)ℓ∗(ti, t)|
+�
+hr
+Thus,
+max
+0≤i≤n |K′
+m(ϕ)(I − Pn)v(ti)| ≤
+n
+�
+j=1
+∥(I − Pn,j)ℓm,s∥∆j,∞ ∥(I − Pn,j)v∥∆j,∞h
+≤ C6 ∥(I − Pn)v∥∞ hr,
+
+Section 4. Asymptotic Error Analysis
+17
+where C6 is a constant independent of h. Now, from (3.16), (4.17) and the above estimate,
+we obtain
+max
+0≤i≤n |K′
+m(ϕm)(I − Pn)v(ti)| ≤ C7 ∥(I − Pn)v∥∞ max
+�
+hr, ˜h2�
+,
+where C7 = C4 + C6. Since hr ≥ ˜h2, the result follows.
+□
+Recall that Lm = [I − K′
+m(ϕm)]−1 K′
+m(ϕm). Therefore, the proof of the following result
+is similar to that of the above lemma.
+Corollary 4.2. If v ∈ X , then for r ≥ 1,
+max
+0≤i≤n|Lm(I − Pn)v(ti)|
+≤
+C8 ∥(I − Pn)v∥∞ hr,
+where C8 is a constant independent of h.
+Lemma 4.5. Let v ∈ X . If r = 1, that is, when the range of Pn is the space of piecewise
+polynomials of degree zero, then
+∥(I − Pn)K′′
+m(ϕ)(v, v)∥∞ ≤ C9h ∥v∥2
+∞ ,
+where C9 is a constant independent of h.
+Proof. Given that Xn is the space of piecewise constant functions with respect to the parti-
+tion (3.2). Note that
+∥(I − Pn)K′′
+m(ϕ)(v, v)∥∞
+=
+max
+1≤j≤n
+sup
+s∈[tj−1,tj]
+|(I − Pn,j)K′′
+m(ϕ)(v, v)(s)|.
+Let s ∈ [tj−1, tj]. Since the Legendre polynomial of zero L0(t) = 1 for all t ∈ [0, 1], we
+have from (3.11),
+(4.18)
+(I − Pn)K′′
+m(ϕ)(v, v)(s)
+= 1
+p
+p
+�
+ν=1
+ρ
+�
+q=1
+wq [K′′
+m(ϕ)(v, v)(s) − K′′
+m(ϕ)(v, v)(tj−1 + µqνh)] .
+We also have
+K′′
+m(ϕ)(v, v)(s) − K′′
+m(ϕ)(v, v)(tj−1 + µqνh)
+= h
+p
+n
+�
+k=1
+ρ
+�
+q=1
+p
+�
+ν=1
+wq
+�∂2κ
+∂u2
+�
+s, µk
+qν, ϕ
+�
+µk
+qν
+��
+− ∂2κ
+∂u2
+�
+µj
+qν, µk
+qν, ϕ
+�
+µk
+qν
+���
+v2�
+µk
+qν
+�
+,
+where µj
+qν = tj−1 + µqνh. For fixed s ∈ [0, 1], let
+λ∗,s(t) = λ∗(s, t) = λ(s, t, ϕ(t)) = ∂2κ
+∂u2 (s, t, ϕ(t)),
+t ∈ [0, 1].
+Then, by (3.5)
+K′′
+m(ϕ)(v, v)(s) − K′′
+m(ϕ)(v, v)(µj
+qν) =
+n
+�
+k=1
+�
+λ∗,s − λ∗,µiqν, v2�
+∆k,m
+=
+n
+�
+k=1
+k̸=j
+�
+λ∗,s − λ∗,µj
+qν , v2�
+∆k,m +
+�
+λ∗,s − λ∗,µj
+qν , v2�
+∆j,m .
+(4.19)
+
+18
+G. RAKSHIT
+First consider the case when k ̸= j. Applying Mean Value Theorem on the first component
+of λ∗(·, ·) in the interval [s, µj
+qν], we obtain
+�
+λ∗,s − λ∗,µj
+qν , v2�
+∆k,m = (s − µj
+qν)
+�
+D(1,0)λ∗(θj
+qν, ·) , v2�
+∆k,m ,
+for some θj
+qν ∈ (tj−1, tj), and the function D(1,0)λ∗(s, t), t ∈ [tk−1, tk] is given by
+D(1,0)λ∗(s, t) =
+�
+D(1,0)λ1,∗(s, t) =
+∂
+∂sλ1(s, t, ϕ(t)),
+0 ≤ t ≤ s ≤ 1,
+D(1,0)λ2,∗(s, t) =
+∂
+∂sλ2(s, t, ϕ(t)),
+0 ≤ s ≤ t ≤ 1.
+Therefore, for k ̸= j,
+����
+�
+λ∗,s − λ∗,µj
+qν , v2�
+∆k,m
+����
+≤
+��s − µj
+qν
+��
+�
+sup
+s̸=t
+��D(1,0)λ∗(s, t)
+��
+�
+∥v∥2
+∞ h
+≤
+�
+sup
+s̸=t
+��D(1,0)λ∗(s, t)
+��
+�
+∥v∥2
+∞ h2,
+where
+sup
+s̸=t
+|D(1,0)λ∗(s, t)|
+= max
+�
+sup
+0≤t<s≤1
+|D(1,0)λ1,∗(s, t)| ,
+sup
+0≤s<t≤1
+|D(1,0)λ2,∗(s, t)|
+�
+= max
+
+
+
+
+
+sup
+0≤t<s≤1
+|u|≤∥ϕ∥∞
+|D(1,0,2)κ1(s, t, u)| ,
+sup
+0≤s<t≤1
+|u|≤∥ϕ∥∞
+|D(1,0,2)κ2(s, t, u)|
+
+
+
+
+
+.
+On the other hand,
+|
+�
+λ∗,s − λ∗,µj
+qν , v2�
+∆j,m| ≤ 2
+�
+sup
+0≤s,t≤1
+|λ∗(s, t)|
+�
+∥v∥2
+∞ h,
+where
+sup
+0≤s,t≤1
+|λ∗(s, t)| =
+sup
+0≤s,t≤1
+|u|≤∥ϕ∥∞
+|D(0,0,2)κ(s, t, u)|. Then, from (4.19) we obtain
+|K′′
+m(ϕ)(v, v)(s) − K′′
+m(ϕ)(v, v)(µj
+qν)|
+≤
+
+
+
+n
+�
+k=1
+k̸=j
+�
+sup
+s̸=t
+|D(1,0)λ∗(s, t)|
+�
+∥v∥2
+∞ h2
+
+
+ + 2
+�
+sup
+0≤s,t≤1
+|λ∗(s, t)|
+�
+∥v∥2
+∞ h.
+It follows that
+|K′′
+m(ϕ)(v, v)(s) − K′′
+m(ϕ)(v, v)(µj
+qν)| ≤ C9 ∥v∥2 h,
+where C9 =
+�
+sup
+s̸=t
+|D(1,0)λ∗(s, t)|
+�
++ 2
+�
+sup
+0≤s,t≤1
+|λ∗(s, t)|
+�
+.
+The result now follows from (4.18) and the above estimate.
+□
+Corollary 4.3. Let v ∈ X . If r = 1, that is, when the range of Pn is the space of piecewise
+polynomials of degree zero, then
+∥(I − Pn)K′
+m(ϕ)v∥∞ ≤ C10h ∥v∥∞ ,
+where C10 is a constant independent of h.
+
+Section 4. Asymptotic Error Analysis
+19
+Proof. The proof is similar to that of Lemma 4.5.
+□
+Proposition 4.3. Let ti be any point of the partition ∆(n) defined by (3.2). Then
+Lm(I − Pn)
+�
+Km(zG
+n ) − Km(ϕm) − K′
+m(ϕm)(zG
+n − ϕm)
+�
+(ti)
+=
+
+
+
+O (h4) ,
+r = 1,
+O
+�
+max
+�
+h3r, hr˜h4��
+,
+r ≥ 2.
+Proof. Generalized Taylor’s series expansion gives
+(4.20)
+Lm(I − Pn)
+�
+Km(zG
+n ) − Km(ϕm) − K′
+m(ϕm)(zG
+n − ϕm)
+�
+= 1
+2Lm(I − Pn)K′′
+m(ϕm)(zG
+n − ϕm)2 + Lm(I − Pn)R3,m
+�
+zG
+n − ϕm
+�
+,
+where
+R3,m
+�
+zG
+n − ϕm
+�
+=
+� 1
+0
+(1 − θ)2
+2!
+K(3)
+m
+�
+ϕm + θ(zG
+n − ϕm)
+�
+(zG
+n − ϕm)3 dθ.
+It follows that
+��R3,m
+�
+zG
+n − ϕm
+���
+∞ ≤ 1
+6
+
+
+
+sup
+s,t∈[0,1]
+|u|≤∥ϕ∥∞+ǫ
+����
+∂3κ
+∂u3 (s, t, u)
+����
+
+
+
+��zG
+n − ϕm
+��3
+∞ .
+Therefore, by (1.4)
+��R3,m
+�
+zG
+n − ϕm
+���
+∞ = O
+�
+max
+�
+h3r, ˜h6��
+.
+Since ∥I − Pn∥ ≤ 1 + ∥Pn∥ < ∞, from Corollary 4.2, it is easy to see that
+Lm(I − Pn)R3,m
+�
+zG
+n − ϕm
+�
+(ti) = O
+�
+max
+�
+h4r, hr˜h6��
+.
+(4.21)
+First consider the case r ≥ 2. Since ∥K′′
+m(ϕm)∥ < ∞ and ∥I − Pn∥∞ < ∞, by (1.4) and
+the Corollary 4.2, we have
+1
+2Lm(I − Pn)K′′
+m(ϕm)(zG
+n − ϕm)2(ti) = O
+�
+max
+�
+h3r, hr˜h4��
+,
+r ≥ 2.
+When r = 1, we write
+Lm(I − Pn)K′′
+m(ϕm)(zG
+n − ϕm)2
+= Lm(I − Pn)
+�
+K′′
+m(ϕm) − K′′
+m(ϕ)
+�
+(zG
+n − ϕm)2 + Lm(I − Pn)K′′
+m(ϕ)(zG
+n − ϕm)2.
+By (1.4), (3.13) and the Lemma 3.3, we have
+Lm(I − Pn) [K′′
+m(ϕm) − K′′
+m(ϕ)] (zG
+n − ϕm)2 = O
+�
+h4�
+.
+On the other hand
+Lm(I − Pn)K′′
+m(ϕ)(zG
+n − ϕm)2(ti) =
+n
+�
+j=1
+�
+ℓm,ti , (I − Pn,j)K′′
+m(ϕ)(zG
+n − ϕm)2�
+∆j,m .
+
+20
+G. RAKSHIT
+Since I − Pn,j is self-adjoint,
+Lm(I − Pn)K′′
+m(ϕ)(zG
+n − ϕm)2(ti)
+=
+n
+�
+j=1
+�
+(I − Pn,j)ℓm,ti , (I − Pn,j)K′′
+m(ϕ)(zG
+n − ϕm)2�
+∆j,m .
+It follows that
+max
+0≤i≤n
+��Lm(I − Pn)K′′
+m(ϕ)(zG
+n − ϕm)2(ti)
+��
+≤
+n
+�
+j=1
+∥(I − Pn,j)ℓm,ti∥∆j,∞
+��(I − Pn,j)K′′
+m(ϕ)(zG
+n − ϕm)2��
+∆j,∞ h.
+By Corollary 4.2 and Lemma 4.5, we obtain
+max
+0≤i≤n|Lm(I − Pn)K′′
+m(ϕ)(zG
+n − ϕm)2(ti)| = O
+�
+h4�
+,
+r = 1.
+Therefore
+1
+2Lm(I − Pn)K′′
+m(ϕm)(zG
+n − ϕm)2(ti) =
+
+
+
+O (h4) ,
+r = 1,
+O
+�
+max
+�
+h3r, hr˜h4��
+,
+r ≥ 2.
+(4.22)
+Then combing (4.20), (4.21) and (4.22), we obtain
+Lm(I − Pn)
+�
+Km(zG
+n ) − Km(ϕm) − K′
+m(ϕm)(zG
+n − ϕm)
+�
+(ti)
+=
+
+
+
+O (h4) ,
+r = 1,
+O
+�
+max
+�
+h3r, hr˜h4��
+,
+r ≥ 2.
+This follows the result.
+□
+We quote the following result from By [17, Proposition 1, Proposition 6], which will be
+used in the next proposition.
+(4.23)
+∥(I − Pn)K′
+m(ϕ)(I − Pn)ϕ∥∞ =
+�
+O (h3) ,
+r = 1,
+O (hr+2) ,
+r ≥ 2.
+Proposition 4.4. If ϕm and zG
+n are respectively the Nyström and the discrete Galerkin ap-
+proximation of ϕ, then
+Lm(I − Pn)K′
+m(ϕm)(zG
+n − ϕm)(ti) = O
+�
+max
+�
+h2r+2, ˜h2��
+.
+Proof. Adding and subtracting K′
+m(ϕ), we have
+Lm(I − Pn)K′
+m(ϕm)(zG
+n − ϕm)
+= Lm(I − Pn) [K′
+m(ϕm) − K′
+m(ϕ)] (zG
+n − ϕm) + Lm(I − Pn)K′
+m(ϕ)(zG
+n − ϕm).
+Then, using (1.4), (3.16) and the Corollary 4.2, we obtain for r ≥ 1,
+(4.24)
+��Lm(I − Pn) [K′
+m(ϕm) − K′
+m(ϕ)] (zG
+n − ϕm)(ti)
+��
+≤ C4C8 (1 + ∥Pn∥) hr˜h2 �
+max
+�
+hr, ˜h2��
+
+Section 4. Asymptotic Error Analysis
+21
+Note that
+zG
+n − ϕm = PnzS
+n − ϕm = PnzS
+n − Pnϕ + Pnϕ − ϕ + ϕ − ϕm
+= Pn
+�
+zS
+n − ϕ
+�
+− (I − Pn) ϕ + (ϕ − ϕm) .
+Then
+Lm(I − Pn)K′
+m(ϕ)(zG
+n − ϕm) = Lm(I − Pn)K′
+m(ϕ)Pn
+�
+zS
+n − ϕ
+�
+− Lm(I − Pn)K′
+m(ϕ) (I − Pn) ϕ
++ Lm(I − Pn)K′
+m(ϕ) (ϕ − ϕm) .
+By the Corollary 4.2
+��Lm(I − Pn)K′
+m(ϕ)Pn
+�
+zS
+n − ϕ
+�
+(ti)
+�� ≤ C8
+��(I − Pn) K′
+m(ϕ)Pn
+�
+zS
+n − ϕ
+��� hr,
+then by (1.5) and Corollary 4.3, we obtain
+��Lm(I − Pn)K′
+m(ϕ)Pn
+�
+zS
+n − ϕ
+�
+(ti)
+�� = O
+�
+max
+�
+h2r+2, ˜h2��
+,
+for r ≥ 1.
+Also, the Corollary 4.2 and (4.23) implies
+Lm(I − Pn)K′
+m(ϕ) (I − Pn) ϕ(ti) = O
+�
+h2r+2�
+,
+for r ≥ 1.
+It is easy to see (from (3.13) and Corollary 4.2) that
+Lm(I − Pn)K′
+m(ϕ) (ϕ − ϕm) = O
+�
+hr˜h2�
+,
+for r ≥ 1.
+Therefore, for r ≥ 1,
+Lm(I − Pn)K′
+m(ϕ)(zG
+n − ϕm)(ti) = O
+�
+max
+�
+h2r+2, ˜h2��
+.
+Hence, the result follows from (4.24) and the above equation.
+□
+We prove the main theorem as follows.
+Theorem 4.1. Let K be the Urysohn integral operator with Green’s function type kernel κ,
+defined by (1.2). Let ϕ be the unique solution of the equation (1.1). Assume that 1 is not
+an eigenvalue of K′(ϕ). Let Xn be the space of piecewise polynomials of degree ≤ r − 1
+with respect to the partition ∆(n) := 0 = t0 < t1 < · · · < tn = 1 defined by (3.2). Let
+Pn : L∞[0, 1] → Xn be the discrete orthogonal projection defined by (3.8) and ˜zS
+n be the
+discrete iterated Galerkin approximation of ϕ. Then
+�
+zS
+n − ϕ
+�
+(ti) =
+�
+E2r(ϕ)(ti) + 1
+2T (ϕ)(ti)
+�
+h2r + O
+�
+max
+�
+h2r+2, ˜h2��
+,
+where the operators E2r and T are respectively defined by (3.12) and (4.5).
+Proof. We have from (4.1)
+zS
+n − ϕ = [I − K′
+m(ϕm)]−1 �
+Km(zG
+n ) − Km(ϕm) − K′
+m(ϕm)(zG
+n − ϕm)
+�
+− Lm(I − Pn)
+�
+Km(zG
+n ) − Km(ϕm) − K′
+m(ϕm)(zG
+n − ϕm)
+�
+− Lm(I − Pn)K′
+m(ϕm)(zG
+n − ϕm)
+− Lm(I − Pn)ϕm
++ ϕm − ϕ.
+
+22
+G. RAKSHIT
+The result now follows from (3.13), Proposition 4.1, Proposition 4.2, Proposition 4.3 and
+Proposition 4.4.
+□
+We now apply Richardson extrapolation to obtain an approximation of ϕ with higher
+order of convergence. Define
+zEX
+n
+= 24rzS
+2n − zS
+n
+24r − 1
+.
+We choose the partitions ∆(m) and ∆(n) such that m2 ≥ n2r+2. Then, it is easy to see from
+the Theorem 4.1, that
+(4.25)
+�
+zEX
+n
+− ϕ
+�
+(ti) = O
+�
+h2r+2�
+,
+for all i = 1, 2, . . . , n.
+5 Numerical results
+For the numerical results, we consider the following example from [19]. Consider
+(5.1)
+ϕ(s) −
+� 1
+0
+κ(s, t) [ψ (t, ϕ(t))] dt = f(s),
+0 ≤ s ≤ 1,
+where
+κ(s, t) =
+1
+γ sinh γ
+�
+sinh γs sinh γ(1 − t),
+0 ≤ t ≤ s ≤ 1,
+γ(1 − s) sinh γt,
+0 ≤ s ≤ t ≤ 1,
+with γ =
+√
+12, and
+ψ(t, ϕ(t)) = γ2ϕ(t) − 2 (ϕ(t))3 ,
+t ∈ [0, 1].
+We have f(s) =
+1
+sinh γ
+�
+2 sinh γ(1 − s) + 2
+3 sinh γs
+�
+. The exact solution of (5.1) is given
+by
+ϕ(s) =
+2
+2s + 1,
+s ∈ [0, 1].
+Let Xn be the space of piecewise constant functions with respect to the uniform partition
+∆(n) of the interval [0, 1]. Let Pn : L∞[0, 1] → Xn be the discrete orthogonal projection
+defined by (3.8).
+Let ti = i−1
+20 , i = 1, 2, . . . , 21 be the partition points with step size h =
+1
+20. The numerical
+quadrature is chosen to be the composite 2 point Gaussian quadrature rule with respect to
+partition ∆(m) with m = n2 subintervals. Then ˜h = h2. Therefore, it is expected from the
+Theorem 4.1 and equation (4.25), that
+ǫS
+n(ti) = |ϕ(ti) − zS
+n(ti)| = O
+�
+h2�
+and ǫEX
+n
+(ti) = |ϕ(ti) − zEX
+n
+(ti)| = O
+�
+h4�
+,
+where
+zEX
+n
+(ti) = 4zS
+2n(ti) − zS
+n(ti)
+3
+.
+Let δS and δEX be respectively the orders of convergence of zS
+n and zEX
+n
+at the partition
+points. We expect δS = 2 and δEX = 4.
+
+Section 5. Numerical results
+23
+Table 1
+ti
+ǫS
+n(ti) : n = 20
+δS
+ǫEX
+n
+(ti) : n = 20
+δEX
+0.05
+8.6 × 10−3
+2.00
+2.98 × 10−6
+3.99
+0.1
+7.56 × 10−3
+2.00
+2.23 × 10−6
+3.99
+0.15
+6.79 × 10−3
+2.00
+1.59 × 10−6
+3.99
+0.2
+6.22 × 10−3
+2.00
+1.09 × 10−6
+3.97
+0.25
+5.78 × 10−3
+2.00
+7.13 × 10−7
+3.96
+0.3
+5.45 × 10−3
+2.00
+4.46 × 10−7
+3.94
+0.35
+5.19 × 10−3
+2.00
+2.7 × 10−7
+3.91
+0.4
+4.98 × 10−3
+2.00
+1.69 × 10−7
+3.86
+0.45
+4.82 × 10−3
+2.00
+1.3 × 10−7
+3.83
+0.5
+4.68 × 10−3
+2.00
+1.41 × 10−7
+3.85
+0.55
+4.55 × 10−3
+2.00
+1.91 × 10−7
+3.89
+0.6
+4.44 × 10−3
+2.00
+2.72 × 10−7
+3.93
+0.65
+4.33 × 10−3
+2.00
+3.75 × 10−7
+3.95
+0.7
+4.22 × 10−3
+2.00
+4.95 × 10−7
+3.97
+0.75
+4.10 × 10−3
+2.00
+6.26 × 10−7
+3.98
+0.8
+3.98 × 10−3
+2.00
+7.6 × 10−7
+3.99
+0.85
+3.84 × 10−3
+2.00
+8.94 × 10−7
+3.99
+0.9
+3.69 × 10−3
+2.00
+1.02 × 10−6
+3.99
+0.95
+3.52 × 10−3
+2.00
+1.14 × 10−6
+4
+From the above table, it is clear that the obtained orders of convergence match well with
+the theoretical orders of convergence. Also the order of convergence of the extrapolated
+solution improves upon the discrete iterated Galerkin solution.
+5 References
+[1] K. E. Atkinson. The numerical solutions of integral equations of the second kind, Cambridge University
+Press, Cambridge, (1997).
+[2] K. E. Atkinson. The numerical evaluation of fixed points for completely continuous operators. SIAM Jour-
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+[4] K. E. Atkinson and F. A. Potra. On the discrete Galerkin method for Fredholm integral equations of the
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+SIAM Journal on Numerical Analysis, 24(6), 1352-1373 (1987).
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+holm integral equation of the second kind and the corresponding eigenvalue problem. Journal of Integral
+Equations, 6(1), 71-91 (1984).
+
+24
+G. RAKSHIT
+[11] W. F. Ford, J. A. Pennline, Y. Xu and Y. Zhao. Asymptotic error analysis of a quadrature method for
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+[12] M. A. Krasnoselskii. Topological Methods in the Theory of Nonlinear Integral Equations. Pergamon
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+for nonlinear integral equations. Journal of Integral Equations and Applications, 27(1), 67-101 (2015).
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+Green’s function type kernels. Mathematical Modelling and Analysis, 25(3), 421 - 440 (2020).
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+smooth kernels. Applied Numerical Mathematics, 126, 180-198 (2018).
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+equations with Green’s function Type Kernels. Mathematical Modelling and Analysis, 19(1), 127-143 (2014).
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+[22] W. McLean. Asymptotic error expansions for numerical solutions of integral equations. IMA Journal of
+Numerical Analysis, 9(3), 373-384 (1989).
+[23] G. Rakshit and A. S. Rane. Asymptotic expansion of iterated Galerkin solution of Fredholm integral
+equations of the second kind with Green’s kernel. Journal of Integral Equations and Applications, 32(4),
+495-507 (2020).
+[24] G. Rakshit, A. S. Rane and K. Patil. Richardson extrapolation for the iterated Galerkin solution of
+Urysohn integral equations with Green’s kernels. International Journal of Computer Mathematics, 19(8),
+1538 - 1556 (2022).
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+DEPARTMENT OF MATHEMATICAL SCIENCES, RAJIV GANDHI INSTITUTE OF PETROLEUM TECHNOL-
+OGY, JAIS CAMPUS, UTTAR PRADESH 229304, INDIA., ORCID ID : 0000-0002-5813-4656
+Email address: g.rakshit@rgipt.ac.in
+
diff --git a/5tE1T4oBgHgl3EQfmgTC/content/tmp_files/load_file.txt b/5tE1T4oBgHgl3EQfmgTC/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..8f7fd148636212fd23821162f32db6d1711f3e04
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@@ -0,0 +1,934 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf,len=933
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='03299v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='NA]  9 Jan 2023  ASYMPTOTIC ERROR ANALYSIS FOR THE DISCRETE ITERATED  GALERKIN SOLUTION OF URYSOHN INTEGRAL EQUATIONS WITH  GREEN’S KERNELS  GOBINDA RAKSHIT  ABSTRACT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Consider a Urysohn integral equation x − K(x) = f, where f and the integral  operator K with kernel of the type of Green’s function are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' In the computation of  approximate solutions of the given integral equation by Galerkin method, all the integrals  are needed to be evaluated by some numerical integration formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' This gives rise to the  discrete version of the Galerkin method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' For r ≥ 1, a space of piecewise polynomials  of degree ≤ r − 1 with respect to a uniform partition is chosen to be the approximating  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' For the appropriate choice of a numerical integration formula, an asymptotic series  expansion of the discrete iterated Galerkin solution is obtained at the above partition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Richardson extrapolation is used to improve the order of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Using this method  we can restore the rate of convergence when the error is measured in the continuous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' A  numerical example is given to illustrate this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  1 Introduction  Let X = L∞[0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Consider the problem of solving Urysohn integral equation  x(s) −  � 1  0  κ(s, t, x(t)) dt = f(s), s ∈ [0, 1],  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1)  where f ∈ X and κ ∈ C ([0, 1] × [0, 1] × R) are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Let the Urysohn integral operator  K : L∞[0, 1] → C[0, 1] be defined by  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2)  K(x)(s) =  � 1  0  κ(s, t, x(t)) dt,  x ∈ X , s ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Since the kernel κ is continuous, K is compact operator on X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Denoting the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1)  by  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='3)  x − Kx = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  We assume that the above equation has a solution, say ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' We also assume that K is twice  Frechét differentiable and 1 is not an eigenvalue of the compact linear operator K′(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' This  gives us that ϕ is an isolated solution of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' See [12], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' We are looking for Galerkin  approximations of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  For r ≥ 1, consider the approximating space Xn as a space of piecewise polynomials of  degree ≤ r − 1 with respect to a uniform partition, say ∆(n), of [0, 1] with n subintervals  Date: January 10, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' 45G10, 65B05, 65J15, 65R20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Urysohn integral operator, Green’s kernel, Galerkin method, Nyström approxima-  tion, Richardson extrapolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  1  2  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' RAKSHIT  each of length h = 1  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Let πn be the restriction to L∞[0, 1] of the orthogonal projection from  L2[0, 1] to Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Then the Galerkin solution ϕG  n satisfies the following integral equation  ϕG  n − πnK(ϕG  n ) = πnf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Galerkin method for Urysohn integral equation has been studied extensively in research  literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' See [5], [12], [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' The iterated Galerkin solution is defined by  ϕS  n = K(ϕG  n ) + f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  In [5], the following orders of convergence are also obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  ∥ϕG  n − ϕ∥∞ = O (h) ,  ∥ϕS  n − ϕ∥∞ = O  �  h2�  ,  if r = 1,  and  ∥ϕG  n − ϕ∥∞ = O (hr) ,  ∥ϕS  n − ϕ∥∞ = O  �  hr+2�  ,  if r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  It is also shown that the order of convergence of ϕS  n at the points of partition ∆(n), is h2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  If an asymptotic expansion for the error exists, one can apply a well-known techniques  to obtain more accurate approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Richardson extrapolation one such method for  application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' In [24], an asymptotic expansion for the iterated Galerkin solution of Urysohn  integral equation with Green’s function type of kernel, is obtained at the above mentioned  partition points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Then, by [11] and using Richardson extrapolation, an approximate solution  with order of convergence h2r+2 can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  In the computation of of above approximations, various integrals are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' There is  an integral in the definition of the Urysohn integral operator K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' In the definition of the  orthogonal projection πn, the standard inner product on L2[0, 1] comes into picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' In  practice, it is necessary to replace all these integrals by a numerical quadrature formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  This gives rise to the discrete versions of the projection methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' The discrete versions of  the Galerkin methods for Urysohn integral with Green’s kernel, are investigated in [6], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Whereas, in [17], a different version of discrete projection method is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  In this article, we consider the Urysohn integral equation with Green’s kernel, and discrete  Galerkin method is applied for approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Then, an asymptotic expansion for the  discrete iterated Galerkin solution is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  We choose a fine partition of [0, 1] with m subintervals each of length ˜h = 1  m and define a  composite numerical quadrature formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Replacing the integrals in the definition of K and  πn, we define the Nyström operator Km and the discrete orthogonal projection Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Then the  discrete Galerkin and the discrete iterated Galerkin equations are given by  zG  n − PnKm(zG  n ) = Pnf and zS  n − Km(PnzS  n) = f  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' If ϕ ∈ Cr+2[0, 1], then from [6] and [17], we have  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='4)  ��zG  n − ϕ  ��  ∞ = O  �  max  �  hr, ˜h2��  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='5)  ��zS  n − ϕ  ��  ∞ =  \uf8f1  \uf8f2  \uf8f3  O  �  max  �  h2, ˜h2��  ,  r = 1,  O  �  max  �  hr+2, ˜h2��  , r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Preliminaries  3  In this article, first we find an asymptotic error expansion due to the discrete orthogonal  projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Then using this, the following asymptotic expansion is obtained:  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='6)  zS  n(ti) = ϕ(ti) + γ(ti)h2r + O  �  max  �  h2r+2, ˜h2��  ,  where the function γ is independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' If we choose m such that ˜h ≤ h2r+2, then using  the Richardson extrapolation, an approximation of ϕ of the order of h2r+2 could be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  See [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  This article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Definitions, notations and some preliminary results  are given in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' In Section 3, a quadrature rule is defined, and using it the discrete  orthogonal projection and the Nyström approximations of the integral operators are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Section 4 contains the asymptotic error analysis for the approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Numerical example  is given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  2 Preliminaries  For an integer α ≥ 0, let Cα[0, 1] denotes the space of all real valued α-times continuously  differentiable functions on [0, 1] with the norm  ∥x∥α,∞ = max  0≤j≤α  ��x(j)��  ∞ ,  where x(j) is the jth derivative of the function x, and  ���x(j)���  ∞ = sup  0≤t≤1  |x(j)(t)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Define  ∥κ∥α,∞ =  max  0≤i+j+k≤α  ���D(i,j,k)κ(s, t, u)  ���  ∞ ,  where  D(i,j,k)κ(s, t, u) =  ∂i+j+kκ  ∂si∂tj∂uk (s, t, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='0 Green’s function type kernel  Let r ≥ 1 be an integer and assume that the kernel κ has the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  (1) For i = 1, 2, 3, 4, the functions κ, ∂iκ  ∂ui ∈ C(Ω), where C(Ω) denotes the space of all  real valued continuous function on Ω = [0, 1] × [0, 1] × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  (2) Let Ω1 = {(s, t, u) : 0 ≤ t ≤ s ≤ 1, u ∈ R} and Ω2 = {(s, t, u) : 0 ≤ s ≤ t ≤  1, u ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' There are two functions κj ∈ Cr(Ωj), j = 1, 2, such that  κ(s, t, u) =  �  κ1(s, t, u),  (s, t, u) ∈ Ω1,  κ2(s, t, u),  (s, t, u) ∈ Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  (3) Denote ℓ(s, t, u) = ∂κ  ∂u(s, t, u) and λ(s, t, u) = ∂2κ  ∂u2(s, t, u), (s, t, u) ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' The partial  derivatives of ℓ(s, t, u) and λ(s, t, u) with respect to s and t have jump discontinu-  ities on s = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  (4) There are functions ℓj, λj ∈ Cr(Ωj), j = 1, 2, with  ℓ(s, t, u) =  �  ℓ1(s, t, u),  (s, t, u) ∈ Ω1,  ℓ2(s, t, u),  (s, t, u) ∈ Ω2,  λ(s, t, u) =  �  λ1(s, t, u),  (s, t, u) ∈ Ω1,  λ2(s, t, u),  (s, t, u) ∈ Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  4  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' RAKSHIT  Under the above assumptions, the operator K is four times Fréchet differentiable, and its  Fréchet derivatives at x ∈ X are given by  K′(x)v1(s) =  � 1  0  ∂κ  ∂u (s, t, x(t)) v1(t) dt,  K(i)(x)(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , vi)(s) =  � 1  0  ∂iκ  ∂ui (s, t, x(t)) v1(t) · · ·vi(t) dt,  i = 2, 3, 4,  where  ∂iκ  ∂ui (s, t, x(t)) = ∂iκ  ∂ui (s, t, u)|u=x(t),  i = 1, 2, 3, 4  and v1, v2, v3, v4 ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Note that K′(x) : X → X is linear and K(i)(x) : X i → X are multi-  linear operators, where X i is the cartesian product of i copies of X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' See [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' The norms of  these operators are defined by  ��K(i)(x)  �� = sup  ���K(i)(x)(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , vi)  ��  ∞ : ∥vj∥∞ ≤ 1, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , i  �  for i = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' It follows that  ��K(i)(x)  ��  ≤  sup  0≤s,t≤1  ����  ∂iκ  ∂ui (s, t, x(t))  ���� ,  i = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Note that, if f ∈ Cα[0, 1] for any positive integer α, then ϕ ∈ Cα[0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' See [5, Corollary  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2], [6, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  3 Discretization of Integrals by numerical quadrature  Rule  In this section, first we consider a numerical integration formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' We replace the integral  in the standard inner product of L2[0, 1] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' ⟨x , y⟩ =  � 1  0 x(t)y(t)dt) by the quadrature rule  and define a discrete inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Subsequently, the corresponding discrete orthogonal  projection is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' After that, an asymptotic error expansion for the discrete orthogonal  projection is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Next we define the Nyström approximations of the integral operator  K and its Fréchet derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Consider a basic numerical integration formula by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1)  � 1  0  x(t)dt ≈  ρ  �  q=1  wq x(µq),  which is exact at least for polynomials of degree ≤ 3r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' If r = 0, then it is assumed that the  quadrature rule is exact atleast for linear polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' It follows that �ρ  q=1 wq = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Let n ∈ N and consider the following uniform partition of [0, 1] :  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2)  ∆(n) :  0 < 1  n < · · · < n − 1  n  < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Define tj = j  n, ∆j = [tj−1, tj] and h = tj − tj−1 = 1  n, j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Define the subspace  Cα  ∆(n)[0, 1] = {x ∈ X : x ∈ Cα[tj−1, tj], j = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=', n} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' For r ≥ 1, the approximating  space  Xn =  �  x ∈ X : x|∆j is a polynomial of degree ≤ r − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Discretization of Integrals by numerical quadrature Rule  5  Let p be a positive integer and m = pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Consider the following uniform partition of [0, 1] :  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='3)  ∆(m) :  0 < 1  m < · · · < m − 1  m  < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Let ˜h = 1  m  and  si = i  m,  i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Note : As our goal to find the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='6), where the higher order term is max  �  h2r+2, ˜h2�  ,  we choose the partition ∆(m) such that ˜h2 ≤ hr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  A composite integration rule with respect to the partition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='3) is then defined as  � 1  0  x(t) dt  =  m  �  i=1  � si  si−1  x(t) dt ≈ ˜h  m  �  i=1  ρ  �  q=1  wq x(si−1 + µq˜h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Thus,  � tj  tj−1  x(t) dt =  � jh  (j−1)h  x(t) dt =  p  �  ν=1  � (j−1)h+ν˜h  (j−1)h+(ν−1)˜h  x(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Since h = p˜h,  � tj  tj−1  x(t)dt  =  p  �  ν=1  �  (j−1)p+ν  p  h  (j−1)p+ν−1  p  h  x(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Substituting t =  (j−1)p+ν−1  p  h + ˜hσ =  (j−1)p+ν−1+σ  p  h in the above equation, we obtain  � tj  tj−1  x(t) dt  =  h  p  p  �  ν=1  � 1  0  x  �(j − 1)p + ν − 1 + σ  p  h  �  dσ  =  h  p  p  �  ν=1  � 1  0  x  �  tj−1 + ν − 1 + σ  p  h  �  dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Note that ν−1+σ  p  ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Now using the numerical quadrature formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1), we obtain  � tj  tj−1  x(t) dt  ≈  h  p  p  �  ν=1  ρ  �  q=1  wq x  �  tj−1 + ν − 1 + µq  p  h  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Let  µqν = ν − 1 + µq  p  ,  q = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' ν = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Then,  � tj  tj−1  x(t) dt  ≈  h  p  p  �  ν=1  ρ  �  q=1  wq x(tj−1 + µqνh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='4)  We prove the following lemma which will be used to find an asymptotic error expansion for  the discrete orthogonal projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Let Lη be the Legendre polynomial of degree η ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , r − 1} defined on  [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Then for any k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , 2r − 1,  1  p  r−1  �  η=0  p  �  ν=1  ρ  �  q=1  wq Lη(µqν)Lη(τ)(µqν − τ)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  =  � 1  0  Λr(τ, s)(s − τ)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  ds,  6  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' RAKSHIT  where  r−1  �  η=0  Lη(τ)Lη(s) = Λr(τ, s), for τ, s ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Since Lη is a polynomial of degree 0 ≤ η ≤ r − 1,  1  p  p  �  ν=1  ρ  �  q=1  wq Lη  �ν − 1 + µq  p  �  = 1  p  p  �  ν=1  � 1  0  Lη  �ν − 1 + t  p  �  dt  =  p  �  ν=1  �  ν  p  ν−1  p  Lη(s) ds  =  � 1  0  Lη(s) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Since the basic quadrature formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1) is exact for polynomials of degree ≤ 3r,  1  p  p  �  ν=1  ρ  �  q=1  wq Lη  �ν − 1 + µq  p  � �ν − 1 + µq  p  − τ  �k  =  � 1  0  Lη(s) (s − τ)k ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  It follows that  1  p  p  �  ν=1  ρ  �  q=1  wq Lη(µqν) (µqν − τ)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  =  � 1  0  Lη(s) (s − τ)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  ds,  where µqν = ν − 1 + µq  p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' This gives  1  p  r−1  �  η=0  p  �  ν=1  ρ  �  q=1  wq Lη(τ)Lη(µqν) (µqν − τ)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  =  r−1  �  η=0  Lη(τ)  � 1  0  Lη(s) (s − τ)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Let  r−1  �  η=0  Lη(τ)Lη(s) = Λr(τ, s),  τ, s ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Then,  1  p  r−1  �  η=0  p  �  ν=1  ρ  �  q=1  wq Lη(τ)Lη(µqν) (µqν − τ)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  =  � 1  0  Λr(τ, s) (s − τ)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Hence the required result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1 Discrete Orthogonal Projection  Let j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , n} and x, y ∈ C(∆j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Define a discrete inner product on ∆j by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='5)  ⟨x , y⟩∆j,m = ˜h  p  �  ν=1  ρ  �  q=1  wq x(tj−1 + µqνh) y(tj−1 + µqνh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Note that, this is an indefinite inner product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' For more details on indefinite inner product  spaces, see [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' However, the properties which we need to define a discrete orthogonal  projection, hold true for (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' For η = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , r−1, let Lη denote the Legendre polynomial  of degree η on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' For j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , n, and for η = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , r − 1, define  ϕj,η(t)  =  � �  1  hLη  �  t−tj−1  h  �  ,  t ∈ (tj−1, tj],  0,  otherwise  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1  Discrete Orthogonal Projection  7  and, ϕ1,η(t) =  �  1  hLη  � t−t0  h  �  if t ∈ [t0, t1] and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Note that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='6)  ϕj,η (tj−1 + µqνh) = h− 1  2Lη(µqν)  for all j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Note that {ϕj,η : j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , n, η = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , r − 1} be a set of orthonormal basis for Xn,  where ϕj,η is the Legendre polynomial of degree η defined on [tj−1, tj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Since the basic  numerical integration (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1) has degree of precision 3r, the set {ϕj,η} is also orthonormal  with respect to the discrete inner product (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Let Pr,∆j be the space of polynomials of  degree ≤ r − 1 on ∆j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Define the discrete orthogonal projection Pn,j : C[tj−1, tj] → Pr,∆j  as follows:  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='7)  Pn,jx =  r−1  �  η=0  ⟨x , ϕj,η⟩∆j ϕj,η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  See [4], [6] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' A discrete orthogonal projection Pn : C[0, 1] → Xn is defined  by  Pnx =  n  �  j=1  Pn,jx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='8)  It follows that Pnx(t) = Pn,jx(t), for all t ∈ [tj−1, tj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' We also have the following error  bound:  ∥Pn∥ < ∞ and also, if x ∈ Cr[tj−1, tj], then  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='9)  ∥x − Pn,jx∥∆j,∞ ≤ C1  ��x(r)��  ∆j,∞ hr,  if x ∈ Cr[0, 1], then  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='10)  ∥x − Pnx∥∆j,∞ ≤ C1  ��x(r)��  ∞ hr,  where ∥x∥∆j,∞ =  sup  t∈[tj−1,tj]  |x(t)| and, C1 is a constant independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' For details see  [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  In (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='10) we have a error bound for the discrete orthogonal projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' But, by the follow-  ing lemma we obtain an asymptotic error expansion for the discrete orthogonal projection,  which is more stronger result than (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Let Pn be the discrete orthogonal projection defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='7) - (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Let  x ∈ C2r+2  ∆(n) [0, 1] and t = tj−1 + τh with τ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Then  Pnx(t) − x(t) =  2r+1  �  k=1  Jk(τ) x(k)(tj−1 + τh) hk + O  �  h2r+2�  ,  where Jk(τ) =  � 1  0  Λr(τ, s)(s − τ)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  ds,  k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , 2r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Define a function vj : [tj−1, tj] → R by  vj(t) = 1,  t ∈ [tj−1, tj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  For τ ∈ [0, 1], let t = tj−1 + hτ ∈ [tj−1, tj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='7) it is easy to see that  Pn,jvj = vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  8  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' RAKSHIT  It follows that  r−1  �  η=0  ⟨vj , ϕj,η⟩∆j ϕj,η(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Since ϕj,η is a polynomial of degree 0 ≤ η ≤ r − 1 on [tj−1, tj],  ⟨vj , ϕj,η⟩∆j =  � tj  tj−1  ϕj,η(s) ds  =  h  p  p  �  ν=1  ρ  �  q=1  wq ϕj,η(tj−1 + µqνh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Thus for any function x : [tj−1, tj] → R, we have  x(t) = x(t) h  p  r−1  �  η=0  p  �  ν=1  ρ  �  q=1  wq ϕj,η(tj−1 + µqνh) ϕj,η(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  It follows that  Pn,jx(t) − x(t)  = h  p  r−1  �  η=0  p  �  ν=1  ρ  �  q=1  wq ϕj,η(tj−1 + µqνh) [x(tj−1 + µqνh) − x(t)] ϕj,η(t),  where t = tj−1 + hτ ∈ [tj−1, tj] and τ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='6), we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='11)  Pn,jx(t) − x(t) = 1  p  r−1  �  η=0  p  �  ν=1  ρ  �  q=1  wq Lη(µqν) [x(tj−1 + µqνh) − x(t)] Lη(τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Since x ∈ C2r+2[tj−1, tj], using Taylor series expansion we obtain  x(tj−1 + µqνh) − x(tj−1 + hτ) =  2r+1  �  k=1  x(k)(tj−1 + τh) (µqν − τ)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  hk + O  �  h2r+2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Thus  Pnx(t) − x(t) = Pn,jx(t) − x(t)  =  2r+1  �  k=1  x(k)(tj−1 + τh) hk  �  1  p  r−1  �  η=0  p  �  ν=1  ρ  �  q=1  wqLη(µqν)Lη(τ)(µqν − τ)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  �  + O  �  h2r+2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Let  Jk(τ) = 1  p  r−1  �  η=0  p  �  ν=1  ρ  �  q=1  wq Lη(µqν) Lη(τ) (µqν − τ)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  ,  τ ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1, we can write Jk(τ) =  � 1  0 Λr(τ, s) (s−τ)k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Hence  Pnx(t) − x(t) =  2r+1  �  k=1  Jk(τ) x(k)(tj−1 + τh) hk + O  �  h2r+2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2  Approximation of the Integral Operator  9  Let  L = (I − K′(ϕ))−1 K′(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Then L is a compact linear integral operator with kernel ˜ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Note that the smoothness of ˜ℓ is  same as the kernel ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' See [26], [5, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' It follows that  LPnx(s) − Lx(s) =  n  �  j=1  � tj  tj−1  ˜ℓ(s, t) (Pnx(t) − x(t)) dt  ∀x ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Then using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2, and following the proofs of [22, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1] and [23, Theorem  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2], it can be shown that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='12)  L(I − Pn)ϕ(ti) = E2r(ϕ)(ti)h2r + O  �  h2r+2�  ,  i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , n,  where  E2r(ϕ)(ti) = ¯b2r,2r  � 1  0  ˜ℓ(ti, t)(t) ϕ(2r)(t) dt  +  2r−1  �  p=1  ¯b2r,p  � �� ∂  ∂t  �2r−p−1 �  ˜ℓ(ti, t)ϕ(p)(t)  ��t=1  t=0  −  �� ∂  ∂t  �2r−p−1 �  ˜ℓ(ti, t)ϕ(p)(t)  ��t=ti+  t=ti−  �  with  ¯b2r,p =  � 1  0  � 1  0  Λr(τ, s)(τ − s)p  p!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  B2r−p(s)  (2r − p)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' dτ ds  and Bk is the Bernoulli polynomial of degree k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2 Approximation of the Integral Operator  Let x ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Recall that  K(x)(s) =  � 1  0  κ(s, t, x(t)) dt,  s ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Replacing the above integral by the numerical quadrature rule (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='4), we define the Nyström  approximation of K by  Km(x)(s) = h  p  n  �  j=1  ρ  �  q=1  p  �  ν=1  wq κ(s, tj−1 + µqνh, x(tj−1 + µqνh)),  s ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Let {µj  qν = tj−1 + µqνh : j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' q = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , ρ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' ν = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , p} denotes the set  of all quadrature nodes in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Then  Km(x)(s) = h  p  n  �  j=1  ρ  �  q=1  p  �  ν=1  wq κ  �  s, µj  qν, x  �  µj  qν  ��  ,  s ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  The Nyström method for solving (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1) is to find the element xm for which  xm − Km(xm) = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  For sufficiently large m, the above equation has a unique solution ϕm in a neighborhood  B(ϕ, ǫ) of ϕ, and  ∥ϕ − ϕm∥∞  ≤  C2 ∥K(ϕ) − Km(ϕ)∥∞ = O  �  ˜h2�  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='13)  10  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' RAKSHIT  where C2 is a constant independent of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' See [2, Theorem 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' We write  (I − Pn) ϕm = (I − Pn) (ϕm − ϕ) + (I − Pn) ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Then from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='10), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='13), we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='14)  (I − Pn) ϕm = O  �  max  �  hr, ˜h2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Let v1, v2 ∈ X and x ∈ B(ϕ, ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Then the Fréchet derivatives of Km at x are given by  K′  m(x)v1(s) = h  p  n  �  j=1  ρ  �  q=1  p  �  ν=1  wq D(0,0,1)κ  �  s, µj  qν, x  �  µj  qν  ��  v1  �  µj  qν  �  ,  s ∈ [0, 1],  K′′  m(x) (v1, v2) (s) = h  p  n  �  j=1  ρ  �  q=1  p  �  ν=1  wq  ∂2κ  ∂u2  �  s, µj  qν, x  �  µj  qν  ��  v1  �  µj  qν  �  v2  �  µj  qν  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  It follows that  ∥K′′  m(x) (v1, v2)∥∞ ≤  \uf8eb  \uf8ec  \uf8ed  sup  s,t∈[0,1]  |u|≤∥ϕ∥∞+ǫ  ����  ∂2κ  ∂u2(s, t, u)  ����  \uf8f6  \uf8f7  \uf8f8 ∥v1∥∞ ∥v2∥∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  This implies  ∥K′′  m(x)∥ < ∞  Similarly, it can be shown that  ���K(3)  m (x)  ��� ≤  \uf8eb  \uf8ec  \uf8ed  sup  s,t∈[0,1]  |u|≤∥ϕ∥∞+ǫ  ����  ∂3κ  ∂u3 (s, t, u)  ����  \uf8f6  \uf8f7  \uf8f8 = C3 < ∞  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Let x1, x2 ∈ B(ϕ, ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' If D(0,0,3)κ ∈ C(Ω) then  ∥K′′  m(x1) − K′′  m(x2)∥ ≤ C3 ∥x1 − x2∥∞ ,  where C3 is constant independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' For v1, v2 ∈ X , we have  [K′′  m(x1) − K′′  m(x2)] (v1, v2)(s)  = h  p  n  �  j=1  ρ  �  q=1  p  �  ν=1  wq  �∂2κ  ∂u2  �  s, µj  qν, x1  �  µj  qν  ��  − ∂2κ  ∂u2  �  s, µj  qν, x2  �  µj  qν  ���  v1  �  µj  qν  �  v2  �  µj  qν  �  for all s ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Since D(0,0,3)κ ∈ C(Ω), applying mean value theorem on ∂2κ  ∂u2 with respect  to its third variable u, we obtain  ∂2κ  ∂u2  �  s, µj  qν, x1  �  µj  qν  ��  −∂2κ  ∂u2  �  s, µj  qν, x2  �  µj  qν  ��  =  �  x1  �  µj  qν  �  − x2  �  µj  qν  �� ∂3κ  ∂u3  �  s, µj  qν, ζj  qν  �  ,  where ζj  qν lies in the line segment joining the points x1  �  µj  qν  �  and x2  �  µj  qν  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Then  ����  ∂2κ  ∂u2  �  s, µj  qν, x1  �  µj  qν  ��  − ∂2κ  ∂u2  �  s, µj  qν, x2  �  µj  qν  ������ ≤ C3 ∥x1 − x2∥∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Hence  ∥[K′′  m(x1) − K′′  m(x2)] (v1, v2)∥∞ ≤ C3 ∥x1 − x2∥∞ ∥v1∥∞ ∥v2∥∞ ,  which follows the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  □  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Asymptotic Error Analysis  11  We will now quote some error estimates for the Nyström approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  For α ≥ 0, if v1, v2 ∈ Cα  ∆(m)[0, 1], then from [27] or [4, Corollary 1], we obtain the  following errors for numerical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  ∥[K′  m(ϕ) − K′(ϕ)] v1∥∞ = O  �  ˜h2�  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='15)  ∥[K′′  m(ϕ) − K′′(ϕ)] (v1, v2)∥∞ = O  �  ˜h2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Also from [18, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='3], we have  ∥K′  m(ϕm) − K′  m(ϕ)∥ ≤ C4 ∥ϕm − ϕ∥∞ = O  �  ˜h2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='16)  Therefore combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='15) and the above equation, we obtain  ∥[K′  m(ϕm) − K′(ϕ)] v∥∞ = O  �  ˜h2�  ,  for all v ∈ Cν  ∆m[0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='17)  Similarly,  ∥[K′′  m(ϕm) − K′′(ϕ)] (v1, v2)∥∞ = O  �  ˜h2�  ,  ∀v1, v2 ∈ Cν  ∆m[0, 1],  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='18)  for all v1, v2, v3 ∈ Cν  ∆m[0, 1] implies  ���  K(3)  m (ϕm) − K(3)(ϕ)  �  (v1, v2, v3)  ��  ∞ = O  �  ˜h2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='19)  4 Asymptotic Error Analysis  Replacing K by Km and πn by Pn in the Galerkin equation x − πnK(x) = πnf, the  discrete Galerkin equation is defined by zG  n − PnKm(zG  n ) = Pnf, where zG  n is the discrete  Galerkin solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Then the discrete iterated Galerkin solution is defined by  zS  n = Km(zG  n ) + f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Note that PnzS  n = zG  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' From the equations ϕm − Km(ϕm) = f and zS  n − Km(PnzS  n) = f,  we obtain the following error term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  zS  n − ϕm = [I − K′  m(ϕm)]−1 �  Km(zG  n ) − Km(ϕm) − K′  m(ϕm)(zG  n − ϕm)  �  − Lm(I − Pn)  �  Km(zG  n ) − Km(ϕm) − K′  m(ϕm)(zG  n − ϕm)  �  − Lm(I − Pn)K′  m(ϕm)(zG  n − ϕm)  − Lm(I − Pn)ϕm,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1)  where  Lm = [I − K′  m(ϕm)]−1 K′  m(ϕm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Using the Resolvent Identity, we get  (I − K′  m(ϕm))−1 − (I − K′(ϕ))−1  = (I − K′(ϕ))−1 [K′  m(ϕm) − K′(ϕ)] (I − K′  m(ϕm))−1  12  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' RAKSHIT  Therefore  (I − K′  m(ϕm))−1 = (I − K′(ϕ))−1  + (I − K′(ϕ))−1 [K′  m(ϕm) − K′  m(ϕ)] (I − K′  m(ϕm))−1  + (I − K′(ϕ))−1 [K′  m(ϕ) − K′(ϕ)] (I − K′  m(ϕm))−1  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2)  Now, we will analyze each of the terms appearing in the RHS of the equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Error  estimates for each of the said terms will be obtained by the following propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Let {ti : i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , n} be the set of partition points of [0, 1] defined by  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2), then  Lm(I − Pn)ϕm(ti) = E2r(ϕ)(ti)h2r + O  �  max  �  h2r+2, ˜h2��  ,  where E2r is defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' It can be easily verified that (using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2))  Lm(I − Pn)ϕm =  �  I − K′  m(ϕm)  �−1 K′  m(ϕm)(I − Pn)ϕm  =  �  I − K′(ϕ)  �−1 K′  m(ϕm)(I − Pn)ϕm  +  �  I − K′(ϕ)  �−1 �  K′  m(ϕm) − K′  m(ϕ)  � �  I − K′  m(ϕm)  �−1 K′  m(ϕm)(I − Pn)ϕm  +  �  I − K′(ϕ)  �−1 �  K′  m(ϕ) − K′(ϕ)  � �  I − K′  m(ϕm)  �−1 K′  m(ϕm)(I − Pn)ϕm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='3)  Consider the first term of the above equation, we have  (I − K′(ϕ))−1K′  m(ϕm)(I − Pn)ϕm  = (I − K′(ϕ))−1 K′(ϕ)(I − Pn)ϕ  + (I − K′(ϕ))−1 K′(ϕ)(I − Pn)(ϕm − ϕ)  + (I − K′(ϕ))−1 [K′  m(ϕm) − K′(ϕ)] (I − Pn)ϕm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='12), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='17), we obtain  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='4)  (I − K′(ϕ))−1 K′  m(ϕm)(I −Pn)ϕm(ti) = E2r(ϕ)(ti)h2r +O  �  max  �  h2r+2, ˜h2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Note that  ∥K′  m(ϕm)∥ ≤  sup  s,t∈[0,1]  |u|≤∥ϕ∥∞+ǫ  |κu(s, t, u)|  and from [18, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2], we have  ��(I − K′  m(ϕm))−1�� < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Thus, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='15) and  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='16), we have the followings  ���  �  I − K′(ϕ)  �−1 �  K′  m(ϕm) − K′  m(ϕ)  � �  I − K′  m(ϕm)  �−1 K′  m(ϕm)(I − Pn)ϕm  ���  ∞ = O  �  ˜h2�  ,  ���  �  I − K′(ϕ)  �−1 �  K′  m(ϕm) − K′  m(ϕ)  � �  I − K′  m(ϕm)  �−1 K′  m(ϕm)(I − Pn)ϕm  ���  ∞ = O  �  ˜h2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Hence the required result follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='3), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='4) and the above two estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  □  Before each of the following propositions, we prove lemmas and its corollaries which are  used to prove next propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Asymptotic Error Analysis  13  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Let Pn be the discrete orthogonal projection defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' If D(0,0,3)κ ∈  C(Ω) and v ∈ Cr+2([0, 1]), then for r ≥ 1  K′′(ϕ)(Pnv − v)2 = T(v)h2r + O(h2r+2),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='5)  where  T(v) =  �� 1  0  Jr(τ)2 dτ  �  K′′(ϕ)  �  v(r)�2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Furthermore, when r = 1, then  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='6)  K(3)(ϕ)(Pnv − v)3 = O  �  h4�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' The proofs of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='6) follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2, [16, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='4] and [16,  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='4] respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  □  Given that (I − K′(ϕ))−1 is a bounded linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Let  M = (I − K′(ϕ))−1 K′′(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Note that M is a compact bi-linear integral operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Also the smoothness of the kernel of  M, is same as the kernels of K′′(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' See [4], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  As a consequence of the above lemma, we get the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' For r ≥ 1,  M(Pnv − v)2 = T (v)h2r + O(h2r+2),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='7)  where  T (v) =  �� 1  0  Jr(τ)2 dτ  �  M  �  v(r)�2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' If ϕ ∈ Cr+2([0, 1]), then for r ≥ 1,  K′′  m(ϕm)(zG  n − ϕm)2 = T(ϕ)h2r + O  �  max  �  h2r+2, ˜h2��  ,  where T is defined in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Note that  zG  n − ϕm = PnzS  n − Pnϕm − ϕm + Pnϕm + ϕ  = Pn  �  zS  n − ϕm  �  − (I − Pn) ϕm  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='8)  Thus,  K′′  m(ϕm)(zG  n − ϕm)2  = K′′  m(ϕm)  �  Pn  �  zS  n − ϕm  ��2 − 2K′′  m(ϕm)  �  Pn  �  zS  n − ϕm  �  , (I − Pn) ϕm  �  + K′′  m(ϕm) ((I − Pn) ϕm)2  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='9)  Since ∥K′′  m(ϕm)∥ < ∞ and ∥Pn∥ < ∞, from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='4) it is easy to see that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='10)  ���K′′  m(ϕm)  �  Pn  �  zS  n − ϕm  ��2���  ∞ = O  �  max  �  h2r+4, ˜h4��  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='11)  ��K′′  m(ϕm)  �  Pn  �  zS  n − ϕm  �  , (I − Pn) ϕm  ���  ∞ = O  �  ˜h2 max  �  hr+2, ˜h2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  14  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' RAKSHIT  Now we write  K′′  m(ϕm) ((I − Pn) ϕm)2 =  �  K′′  m(ϕm) − K′′(ϕ)  �  ((I − Pn) ϕm)2  + K′′(ϕ) ((I − Pn) ϕm)2  = K′′(ϕ) ((I − Pn) ϕ)2 + K′′(ϕ) ((I − Pn) (ϕ − ϕm))2  +  �  K′′  m(ϕm) − K′′(ϕ)  �  ((I − Pn) ϕm)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Since ∥K′′  m(ϕm)∥ < ∞ and ∥Pn∥ < ∞, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='13), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='18) and the above estimate, we  obtain  K′′  m(ϕm) ((I − Pn) ϕm)2 = K′′(ϕ) ((I − Pn) ϕ)2 + O  �  ˜h2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Therefore, from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='5) we obtain  K′′  m(ϕm) ((I − Pn) ϕm)2 = T(ϕ)h2r + O  �  max  �  h2r+2, ˜h2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Hence, the required result follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='9), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='10), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='11) and the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  □  From the above lemma, we obtain  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='12)  (I − K′(ϕ))−1 K′′  m(ϕm)(zG  n − ϕm)2 = T (ϕ)h2r + O  �  max  �  h2r+2, ˜h2��  ,  where T is defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' If ϕ ∈ Cr+2([0, 1]), then  [I − K′  m(ϕm)]−1 K(3)  m (ϕm)(zG  n − ϕm)3 =  \uf8f1  \uf8f2  \uf8f3  O  �  max  �  h4, ˜h2��  ,  r = 1,  O  �  max  �  h3r, ˜h6��  , r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' First, we consider the case when r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Since  ��[I − K′  m(ϕm)]−1�� < ∞ and  ���K(3)  m (ϕm)  ��� < ∞, from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='4) we obtain  ���[I − K′  m(ϕm)]−1 K(3)  m (ϕm)(zG  n − ϕm)3���  ∞ = O  �  max  �  h3r, ˜h6��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Now consider the case, when r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' We rewrite (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='8) as  (zG  n − ϕm)3 =  �  Pn  �  zS  n − ϕm  �  − (I − Pn) ϕm  �3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Thus  K(3)  m (ϕm)(zG  n − ϕm)3 = K(3)  m (ϕm)  �  Pn  �  zS  n − ϕm  ��3 − K(3)  m (ϕm) ((I − Pn) ϕm)3  − K(3)  m (ϕm)  ��  Pn  �  zS  n − ϕm  ��2 , (I − Pn) ϕm  �  + K(3)  m (ϕm)  �  Pn  �  zS  n − ϕm  �  , ((I − Pn) ϕm)2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='13)  Since  ���K(3)  m (ϕm)  ��� < ∞ and ∥Pn∥ < ∞, from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='5) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='14) we obtain  K(3)  m (ϕm)  �  Pn  �  zS  n − ϕm  ��3 = O  �  h6�  ,  K(3)  m (ϕm)  ��  Pn  �  zS  n − ϕm  ��2 , (I − Pn) ϕm  �  = O  �  h5�  ,  K(3)  m (ϕm)  �  Pn  �  zS  n − ϕm  �  , ((I − Pn) ϕm)2�  = O  �  h4�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Asymptotic Error Analysis  15  Note that, we have used the fact ˜h ≤ h in the above three expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' On the other hand,  from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='13), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='19) we have  K(3)  m (ϕm) ((I − Pn) ϕm)3 = K(3)(ϕ) ((I − Pn) ϕ)3 + O  �  ˜h2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  From (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='6), it follows that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='14)  K(3)  m (ϕm) ((I − Pn) ϕm)3 = O  �  max  �  h4, ˜h2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Now, combining the results (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='13) - (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='14), we obtain  K(3)  m (ϕm)(zG  n − ϕm)3 = O  �  max  �  h4, ˜h2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Therefore  [I − K′  m(ϕm)]−1 K(3)  m (ϕm)(zG  n − ϕm)3 = O  �  max  �  h4, ˜h2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Hence follows the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  □  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Let ϕ ∈ Cr+2([0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Then for r ≥ 1,  [I − K′  m(ϕm)]−1 �  Km(zG  n ) − Km(ϕm) − K′  m(ϕm)(zG  n − ϕm)  �  (s)  = 1  2T (ϕ)(s)h2r + O  �  max  �  h2r+2, ˜h2��  ,  for all s ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Applying the generalized Taylor’s series expansion of Km about ϕm in the neigh-  bourhood B(ϕ, ǫ), we obtain  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='15)  Km(zG  n ) − Km(ϕm) − K′  m(ϕm)(zG  n − ϕm)  = 1  2K′′  m(ϕm)(zG  n − ϕm)2 + 1  6K(3)  m (ϕm)(zG  n − ϕm)3 + R4,m  �  zG  n − ϕm  �  ,  where  R4,m  �  zG  n − ϕm  �  =  � 1  0  (1 − θ)3  3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  K(4)  m  �  ϕm + θ(zG  n − ϕm)  �  (zG  n − ϕm)4 dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Note that for any x ∈ B(ϕ, ǫ), v ∈ X ,  K(4)  m (x)v4(s) = h  p  n  �  j=1  ρ  �  q=1  p  �  ν=1  wq  ∂4κ  ∂u4  �  s, µj  qν, x  �  µj  qν  ��  v4�  µj  qν  �  ,  s ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  It follows that  ��K(4)  m (x)v4��  ∞ ≤  \uf8eb  \uf8ec  \uf8ed  sup  s,t∈[0,1]  |u|≤∥ϕ∥∞+ǫ  ����  ∂4κ  ∂u4 (s, t, u)  ����  \uf8f6  \uf8f7  \uf8f8 ∥v∥4  ∞ = C5 ∥v∥4  ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Since ϕm and zG  n ∈ B(ϕ, ǫ), ϕm + θ(zG  n − ϕm) ∈ B(ϕ, ǫ) and therefore  ��K(4)  m  �  ϕm + θ(zG  n − ϕm)  �  (zG  n − ϕm)4��  ∞ ≤ C5  ��zG  n − ϕm  ��4  ∞ = O  �  h4r�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  It follows that  R4,m  �  zG  n − ϕm  �  = O  �  h4r�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  16  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' RAKSHIT  Using the resolvent identity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='17) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2), we obtain  [I − K′  m(ϕm)]−1 K′′  m(ϕm)(zG  n −ϕm)2 = [I − K′(ϕ)]−1 K′′  m(ϕm)(zG  n −ϕm)2 +O  �  ˜h2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='12), it follows that  [I − K′  m(ϕm)]−1 K′′  m(ϕm)(zG  n − ϕm)2 = T (ϕ)h2r + O  �  max  �  h2r+2, ˜h2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  From the Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='3, we have  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='16)  [I − K′  m(ϕm)]−1 K(3)  m (ϕm)(zG  n − ϕm)3 =  \uf8f1  \uf8f2  \uf8f3  O  �  max  �  h4, ˜h2��  ,  r = 1,  O  �  max  �  h3r, ˜h6��  , r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Combining the results from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='15) to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='16), we obtain for r ≥ 1,  [I − K′  m(ϕm)]−1 �  Km(zG  n ) − Km(ϕm) − K′  m(ϕm)(zG  n − ϕm)  �  = 1  2T (ϕ)h2r + O  �  max  �  h2r+2, ˜h2��  ,  which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  □  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' If v ∈ X , then for r ≥ 1,  max  0≤i≤n |K′  m(ϕm)(I − Pn)v(ti)|  ≤  C7 ∥(I − Pn)v∥∞ hr,  where C7 is a constant independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' We write  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='17)  K′  m(ϕm)(I − Pn)v = K′  m(ϕ)(I − Pn)v + [K′  m(ϕm) − K′  m(ϕ)](I − Pn)v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  For fixed s ∈ [0, 1], let  ℓ∗,s(t) = ℓ∗(s, t) = ℓ(s, t, ϕ(t)) = ∂κ  ∂u(s, t, ϕ(t)),  t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  From the definition of K′  m(ϕ) and the discrete inner product, we have  K′  m(ϕ)(I − Pn)v(s)  =  n  �  j=1  ⟨ℓ∗,s , (I − Pn,j)v⟩∆j,m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Since Pn,j is self-adjoint on C(∆j), so as I − Pn,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Therefore  K′  m(ϕ)(I − Pn)v(s)  =  n  �  j=1  ⟨(I − Pn,j)ℓ∗,s , (I − Pn,j)v⟩∆j,m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Note that, if s = ti for some i ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , n}, then ℓ∗,s ∈ Cr[tj−1, tj] for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Hence from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='9),  ∥(I − Pn,j)ℓ∗,ti∥∆j,∞  ≤  C1  �  sup  t∈[tj−1,tj]  |D(0,r)ℓ∗(ti, t)|  �  hr  Thus,  max  0≤i≤n |K′  m(ϕ)(I − Pn)v(ti)| ≤  n  �  j=1  ∥(I − Pn,j)ℓm,s∥∆j,∞ ∥(I − Pn,j)v∥∆j,∞h  ≤ C6 ∥(I − Pn)v∥∞ hr,  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Asymptotic Error Analysis  17  where C6 is a constant independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Now, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='16), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='17) and the above estimate,  we obtain  max  0≤i≤n |K′  m(ϕm)(I − Pn)v(ti)| ≤ C7 ∥(I − Pn)v∥∞ max  �  hr, ˜h2�  ,  where C7 = C4 + C6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Since hr ≥ ˜h2, the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  □  Recall that Lm = [I − K′  m(ϕm)]−1 K′  m(ϕm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Therefore, the proof of the following result  is similar to that of the above lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' If v ∈ X , then for r ≥ 1,  max  0≤i≤n|Lm(I − Pn)v(ti)|  ≤  C8 ∥(I − Pn)v∥∞ hr,  where C8 is a constant independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Let v ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' If r = 1, that is, when the range of Pn is the space of piecewise  polynomials of degree zero, then  ∥(I − Pn)K′′  m(ϕ)(v, v)∥∞ ≤ C9h ∥v∥2  ∞ ,  where C9 is a constant independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Given that Xn is the space of piecewise constant functions with respect to the parti-  tion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Note that  ∥(I − Pn)K′′  m(ϕ)(v, v)∥∞  =  max  1≤j≤n  sup  s∈[tj−1,tj]  |(I − Pn,j)K′′  m(ϕ)(v, v)(s)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Let s ∈ [tj−1, tj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Since the Legendre polynomial of zero L0(t) = 1 for all t ∈ [0, 1], we  have from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='11),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='18)  (I − Pn)K′′  m(ϕ)(v, v)(s)  = 1  p  p  �  ν=1  ρ  �  q=1  wq [K′′  m(ϕ)(v, v)(s) − K′′  m(ϕ)(v, v)(tj−1 + µqνh)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  We also have  K′′  m(ϕ)(v, v)(s) − K′′  m(ϕ)(v, v)(tj−1 + µqνh)  = h  p  n  �  k=1  ρ  �  q=1  p  �  ν=1  wq  �∂2κ  ∂u2  �  s, µk  qν, ϕ  �  µk  qν  ��  − ∂2κ  ∂u2  �  µj  qν, µk  qν, ϕ  �  µk  qν  ���  v2�  µk  qν  �  ,  where µj  qν = tj−1 + µqνh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' For fixed s ∈ [0, 1], let  λ∗,s(t) = λ∗(s, t) = λ(s, t, ϕ(t)) = ∂2κ  ∂u2 (s, t, ϕ(t)),  t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Then, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='5)  K′′  m(ϕ)(v, v)(s) − K′′  m(ϕ)(v, v)(µj  qν) =  n  �  k=1  �  λ∗,s − λ∗,µiqν, v2�  ∆k,m  =  n  �  k=1  k̸=j  �  λ∗,s − λ∗,µj  qν , v2�  ∆k,m +  �  λ∗,s − λ∗,µj  qν , v2�  ∆j,m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='19)  18  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' RAKSHIT  First consider the case when k ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Applying Mean Value Theorem on the first component  of λ∗(·, ·) in the interval [s, µj  qν], we obtain  �  λ∗,s − λ∗,µj  qν , v2�  ∆k,m = (s − µj  qν)  �  D(1,0)λ∗(θj  qν, ·) , v2�  ∆k,m ,  for some θj  qν ∈ (tj−1, tj), and the function D(1,0)λ∗(s, t), t ∈ [tk−1, tk] is given by  D(1,0)λ∗(s, t) =  �  D(1,0)λ1,∗(s, t) =  ∂  ∂sλ1(s, t, ϕ(t)),  0 ≤ t ≤ s ≤ 1,  D(1,0)λ2,∗(s, t) =  ∂  ∂sλ2(s, t, ϕ(t)),  0 ≤ s ≤ t ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Therefore, for k ̸= j,  ����  �  λ∗,s − λ∗,µj  qν , v2�  ∆k,m  ����  ≤  ��s − µj  qν  ��  �  sup  s̸=t  ��D(1,0)λ∗(s, t)  ��  �  ∥v∥2  ∞ h  ≤  �  sup  s̸=t  ��D(1,0)λ∗(s, t)  ��  �  ∥v∥2  ∞ h2,  where  sup  s̸=t  |D(1,0)λ∗(s, t)|  = max  �  sup  0≤t<s≤1  |D(1,0)λ1,∗(s, t)| ,  sup  0≤s<t≤1  |D(1,0)λ2,∗(s, t)|  �  = max  \uf8f1  \uf8f4  \uf8f2  \uf8f4  \uf8f3  sup  0≤t<s≤1  |u|≤∥ϕ∥∞  |D(1,0,2)κ1(s, t, u)| ,  sup  0≤s<t≤1  |u|≤∥ϕ∥∞  |D(1,0,2)κ2(s, t, u)|  \uf8fc  \uf8f4  \uf8fd  \uf8f4  \uf8fe  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  On the other hand,  |  �  λ∗,s − λ∗,µj  qν , v2�  ∆j,m| ≤ 2  �  sup  0≤s,t≤1  |λ∗(s, t)|  �  ∥v∥2  ∞ h,  where  sup  0≤s,t≤1  |λ∗(s, t)| =  sup  0≤s,t≤1  |u|≤∥ϕ∥∞  |D(0,0,2)κ(s, t, u)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Then, from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='19) we obtain  |K′′  m(ϕ)(v, v)(s) − K′′  m(ϕ)(v, v)(µj  qν)|  ≤  \uf8eb  \uf8ec  \uf8ed  n  �  k=1  k̸=j  �  sup  s̸=t  |D(1,0)λ∗(s, t)|  �  ∥v∥2  ∞ h2  \uf8f6  \uf8f7  \uf8f8 + 2  �  sup  0≤s,t≤1  |λ∗(s, t)|  �  ∥v∥2  ∞ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  It follows that  |K′′  m(ϕ)(v, v)(s) − K′′  m(ϕ)(v, v)(µj  qν)| ≤ C9 ∥v∥2 h,  where C9 =  �  sup  s̸=t  |D(1,0)λ∗(s, t)|  �  + 2  �  sup  0≤s,t≤1  |λ∗(s, t)|  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  The result now follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='18) and the above estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  □  Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Let v ∈ X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' If r = 1, that is, when the range of Pn is the space of piecewise  polynomials of degree zero, then  ∥(I − Pn)K′  m(ϕ)v∥∞ ≤ C10h ∥v∥∞ ,  where C10 is a constant independent of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Asymptotic Error Analysis  19  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' The proof is similar to that of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  □  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Let ti be any point of the partition ∆(n) defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Then  Lm(I − Pn)  �  Km(zG  n ) − Km(ϕm) − K′  m(ϕm)(zG  n − ϕm)  �  (ti)  =  \uf8f1  \uf8f2  \uf8f3  O (h4) ,  r = 1,  O  �  max  �  h3r, hr˜h4��  ,  r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Generalized Taylor’s series expansion gives  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='20)  Lm(I − Pn)  �  Km(zG  n ) − Km(ϕm) − K′  m(ϕm)(zG  n − ϕm)  �  = 1  2Lm(I − Pn)K′′  m(ϕm)(zG  n − ϕm)2 + Lm(I − Pn)R3,m  �  zG  n − ϕm  �  ,  where  R3,m  �  zG  n − ϕm  �  =  � 1  0  (1 − θ)2  2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  K(3)  m  �  ϕm + θ(zG  n − ϕm)  �  (zG  n − ϕm)3 dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  It follows that  ��R3,m  �  zG  n − ϕm  ���  ∞ ≤ 1  6  \uf8eb  \uf8ec  \uf8ed  sup  s,t∈[0,1]  |u|≤∥ϕ∥∞+ǫ  ����  ∂3κ  ∂u3 (s, t, u)  ����  \uf8f6  \uf8f7  \uf8f8  ��zG  n − ϕm  ��3  ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Therefore, by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='4)  ��R3,m  �  zG  n − ϕm  ���  ∞ = O  �  max  �  h3r, ˜h6��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Since ∥I − Pn∥ ≤ 1 + ∥Pn∥ < ∞, from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2, it is easy to see that  Lm(I − Pn)R3,m  �  zG  n − ϕm  �  (ti) = O  �  max  �  h4r, hr˜h6��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='21)  First consider the case r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Since ∥K′′  m(ϕm)∥ < ∞ and ∥I − Pn∥∞ < ∞, by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='4) and  the Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2, we have  1  2Lm(I − Pn)K′′  m(ϕm)(zG  n − ϕm)2(ti) = O  �  max  �  h3r, hr˜h4��  ,  r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  When r = 1, we write  Lm(I − Pn)K′′  m(ϕm)(zG  n − ϕm)2  = Lm(I − Pn)  �  K′′  m(ϕm) − K′′  m(ϕ)  �  (zG  n − ϕm)2 + Lm(I − Pn)K′′  m(ϕ)(zG  n − ϕm)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  By (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='4), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='13) and the Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='3, we have  Lm(I − Pn) [K′′  m(ϕm) − K′′  m(ϕ)] (zG  n − ϕm)2 = O  �  h4�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  On the other hand  Lm(I − Pn)K′′  m(ϕ)(zG  n − ϕm)2(ti) =  n  �  j=1  �  ℓm,ti , (I − Pn,j)K′′  m(ϕ)(zG  n − ϕm)2�  ∆j,m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  20  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' RAKSHIT  Since I − Pn,j is self-adjoint,  Lm(I − Pn)K′′  m(ϕ)(zG  n − ϕm)2(ti)  =  n  �  j=1  �  (I − Pn,j)ℓm,ti , (I − Pn,j)K′′  m(ϕ)(zG  n − ϕm)2�  ∆j,m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  It follows that  max  0≤i≤n  ��Lm(I − Pn)K′′  m(ϕ)(zG  n − ϕm)2(ti)  ��  ≤  n  �  j=1  ∥(I − Pn,j)ℓm,ti∥∆j,∞  ��(I − Pn,j)K′′  m(ϕ)(zG  n − ϕm)2��  ∆j,∞ h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  By Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2 and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='5, we obtain  max  0≤i≤n|Lm(I − Pn)K′′  m(ϕ)(zG  n − ϕm)2(ti)| = O  �  h4�  ,  r = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Therefore  1  2Lm(I − Pn)K′′  m(ϕm)(zG  n − ϕm)2(ti) =  \uf8f1  \uf8f2  \uf8f3  O (h4) ,  r = 1,  O  �  max  �  h3r, hr˜h4��  ,  r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='22)  Then combing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='20), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='22), we obtain  Lm(I − Pn)  �  Km(zG  n ) − Km(ϕm) − K′  m(ϕm)(zG  n − ϕm)  �  (ti)  =  \uf8f1  \uf8f2  \uf8f3  O (h4) ,  r = 1,  O  �  max  �  h3r, hr˜h4��  ,  r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  This follows the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  □  We quote the following result from By [17, Proposition 1, Proposition 6], which will be  used in the next proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='23)  ∥(I − Pn)K′  m(ϕ)(I − Pn)ϕ∥∞ =  �  O (h3) ,  r = 1,  O (hr+2) ,  r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' If ϕm and zG  n are respectively the Nyström and the discrete Galerkin ap-  proximation of ϕ, then  Lm(I − Pn)K′  m(ϕm)(zG  n − ϕm)(ti) = O  �  max  �  h2r+2, ˜h2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Adding and subtracting K′  m(ϕ), we have  Lm(I − Pn)K′  m(ϕm)(zG  n − ϕm)  = Lm(I − Pn) [K′  m(ϕm) − K′  m(ϕ)] (zG  n − ϕm) + Lm(I − Pn)K′  m(ϕ)(zG  n − ϕm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Then, using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='4), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='16) and the Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2, we obtain for r ≥ 1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='24)  ��Lm(I − Pn) [K′  m(ϕm) − K′  m(ϕ)] (zG  n − ϕm)(ti)  ��  ≤ C4C8 (1 + ∥Pn∥) hr˜h2 �  max  �  hr, ˜h2��  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Asymptotic Error Analysis  21  Note that  zG  n − ϕm = PnzS  n − ϕm = PnzS  n − Pnϕ + Pnϕ − ϕ + ϕ − ϕm  = Pn  �  zS  n − ϕ  �  − (I − Pn) ϕ + (ϕ − ϕm) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Then  Lm(I − Pn)K′  m(ϕ)(zG  n − ϕm) = Lm(I − Pn)K′  m(ϕ)Pn  �  zS  n − ϕ  �  − Lm(I − Pn)K′  m(ϕ) (I − Pn) ϕ  + Lm(I − Pn)K′  m(ϕ) (ϕ − ϕm) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  By the Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2  ��Lm(I − Pn)K′  m(ϕ)Pn  �  zS  n − ϕ  �  (ti)  �� ≤ C8  ��(I − Pn) K′  m(ϕ)Pn  �  zS  n − ϕ  ��� hr,  then by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='5) and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='3, we obtain  ��Lm(I − Pn)K′  m(ϕ)Pn  �  zS  n − ϕ  �  (ti)  �� = O  �  max  �  h2r+2, ˜h2��  ,  for r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Also, the Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='23) implies  Lm(I − Pn)K′  m(ϕ) (I − Pn) ϕ(ti) = O  �  h2r+2�  ,  for r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  It is easy to see (from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='13) and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2) that  Lm(I − Pn)K′  m(ϕ) (ϕ − ϕm) = O  �  hr˜h2�  ,  for r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Therefore, for r ≥ 1,  Lm(I − Pn)K′  m(ϕ)(zG  n − ϕm)(ti) = O  �  max  �  h2r+2, ˜h2��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Hence, the result follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='24) and the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  □  We prove the main theorem as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Let K be the Urysohn integral operator with Green’s function type kernel κ,  defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Let ϕ be the unique solution of the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Assume that 1 is not  an eigenvalue of K′(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Let Xn be the space of piecewise polynomials of degree ≤ r − 1  with respect to the partition ∆(n) := 0 = t0 < t1 < · · · < tn = 1 defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Let  Pn : L∞[0, 1] → Xn be the discrete orthogonal projection defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='8) and ˜zS  n be the  discrete iterated Galerkin approximation of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Then  �  zS  n − ϕ  �  (ti) =  �  E2r(ϕ)(ti) + 1  2T (ϕ)(ti)  �  h2r + O  �  max  �  h2r+2, ˜h2��  ,  where the operators E2r and T are respectively defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='12) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' We have from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1)  zS  n − ϕ = [I − K′  m(ϕm)]−1 �  Km(zG  n ) − Km(ϕm) − K′  m(ϕm)(zG  n − ϕm)  �  − Lm(I − Pn)  �  Km(zG  n ) − Km(ϕm) − K′  m(ϕm)(zG  n − ϕm)  �  − Lm(I − Pn)K′  m(ϕm)(zG  n − ϕm)  − Lm(I − Pn)ϕm  + ϕm − ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  22  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' RAKSHIT  The result now follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='13), Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='3 and  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  □  We now apply Richardson extrapolation to obtain an approximation of ϕ with higher  order of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Define  zEX  n  = 24rzS  2n − zS  n  24r − 1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  We choose the partitions ∆(m) and ∆(n) such that m2 ≥ n2r+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Then, it is easy to see from  the Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1, that  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='25)  �  zEX  n  − ϕ  �  (ti) = O  �  h2r+2�  ,  for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  5 Numerical results  For the numerical results, we consider the following example from [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Consider  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1)  ϕ(s) −  � 1  0  κ(s, t) [ψ (t, ϕ(t))] dt = f(s),  0 ≤ s ≤ 1,  where  κ(s, t) =  1  γ sinh γ  �  sinh γs sinh γ(1 − t),  0 ≤ t ≤ s ≤ 1,  γ(1 − s) sinh γt,  0 ≤ s ≤ t ≤ 1,  with γ =  √  12, and  ψ(t, ϕ(t)) = γ2ϕ(t) − 2 (ϕ(t))3 ,  t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  We have f(s) =  1  sinh γ  �  2 sinh γ(1 − s) + 2  3 sinh γs  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' The exact solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1) is given  by  ϕ(s) =  2  2s + 1,  s ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Let Xn be the space of piecewise constant functions with respect to the uniform partition  ∆(n) of the interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Let Pn : L∞[0, 1] → Xn be the discrete orthogonal projection  defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Let ti = i−1  20 , i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' , 21 be the partition points with step size h =  1  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' The numerical  quadrature is chosen to be the composite 2 point Gaussian quadrature rule with respect to  partition ∆(m) with m = n2 subintervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Then ˜h = h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Therefore, it is expected from the  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1 and equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='25), that  ǫS  n(ti) = |ϕ(ti) − zS  n(ti)| = O  �  h2�  and ǫEX  n  (ti) = |ϕ(ti) − zEX  n  (ti)| = O  �  h4�  ,  where  zEX  n  (ti) = 4zS  2n(ti) − zS  n(ti)  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Let δS and δEX be respectively the orders of convergence of zS  n and zEX  n  at the partition  points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' We expect δS = 2 and δEX = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Numerical results  23  Table 1  ti  ǫS  n(ti) : n = 20  δS  ǫEX  n  (ti) : n = 20  δEX  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='05  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='6 × 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='98 × 10−6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='1  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='56 × 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='23 × 10−6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='15  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='79 × 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='59 × 10−6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='2  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='22 × 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='09 × 10−6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='97  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='25  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='78 × 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='00  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='13 × 10−7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='96  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='3  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='45 × 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='00  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='46 × 10−7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='94  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='35  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='19 × 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='7 × 10−7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='91  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='98 × 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='69 × 10−7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='86  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='45  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='82 × 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='3 × 10−7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='83  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='68 × 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='41 × 10−7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='85  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='55  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='55 × 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='91 × 10−7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='89  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='44 × 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='72 × 10−7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='93  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='65  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='33 × 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='00  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='75 × 10−7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='22 × 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='00  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='95 × 10−7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='97  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='75  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='10 × 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='00  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='26 × 10−7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='98  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='98 × 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='00  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='6 × 10−7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='85  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='84 × 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='00  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='94 × 10−7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='9  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='69 × 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='02 × 10−6  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='99  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='95  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='52 × 10−3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='14 × 10−6  4  From the above table, it is clear that the obtained orders of convergence match well with  the theoretical orders of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Also the order of convergence of the extrapolated  solution improves upon the discrete iterated Galerkin solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  5 References  [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Atkinson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' The numerical solutions of integral equations of the second kind, Cambridge University  Press, Cambridge, (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  [2] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Atkinson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' The numerical evaluation of fixed points for completely continuous operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' SIAM Jour-  nal on Numerical Analysis, 10(5) , 799–807 (1973).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  [3] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Atkinson and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Bogomolny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' The discrete Galerkin method for integral equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Mathematics of  Computation, 48(178), 595–616 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Atkinson and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Potra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' On the discrete Galerkin method for Fredholm integral equations of the  second kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' IMA Journal of Numerical Analysis, 9(3), 385-403 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
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+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Atkinson and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Potra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Projection and iterated projection methods for nonlinear integral equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  SIAM Journal on Numerical Analysis, 24(6), 1352-1373 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  [6] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
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+page_content=' Atkinson and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Potra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' The discrete Galerkin method for nonlinear integral equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Journal of  Integral Equations and Applications, 1(1), 17-54 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  [7] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Baker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' The numerical treatment of integral equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Oxford University Press, (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
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+page_content=' Bognár.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Indefinite inner product spaces (Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' 78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Springer Science & Business Media, (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
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+page_content=' Brunner, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Lin and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Higher accuracy methods for second-kind Volterra integral equations  based on asymptotic expansions of iterated Galerkin methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Journal of Integral Equations and Applica-  tions, 10(4), 375-396 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  [10] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
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+page_content=' Lebbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Superconvergence results for the iterated projection method applied to a Fred-  holm integral equation of the second kind and the corresponding eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Journal of Integral  Equations, 6(1), 71-91 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content='  24  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' RAKSHIT  [11] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Ford, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Pennline, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Xu and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Zhao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Asymptotic error analysis of a quadrature method for  integral equations with Green’s function kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
+page_content=' Journal of Integral Equations and Applications, 12(4), 349-  384 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5tE1T4oBgHgl3EQfmgTC/content/2301.03299v1.pdf'}
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diff --git a/69AyT4oBgHgl3EQfpvg-/content/tmp_files/2301.00530v1.pdf.txt b/69AyT4oBgHgl3EQfpvg-/content/tmp_files/2301.00530v1.pdf.txt
new file mode 100644
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@@ -0,0 +1,927 @@
+Test Reuse Based on Adaptive Semantic Matching across Android Mobile Applications
+Shuqi Liu1, Yu Zhou1,∗, Tingting Han2, and Taolue Chen2
+1College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, China
+2Department of Computer Science and Data Science, Birkbeck, University of London, UK
+{liushuqi, zhouyu}@nuaa.edu.cn, {t.han, t.chen}@bbk.ac.uk
+*corresponding author
+Abstract—Automatic test generation can help verify and de-
+velop the behavior of mobile applications. Test reuse based on
+semantic similarities between applications of the same category
+has been utilized to reduce the manual effort of Graphical User
+Interface (GUI) testing. However, most of the existing studies fail
+to solve the semantic problem of event matching, which leads to
+the failure of test reuse. To overcome this challenge, we propose
+TRASM (Test Reuse based on Adaptive Semantic Matching), a
+test reuse approach based on adaptive strategies to find a better
+event matching across android mobile applications. TRASM
+first performs GUI events deduplication on the initial test set
+obtained from test generation, and then employs an adaptive
+strategy to find better event matching, which enables reusing
+the existing test. Preliminary experiments with comparison to
+baseline methods on 15 applications demonstrate that TRASM
+can improve the precision of GUI event matching while reducing
+the failure of test reuse and the running time required for test
+reuse.
+Keywords—adaptive semantic matching; android mobile appli-
+cations; GUI event; test reuse; oracle generation
+I. INTRODUCTION
+Graphical User Interface (GUI) testing is commonly em-
+ployed to verify and develop the behaviors of applications by
+designing and executing test cases of GUI applications [1].
+However, with the ever increasing functionalities in mobile
+applications, it takes more effort for developers to manually
+design GUI test cases (GUI test in short) [2–4], which in turn
+decreases the efficiency of testing processes.
+Considering the necessity of reducing time consumption,
+many researchers have conducted a series of investigation on
+automatic test generation [5–15] in GUI testing. Recently,
+some researchers observed that test reuse [16–24] could
+be achieved by exploiting the semantic similarity of GUIs
+between similar applications to generate tests automatically.
+Figure 1 shows a simple example in which the existing test (a)
+of application To-Do List is successfully reused to application
+Minimal, and the reused test (b) is obtained. As Figure 1
+shows, events em
+1 , em
+2 , em
+3 , and em
+4 in test (b) are similar to
+et
+1, et
+2, et
+3, and et
+4 in test (a), respectively.
+Existing research mainly focuses on how to accurately select
+specific characteristics of widgets in GUIs such as ‘text’
+and ‘resource-id’. Combining the selected characteristics, they
+design the semantic similarity calculation method between
+widgets to generate meaningful tests. They attempt to select
+widgets with high similarity in a similar application for match-
+ing each event of the existing test. However, little attention
+has been paid to optimizing the matching process. Taking
+Figure 1 as an example, the widget wt
+3 of To-Do List and
+the correctly similar widget wm
+3
+of the application Minimal
+are laid out differently in the GUI. When reusing test (b) to
+the application To-Do List, adopting the existing approach may
+always incorrectly match the widget wm
+3 with other widgets.
+This may cause subsequent events to match incorrectly or even
+result in failed test reuse. In cases where the existing methods
+do not work well, it is necessary to adopt other corresponding
+measures. The lower the similarity of the generated event,
+the more likely the match is inappropriate. Hence, mining
+such events and exploring other widgets with more similar
+semantics to form events for substitution is considered.
+In addition, in Figure 1, the application To-Do List needs
+to skip the boot page that the application Minimal does not
+before entering the home page. And it is assumed that the
+event step is et
+0 = click(wt
+0). Under this assumption, the
+widget wt
+0 will match the widget with the highest similarity
+on the home page of the application Minimal. Obviously, the
+event produced by this step is redundant in the generated test.
+This simple example explains that we need to solve the event
+redundancy issue in the process of test reuse caused by some
+particular functionality in the existing test.
+(a) The existing test for To-Do List
+(b) The reused test for Minimal
+Figure 1. A simple example of test reuse. The test (b) is obtained by reusing
+the existing test (a).
+Inspired by the above observation, in this paper, we propose
+a novel approach TRASM (Test Reuse based on Adaptive
+Semantic Matching) to reuse the existing tests across android
+arXiv:2301.00530v1  [cs.SE]  2 Jan 2023
+
+自7:36
+47:36
+7:37
+7:37
+三
+To-Do List
+Q
+ ＋
+To-Do List
+三
+To-Do List
+Q+"
+：
+Q
+CLICK+FOR NEW LIST
+CLICK+FORNEWLIST
+No Deadline
+New To-Do task
+e’ = exist(Test')(w')
+e, = fill(w)
+Welcome!
+Test
+8
+scription
+菌 Deadline
+@Reminder
+Notasks available
+Progress:
+0%
+Thisappdoesnotuseanypermissions
+Priority:
+Medium
+List:
+Click to select!
+CANCEL
+OKAY
+e, = click(w')
+e. = click(w²)
++
++
+ADDNEWTASK>
+×
+ADDNEWTASK>
+sKIP
+NEXT日7:35
+47:35
+日7:36
+ Minimal
+X
+Minimal
+itl
+Test
+e" = exist(Test')(w")
+e’ = fill(wz)
+You don't have any todos
+" = click(w"mobile applications. In addition, we carry out comparative
+experiments with the-state-of-the-art baseline approaches to
+evaluate our work. Overall, our main contributions are as
+follows:
+1) We propose a novel approach TRASM, which utilizes an
+adaptive strategy to reuse more existing tests. TRASM
+can get more semantic matches in the generated test.
+2) TRASM includes a GUI events deduplication method,
+which could eliminate duplicated events caused by
+reusing particular functionality contained in the existing
+test to improve the quality of the generated test.
+3) We carry out extensive experiments which confirm that
+TRASM improves the accuracy of GUI event matching
+while reducing test reuse failures and reduces the run-
+ning time required for test reuse.
+The rest of this paper is organized as follows. Section
+II introduces related work. Section III describes the main
+idea and the proposed approach in detail. Section IV carries
+out experimental evaluation. Finally, Section V concludes the
+paper and outlines future research.
+II. RELATED WORK
+A. Test Generation
+In order to improve the efficiency of developers, based on
+different exploration strategies, several studies on automatic
+test generation have been proposed, which has laid a solid
+foundation.
+Sapienz [5] combined random fuzzing, systematic and
+search-based exploration, exploiting seeding and multi-level
+instrumentation to explore and optimize test sequences auto-
+matically. Gu et al. [7] dynamically abstracted the model by
+leveraging decision tree-based representation and updated the
+model by utilizing the evolution mechanism, which balances
+the accuracy and size of the model. ConmboDroid [15] ob-
+tained the use cases for verifying the unique functions of the
+application and then systematically enumerates and combines
+them to generate higher quality input. The advantage of their
+work is that they can mine more hidden bugs or achieve as high
+coverage as possible. Nevertheless, the test generated by their
+method is seldom standardized for verifying the application’s
+functionality.
+Different from their purpose and inspired by their explo-
+ration method, we focus on generating more meaningful tests
+based on semantic information.
+B. Test Reuse
+Test reuse, as an alternative method to automatically gener-
+ate GUI test, makes full use of existing resources to provide
+convenience for verifying the application’s behavior.
+Lin et al. proposed CRAFTDROID [19], an approach of
+test transfer across applications, which utilizes the GUI model
+extracted by static analysis to match event sequences similar to
+the semantics of the existing test in order. They realized the
+successful transfer of GUI and oracle events, which guides
+for improving test transfer. To more accurately express the
+similarity of widgets in test events, Mao et al. [20] raised a
+semantic-based event fuzzy mapping strategy when matching
+candidate widgets to generate target events. They always
+greedily preferentially explore and match the widgets with the
+highest similarity. Unfortunately, when their similarity calcu-
+lation method does not work well, the correctness of event
+matching will be threatened. Considering that the success
+of test reuse heavily depends on the semantic matching of
+test events, there is still space for improvement by adopting
+appropriate strategies to increase the quality of reused tests
+from the perspective of application functionality.
+Leonardo et al. [21] conducted extensive research and
+pointed out that some attributes representing widgets play
+a negative role and how designing the semantic matching
+process is the most influential component to matched results.
+Their key findings point to an entry point for better reuse of
+test. Up to now, there is still no effective method to solve
+semantic problems [25, 26]. Trying to optimize the generated
+test sequence to ensure the quality of reused tests should be
+an optional strategy.
+III. OUR APPROACH
+Figure 2 shows an overview of the proposed test reuse
+approach TRASM. Based on semantic matching of events,
+TRASM considers the test (source test) of the existing appli-
+cation (source app), and the new application (target app) as
+inputs and outputs target test. TRASM employs two significant
+phases to implement test reuse: preliminary preparation and
+source test reuse. For the former, the existing data is processed
+through test augmentation and model extraction to facilitate
+the implementation of source test reuse. For the latter, the
+processed data obtained by the former is used together to reuse
+the source test on the target app.
+Test augmentation and model extraction are preliminary
+preparation steps that follow existing work [19], and we will
+only briefly introduce them. In detail, we focus on introducing
+our main contributions.
+A. Test augmentation
+The main task of test augmentation is to extract semantic
+information of widgets during the execution of collected
+source tests. The semantically represented widgets, together
+with actions, compose augmented tests, which are used to
+match widgets in the GUI of the target app.
+After the source app executes each event, the adb tool1 is
+used to extract the semantic information of the corresponding
+widget in the executed event according to the reached GUI
+state. Multiple attributes (including class, resource-id, text,
+content-desc, clickable, password, parent text, sibling text,
+activity, and package) uniquely represent a widget in the GUI.
+These non-empty attributes and their values constitute the
+widget’s semantic information. For example, for widget wt
+1
+of test (a) in Figure 1, the collected semantic information is
+shown in Table 1.
+1https://developer.android.com/studio/command-line/adb.html
+
+Figure 2.
+The overview of TRASM.
+Table 1. The semantic information of widget wt
+1.
+Attribute
+Value
+class
+android.widget.ImageButton
+resource-id
+fab new task
+clickable
+true
+password
+false
+activity
+.view.MainActivity
+package
+org.secuso.privacyfriendlytodolist
+B. Model extraction
+The model extraction aims to statically analyze the source
+code obtained by the target app to obtain the window transition
+graph (WTG). Following with existing work, we employ tool2
+to obtain the WTG. And the steps of constructing WTG can
+refer to literature [19].
+The WTG can visually represent the interaction between
+application activities and is composed of node sets and edges.
+Among WTG, the node represents the activity of the applica-
+tion, and the directed edge represents the activity transition
+that the event can trigger. For example, for the test (b)
+of application Minimal in Figure 1, the window transition
+triggered by the execution of the event is shown in Figure 3,
+where the nodes Main and AddToDo respectively represent
+the two activities of the application Minimal. By triggering
+the event on edge, state transition occurs between activities.
+The obtained WTG can provide matching candidate widgets
+for widgets in the source test. However, the WTG obtained
+may be incomplete. More fully, we adopt updating the WTG
+based on the feedback running information when executing
+the application.
+C. Test generation
+The purpose of test generation is to generate the initial test
+on the target app according to the semantic information of the
+augmented test and the WTG of the target app.
+Every event in the augmented test iteratively matches the
+corresponding events in the target app. The candidate widgets
+are first obtained by similarity calculation to match the event.
+2https://drive.google.com/file/d/1HEFS96c5nNKnzBPkWlRdwBiunOHgOs-/
+view?usp=sharing
+Figure 3.
+Window transition triggered by test (b).
+Similarity calculation. For each widget, the semantic in-
+formation is captured from the current GUI page of the
+target app to build a word list of attributes. For example,
+the attribute ‘resource-id’ of widget wt
+2 and widget wm
+2
+are ‘et new task name’ and ‘userToDoEditText’ respec-
+tively. After preprocessing [10, 19], we get two word lists
+a=[‘edit’, ‘text’, ‘new’, ‘task’, ‘name’] and a′=[‘user’, ‘todo’,
+‘edit’, ‘text’]. For any word w ∈ a and w′
+∈ a′, the
+highest similarity max sim(w, w′) between words w and w′
+is synthesized as the similarity of attributes:
+sim(a, a′) =
+�
+w∈a maxw′∈a′ sim(w, w′)
+|a|
+(1)
+where a and a′ represent the attributes corresponding to
+widget S in source test and widget T in target app respectively,
+sim(w, w′) expresses the cosine distance of the word vectors
+−→
+Vw and −→
+Vw′, obtained by the Word2Vec model[27]:
+sim(w, w′) =
+−→
+Vw · −→
+Vw′
+|−→
+Vw||−→
+Vw′|
+(2)
+Based on Equation (1), we get the similarity between widget
+S and widget T:
+sim(S, T) =
+�
+a∈S sim(a, a′) ∗ wg(a)
+|S|
+(3)
+where, wg(a) represents the weight of attribute a among
+all attributes. Based above calculation, we build candidate
+
+Source Test
+Preliminary preparation
+② Source test reuse
+Candidate
+Widgets
+口
+A. Test
+E. Test
+adaptation
+augmentation
+口
+APP
+Augmented
+Processed Test
+Source Test
+Target Test
+Similarity
+Source App
+Window
+calculation
+Transition Graph
+B. Model
+APP
+C. Test
+D. GUI events
+extraction
+generation
+deduplication
+Target App
+-
+Initial Test= click(w") em = fill(w²)
+em
+m
+= click(wm) em = exist(Test')(wm)
+m
+m
+Main
+AddToDo
+m
+mwidgets by selecting several widgets in the target app with
+high similarity.
+We identify a reachable widget based on the obtained
+candidate widgets and assign an action to form an event.
+All the paths from the current activity to the activity of
+each candidate widget can be queried from the WTG. These
+paths are executed to identify the reachable widget and return
+leading events. In addition, to avoid repeated path exploration,
+we adopt a strategy to preserve the path that has been explored
+and the corresponding leading events. For example, when
+matching event et
+2 in test (a), the application Minimal reaches
+the AddToDo activity after executing event em
+1 as shown in
+Figure 3, and then candidate widgets on the current page are
+collected. From the stored explored paths, it is found that there
+is a reachable path between activity AddToDo and activity
+AddToDo. Widget wm
+2 is located in the reachable path, which
+is identified as a reachable widget. Finally, according to the
+source event, the action is allocated to the widget wm
+2 .
+D. GUI events deduplication
+Invalid repeated events will increase the complexity of test
+execution. Although repeated events in the test will not affect
+the triggering of the behavior of an application, GUI events
+deduplication intends to reduce the time consumption occupied
+by such events.
+Since the GUIs of the two applications are different, the
+target app may not have the special functionality contained
+in the source test. As explained in Section
+I, the reuse of
+special functionality in test (a), that is, the matching of event
+et
+0 on application Minimal, is meaningless. To remove such
+GUI events in the test sequence, deduplication is performed.
+However, it is a challenge to identify the meaningless events
+in the test. We take the operation of detecting and deleting
+duplicate events unrelated to the generated initial test. Con-
+sidering the variety of possible duplicate event patterns, we
+set two rules to distinguish them. First, if only a single event
+is repeated in the initial test, we delete the repeated events
+at the beginning of the test sequence. Second, if the test
+sequence starts with ⟨en0, en1⟩ and also contains ⟨en1, en0⟩
+such events, we delete the pair of events. After this operation,
+to maintain the correctness, we check whether the test after
+deduplication, that is, the processed test, can maintain the
+functionality as the initial test. If not, we will give up the
+GUI events deduplication.
+E. Test adaptation
+The goal of test adaptation is to explore whether there is a
+better test sequence than the processed test using the designed
+adaptation strategy.
+Test generation always prioritizes the widget with the high-
+est similarity for matching. When the method of calculating
+similarity does not work well, it may not be possible to
+distinguish the best widgets to match, which will affect the
+accuracy of the result. The design idea of test adaptation
+is to find indexes that may have more semantically similar
+events in the processed test and then rematch them. However,
+determining such indexes in the sequences of the processed
+test is a challenging task. In this paper, we first record the
+indexes for which widgets in the processed test have higher
+similarity to another widget in the augmented test, except
+for the current matching event. Then, we choose the indexes
+with the lowest similarity of matching events in the processed
+test, which tries to mine the event with the incorrect match.
+After these two processing stages, we obtain the index sets
+of events that can perform rematching. Based on the above,
+we successively rematch the events of each index set from
+the candidate widgets obtained by Section III-C. We set the
+early termination condition to obtain a new test sequence that
+is more semantically similar than the original ones.
+We explain how test adaptation solves the problem of
+reusing test (b) to application To-Do List in Figure
+1. As
+mentioned in Section
+I, different GUI designs make the
+similarity between the correct widget and the source widget
+low, resulting in the incorrect match of event em
+3 . Through the
+strategies mentioned above, we get the index of event em
+3 to be
+rematched. Then, combined with the WTG obtained from the
+model extraction, the correct reachable widget wt
+3 is searched
+again from the obtained candidate widgets on this index to
+form event et
+3. Finally, the process ends after the oracle event
+et
+4 matching.
+IV. EXPERIMENTAL EVALUATION
+We implement our approach TRASM as a tool. Moreover,
+we compared TRASM with the baseline approach CRAFT-
+DROID [19], a test transfer method across mobile applications
+through semantic mapping, to verify the effectiveness and
+efficiency of TRASM. In this section, we introduce the exper-
+imental setup and experimental results to evaluate TRASM.
+A. Experimental setup
+For consistency, we reused the dataset3 of [19] to evaluate
+the proposed TRASM. Following the steps of the baseline, we
+conducted reuse tests on 15 applications in three categories,
+including browser, Tip Calculator, and To-Do List. These
+applications come from Google Play and F-Droid, which are
+often used in the GUI testing field to explore the functionalities
+of application [5, 24, 25]. Concretely, Table 2 details the
+category, name (version), and source of each application.
+Specifically, for each application category, two typical func-
+tionalities are selected, and the corresponding tests of each
+application are collected according to the functionalities. To
+achieve the goal of verifying the implemented functionality,
+the last event of each test case is set as an oracle. In general,
+there are six functionalities in three categories of applications,
+as shown in Table 3. Table 3 lists the number of test cases for
+each functionality and the average number of GUI and oracle
+events.
+Our experiment was implemented on a Nexus 5X Emulator
+running Android 6.0 (API 23) installed on a Ubuntu desktop
+with a 3.4 GHz Intel Core i7 CPU and 32 GB RAM.
+3https://sites.google.com/view/craftdroid/
+
+Table 2. The specific information of applications.
+Category
+Application (version)
+Source
+a1-Browser
+a11-Lightning (4.5.1)
+F-Droid
+a12-Browser for Android (6.0)
+Google Play
+a13-Privacy Browser (2.10)
+F-Droid
+a14-FOSS Browser (5.8)
+F-Droid
+a15-Firefox Focus (6.0)
+Google Play
+a2-Tip Calculator
+a21-Tip Calculator (1.1)
+Google Play
+a22-Tip Calc (1.11)
+Google Play
+a23-Simple Tip Calculator (1.2)
+Google Play
+a24-Tip Calculator Plus (2.0)
+Google Play
+a25-Free Tip Calculator (1.0.0.9)
+Google Play
+a3-To Do List
+a31-Minimal (1.2)
+F-Droid
+a32-Clear List (1.5.6)
+F-Droid
+a33-To-Do List (2.1)
+F-Droid
+a34-Simply Do (0.9.1)
+F-Droid
+a35-Shopping List (0.10.1)
+F-Droid
+Table 3. Tests for the typical functionalities.
+Functionality
+Test
+Avg
+Avg
+Cases
+GUIs
+Oracles
+b11-Access website by URL
+5
+3.4
+1
+b12-Back button
+5
+7.4
+3
+b21-Calculate total bill with tip
+5
+3.8
+1
+b22-Split bill
+5
+4.8
+1
+b31-Add task
+5
+4
+1
+b32-Remove task
+5
+6.8
+2
+Total
+30
+5.1
+1.5
+B. Experimental results
+This
+subsection
+presents
+the
+experimental
+results
+of
+TRASM and the baseline approach CRAFTDROID under
+the same evaluation metrics. For each functionality of each
+category, we reuse the test of one application on the remaining
+four applications respectively, and the total number of test
+reuse is 5(test cases) × 4(target applications) = 20. This paper
+shows the average result of 20 different test reuses. In order
+to avoid randomness, for each test reuse, we take the average
+of the multiple results recorded.
+Effectiveness. By comparison, the tests reused by TRASM
+perform higher usability than CRAFTDROID. The following
+two aspects, including the evaluation of successful reuse and
+the evaluation of matching events, can support the usability of
+the TRASM approach reuse test.
+Regarding reuse success rate, TRASM has significantly
+improved test reuse in 3 of the six functionalities, as shown
+in the last column of Table 4. For functionalities b21 and
+b22, successfully reused tests achieved a 10% increase. For
+functionality b32, successful reuse also increased from 20%
+to 25%. In addition, 2 of the six functionalities, namely b11
+and b12, have shown the highest successful reuse, i.e., 100%,
+no matter whether it is approach CRAFTDROID or TRASM.
+For the evaluation of matching events, the third and fourth
+columns of Table 4 list the precision and recall of GUI
+events and oracle events, respectively. As shown in the table,
+compared with CRAFTDROID, TRASM improves the preci-
+sion of the GUI by 5% to 15% in different functionalities.
+Unfortunately, while improving the precision, the recall rate
+of GUI events for functionalities b22 and b32 has decreased
+slightly by 2% and 3%, respectively. The success of the reused
+test depends on whether the match of the last oracle event
+in the test sequence is correct. Therefore, the improvement
+of successful reuse also represents the increase in the recall
+rate of oracle events, as listed in Table 4. Among them, the
+most significant is that for functionalities b21 and b22, the
+recall rate is improved by 10%. In general, the improvement
+in the precision and the recall of oracle events shows that
+the proposed TRASM indeed increases the availability of the
+reused test.
+Efficiency. Figure
+4 lists the average test reuse time on
+each functionality. It is obvious that the average time spent on
+reuse testing of TRASM is less than that of CRAFTDROID
+for each functionality. Even the most significant effect is that
+for functionality b21, the average reuse test of CRAFTDROID
+takes 2581 seconds (43 minutes), while our TRASM only takes
+890 seconds (15 minutes), which is close to 35% of the time
+of CRAFTDROID. In summary, the results are attributed to
+two factors. One is that the storage of explored paths avoids
+repeated time consumption, and the other is that the adaptive
+strategy improves the efficiency of widget matching. The
+above results prove that we can break through the limitation
+on efficiency in CRAFTDROID.
+Figure 4. The average test reuse time on each functionality.
+While evaluating the efficiency of TRASM, an important
+finding is that implementing an adaptive strategy can improve
+the success of some reused tests. Inevitably, the potential
+drawback is that more suitable event matching can not be
+found will bring additional time consumption. We need to
+address this crucial point further to balance efficiency and
+effectiveness.
+V. CONCLUSION
+Test reuse as an alternative method of test generation
+can help developers verify the behavior of applications. In
+this paper, a novel test reuse approach has been proposed
+to alleviate the challenge of semantic problems in event
+matching. From the initial test set, we have extended GUI
+events deduplication and test adaptation to build up target tests.
+The experimental results indicate that our proposed approach
+achieves better performance than the baseline approaches with
+increased usability of the reused tests.
+
+12000
+CRAFTDROID
+TRASM
+10000
+(sec)
+8000
+time
+reuse
+6000
+Average
+4000
+2000
+0
+b11
+b12
+b21
+b22
+b31
+b32
+FunctionalityTable 4. Effectiveness and Efficiency Evaluation
+Functionality
+Approach
+GUI Event
+Oracle Event
+Successful Reuse
+Precision
+Recall
+Precision
+Recall
+b11
+CRAFTDROID
+79%
+100%
+100%
+100%
+20/20(100%)
+TRASM
+100%
+100%
+100%
+100%
+20/20(100%)
+b12
+CRAFTDROID
+85%
+100%
+100%
+100%
+20/20(100%)
+TRASM
+100%
+100%
+100%
+100%
+20/20(100%)
+b21
+CRAFTDROID
+82%
+100%
+100%
+80%
+16/20(80%)
+TRASM
+93%
+100%
+100%
+90%
+18/20(90%)
+b22
+CRAFTDROID
+80%
+100%
+100%
+65%
+13/20(65%)
+TRASM
+85%
+98%
+100%
+75%
+15/20(75%)
+b31
+CRAFTDROID
+78%
+100%
+85%
+100%
+17/20(85%)
+TRASM
+87%
+100%
+85%
+100%
+17/20(85%)
+b32
+CRAFTDROID
+69%
+100%
+85%
+80%
+11/20(55%)
+TRASM
+81%
+97%
+93%
+81%
+12/20(60%)
+We believe that matching events are a promising direction,
+and we plan to study how to improve the matching strategy
+further in the future. In addition, we plan to verify the
+generalization of the method and further explore the effect
+of test reuse on more applications.
+ACKNOWLEDGMENTS
+The work is partially supported by the National Natural
+Science Foundation of China (No. 61972197), the Natural Sci-
+ence Foundation of Jiangsu Province (No. BK20201292), and
+the Collaborative Innovation Center of Novel Software Tech-
+nology and Industrialization. T. Chen is partially supported
+by an oversea grant from the State Key Laboratory of Novel
+Software Technology, Nanjing University (KFKT2022A03),
+Birkbeck BEI School Project (EFFECT), National Natural
+Science Foundation of China (NSFC) under Grants (No.
+62072309, 62272397).
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf,len=583
+page_content='Test Reuse Based on Adaptive Semantic Matching across Android Mobile Applications  Shuqi Liu1, Yu Zhou1,∗, Tingting Han2, and Taolue Chen2  1College of Computer Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing, China  2Department of Computer Science and Data Science, Birkbeck, University of London, UK  {liushuqi, zhouyu}@nuaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='cn, {t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='han, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='chen}@bbk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='uk  corresponding author  Abstract—Automatic test generation can help verify and de-  velop the behavior of mobile applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Test reuse based on  semantic similarities between applications of the same category  has been utilized to reduce the manual effort of Graphical User  Interface (GUI) testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' However, most of the existing studies fail  to solve the semantic problem of event matching, which leads to  the failure of test reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' To overcome this challenge, we propose  TRASM (Test Reuse based on Adaptive Semantic Matching), a  test reuse approach based on adaptive strategies to find a better  event matching across android mobile applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' TRASM  first performs GUI events deduplication on the initial test set  obtained from test generation, and then employs an adaptive  strategy to find better event matching, which enables reusing  the existing test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Preliminary experiments with comparison to  baseline methods on 15 applications demonstrate that TRASM  can improve the precision of GUI event matching while reducing  the failure of test reuse and the running time required for test  reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Keywords—adaptive semantic matching;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' android mobile appli-  cations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' GUI event;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' test reuse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' oracle generation  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' INTRODUCTION  Graphical User Interface (GUI) testing is commonly em-  ployed to verify and develop the behaviors of applications by  designing and executing test cases of GUI applications [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  However, with the ever increasing functionalities in mobile  applications, it takes more effort for developers to manually  design GUI test cases (GUI test in short) [2–4], which in turn  decreases the efficiency of testing processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Considering the necessity of reducing time consumption,  many researchers have conducted a series of investigation on  automatic test generation [5–15] in GUI testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Recently,  some researchers observed that test reuse [16–24] could  be achieved by exploiting the semantic similarity of GUIs  between similar applications to generate tests automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Figure 1 shows a simple example in which the existing test (a)  of application To-Do List is successfully reused to application  Minimal, and the reused test (b) is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' As Figure 1  shows, events em  1 , em  2 , em  3 , and em  4 in test (b) are similar to  et  1, et  2, et  3, and et  4 in test (a), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Existing research mainly focuses on how to accurately select  specific characteristics of widgets in GUIs such as ‘text’  and ‘resource-id’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Combining the selected characteristics, they  design the semantic similarity calculation method between  widgets to generate meaningful tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' They attempt to select  widgets with high similarity in a similar application for match-  ing each event of the existing test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' However, little attention  has been paid to optimizing the matching process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Taking  Figure 1 as an example, the widget wt  3 of To-Do List and  the correctly similar widget wm  3  of the application Minimal  are laid out differently in the GUI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' When reusing test (b) to  the application To-Do List, adopting the existing approach may  always incorrectly match the widget wm  3 with other widgets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  This may cause subsequent events to match incorrectly or even  result in failed test reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' In cases where the existing methods  do not work well, it is necessary to adopt other corresponding  measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' The lower the similarity of the generated event,  the more likely the match is inappropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Hence, mining  such events and exploring other widgets with more similar  semantics to form events for substitution is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  In addition, in Figure 1, the application To-Do List needs  to skip the boot page that the application Minimal does not  before entering the home page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' And it is assumed that the  event step is et  0 = click(wt  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Under this assumption, the  widget wt  0 will match the widget with the highest similarity  on the home page of the application Minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Obviously, the  event produced by this step is redundant in the generated test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  This simple example explains that we need to solve the event  redundancy issue in the process of test reuse caused by some  particular functionality in the existing test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  (a) The existing test for To-Do List  (b) The reused test for Minimal  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' A simple example of test reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' The test (b) is obtained by reusing  the existing test (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Inspired by the above observation, in this paper, we propose  a novel approach TRASM (Test Reuse based on Adaptive  Semantic Matching) to reuse the existing tests across android  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='00530v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='SE]  2 Jan 2023  自7:36  47:36  7:37  7:37  三  To-Do List  Q  ＋  To-Do List  三  To-Do List  Q+"  ：  Q  CLICK+FOR NEW LIST  CLICK+FORNEWLIST  No Deadline  New To-Do task  e’ = exist(Test\')(w\')  e, = fill(w)  Welcome!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Test  8  scription  菌 Deadline  @Reminder  Notasks available  Progress:  0%  Thisappdoesnotuseanypermissions  Priority:  Medium  List:  Click to select!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content="  CANCEL  OKAY  e, = click(w')  e." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' = click(w²)  +  +  ADDNEWTASK>  ×  ADDNEWTASK>  sKIP  NEXT日7:35  47:35  日7:36  Minimal  X  Minimal  itl  Test  e" = exist(Test\')(w")  e’ = fill(wz)  You don\'t have any todos  " = click(w"mobile applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' In addition, we carry out comparative  experiments with the-state-of-the-art baseline approaches to  evaluate our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Overall, our main contributions are as  follows:  1) We propose a novel approach TRASM, which utilizes an  adaptive strategy to reuse more existing tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' TRASM  can get more semantic matches in the generated test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  2) TRASM includes a GUI events deduplication method,  which could eliminate duplicated events caused by  reusing particular functionality contained in the existing  test to improve the quality of the generated test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  3) We carry out extensive experiments which confirm that  TRASM improves the accuracy of GUI event matching  while reducing test reuse failures and reduces the run-  ning time required for test reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Section  II introduces related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Section III describes the main  idea and the proposed approach in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Section IV carries  out experimental evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Finally, Section V concludes the  paper and outlines future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' RELATED WORK  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Test Generation  In order to improve the efficiency of developers, based on  different exploration strategies, several studies on automatic  test generation have been proposed, which has laid a solid  foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Sapienz [5] combined random fuzzing, systematic and  search-based exploration, exploiting seeding and multi-level  instrumentation to explore and optimize test sequences auto-  matically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Gu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' [7] dynamically abstracted the model by  leveraging decision tree-based representation and updated the  model by utilizing the evolution mechanism, which balances  the accuracy and size of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' ConmboDroid [15] ob-  tained the use cases for verifying the unique functions of the  application and then systematically enumerates and combines  them to generate higher quality input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' The advantage of their  work is that they can mine more hidden bugs or achieve as high  coverage as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Nevertheless, the test generated by their  method is seldom standardized for verifying the application’s  functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Different from their purpose and inspired by their explo-  ration method, we focus on generating more meaningful tests  based on semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Test Reuse  Test reuse, as an alternative method to automatically gener-  ate GUI test, makes full use of existing resources to provide  convenience for verifying the application’s behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' proposed CRAFTDROID [19], an approach of  test transfer across applications, which utilizes the GUI model  extracted by static analysis to match event sequences similar to  the semantics of the existing test in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' They realized the  successful transfer of GUI and oracle events, which guides  for improving test transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' To more accurately express the  similarity of widgets in test events, Mao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' [20] raised a  semantic-based event fuzzy mapping strategy when matching  candidate widgets to generate target events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' They always  greedily preferentially explore and match the widgets with the  highest similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Unfortunately, when their similarity calcu-  lation method does not work well, the correctness of event  matching will be threatened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Considering that the success  of test reuse heavily depends on the semantic matching of  test events, there is still space for improvement by adopting  appropriate strategies to increase the quality of reused tests  from the perspective of application functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Leonardo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' [21] conducted extensive research and  pointed out that some attributes representing widgets play  a negative role and how designing the semantic matching  process is the most influential component to matched results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Their key findings point to an entry point for better reuse of  test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Up to now, there is still no effective method to solve  semantic problems [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Trying to optimize the generated  test sequence to ensure the quality of reused tests should be  an optional strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' OUR APPROACH  Figure 2 shows an overview of the proposed test reuse  approach TRASM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Based on semantic matching of events,  TRASM considers the test (source test) of the existing appli-  cation (source app), and the new application (target app) as  inputs and outputs target test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' TRASM employs two significant  phases to implement test reuse: preliminary preparation and  source test reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' For the former, the existing data is processed  through test augmentation and model extraction to facilitate  the implementation of source test reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' For the latter, the  processed data obtained by the former is used together to reuse  the source test on the target app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Test augmentation and model extraction are preliminary  preparation steps that follow existing work [19], and we will  only briefly introduce them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' In detail, we focus on introducing  our main contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Test augmentation  The main task of test augmentation is to extract semantic  information of widgets during the execution of collected  source tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' The semantically represented widgets, together  with actions, compose augmented tests, which are used to  match widgets in the GUI of the target app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  After the source app executes each event, the adb tool1 is  used to extract the semantic information of the corresponding  widget in the executed event according to the reached GUI  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Multiple attributes (including class, resource-id, text,  content-desc, clickable, password, parent text, sibling text,  activity, and package) uniquely represent a widget in the GUI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  These non-empty attributes and their values constitute the  widget’s semantic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' For example, for widget wt  1  of test (a) in Figure 1, the collected semantic information is  shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  1https://developer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='android.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='com/studio/command-line/adb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='html  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  The overview of TRASM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' The semantic information of widget wt  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Attribute  Value  class  android.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='widget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='ImageButton  resource-id  fab new task  clickable  true  password  false  activity  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='MainActivity  package  org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='secuso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='privacyfriendlytodolist  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Model extraction  The model extraction aims to statically analyze the source  code obtained by the target app to obtain the window transition  graph (WTG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Following with existing work, we employ tool2  to obtain the WTG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' And the steps of constructing WTG can  refer to literature [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  The WTG can visually represent the interaction between  application activities and is composed of node sets and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Among WTG, the node represents the activity of the applica-  tion, and the directed edge represents the activity transition  that the event can trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' For example, for the test (b)  of application Minimal in Figure 1, the window transition  triggered by the execution of the event is shown in Figure 3,  where the nodes Main and AddToDo respectively represent  the two activities of the application Minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' By triggering  the event on edge, state transition occurs between activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  The obtained WTG can provide matching candidate widgets  for widgets in the source test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' However, the WTG obtained  may be incomplete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' More fully, we adopt updating the WTG  based on the feedback running information when executing  the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Test generation  The purpose of test generation is to generate the initial test  on the target app according to the semantic information of the  augmented test and the WTG of the target app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Every event in the augmented test iteratively matches the  corresponding events in the target app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' The candidate widgets  are first obtained by similarity calculation to match the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  2https://drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='com/file/d/1HEFS96c5nNKnzBPkWlRdwBiunOHgOs-/  view?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='usp=sharing  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Window transition triggered by test (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Similarity calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' For each widget, the semantic in-  formation is captured from the current GUI page of the  target app to build a word list of attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' For example,  the attribute ‘resource-id’ of widget wt  2 and widget wm  2  are ‘et new task name’ and ‘userToDoEditText’ respec-  tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' After preprocessing [10, 19], we get two word lists  a=[‘edit’, ‘text’, ‘new’, ‘task’, ‘name’] and a′=[‘user’, ‘todo’,  ‘edit’, ‘text’].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' For any word w ∈ a and w′  ∈ a′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' the  highest similarity max sim(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' w′) between words w and w′  is synthesized as the similarity of attributes:  sim(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' a′) =  �  w∈a maxw′∈a′ sim(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' w′)  |a|  (1)  where a and a′ represent the attributes corresponding to  widget S in source test and widget T in target app respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  sim(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' w′) expresses the cosine distance of the word vectors  −→  Vw and −→  Vw′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' obtained by the Word2Vec model[27]:  sim(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' w′) =  −→  Vw · −→  Vw′  |−→  Vw||−→  Vw′|  (2)  Based on Equation (1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' we get the similarity between widget  S and widget T:  sim(S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' T) =  �  a∈S sim(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' a′) ∗ wg(a)  |S|  (3)  where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' wg(a) represents the weight of attribute a among  all attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Based above calculation, we build candidate  Source Test  Preliminary preparation  ② Source test reuse  Candidate  Widgets  口  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Test  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Test  adaptation  augmentation  口  APP  Augmented  Processed Test  Source Test  Target Test  Similarity  Source App  Window  calculation  Transition Graph  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Model  APP  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Test  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' GUI events  extraction  generation  deduplication  Target App Initial Test= click(w") em = fill(w²)  em  m  = click(wm) em = exist(Test\')(wm)  m  m  Main  AddToDo  m  mwidgets by selecting several widgets in the target app with  high similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  We identify a reachable widget based on the obtained  candidate widgets and assign an action to form an event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  All the paths from the current activity to the activity of  each candidate widget can be queried from the WTG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' These  paths are executed to identify the reachable widget and return  leading events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' In addition, to avoid repeated path exploration,  we adopt a strategy to preserve the path that has been explored  and the corresponding leading events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' For example, when  matching event et  2 in test (a), the application Minimal reaches  the AddToDo activity after executing event em  1 as shown in  Figure 3, and then candidate widgets on the current page are  collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' From the stored explored paths, it is found that there  is a reachable path between activity AddToDo and activity  AddToDo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Widget wm  2 is located in the reachable path, which  is identified as a reachable widget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Finally, according to the  source event, the action is allocated to the widget wm  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' GUI events deduplication  Invalid repeated events will increase the complexity of test  execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Although repeated events in the test will not affect  the triggering of the behavior of an application, GUI events  deduplication intends to reduce the time consumption occupied  by such events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Since the GUIs of the two applications are different, the  target app may not have the special functionality contained  in the source test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' As explained in Section  I, the reuse of  special functionality in test (a), that is, the matching of event  et  0 on application Minimal, is meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' To remove such  GUI events in the test sequence, deduplication is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  However, it is a challenge to identify the meaningless events  in the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' We take the operation of detecting and deleting  duplicate events unrelated to the generated initial test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Con-  sidering the variety of possible duplicate event patterns, we  set two rules to distinguish them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' First, if only a single event  is repeated in the initial test, we delete the repeated events  at the beginning of the test sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Second, if the test  sequence starts with ⟨en0, en1⟩ and also contains ⟨en1, en0⟩  such events, we delete the pair of events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' After this operation,  to maintain the correctness, we check whether the test after  deduplication, that is, the processed test, can maintain the  functionality as the initial test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' If not, we will give up the  GUI events deduplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Test adaptation  The goal of test adaptation is to explore whether there is a  better test sequence than the processed test using the designed  adaptation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Test generation always prioritizes the widget with the high-  est similarity for matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' When the method of calculating  similarity does not work well, it may not be possible to  distinguish the best widgets to match, which will affect the  accuracy of the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' The design idea of test adaptation  is to find indexes that may have more semantically similar  events in the processed test and then rematch them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' However,  determining such indexes in the sequences of the processed  test is a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' In this paper, we first record the  indexes for which widgets in the processed test have higher  similarity to another widget in the augmented test, except  for the current matching event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Then, we choose the indexes  with the lowest similarity of matching events in the processed  test, which tries to mine the event with the incorrect match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  After these two processing stages, we obtain the index sets  of events that can perform rematching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Based on the above,  we successively rematch the events of each index set from  the candidate widgets obtained by Section III-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' We set the  early termination condition to obtain a new test sequence that  is more semantically similar than the original ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  We explain how test adaptation solves the problem of  reusing test (b) to application To-Do List in Figure  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' As  mentioned in Section  I, different GUI designs make the  similarity between the correct widget and the source widget  low, resulting in the incorrect match of event em  3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Through the  strategies mentioned above, we get the index of event em  3 to be  rematched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Then, combined with the WTG obtained from the  model extraction, the correct reachable widget wt  3 is searched  again from the obtained candidate widgets on this index to  form event et  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Finally, the process ends after the oracle event  et  4 matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' EXPERIMENTAL EVALUATION  We implement our approach TRASM as a tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Moreover,  we compared TRASM with the baseline approach CRAFT-  DROID [19], a test transfer method across mobile applications  through semantic mapping, to verify the effectiveness and  efficiency of TRASM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' In this section, we introduce the exper-  imental setup and experimental results to evaluate TRASM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Experimental setup  For consistency, we reused the dataset3 of [19] to evaluate  the proposed TRASM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Following the steps of the baseline, we  conducted reuse tests on 15 applications in three categories,  including browser, Tip Calculator, and To-Do List.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' These  applications come from Google Play and F-Droid, which are  often used in the GUI testing field to explore the functionalities  of application [5, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Concretely, Table 2 details the  category, name (version), and source of each application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Specifically, for each application category, two typical func-  tionalities are selected, and the corresponding tests of each  application are collected according to the functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' To  achieve the goal of verifying the implemented functionality,  the last event of each test case is set as an oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' In general,  there are six functionalities in three categories of applications,  as shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Table 3 lists the number of test cases for  each functionality and the average number of GUI and oracle  events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Our experiment was implemented on a Nexus 5X Emulator  running Android 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='0 (API 23) installed on a Ubuntu desktop  with a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='4 GHz Intel Core i7 CPU and 32 GB RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  3https://sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='com/view/craftdroid/  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' The specific information of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Category  Application (version)  Source  a1-Browser  a11-Lightning (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='1)  F-Droid  a12-Browser for Android (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='0)  Google Play  a13-Privacy Browser (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='10)  F-Droid  a14-FOSS Browser (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='8)  F-Droid  a15-Firefox Focus (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='0)  Google Play  a2-Tip Calculator  a21-Tip Calculator (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='1)  Google Play  a22-Tip Calc (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='11)  Google Play  a23-Simple Tip Calculator (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='2)  Google Play  a24-Tip Calculator Plus (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='0)  Google Play  a25-Free Tip Calculator (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='9)  Google Play  a3-To Do List  a31-Minimal (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='2)  F-Droid  a32-Clear List (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='6)  F-Droid  a33-To-Do List (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='1)  F-Droid  a34-Simply Do (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='1)  F-Droid  a35-Shopping List (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='1)  F-Droid  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Tests for the typical functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Functionality  Test  Avg  Avg  Cases  GUIs  Oracles  b11-Access website by URL  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='4  1  b12-Back button  5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='4  3  b21-Calculate total bill with tip  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='8  1  b22-Split bill  5  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='8  1  b31-Add task  5  4  1  b32-Remove task  5  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='8  2  Total  30  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='5  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Experimental results  This  subsection  presents  the  experimental  results  of  TRASM and the baseline approach CRAFTDROID under  the same evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' For each functionality of each  category, we reuse the test of one application on the remaining  four applications respectively, and the total number of test  reuse is 5(test cases) × 4(target applications) = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' This paper  shows the average result of 20 different test reuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' In order  to avoid randomness, for each test reuse, we take the average  of the multiple results recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' By comparison, the tests reused by TRASM  perform higher usability than CRAFTDROID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' The following  two aspects, including the evaluation of successful reuse and  the evaluation of matching events, can support the usability of  the TRASM approach reuse test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Regarding reuse success rate, TRASM has significantly  improved test reuse in 3 of the six functionalities, as shown  in the last column of Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' For functionalities b21 and  b22, successfully reused tests achieved a 10% increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' For  functionality b32, successful reuse also increased from 20%  to 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' In addition, 2 of the six functionalities, namely b11  and b12, have shown the highest successful reuse, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=', 100%,  no matter whether it is approach CRAFTDROID or TRASM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  For the evaluation of matching events, the third and fourth  columns of Table 4 list the precision and recall of GUI  events and oracle events, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' As shown in the table,  compared with CRAFTDROID, TRASM improves the preci-  sion of the GUI by 5% to 15% in different functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Unfortunately, while improving the precision, the recall rate  of GUI events for functionalities b22 and b32 has decreased  slightly by 2% and 3%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' The success of the reused  test depends on whether the match of the last oracle event  in the test sequence is correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Therefore, the improvement  of successful reuse also represents the increase in the recall  rate of oracle events, as listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Among them, the  most significant is that for functionalities b21 and b22, the  recall rate is improved by 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' In general, the improvement  in the precision and the recall of oracle events shows that  the proposed TRASM indeed increases the availability of the  reused test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Figure  4 lists the average test reuse time on  each functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' It is obvious that the average time spent on  reuse testing of TRASM is less than that of CRAFTDROID  for each functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Even the most significant effect is that  for functionality b21, the average reuse test of CRAFTDROID  takes 2581 seconds (43 minutes), while our TRASM only takes  890 seconds (15 minutes), which is close to 35% of the time  of CRAFTDROID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' In summary, the results are attributed to  two factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' One is that the storage of explored paths avoids  repeated time consumption, and the other is that the adaptive  strategy improves the efficiency of widget matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' The  above results prove that we can break through the limitation  on efficiency in CRAFTDROID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' The average test reuse time on each functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  While evaluating the efficiency of TRASM, an important  finding is that implementing an adaptive strategy can improve  the success of some reused tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Inevitably, the potential  drawback is that more suitable event matching can not be  found will bring additional time consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' We need to  address this crucial point further to balance efficiency and  effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' CONCLUSION  Test reuse as an alternative method of test generation  can help developers verify the behavior of applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' In  this paper, a novel test reuse approach has been proposed  to alleviate the challenge of semantic problems in event  matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' From the initial test set, we have extended GUI  events deduplication and test adaptation to build up target tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  The experimental results indicate that our proposed approach  achieves better performance than the baseline approaches with  increased usability of the reused tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  12000  CRAFTDROID  TRASM  10000  (sec)  8000  time  reuse  6000  Average  4000  2000  0  b11  b12  b21  b22  b31  b32  FunctionalityTable 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Effectiveness and Efficiency Evaluation  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='Functionality  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='Approach  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='GUI Event  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='Oracle Event  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='Successful Reuse  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='Precision  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='Recall  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='Precision  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='Recall  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='b11  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='CRAFTDROID  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='79%  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
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+page_content='100%  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
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+page_content='b12  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
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+page_content='85%  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
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+page_content='100%  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
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+page_content='We believe that matching events are a promising direction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  and we plan to study how to improve the matching strategy  further in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' In addition, we plan to verify the  generalization of the method and further explore the effect  of test reuse on more applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The work is partially supported by the National Natural  Science Foundation of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' 61972197), the Natural Sci-  ence Foundation of Jiangsu Province (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' BK20201292), and  the Collaborative Innovation Center of Novel Software Tech-  nology and Industrialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Chen is partially supported  by an oversea grant from the State Key Laboratory of Novel  Software Technology, Nanjing University (KFKT2022A03),  Birkbeck BEI School Project (EFFECT), National Natural  Science Foundation of China (NSFC) under Grants (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  62072309, 62272397).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  REFERENCES  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Anand, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Naik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Harrold, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Yang, “Au-  tomated concolic testing of smartphone apps,” Proceed-  ings of the ACM SIGSOFT 20th International Symposium  on the Foundations of Software Engineering, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' 1–11,  November 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
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+page_content=' Joorabchi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Mesbah, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Kruchten, “Real  challenges in mobile app development,” 2013 ACM/IEEE  International Symposium on Empirical Software Engineer-  ing and Measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  IEEE, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' 15–24, October 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content='  [3] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Kochhar, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
+page_content=' Thung, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/69AyT4oBgHgl3EQfpvg-/content/2301.00530v1.pdf'}
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+Performance of the r2SCAN functional in transition metal oxides
+S. Swathilakshmi1, Reshma Devi1, and Gopalakrishnan Sai Gautam1,*
+1Department of Materials Engineering, Indian Institute of Science, Bengaluru, 560012,
+India
+*Email: saigautamg@iisc.ac.in
+Abstract
+We assess the accuracy and computational efficiency of the recently developed meta-generalized gra-
+dient approximation (metaGGA) functional, the restored regularized strongly constrained and appropri-
+ately normed (r2SCAN), in transition metal oxide (TMO) systems and compare its performance against
+SCAN. Specifically, we benchmark the r2SCAN-calculated oxidation enthalpies, lattice parameters, on-
+site magnetic moments, and band gaps of binary 3d TMOs against the SCAN-calculated and experimental
+values. Additionally, we evaluate the optimal Hubbard U correction required for each transition metal
+(TM) to improve the accuracy of the r2SCAN functional, based on experimental oxidation enthalpies,
+and verify the transferability of the U values by comparing against experimental properties on other
+TM-containing oxides. Notably, including the U -correction to r2SCAN increases the lattice parameters,
+on-site magnetic moments and band gaps of TMOs, apart from an improved description of the ground
+state electronic state in narrow band gap TMOs. The r2SCAN and r2SCAN+U calculated oxidation en-
+thalpies follow the qualitative trends of SCAN and SCAN+U, with r2SCAN and r2SCAN+U predicting
+marginally larger lattice parameters, smaller magnetic moments, and lower band gaps compared to SCAN
+and SCAN+U, respectively. We observe that the overall computational time (i.e., for all ionic+electronic
+steps) required for r2SCAN(+U ) to be lower than SCAN(+U ). Thus, the r2SCAN(+U ) framework can
+offer a reasonably accurate description of the ground state properties of TMOs with better computational
+efficiency than SCAN(+U ).
+1
+Introduction
+Density functional theory (DFT [1]) calculations are the bedrock of modern computational materials
+science in terms of predicting thermodynamic and kinetic properties, with such property predictions being
+put to use in subsequent materials discovery [2–7] and understanding underlying physical phenomena. [8–12]
+In recent years, machine learning has been used to augment DFT in property predictions, thereby reducing
+computational cost and accelerating materials discovery. [13–17] Note that a key approximation within
+DFT is the exchange-correlation (XC) functional, the exact form of which is unknown. However, several
+approximations for the XC functional have been proposed over the years, which can be categorized into
+different classes depending on the degree of sophistication and accuracy, and visually represented as rungs
+on the Jacob’s ladder. [1, 2, 18, 19] As with most computational tools, the higher the accuracy (higher up
+Jacob’s ladder) higher is the computational cost.
+1
+arXiv:2301.00535v1  [cond-mat.mtrl-sci]  2 Jan 2023
+
+Most DFT calculations for “large” solid systems (10s to 100s of atoms) are performed using the Perdew-
+Burke-Ernzerhof (PBE) parameterization of the generalized gradient approximation (GGA) XC functional,
+[20] as it offers fair accuracy at reasonable computational cost for a wide variety of materials. [21–23]
+Specifically, GGAs include the local electron density as well as the gradient of the electron density in
+describing the XC. As a semilocal functional of electron density, PBE captures short range interactions but
+fails to capture medium and long-range dispersions and also exhibits large electronic self-interaction errors
+(SIEs), especially in highly correlated systems. [24, 25] Also, PBE typically underestimates the formation
+energies [26,27] and semiconductor band gaps of crystalline solids, [26,28] while overestimating their lattice
+volumes. [26,29]
+As we move higher in the Jacob’s ladder, [19] we obtain metaGGA functionals, which may account for
+medium range dispersions and exhibit lower SIEs. Some metaGGAs consider orbital kinetic energy density
+in addition to the local electron density and its gradient, such as the recently developed strongly constrained
+and appropriately normed (SCAN [30]) functional, which offers better numerical accuracy than PBE and
+satisfies all 17 known constraints for a XC functional (namely, 6 for exchange, 6 for correlation, and 5
+for both). The iso-orbital indicator (α), which includes the kinetic energy density in SCAN, distinguishes
+various bonding environments in a given material and consequently improves the accuracy of SCAN over
+GGA. However, SCAN suffers from numerical instability during self-consistent-field (SCF) calculations [31]
+wherein denser k-grids (than PBE) are required for accurate and consistent predictions. [31–33] Thus it is
+computationally expensive (per SCF step) compared to PBE. [21]
+To overcome the numerical instability and reduce the computational cost of SCAN, Bartok and Yates [34]
+developed regularized SCAN (rSCAN), which satisfies 13 out of the 17 known constraints. The authors
+replaced the non-analytical switching α interpolation function in SCAN with a simple polynomial function,
+which improves computational speed. [35] However, subsequent investigations showed a significant drop in
+numerical accuracy with rSCAN (compared to SCAN), which is attributed to the failure of the polynomial
+α function to fully recover the uniform gas limit. [31, 32] Subsequently, Furness et al. [32] introduced the
+restored regularized SCAN (or r2SCAN), wherein the constraints broken by rSCAN were restored except
+the fourth order gradient expansion constraint for exchange (or GE4X). Furness et al. claimed that the new
+r2SCAN functional combines the numerical accuracy of SCAN and computational speed of rSCAN as the
+smooth polynomial α function of rSCAN is modified to satisfy the uniform gas limit in r2SCAN. [32] Recently,
+Kingsbury et al. [36] demonstrated that r2SCAN functional indeed delivers robust numerical accuracy (i.e.,
+similar to SCAN) and better computational performance (faster and numerically stable) by comparing
+r2SCAN and SCAN for solids using a high-throughput computational workflow. Specifically, the authors [36]
+reported that while r2SCAN predicts a smaller band gap (for most of the strongly-bound materials) and
+larger lattice volumes than SCAN, the mean atomization error with r2SCAN is ∼15-20% lower for most
+solids. However, the performance of r2SCAN in correlated electron systems, i.e., transition metal oxides
+(TMOs) containing open-shell d electrons, remains to be seen and forms the main focus of this work.
+Despite the accuracy of SCAN, it still has shortcomings in TMOs, which can be mitigated by adding an
+on-site Hubbard U correction term for the transition metal (TM) under consideration. [37,38] This approach
+is similar to the one followed to mitigate the SIEs of PBE in TMOs. [39,40] However, the magnitude of the
+U correction required is not known a priori, and there are both theory-based approaches such as density
+functional perturbation theory, [41] linear response theory, [42–44] embedded Hartree-Fock method, [45,46]
+and machine learning based Bayesian optimisation, [47] and experimental-data-based approaches to identify
+the appropriate U values. For example, Gautam et al. [37, 38] used the experimental oxidation enthalpies
+2
+
+among binary TMOs to identify optimal U values across various oxidation states of 3d TMs. A similar
+experimental-data-based Hubbard U correction scheme can be developed in conjunction with r2SCAN as
+well, resulting in a r2SCAN+U framework, in case r2SCAN exhibits similar SIEs as SCAN in TMOs. We
+explore the usefulness of such a r2SCAN+U framework also in this work.
+Here, we verify the numerical accuracy and computational efficiency of the r2SCAN and r2SCAN+U
+frameworks in comparison to SCAN and SCAN+U, respectively, in describing material properties such as
+lattice parameters, on-site magnetic moments, and band gaps of binary 3d TMOs, including Ti, V, Cr,
+Mn, Fe, Co, Ni, and Cu. As necessary, we evaluate the optimal Hubbard U correction with r2SCAN for
+each TM by using the experimental-data-based approach employed in previous works. [37,38] We find that
+r2SCAN predicts marginally larger lattice constants and smaller on-site magnetic moments than SCAN for
+most of the TMOs considered. On addition of the U -correction to both SCAN and r2SCAN, we observe an
+increase in the calculated lattice constants, on-site magnetic moments and band gaps. In the case of narrow
+band gap TMOs, SCAN+U and r2SCAN+U generally estimate a non-zero band gap, with r2SCAN+U ’s
+band gap in better agreement with experiments. Also, we perform transferability checks for the optimal U
+values derived in this work for each TM, by benchmarking various properties in oxides that were not used
+in obtaining the U values. Finally, we compare the computational performance of r2SCAN/r2SCAN+U
+relative to SCAN/SCAN+U to explore the accuracy-cost trade-off. We report that r2SCAN/r2SCAN+U
+is computationally less expensive than SCAN and SCAN+U, when all required ionic and electronic steps
+are taken into account for convergence during structure relaxations. We hope that our work will provide a
+foundational basis for further studies on understanding material behavior and computationally discovering
+new materials in the near future.
+2
+Methods
+2.1
+Computational Methods
+We used the Vienna ab-initio simulation package (VASP 6.2.1) [48–50] for all the spin-polarized DFT
+calculations, where the frozen-core PBE-based projector augmented wave (PAW) [51] potentials employed
+were identical to previous work. [37, 38] The plane waves for each system were expanded up to a kinetic
+energy of 520 eV, with each structure converged until the total energy differences and atomic forces became
+<0.01 meV and <|0.01| eV/˚A, respectively.
+We adopted a Γ-centered Monkhorst-Pack [52] grid with a
+density of 48 k-points per ˚A for all systems.
+The conjugate gradient algorithm was used to relax the
+structures (i.e., cell shapes, volumes, and ionic positions), without preserving any underlying symmetry. An
+‘accurate’ level of precision was maintained while projecting the wavefunctions in the reciprocal space. The
+Fermi surface of each system was integrated with a Gaussian smearing of partial occupancies, with a width
+of 0.05 eV. In terms of DFT+U calculations, we used the Dudarev framework [53] for adding a effective
+U correction on the d orbitals of TM atoms. All U values used in SCAN+U calculations were taken from
+previous work (see Table S1 of the Supporting Information –SI). [37,38] Since we used different computing
+systems to perform our structure relaxations for different systems, we normalized the computational time
+with the number of cores used in each calculation to compare the computational efficiency of the different
+XC functionals considered.
+For calculating band gaps, GGA functionals typically use the Kohn Sham potential as a multiplicative
+term, which typically underestimates the band gap of solids even at the SCAN level. [54, 55] Here, we
+3
+
+use the generalized Kohn Sham technique to determine the band gaps by calculating the density of states
+(DOS) for all systems considered. For each DOS calculation, we used the optimized structure and the initial
+charge density from a previous structure relaxation. Subsequently, we introduced a set of zero-weighted k-
+points, corresponding to a density of 96 k-points per ˚A, where the k-points that were used for the structure
+relaxation retained their original weights (as determined by VASP). Finally, we performed a single-SCF
+calculation where the DOS was sampled between energies of -20 to 20 eV in steps of 0.005 eV.
+2.2
+Structures and magnetic configurations
+We considered the binary oxides of each TM, i.e., Ti, V, Cr, Mn, Fe, Co, Ni, and Cu with different
+oxidation states, similar to previous studies. [37, 38] The main criteria in selection of these metal oxides
+are the availability of reliable thermodynamic data (i.e., formation energies [56–58]) and the experimentally-
+determined ground-state structures that are compiled in the inorganic crystal structure database (ICSD) [59]
+Note that the structures from the ICSD were the initial structures in all our DFT structure relaxations,
+including the systems used as transferability checks. In the case of Ni oxides, we chose NiO and LiNiO2
+(similar to previous work, [38]), as reliable thermodynamic data is not available for higher-oxidation-state
+binary Ni oxides (e.g., Ni2O3 and NiO2). The TM in all oxides, except select Co and Ni compounds, was
+initialized in its high-spin configuration (e.g., high-spin configuration of Fe3+ consists of five unpaired d
+electrons). A detailed description of all structures utilised in this work is provided in the SI, under the
+’Crystal Structures’ section, with the magnetic configurations depicted in Figure S1.
+The magnetic configuration of each TMO considered (see Figure S1) was initialized to its appropriate
+(in several cases, experimentally-known) ground state configuration during the structural relaxation. For
+example, we considered the ferromagnetic (FM) ground state configuration for CrO2 and VO2, given that
+CrO2 is metallic [60] and VO2 undergoes a metal-to-insulator transition (MIT) below 341 K. [61] The
+rocksalt (RS) TMOs, namely, VO, MnO, FeO, CoO, and NiO were initialized with their experimentally-
+known type-II antiferromagnetic (AFM) configuration. [62–67] Each Ni’s spin in NiO was initialized with
+two unpaired d electrons (i.e., its high-spin configuration). In CuO, we arranged the magnetic moments of
+Cu2+ antiferromagnetically along the Cu-O-Cu chains in the [10¯1] direction. [68,69]
+We initialized α-Mn2O3 (bixbyite structure) in a FM configuration as this configuration was found to
+be the most stable in previous work. [37] AFM configurations were utilized for rutile-MnO2, [70], and the
+other TM2O3 oxides, namely, V2O3, Fe2O3, Ti2O3, and Cr2O3.
+Note that V2O3 becomes AFM below
+its MIT temperature, [71–73] while Fe2O3 displays an AFM configuration with the magnetic moment of
+Fe alternating every two consecutive layers along the c-axis. [74] Cr2O3 and Ti2O3 exhibit ↑↓↑↓ and ↑↓↓↑
+magnetic configurations, respectively, on the TM centers along the a-axis. [75,76]
+In case of spinels, we used different ferrimagnetic (FIM) configurations, as per experimental observations.
+For example, spinel-Fe3O4 contains both Fe3+ and Fe2+, with up-spin Fe3+ occupying tetrahedral sites and
+down-spin Fe3+ occupying half the octahedral sites. The remaining octahedral sites in Fe3O4 are occupied
+by up-spin Fe2+. [77, 78] In Co3O4, no-spin Co3+ occupies octahedral sites, while high-spin Co2+ (three
+unpaired d electrons) occupies tetrahedral sites in an AFM configuration. [79–81] For Mn3O4, we adopted
+the ”FIM6” configuration, as this was found to be the ground state in previous work. [37] TiO2, CrO3,
+and V2O5 are diamagnetic, since they contain TMs with empty 3d orbitals. Similarly, Cu2O is diamagnetic
+owing to the completely-filled 3d orbitals of Cu.
+4
+
+2.3
+Determining U
+We determined the required U value, with r2SCAN, for each binary TMO oxidation reaction (e.g., Ti3+ →
+Ti4+ in 2Ti2O3 + O2 → 4TiO2) by comparing the experimental enthalpy (per mole of O2) with the calculated
+(r2SCAN+U ) value that minimizes the error against the experimental value. Note that U = 0 eV in our
+data simply reflects a r2SCAN calculation. In order to obtain the experimental oxidation enthalpy, standard
+enthalpy of formation for all the considered TMOs were taken from the Wagman and/or Kubaschewski
+tables, [56,57] thus ignoring the p − V and entropic contributions, similar to previous works. [37,38,82] The
+overall optimal U value for each TM was obtained by taking the average of the required U for each of the
+available oxidation reactions. In the case of Ni oxides, oxidation of NiO to LiNiO2 by 2Li2O + 4NiO + O2
+→ 4LiNiO2 was considered as a proxy for the Ni2+ → Ni3+ oxidation reaction. [38]
+3
+Results
+3.1
+Oxidation energetics
+Figure 1 displays the variation of the enthalpy of different oxidation reactions among binary TMOs, as a
+function of applied U in the r2SCAN+U framework, for all TMs considered except Cr and Cu. Solid lines
+in each panel of Figure 1 represent DFT-calculated oxidation enthalpies, with each color corresponding to
+different oxidation reactions for the TM. For instance in V oxides (Figure 1b), the solid black line corresponds
+to the oxidation reaction, VO → V2O3, while the solid red and green lines indicate V2O3 → VO2 and VO2 →
+V2O5, respectively. Similarly, the experimental enthalpy of each oxidation reaction is represented by dashed
+horizontal line of the same color. For example, the black dashed line in Figure 1b indicates the experimental
+oxidation enthalpy (-7.36 eV) of VO → V2O3. Also, dotted vertical line of a given color highlights the
+required U value to minimize the error between DFT-calculated and experimental value for the oxidation
+reaction enthalpy indicated by the same color. The dotted blue line in each panel signifies the overall optimal
+U for the TM that is averaged across all available oxidation reactions.
+We report an optimal U value of 2.3, 1.0, 1.8, 3.1, 1.8, and 2.1 eV, respectively, for Ti, V, Mn, Fe,
+Co, and Ni oxides, within the r2SCAN+U framework (Figure 1). Notably, the optimal U obtained with
+r2SCAN is less than that reported previously for SCAN functional (Table S1) for all 3d TMs considered
+(except V and Fe), which can be attributed to better accuracy of r2SCAN compared to SCAN, as observed
+in non-TMOs. [36] For V oxides, the required U value for VO2 → V2O5, V2O3 → VO2, VO → V2O3 is
+0.0, 0.7, and 2.2 eV, respectively. Thus, the optimal U value for V is 1.0 eV (average of the three required
+U values), which is identical to the U correction required with SCAN. [38] The decreasing required U with
+increasing oxidation state of V in V oxides is expected due to the decrease in the strength of exchange
+interactions among the d electrons as oxidation state increases. In the case of Fe, FeO → Fe2O3 and FeO
+→ Fe3O4 reactions require a U of 2.9 and 3.3 eV, respectively, resulting in an optimal U of 3.1 eV, which
+is also identical to the optimal U with SCAN. [37] Moreover, we obtain the highest optimal U of 3.1 eV
+for Fe, among all TMs considered in this work, which is consistent with the fact that Fe3+ has the highest
+number of unpaired d electrons resulting in the strongest exchange interactions.
+For Ti and Ni, we observe a marginal improvement in the U -value for r2SCAN when compared to SCAN.
+Specifically, we obtain an optimal U of 2.3 eV and 2.1 eV for Ti and Ni, respectively, versus 2.5 eV for
+both elements with SCAN. We find an optimal U value of 1.8 eV for both Mn (2.7 eV with SCAN) and
+Co (3.0 eV with SCAN). In Mn-oxides, the required U for the oxidation of Mn2O3 → MnO2, and MnO →
+5
+
+Mn2O3 are 1.5 and 2.1 eV, respectively. The optimal U for Mn is transferable to other Mn oxides as well,
+indicated by the robust agreement between r2SCAN+U -calculated and experimental oxidation enthalpy for
+MnO → Mn3O4 (green lines in Figure 1c).
+For Cr and Cu oxides, we obtain reasonable agreement with experimental data without a U correction
+(Figure S2), similar to our observation with SCAN. [38] In fact, for Cu, introducing U -correction worsens the
+error in the calculated oxidation enthalpy for Cu2O → CuO versus experiment, similar to our observation
+with SCAN(+U ) as well, which can be attributed to PAW potentials derived at the PBE-level. [38] However,
+the magnitude of error (versus experiment) is smaller with r2SCAN (≈13.1%) than with SCAN (≈25.7%).
+In case of Cr, the oxidation reaction of CrO2 → CrO3 requires U ∼ 0.9 eV, but introducing a U correction
+worsens any agreement with experiment for Cr2O3 → CrO2 (where required U = 0 eV). Thus, the optimal
+U for Cr oxides is 0.45 eV (<0.5 eV), which only provides a marginal improvement in describing oxidation
+enthalpies. Hence, we recommend using only r2SCAN for calculating any Cr oxide framework.
+0
+1
+2
+3
+4
+U(eV)
+-9
+-8
+-7
+-6
+-5
+-4
+Reaction Enthalpy (eV per O2)
+FeO/Fe2O3
+FeO/Fe3O4
+Experimental
+2.9 eV
+3.3 eV
+3.1 eV
+(d)
+(e)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+U(eV)
+-7
+-6
+-5
+-4
+-3
+-2
+-1
+Reaction Enthalpy (eV per O2)
+CoO/Co3O4
+Experimental
+1.8 eV
+0
+0.5
+1
+1.5
+2
+2.5
+U(eV)
+-4
+-3.5
+-3
+-2.5
+-2
+-1.5
+Reaction Enthalpy (eV per O2)
+NiO/LiNiO2
+Experimental
+2.1 eV
+(f)
+(a)
+0
+0.5
+1
+1.5
+2
+2.5
+U (eV)
+-8.4
+-8.2
+-8
+-7.8
+-7.6
+-7.4
+Reaction Enthalpy (eV per O2)
+Ti2O3/TiO2
+Experimental
+2.3 eV
+(b)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+U (eV)
+-8
+-6
+-4
+-2
+0
+2
+Reaction Enthalpy (eV per O2)
+VO/V2O3
+V2O3/VO2
+VO2/V2O5
+Experimental
+0.0 eV
+0.7 eV
+1.0 eV
+2.2 eV
+(c)
+0
+0.5
+1
+1.5
+2
+2.5
+U(eV)
+-7
+-6
+-5
+-4
+-3
+-2
+-1
+0
+1
+Reaction Enthalpy (eV per O2)
+MnO/Mn2O3
+Mn2O3/MnO2
+Experimental
+MnO/Mn3O4
+1.5 eV
+1.8 eV
+2.1 eV
+Figure 1: Calculated oxidation enthalpy versus the magnitude of U correction within r2SCAN+U framework
+for (a) Ti, (b) V, (c) Mn, (d) Fe, (e) Co, and (f) Ni oxides. Solid, dashed, and dotted lines of a given color
+indicate calculated, experimental, and required U values for a given oxidation reaction. Optimal U for each
+TM is indicated by the dotted blue line in each panel.
+3.2
+Lattice parameters
+All r2SCAN(+U ) and SCAN(+U ) calculated lattice parameters, on-site magnetic moments, and band
+gaps for each TMO are tabulated in Table S2. Additionally, the calculated lattice volumes by the four XC
+functionals are plotted against experimental data in Figure 2a for all oxides. Generally, both SCAN (green
+squares in Figure 2a) and r2SCAN (blue symbols) offer < 2.8% deviation from the experimental lattice
+parameters for all the TMOs considered, except VO, FeO, CuO, and LiNiO2, indicating robust agreement
+with experiments for both functionals. In VO, SCAN and r2SCAN overestimate (by ∼8%) the experimental
+lattice constants, while the deviation in FeO and CuO is ∼3-4% and ∼8-10%, respectively.
+In LiNiO2,
+6
+
+SCAN’s β angle evaluation is ∼4.1% different from experiment.
+Notably, SCAN and r2SCAN do show qualitative differences in their calculated lattice parameters (when
+compared against experiments) across TMOs. For instance, both functionals overestimate the experimental
+lattice constants in TiO2, Ti2O3, and VO, while they underestimate in CrO2, CrO3, MnO2, and Fe3O4.
+There are also examples (MnO and Mn2O3) where SCAN underestimates the experimental lattice constants
+while r2SCAN overestimates. Overall, there are cases where SCAN’s errors in lattice parameter estimations
+are lower versus experiments (e.g., Cr2O3, CoO), r2SCAN’s errors are lower (e.g., CrO2, CrO3, MnO2,
+Fe3O4), and both functionals exhibit identical errors (e.g., TiO2, Co3O4, NiO, Cu2O), signifying that both
+functionals offer similar performance in terms of geometrical properties.
+Comparing r2SCAN and SCAN, we find that r2SCAN’s lattice constants are generally larger than SCAN
+across TMOs (e.g., Ti2O3, Cr2O3, CrO3, VO2, etc.). As a range, r2SCAN estimates lattice constants that
+are a maximum of ∼1.5% larger than SCAN (in CrO3) and a minimum of ∼0.1% larger than SCAN (in
+Mn2O3). Having said that, there are instances where r2SCAN’s lattice constant evaluations are lower than
+SCAN (VO, CoO, CuO, and LiNiO2) and cases where both functionals are identical (TiO2, Co3O4, NiO,
+and Cu2O). In specific TMOs, SCAN and r2SCAN calculate an identical (individual) lattice constant, while
+the other lattice constants with r2SCAN are larger than SCAN. For example, a and c lattice constants with
+r2SCAN are higher than SCAN in V2O5 while both functionals estimate b = 3.55 ˚A.
+On introducing the optimal U correction, an increase in the value of calculated lattice constants is ob-
+tained for both SCAN and r2SCAN functionals for all TMOs. The lattice constants computed by r2SCAN+U
+(yellow symbols in Figure 2a) is up to 1.3% higher than r2SCAN, except FeO (∼4.2% higher). Similar to the
+comparison of r2SCAN vs. SCAN, there are systems where r2SCAN+U predicts larger, smaller, and identical
+lattice constants compared to SCAN+U (red triangles). For example, r2SCAN+U calculates larger lattice
+constants than SCAN+U in VO2, V2O5, MnO, Mn2O3 and Fe3O4 (maximum of ∼0.5% higher in V2O5),
+while for Ti2O3, CoO and NiO, r2SCAN+U ’s estimations are smaller than SCAN+U (maximum deviation
+of ∼2.1% in Ti2O3). Both SCAN+U and r2SCAN+U functionals evaluate identical lattice parameters for
+TiO2, Co3O4 and LiNiO2.
+Overall, lattice constants calculated by SCAN+U and r2SCAN+U deviate <∼3.3% from experiments
+for all TMOs, except VO and VO2 where deviations of ∼8.5% and ∼4.6% are observed, respectively. Adding
+U improves the agreement with experiment for both SCAN and r2SCAN in Co3O4, while r2SCAN+U gives
+the best estimate of the lattice parameters in FeO (< 1% deviation vs. experiments) compared to SCAN,
+SCAN+U and r2SCAN. Notably, all functionals break the rocksalt symmetry of VO, MnO, and FeO, while
+the cubic symmetry of Fe3O4 is retained only by SCAN. In Ti2O3, the hexagonal symmetry is broken by
+SCAN but the symmetry is preserved by the other frameworks. In summary, we find that the differences in
+lattice parameter estimations to be minimal across the four functionals on average, with notable exceptions
+of a few systems.
+3.3
+On-site magnetic moments
+On-site magnetic moments of the TMOs (Figure 2c and Table S2) computed by SCAN and r2SCAN
+generally underestimate experimental values, with the exception of MnO2, Mn2O3, CrO2, and VO2. Note
+that larger magnetic moments typically indicate stronger localization of d electrons. Comparing r2SCAN
+and SCAN calculations, we find that r2SCAN typically estimates smaller magnetic moments than SCAN
+but with several exceptions, such as MnO, MnO2, Mn2O3, Cr2O3, and VO2. Thus, on average, SCAN’s
+magnetic moment predictions are in better agreement with experiments. However, in terms of magnitude,
+7
+
+Ti2O3
+TiO2
+VO
+V2O3
+VO2
+V2O5
+Cr2O3
+CrO2
+CrO3
+MnO
+Mn3O4
+Mn2O3
+MnO2
+Fe2O3
+Fe3O4
+FeO
+CoO
+Co3O4
+NiO
+LiNiO2
+Cu2O
+CuO
+SCAN
+SCAN+U
+r2SCAN
+r2SCAN+U
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+On-site magnetic moment
+difference (
+B)
+Ti2O3
+TiO2
+VO
+V2O3
+VO2
+V2O5
+Cr2O3
+CrO2
+CrO3
+MnO
+Mn3O4
+Mn2O3
+MnO2
+Fe2O3
+Fe3O4
+FeO
+CoO
+Co3O4
+NiO
+LiNiO2
+Cu2O
+CuO
+SCAN
+SCAN+U
+r2SCAN
+r2SCAN+U
+0.8
+0.6
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+On-site magnetic moment
+difference (
+B)
+SCAN
+SCAN+U
+r2SCAN
+r2SCAN+U
+2.0
+1.5
+1.0
+0.5
+0.0
+0.5
+1.0
+1.5
+Band gap difference (eV)
+(a)
+(c)
+(b)
+0
+100
+200
+300
+400
+500
+600
+700
+800
+900
+Experimental lattice volume (Å3)
+0
+100
+200
+300
+400
+500
+600
+700
+800
+900
+Predicted lattice volume (Å3)
+SCAN
+SCAN+U
+r2SCAN
+r2SCAN+U
+Figure 2: (a) Comparison of calculated and experimental lattice volume (in ˚A3) of all TMOs considered.
+(b) Violin plot capturing the difference between the experimental and computed band gap (in eV) across
+TMO systems using the four XC frameworks. The empty circle and horizontal line in the inner box plot
+corresponds to the mean and median of the calculated band gaps, respectively. (c) Heat map representation
+of the differences between the experimental and calculated on-site magnetic moments (in µB) using the
+four XC functionals and across all TMOs. A value of zero indicates perfect consistency, while red (blue)
+colors indicate overestimation (underestimation) of magnetic moments. Hatched boxes either correspond to
+experimentally undetermined magnetic moments (VO) or calculations not executed with U frameworks (Cr
+and Cu oxides).
+moments predicted by r2SCAN deviate by < 3% from SCAN’s estimates, except CuO (∼ 6.8% deviation),
+CrO2 (∼ 3.5%), and MnO2 (∼ 3.5%), highlighting that the differences in the predictions are marginal.
+Adding optimal U to both SCAN and r2SCAN increases the magnitude of the calculated on-site magnetic
+moments for all TMOs (except VO2, which is predicted to be metallic by all functionals), consistent with
+the expectation that the U correction facilitates d electron localization. r2SCAN+U -calculated data are
+similar to the corresponding SCAN+U values (< 2.3% variation), except LiNiO2 (∼6.3% variation), and
+Ti2O3 (∼3.8%). Similar to r2SCAN versus SCAN, r2SCAN+U estimates smaller magnetic moments than
+SCAN+U, with notable exceptions being VO2, Mn2O3, MnO2 and FeO. Overall, we observe the accuracy
+in calculated on-site magnetic moments versus experiments to follow the order SCAN+U > r2SCAN+U >
+SCAN > r2SCAN for several TMOs. However, there are specific cases where specific XC frameworks offer
+better accuracy in calculating magnetic moments, such as SCAN in CrO2, Mn2O3, MnO2, Fe3O4 and CuO,
+8
+
+r2SCAN in Mn3O4 and Cr2O3, and r2SCAN+U in V2O3. Given the numerically marginal deviations in
+calculated magnetic moments across the XC frameworks (∼10% deviation), we expect an increase/decrease
+in accuracy to be marginal amongst the XC frameworks considered.
+3.4
+Band gaps
+The differences between calculated and experimental band gaps of all TMOs considered are visualized
+as violin plots for SCAN (green violin), SCAN+U (red), r2SCAN (blue), and r2SCAN+U in Figure 2b.
+The top and bottom ends of the individual violins mark the highest and lowest differences in the respective
+calculated data. Note that the mean values (white empty circles) are similar for SCAN and r2SCAN, and
+in turn are lower than their U -corrected versions. In other words, addition of the U-correction reduces
+the error of calculated band gaps compared to experimental values, which is expected given that semi-local
+DFT typically underestimates band gaps.
+Also, we find that SCAN+U displays the lowest mean band
+gap difference among the XC functionals considered, indicating that on-average SCAN+U provides better
+computed band gaps.
+We present calculated electronic DOS of select TMOs, namely CoO (panels a and b), V2O3 (c and d),
+and Mn2O3 (e and f), in Figure 3, to illustrate qualitative trends in computed band gaps. The DOS for
+the remaining TMOs, calculated by the four XC frameworks, are compiled in Figures S3-S19 of the SI. In
+each DOS panel, solid orange and solid green lines correspond to the 2p-states of O and the 3d-states of the
+TM, respectively. Dashed black lines represent Fermi levels in metallic compounds. Dotted vertical lines
+represent valence and conduction band edges in semiconducting/insulating compounds, with the band gaps
+indicated by the text annotation near the conduction band minimum (CBM). The zero of the energy scale
+is set to the valence band maximum (VBM) for TMOs with a band gap and to the Fermi level in metallic
+TMOs.
+We observe that r2SCAN generally calculates a smaller band gap than SCAN for most TMOs (maximum
+of ∼66% lower in MnO2, see Table S2), as illustrated by the case of CoO in panels a and b of Figure 3.
+Notable exceptions do exist to this observation, such as V2O5 (∼1.7% larger), CrO3 (∼3.2%), MnO (∼4.3%),
+and Fe2O3 (∼1.7%), where r2SCAN calculated band gaps are marginally larger than SCAN. Both SCAN
+and r2SCAN incorrectly describe the ground state electronic configuration of narrow band gap TMOs (i.e.,
+experimental band gaps < 1 eV), including Ti2O3 (Figure S4), V2O3(Figure 3c and S3c), VO2 (Figure S7)
+and Fe3O4 (Figure S15) to be metallic, with the exception of MnO2 where both SCAN and r2SCAN estimate
+a narrow gap (Figures S12a and S12c). Additionally, both functionals also calculate the wrong electronic
+structure in the case of a non-narrow-gap semiconductor, Mn2O3 (Figure S3), which exhibits an experimental
+gap of 1.2-1.3 eV. [83,84] However, SCAN and r2SCAN qualitatively describe the right electronic structure in
+the case of wide band gap TMOs such as FeO (Figure S13), Fe2O3 (Figure S14), and NiO (Figure S17), with
+a significant quantitative underestimation of the experimental gaps. In any case, the differences in electronic
+structure predictions between SCAN and r2SCAN in TMOs are minimal, with SCAN being marginally better
+in accuracy.
+Introducing a U correction to SCAN and r2SCAN widens or opens the band gap, especially in narrow
+band gap TMOs, as illustrated by the case of V2O3 (panels c and d in Figure 3). The opening of band
+gap with U correction is expected since localization of d electrons, which form the VBM and/or CBM in
+3d-TMOs, is faciliated with U addition, in turn resulting in a larger gap. However, in the case of VO2
+(Figure S7), adding U does not capture the MIT that occurs at low temperatures (< 341 K [61]) with either
+SCAN or r2SCAN, causing the erroneous prediction of metallic behavior. Generally, SCAN+U calculates
+9
+
+0.36 eV
+(a)
+0.91 eV
+(b)
+0.61 eV
+(c)
+(d)
+0.24 eV
+(e)
+(f)
+Figure 3: DOS for CoO calculated using (a) SCAN and (b) r2SCAN, DOS for V2O3 computed using (c)
+r2SCAN and (d) r2SCAN+U, and DOS for Mn2O3 estimated using (e) SCAN+U and (f) r2SCAN+U.
+a larger band gap than r2SCAN+U (Table S2), as highlighted by the case of Mn2O3 (panels e and f in
+Figure 3). In fact, SCAN+U is the only framework (among those considered) to estimate a band gap in
+10
+
+6
+/eV
+(states)
+4
+2
+Density of states (
+-6
+-4-3　-2　-1
+2
+3
+4
+5
+Energy (eV)6
+/eV
+(states)
+4
+Density of states (
+-6
+.5
+-4　-3-2
+0
+2
+3
+4
+Energy (eV)/eV)
+60
+Mnd
+(states)
+40
+20
+MWN
+Density of states
+-20
+40
+-60
+-6
+-5
+4
+¥-3
+¥-2-101
+2
+3
+4
+Energy (eV)60
+/eV)
+Mnd
+(states)
+40
+20
+Density of states
+-20
+40
+-60
+-6
+-5
+4
+2
+3
+4
+Energy (eV)/ev
+10
+(states)
+5
+Density of states
+0
+-10
+-6
+-5
+-2
+0
+2
+3
+4
+Energy (eV)/eV
+10
+(states)
+Density of states
+Wi
+-10
+-6
+-5
+¥1
+2
+3
+4
+Energy (eV)Mn2O3, which is consistent with experiment. Moreover, SCAN+U ’s evaluations of larger band gaps results
+in better (poorer) quantitative agreement with experiments in wide (narrow) gap materials, such as MnO
+and FeO (V2O3 and MnO2).
+Note that SCAN+U and r2SCAN+U do underestimate the experimental band gaps, similar to SCAN and
+r2SCAN, in wide gap TMOs. The only exception to this observation is CoO, where SCAN+U overestimates
+the band gap versus experiment (Figure S3a and Table S2), as also observed in our previous work. [38] In select
+TMOs, including Fe2O3 and V2O5, r2SCAN+U ’s band gap is larger than SCAN+U, but the magnitude of
+difference (≤ 0.2 eV) is meagre. Thus, for electronic structure predictions, we expect SCAN+U to provide the
+best qualitative and quantitative band gaps across TMOs, among the functionals considered here, especially
+for wide gap semiconductors/insulators. However, the qualitative trends provided by r2SCAN+U are quite
+robust as well and in small gap semiconductors (< 1 eV gap), r2SCAN+U ’s quantitative accuracy is often
+better than SCAN+U.
+3.5
+Transferability checks
+To examine the transferability of the optimal U values determined in this work (with r2SCAN), to oxide
+systems not used for obtaining the values, we perform checks on systems with different oxidation state and/or
+coordination environment for each TM. We compare calculated values against available experimental data,
+such as structural, electronic, magnetic, and/or electrochemical properties. Specifically, we choose Ba2TiO4
+as a check for Ti, BiVO4 for V, K3MnO4, K2MnO4, and Mn2O7 for Mn, SrFeO3 for Fe, LiCoO2-CoO2 for
+Co, and LiNiO2-NiO2 for Ni. Data related to transferability checks are compiled in Figure 4, Table 1, and
+Table S3.
+In the case of Ba2TiO4, we compare the calculated lattice parameters with experimental values (see
+Table S3 and lattice voliume differences plotted in Figure 4). Ba2TiO4 crystallizes in a monoclinic structure
+(space group P21/n) at low temperatures, where the unit cell is composed of four formula units. [85, 86]
+Ti atoms are present in distorted tetrahedra composed of neighbouring oxygen atoms (TiO4) within the
+Ba2TiO4 lattice, which is different from the octahedral environments sampled in TiO2 and Ti2O3. Upon
+structure relaxation, we observe that both r22SCAN and r22SCAN+U functionals marginally overestimate
+(by ∼2%) experimental lattice parameters (Figure 4 and Table S3). Similar to trends observed in Table S2,
+adding U to r2SCAN increases the calculated lattice parameters in Ba2TiO4 (by ∼0.03 ˚A), thereby marginally
+reducing the agreement with experiment.
+We benchmark both structural and electronic properties of BiVO4 as a transferability check for V-based
+systems. Note that BiVO4 transforms from tetragonal (I 41/a) to a monoclinic (I 2/b) ‘scheelite’ phase below
+∼ 528 K, [87, 88] which is a reversible second order ferroelastic transition driven by soft optical phonon
+modes. The BiVO4 unit cell possesses four formula units, with tetrahedrally coordinated V ions, which is
+different from the coordination environments of V in VO, V2O3, VO2, and V2O5. Importantly, monoclinic-
+BiVO4 spontaneously transforms to the tetragonal structure upon structure relaxation with r22SCAN and
+r22SCAN+U, similar to the observation by Liu et al [87] with GGA and hybrid functionals. Thus, neither
+r22SCAN nor r22SCAN+U predict the correct ground state structure. Additionally, BiVO4 possess a band
+gap of 2.4–2.48 eV [89] and is a candidate photocatalyst. [87] Both r22SCAN and r2SCAN+U provide similar
+band gap predictions (2.01-1.98 eV), which is in good qualitative agreement with experiment. Surprisingly,
+r2SCAN+U evaluates a marginally lower band gap than r2SCAN (see panels a and b in Figure 4). However,
+both r22SCAN and r2SCAN+U predict similar states occupying the valence band (Op) and conduction band
+(Vd) edges.
+11
+
+Ba2TiO4
+BiVO4
+K3MnO4
+K2MnO4
+Mn2O7
+SrFeO3
+r2SCAN
+r2SCAN+U
+20
+10
+0
+10
+20
+Lattice volume
+difference (Å3)
+(a)
+1.98 eV
+(b)
+(c)
+2.013 eV
+Figure 4: DOS for BiVO4 calculated using (a) r2SCAN and (b) r2SCAN+U. (c) Difference between experi-
+mental and calculated lattice volumes (using r2SCAN and r2SCAN+U ), plotted as a heatmap, for various
+systems. Red (blue) squares indicate overestimated (underestimated) calculated lattice volumes versus ex-
+periment.
+The rationale behind the choice of K3MnO4, K2MnO4, and Mn2O7 as checks for Mn-based systems is to
+explore the higher, unsampled oxidation states of Mn, namely +5, +6, and +7 in K3MnO4, K2MnO4, and
+Mn2O7, respectively. Also, Mn resides in tetrahedral coordination in these compounds, which is different from
+the octahedral coordination observed in MnO, Mn2O3, and MnO2. Although Mn2+ resides in tetrahedral
+sites in spinel-Mn3O4, we had not used in the spinel structure to obtain our optimal U. We benchmark the
+calculated lattice parameters versus experiments for all Mn-based transferability checks.
+Mn2O7 is a volatile liquid at 298 K and solidifies to a monoclinic crystal structure (P21/c) below ∼ 279 K,
+with the unit cell consisting of 8 formula units of corner sharing tetrahedral MnO4 pairs. [90, 91] Upon
+structural relaxation, both r2SCAN and r2SCAN+U underestimate the lattice constants of monoclinic-
+Mn2O7 by ∼1-3% (Figure 4 and Table S3). In the case of K3MnO4, the tetragonal symmetry (I 42m) [92]
+is broken with r2SCAN functional resulting in an orthorhombic structure, while the symmetry is preserved
+by r2SCAN+U (see Figure
+4 and Table S3). Nonetheless, both r2SCAN and r2SCAN+U significantly
+underestimate the c parameter (by ∼ 13.5%) and overestimate the a or b parameter (∼ 10.2%). K2MnO4 is
+an orthorhombic crystal (Pnma) with four formula units per unit cell. [93] Here, r2SCAN and r2SCAN+U
+predict identical lattice parameters, which marginally underestimate experimental values (by ∼ 0.4-1%, see
+Figure 4 and Table S3).
+The choice of SrFeO3, a cubic perovskite, as a check for Fe is largely motivated by the 4+ oxidation
+state exhibited by Fe in the structure, which is not sampled in FeO, Fe2O3, or Fe3O4. Both r2SCAN and
+r2SCAN+U preserve the cubic symmetry during structure relaxation, with r2SCAN+U ’s lattice parameters
+12
+
+(states/eV
+Bi,
+Bi
+Density of states (
+-6
+-5
+-4　-3
+¥-2
+0
+2
+3
+4
+Energy (eV)(states/eV
+Bip
+Bi
+Density of states (
+-6
+-5
+-4 -3
+¥-2
+0
+2
+3
+4
+Energy (eV)(states/eV
+Bi,
+Bi
+Density of states (
+-6
+-5
+-4　-3
+¥-2
+0
+2
+3
+4
+Energy (eV)(states/eV
+Bi,
+Bi
+Density of states (
+-6
+-5
+-4　-3
+¥-2
+0
+2
+3
+4
+Energy (eV)(states/eV
+Bi,
+Bi
+Density of states (
+-6
+-5
+-4　-3
+¥-2
+0
+2
+3
+4
+Energy (eV)(states/eV
+Bi,
+Bi
+Density of states (
+-6
+-5
+-4　-3
+¥-2
+0
+2
+3
+4
+Energy (eV)identical to experiments and r2SCAN’s parameters being a slight underestimation (∼ 0.5%, see Figure 4
+and Table S3). In terms of magnetic configuration of Fe in SrFeO3, Takeda et al. [94] reported a helical spin
+structure via their neutron diffraction experiments, with competing FM and AFM interactions. However,
+Shein et al. [95] found a FM metallic state to be the ground state of SrFeO3, over a wide range of pressures,
+based on their first principles calculations, which they attributed to stronger FM than AFM interactions. We
+considered a FM configuration of Fe atoms in the SrFeO3 unit cell, and the on-site magnetic moments on Fe
+calculated by both r2SCAN (3.375 µB, Table 1) and r2SCAN+U (3.819 µB) overestimate the experimental
+value (2.7±0.4 µB [94]). However, our calculated magnetic moments do indicate a localization of ∼4 electrons
+on the d orbitals of Fe, consistent with its +4 oxidation state.
+We choose CoO2 (R3m or ‘O3‘ polymorph [96]), and NiO2 (P1m1 or ‘O1’ [97]), both layered structures,
+as transferability checks for Co and Ni, respectively, owing to the unsampled 4+ oxidation states of each
+TM. In terms of experimental property to benchmark, we choose the average Li intercalation voltage in these
+structures, i.e., LiCoO2-CoO2, and LiNiO2-NiO2 pairs, since they have been measured with high precision.
+The reader is referred to previous works on calculating and benchmarking average ‘topotactic’ intercalation
+voltages. [98,99] r2SCAN underestimates the experimental average voltage [96,99–103] in LiNiO2-NiO2 (by
+∼ 8%), while it overestimates the average voltage in LiCoO2-CoO2 (by ∼ 1.7%), similar to trends observed
+with SCAN. [99] The addition of U to r2SCAN leads to an improvement in agreement with the experimental
+voltage in the Ni-system (deviation of ∼ 1.8%), while it worsens the agreement in the Co-system (deviation
+of ∼ 4.4%). Nevertheless, r2SCAN+U does overestimate the average voltage in both Co and Ni systems,
+similar to the behavior of SCAN+U. [99]
+Table 1: Voltage and magnetic moments calculated by r2SCAN, and r2SCAN+U compared against experi-
+mental values (denoted by ‘Expt.’). The U values used with r2SCAN+U are the corresponding optimal U
+values obtained for each TM (from Figure 1).
+Composition
+Source
+Voltage
+Magnetic moment
+(space group)
+(V)
+(µB)
+LiCoO2-CoO2
+Expt.
+4.05
+-
+(R¯3m)
+r2SCAN
+4.12
+-
+r2SCAN+U
+4.23
+-
+LiNiO2-NiO2
+Expt.
+3.85
+-
+(P1m1)
+r2SCAN
+3.54
+-
+r2SCAN+U
+3.92
+-
+SrFeO3
+Expt.
+-
+2.7±0.4
+(Pm¯3m)
+r2SCAN
+-
+3.375
+r2SCAN+U
+-
+3.819
+4
+Discussion
+In this work, we evaluated the performance of the r2SCAN functional among binary TMOs consisting
+of 3d-TMs by calculating the oxidation enthalpies, lattice parameters, on-site magnetic moments, and band
+gaps. Additionally, for each TM-O2 system considered, we calculated the optimal Hubbard-U corrections
+to be used in a r2SCAN+U framework, based on experimental oxidation enthalpies. Although theoretical
+approaches exist to derive U values, [41–47] using oxidation enthalpies nominally gives an “average” correc-
+tion that is suitable across several oxidation states of a given TM. Specifically, our optimal U values are 2.3,
+13
+
+1.0, 1.8, 3.1, 1.8, and 2.1 eV for Ti, V, Mn, Fe, Co, and Ni, respectively, while we don’t deem a U correction
+necessary for Cr and Cu oxides. Interestingly, the optimal U corrections needed with r2SCAN are lower in
+magnitude compared to SCAN for Ti, Mn, Co, and Ni oxides (while the corrections are identical for V and
+Fe oxides), indicating that r2SCAN exhibits lower errors with oxidation enthalpies and possibly lower SIEs
+than SCAN. However, this is not reflected in other physical properties. On an average, we find the accuracy,
+versus experimental values, to be similar for r2SCAN compared to SCAN, and for r2SCAN+U compared to
+SCAN+U, respectively, in lattice parameter, on-site magnetic moment, and band gap evaluations as seen in
+Figure 2.
+The general trends in lattice parameter, magnetic moment, and band gap predictions, across the XC
+frameworks considered, can be summarized as follows. We observe that r2SCAN generates larger lattice
+constants than SCAN and on addition of the U correction to both functionals, the lattice constants further
+increase. Thus, in systems where SCAN underestimates experimental lattice constants (e.g., CrO2, CrO3,
+MnO2), shifting to r2SCAN improves agreement (e.g., error in r2SCAN in CrO3 is 0.8% versus 2.3% with
+SCAN). Also, there are instances where the ground state symmetry of the TMO is not preserved by some
+or all of the XC frameworks considered (i.e., in VO, MnO, FeO, Fe3O4, and Ti2O3), highlighting systematic
+issues in the XC treatment across the four frameworks considered. The calculated on-site magnetic moments
+by r2SCAN (and r2SCAN+U ) are marginally lower than SCAN (SCAN+U ), with the U correction nom-
+inally increasing the calculated moments calculated by r2SCAN and SCAN. However, calculated magnetic
+moments across the four XC frameworks differ by < 10% (except LiNiO2), signifying marginal differences
+in accuracy. Both SCAN and r2SCAN underestimate band gaps across all TMOs (except MnO2), with
+band gaps calculated by r2SCAN typically being lower than SCAN, and adding the U opens/widens the
+gap. Thus, SCAN+U offers the best quantitative accuracy versus experimental band gaps, especially for
+wide gap semiconductors. Note that the qualitative trends from r2SCAN+U are consistent with the trends
+exhibited by SCAN+U and should be reliable in electronic structure predictions in other TM-based oxide
+systems.
+r2SCAN adopts the smooth polynomial interpolation function of rSCAN to maintain numerical stability
+during SCF calculations. Additionally, the reformed gradient expansion for correlation introduced in r2SCAN
+(partially) negates the error introduced to the slowly varying density by the non-vanishing interpolation
+function, [32] which largely accounts for the observed variation in accuracy of r2SCAN versus SCAN. Based
+on our data, we observe that r2SCAN is not systematically more accurate than SCAN across all TMOs
+and for all property predictions. For example, we have lower optimal U values indicating lower SIEs with
+r2SCAN versus SCAN, but also lower on-site magnetic moments (except Mn and Cr oxides) signifying poorer
+d-electron localization with r2SCAN. Further, the smaller band gaps with r2SCAN (versus SCAN) may be
+caused by the residual SIEs, resulting in an underestimation of the CBM across TMOs. Hence, usage of
+r2SCAN(+U ) in TM-based systems must be done with care and efforts should be made to benchmark as
+many available experimental properties as possible before performing “true” computational predictions.
+We considered the transferability of the U values estimated in this work, with r2SCAN, by examining
+systems for each TM with oxidation states and/or coordination environments not sampled while calculating
+the optimal U. In general, we find that r2SCAN or its Hubbard U corrected version estimate similar lattice
+parameters and hence yield similar accuracies on structural properties. Analogously, the calculated on-site
+magnetic moments in SrFeO3 and the band gaps in BiVO4 are similar between r2SCAN and r2SCAN+U. In
+case of electrochemical properties, we do find tangible variations in the calculated average voltages of r2SCAN
+and r2SCAN+U, with r2SCAN+U exhibiting overall lower errors across the Co and Ni systems. Thus, we
+14
+
+SCAN+U
+r2SCAN
+r2SCAN+U
+Ti
+V
+Cr
+Mn
+Fe
+Co
+Ni
+Cu
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
+Ti
+V
+Cr
+Mn
+Fe
+Co
+Ni
+Cu
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+(a)
+Ti
+V
+Cr
+Mn
+Fe
+Co
+Ni
+Cu
+0.6
+0.8
+1
+1.2
+(c)
+(b)
+Rela�ve overall 
+computa�on �me
+Rela�ve computa�on �me
+per ionic step
+Rela�ve computa�on �me
+per electronic step
+Figure 5: (a) Overall computational time (electronic+ionic steps) (b) computational time per ionic step and
+(c) computational time per electronic loop taken for each TM-O2 binary system with SCAN+U, r2SCAN,
+and r2SCAN+U frameworks relative to SCAN. Values greater (smaller) than 1 in each panel indicates that
+a given calculation is slower (faster) than SCAN.
+15
+
+find the optimal U values obtained in this work to be transferable across oxide frameworks not sampled a
+priori. Nevertheless, more benchmarking studies to compare the performance of r2SCAN+U with r2SCAN
+(and experiments) will help in quantifying the reliability and errors associated with using r2SCAN+U.
+Given that r2SCAN(+U ) is not systematically more or less accurate than SCAN(+U ), the computational
+performance and numerical stability of r2SCAN(+U ) is critical in determining its utility in property pre-
+dictions across materials. Thus, we have quantified the computational time of r2SCAN(+U ) and SCAN+U
+relative to SCAN for each TM-O2 system considered in Figure S1. Specifically, panels a, b, and c of Fig-
+ure 5 plot the overall (electronic+ionic steps), per ionic step, and per electronic step computational time,
+respectively, taken by the SCAN+U (blue bars), r2SCAN (red), and r2SCAN+U (yellow) frameworks, rel-
+ative to the computational time taken by the SCAN functional (dotted black lines), for each TM-based
+set of oxides. Details on calculating the computational times used by the functionals is described in the
+‘Computational time’ section of the SI. Note that our objective is not to provide a rigorous quantification of
+computational resources required for each XC framework, but to provide a qualitative understanding of the
+relative computational costs across the frameworks considered.
+For each electronic step, r2SCAN(+U ) is typically faster than SCAN (Figure 5), signifying better numer-
+ical stability than SCAN, with Mn, Ni, and Cu oxides being marginal exceptions. In contrast, on a per-ionic
+step basis, r2SCAN and r2SCAN+U is slower than SCAN, by ∼1.05-1.78× and ∼1.1-1.31×, respectively,
+highlighting that r2SCAN(+U ) takes more electronic steps to converge per ionic step. Importantly, the over-
+all computational time (ionic+electronic steps, Figure 5) required for structural relaxation of TMOs using
+r2SCAN and r2SCAN+U is lower than SCAN, by ∼12.1-61.2% and ∼1.9-34.5%, respectively, except in Fe
+oxides, indicating that r2SCAN(+U ) takes lower number of ionic steps to converge, which possibly indicates
+a better description of atomic forces. The higher overall computation time in Fe oxides with r2SCAN(+U )
+than SCAN is primarily due to the difficulty in converging Fe3O4 with r2SCAN(+U ). Comparing r2SCAN
+and r2SCAN+U, we find that r2SCAN+U takes a higher overall computational time to converge, except
+in Fe and Ni oxides.
+Thus, we expect r2SCAN(+U ) to provide good utility in property predictions in
+TM-containing systems given its better computational performance and reasonable accuracy compared to
+SCAN(+U ).
+5
+Conclusion
+3d-TMs and their compound phases find applications in several fields such as energy storage, solar
+cells, catalysts, thermochemical water splitting, etc., and it is imperative to predict their properties such
+as lattice constants, magnetic moments, reaction enthalpies, and band gaps accurately using DFT-based
+techniques for designing better materials. Recently, the r2SCAN metaGGA XC functional was proposed
+to exhibit the accuracy of its predecessor, SCAN, and the computational performance of rSCAN in main-
+group compounds, but the accuracy of r2SCAN was not rigorously tested on TM-based systems.
+Here,
+we assessed the numerical accuracy and computational performance of r2SCAN in binary 3d-TMOs, in
+calculating the lattice parameters, on-site magnetic moments, binary oxidation enthalpies, and band gaps
+against experimental data. Notably, we observed that r2SCAN exhibited similar qualitative trends as that
+of SCAN, with marginally larger estimations of lattice parameters than SCAN, while the on-site magnetic
+moments and band gap calculations are marginally smaller than SCAN. While both r2SCAN and SCAN
+underestimated the band gaps in wide gap TMOs, with SCAN offering slightly better accuracy, they failed
+to predict the correct ground state electronic configurations of narrow band gap TMOs (e.g., Mn2O3).
+16
+
+On analysing the addition of Hubbard U -correction to improve the accuracy of the r2SCAN functional,
+we observed that a lower optimal U value, based on experimental oxidation enthalpies, was required in
+a r2SCAN+U framework for Ti, Mn, Co and Ni oxides, when compared to a SCAN+U framework. The
+optimal U values were identical in both r2SCAN+U and SCAN+U frameworks for V and Fe oxides, while we
+did not observe the need for a U correction in Cr and Cu oxides with r2SCAN, similar to SCAN. Moreover,
+introducing the U -correction to SCAN and r2SCAN increased the calculated lattice parameters, on-site
+magnetic moments and the band gaps of the TMOs.
+r2SCAN+U and SCAN+U successfully opened a band gap for narrow gap TMOs (except VO2 and
+Mn2O3 with r2SCAN+U ). Upon testing the optimal U values with r2SCAN+U on oxides with different
+oxidation states and/or coordination environments, we found that the U values derived in this work are in
+general transferable to other TM-containing oxides as well. Furthermore, we observed that r2SCAN(+U )
+took less overall computational time (ionic+electronic steps) to converge when compared to SCAN, which
+indicated that r2SCAN(+U ) was computationally more efficient than SCAN(+U ). Since r2SCAN+U offers
+a reasonably accurate prediction of material properties at a lower computational expense than SCAN+U, we
+observe that r2SCAN+U can be used in high-throughput materials discovery, after adequate benchmarking
+tests are done in each new chemical space explored.
+Acknowledgments
+G.S.G. acknowledges the Indian Institute of Science (IISc) Seed Grant, SG/MHRD/20/0020 and SR/MHRD/20/0013
+and the Science and Engineering Research Board (SERB) of the Department of Science and Technology, Gov-
+ernment of India, under sanction numbers SRG/2021/000201 and IPA/2021/000007 for financial support.
+R.D. thanks the Ministry of Human Resource Development, Government of India, for financial assistance.
+S.S. acknowledges financial support from SERB under IPA/2021/000007. All the authors acknowledge the
+computational resources provided by the Supercomputer Education and Research Centre, IISc, for enabling
+some of the density functional theory calculations showcased in this work.
+Author Contributions
+G.S.G. envisioned and designed the work. S.S. and R.D. performed the calculations. All authors con-
+tributed in data analysis and writing the paper.
+Conflicts of Interest
+The authors declare no competing financial or non-financial interests.
+Availability of data
+The data that support the findings of this study are openly available at https://github.com/sai-mat-
+group/r2SCAN-U-benchmarking.
+Supplementary Materials
+Electronic Supporting Information is available online at , with details on the crystal structures used for
+calculations, oxidation energetics of Cr and Cu oxides, densities of states of all systems not showcased in the
+17
+
+main text, and details on computational time calculations.
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf,len=1342
+page_content='Performance of the r2SCAN functional in transition metal oxides  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Swathilakshmi1, Reshma Devi1, and Gopalakrishnan Sai Gautam1,*  1Department of Materials Engineering, Indian Institute of Science, Bengaluru, 560012,  India  Email: saigautamg@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='in  Abstract  We assess the accuracy and computational efficiency of the recently developed meta-generalized gra-  dient approximation (metaGGA) functional, the restored regularized strongly constrained and appropri-  ately normed (r2SCAN), in transition metal oxide (TMO) systems and compare its performance against  SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Specifically, we benchmark the r2SCAN-calculated oxidation enthalpies, lattice parameters, on-  site magnetic moments, and band gaps of binary 3d TMOs against the SCAN-calculated and experimental  values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Additionally, we evaluate the optimal Hubbard U correction required for each transition metal  (TM) to improve the accuracy of the r2SCAN functional, based on experimental oxidation enthalpies,  and verify the transferability of the U values by comparing against experimental properties on other  TM-containing oxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Notably, including the U -correction to r2SCAN increases the lattice parameters,  on-site magnetic moments and band gaps of TMOs, apart from an improved description of the ground  state electronic state in narrow band gap TMOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' The r2SCAN and r2SCAN+U calculated oxidation en-  thalpies follow the qualitative trends of SCAN and SCAN+U, with r2SCAN and r2SCAN+U predicting  marginally larger lattice parameters, smaller magnetic moments, and lower band gaps compared to SCAN  and SCAN+U, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' We observe that the overall computational time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=', for all ionic+electronic  steps) required for r2SCAN(+U ) to be lower than SCAN(+U ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Thus, the r2SCAN(+U ) framework can  offer a reasonably accurate description of the ground state properties of TMOs with better computational  efficiency than SCAN(+U ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  1  Introduction  Density functional theory (DFT [1]) calculations are the bedrock of modern computational materials  science in terms of predicting thermodynamic and kinetic properties, with such property predictions being  put to use in subsequent materials discovery [2–7] and understanding underlying physical phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [8–12]  In recent years, machine learning has been used to augment DFT in property predictions, thereby reducing  computational cost and accelerating materials discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [13–17] Note that a key approximation within  DFT is the exchange-correlation (XC) functional, the exact form of which is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' However, several  approximations for the XC functional have been proposed over the years, which can be categorized into  different classes depending on the degree of sophistication and accuracy, and visually represented as rungs  on the Jacob’s ladder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [1, 2, 18, 19] As with most computational tools, the higher the accuracy (higher up  Jacob’s ladder) higher is the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='00535v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='mtrl-sci]  2 Jan 2023  Most DFT calculations for “large” solid systems (10s to 100s of atoms) are performed using the Perdew-  Burke-Ernzerhof (PBE) parameterization of the generalized gradient approximation (GGA) XC functional,  [20] as it offers fair accuracy at reasonable computational cost for a wide variety of materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [21–23]  Specifically, GGAs include the local electron density as well as the gradient of the electron density in  describing the XC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' As a semilocal functional of electron density, PBE captures short range interactions but  fails to capture medium and long-range dispersions and also exhibits large electronic self-interaction errors  (SIEs), especially in highly correlated systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [24, 25] Also, PBE typically underestimates the formation  energies [26,27] and semiconductor band gaps of crystalline solids, [26,28] while overestimating their lattice  volumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [26,29]  As we move higher in the Jacob’s ladder, [19] we obtain metaGGA functionals, which may account for  medium range dispersions and exhibit lower SIEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Some metaGGAs consider orbital kinetic energy density  in addition to the local electron density and its gradient, such as the recently developed strongly constrained  and appropriately normed (SCAN [30]) functional, which offers better numerical accuracy than PBE and  satisfies all 17 known constraints for a XC functional (namely, 6 for exchange, 6 for correlation, and 5  for both).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' The iso-orbital indicator (α), which includes the kinetic energy density in SCAN, distinguishes  various bonding environments in a given material and consequently improves the accuracy of SCAN over  GGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' However, SCAN suffers from numerical instability during self-consistent-field (SCF) calculations [31]  wherein denser k-grids (than PBE) are required for accurate and consistent predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [31–33] Thus it is  computationally expensive (per SCF step) compared to PBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [21]  To overcome the numerical instability and reduce the computational cost of SCAN, Bartok and Yates [34]  developed regularized SCAN (rSCAN), which satisfies 13 out of the 17 known constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' The authors  replaced the non-analytical switching α interpolation function in SCAN with a simple polynomial function,  which improves computational speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [35] However, subsequent investigations showed a significant drop in  numerical accuracy with rSCAN (compared to SCAN), which is attributed to the failure of the polynomial  α function to fully recover the uniform gas limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [31, 32] Subsequently, Furness et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [32] introduced the  restored regularized SCAN (or r2SCAN), wherein the constraints broken by rSCAN were restored except  the fourth order gradient expansion constraint for exchange (or GE4X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Furness et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' claimed that the new  r2SCAN functional combines the numerical accuracy of SCAN and computational speed of rSCAN as the  smooth polynomial α function of rSCAN is modified to satisfy the uniform gas limit in r2SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [32] Recently,  Kingsbury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [36] demonstrated that r2SCAN functional indeed delivers robust numerical accuracy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=',  similar to SCAN) and better computational performance (faster and numerically stable) by comparing  r2SCAN and SCAN for solids using a high-throughput computational workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Specifically, the authors [36]  reported that while r2SCAN predicts a smaller band gap (for most of the strongly-bound materials) and  larger lattice volumes than SCAN, the mean atomization error with r2SCAN is ∼15-20% lower for most  solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' However, the performance of r2SCAN in correlated electron systems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=', transition metal oxides  (TMOs) containing open-shell d electrons, remains to be seen and forms the main focus of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Despite the accuracy of SCAN, it still has shortcomings in TMOs, which can be mitigated by adding an  on-site Hubbard U correction term for the transition metal (TM) under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [37,38] This approach  is similar to the one followed to mitigate the SIEs of PBE in TMOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [39,40] However, the magnitude of the  U correction required is not known a priori, and there are both theory-based approaches such as density  functional perturbation theory, [41] linear response theory, [42–44] embedded Hartree-Fock method, [45,46]  and machine learning based Bayesian optimisation, [47] and experimental-data-based approaches to identify  the appropriate U values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' For example, Gautam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [37, 38] used the experimental oxidation enthalpies  2  among binary TMOs to identify optimal U values across various oxidation states of 3d TMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' A similar  experimental-data-based Hubbard U correction scheme can be developed in conjunction with r2SCAN as  well, resulting in a r2SCAN+U framework, in case r2SCAN exhibits similar SIEs as SCAN in TMOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' We  explore the usefulness of such a r2SCAN+U framework also in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Here, we verify the numerical accuracy and computational efficiency of the r2SCAN and r2SCAN+U  frameworks in comparison to SCAN and SCAN+U, respectively, in describing material properties such as  lattice parameters, on-site magnetic moments, and band gaps of binary 3d TMOs, including Ti, V, Cr,  Mn, Fe, Co, Ni, and Cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' As necessary, we evaluate the optimal Hubbard U correction with r2SCAN for  each TM by using the experimental-data-based approach employed in previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [37,38] We find that  r2SCAN predicts marginally larger lattice constants and smaller on-site magnetic moments than SCAN for  most of the TMOs considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' On addition of the U -correction to both SCAN and r2SCAN, we observe an  increase in the calculated lattice constants, on-site magnetic moments and band gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In the case of narrow  band gap TMOs, SCAN+U and r2SCAN+U generally estimate a non-zero band gap, with r2SCAN+U ’s  band gap in better agreement with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Also, we perform transferability checks for the optimal U  values derived in this work for each TM, by benchmarking various properties in oxides that were not used  in obtaining the U values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Finally, we compare the computational performance of r2SCAN/r2SCAN+U  relative to SCAN/SCAN+U to explore the accuracy-cost trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' We report that r2SCAN/r2SCAN+U  is computationally less expensive than SCAN and SCAN+U, when all required ionic and electronic steps  are taken into account for convergence during structure relaxations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' We hope that our work will provide a  foundational basis for further studies on understanding material behavior and computationally discovering  new materials in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  2  Methods  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='1  Computational Methods  We used the Vienna ab-initio simulation package (VASP 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='1) [48–50] for all the spin-polarized DFT  calculations, where the frozen-core PBE-based projector augmented wave (PAW) [51] potentials employed  were identical to previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [37, 38] The plane waves for each system were expanded up to a kinetic  energy of 520 eV, with each structure converged until the total energy differences and atomic forces became  <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='01 meV and <|0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='01| eV/˚A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  We adopted a Γ-centered Monkhorst-Pack [52] grid with a  density of 48 k-points per ˚A for all systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  The conjugate gradient algorithm was used to relax the  structures (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=', cell shapes, volumes, and ionic positions), without preserving any underlying symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' An  ‘accurate’ level of precision was maintained while projecting the wavefunctions in the reciprocal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' The  Fermi surface of each system was integrated with a Gaussian smearing of partial occupancies, with a width  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='05 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In terms of DFT+U calculations, we used the Dudarev framework [53] for adding a effective  U correction on the d orbitals of TM atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' All U values used in SCAN+U calculations were taken from  previous work (see Table S1 of the Supporting Information –SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [37,38] Since we used different computing  systems to perform our structure relaxations for different systems, we normalized the computational time  with the number of cores used in each calculation to compare the computational efficiency of the different  XC functionals considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  For calculating band gaps, GGA functionals typically use the Kohn Sham potential as a multiplicative  term, which typically underestimates the band gap of solids even at the SCAN level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [54, 55] Here, we  3  use the generalized Kohn Sham technique to determine the band gaps by calculating the density of states  (DOS) for all systems considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' For each DOS calculation, we used the optimized structure and the initial  charge density from a previous structure relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Subsequently, we introduced a set of zero-weighted k-  points, corresponding to a density of 96 k-points per ˚A, where the k-points that were used for the structure  relaxation retained their original weights (as determined by VASP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Finally, we performed a single-SCF  calculation where the DOS was sampled between energies of -20 to 20 eV in steps of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='005 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='2  Structures and magnetic configurations  We considered the binary oxides of each TM, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=', Ti, V, Cr, Mn, Fe, Co, Ni, and Cu with different  oxidation states, similar to previous studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [37, 38] The main criteria in selection of these metal oxides  are the availability of reliable thermodynamic data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=', formation energies [56–58]) and the experimentally-  determined ground-state structures that are compiled in the inorganic crystal structure database (ICSD) [59]  Note that the structures from the ICSD were the initial structures in all our DFT structure relaxations,  including the systems used as transferability checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In the case of Ni oxides, we chose NiO and LiNiO2  (similar to previous work, [38]), as reliable thermodynamic data is not available for higher-oxidation-state  binary Ni oxides (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=', Ni2O3 and NiO2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' The TM in all oxides, except select Co and Ni compounds, was  initialized in its high-spin configuration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=', high-spin configuration of Fe3+ consists of five unpaired d  electrons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' A detailed description of all structures utilised in this work is provided in the SI, under the  ’Crystal Structures’ section, with the magnetic configurations depicted in Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  The magnetic configuration of each TMO considered (see Figure S1) was initialized to its appropriate  (in several cases, experimentally-known) ground state configuration during the structural relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' For  example, we considered the ferromagnetic (FM) ground state configuration for CrO2 and VO2, given that  CrO2 is metallic [60] and VO2 undergoes a metal-to-insulator transition (MIT) below 341 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [61] The  rocksalt (RS) TMOs, namely, VO, MnO, FeO, CoO, and NiO were initialized with their experimentally-  known type-II antiferromagnetic (AFM) configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [62–67] Each Ni’s spin in NiO was initialized with  two unpaired d electrons (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=', its high-spin configuration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In CuO, we arranged the magnetic moments of  Cu2+ antiferromagnetically along the Cu-O-Cu chains in the [10¯1] direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [68,69]  We initialized α-Mn2O3 (bixbyite structure) in a FM configuration as this configuration was found to  be the most stable in previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [37] AFM configurations were utilized for rutile-MnO2, [70], and the  other TM2O3 oxides, namely, V2O3, Fe2O3, Ti2O3, and Cr2O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Note that V2O3 becomes AFM below  its MIT temperature, [71–73] while Fe2O3 displays an AFM configuration with the magnetic moment of  Fe alternating every two consecutive layers along the c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [74] Cr2O3 and Ti2O3 exhibit ↑↓↑↓ and ↑↓↓↑  magnetic configurations, respectively, on the TM centers along the a-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [75,76]  In case of spinels, we used different ferrimagnetic (FIM) configurations, as per experimental observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  For example, spinel-Fe3O4 contains both Fe3+ and Fe2+, with up-spin Fe3+ occupying tetrahedral sites and  down-spin Fe3+ occupying half the octahedral sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' The remaining octahedral sites in Fe3O4 are occupied  by up-spin Fe2+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [77, 78] In Co3O4, no-spin Co3+ occupies octahedral sites, while high-spin Co2+ (three  unpaired d electrons) occupies tetrahedral sites in an AFM configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [79–81] For Mn3O4, we adopted  the ”FIM6” configuration, as this was found to be the ground state in previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [37] TiO2, CrO3,  and V2O5 are diamagnetic, since they contain TMs with empty 3d orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Similarly, Cu2O is diamagnetic  owing to the completely-filled 3d orbitals of Cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='3  Determining U  We determined the required U value, with r2SCAN, for each binary TMO oxidation reaction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=', Ti3+ →  Ti4+ in 2Ti2O3 + O2 → 4TiO2) by comparing the experimental enthalpy (per mole of O2) with the calculated  (r2SCAN+U ) value that minimizes the error against the experimental value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Note that U = 0 eV in our  data simply reflects a r2SCAN calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In order to obtain the experimental oxidation enthalpy, standard  enthalpy of formation for all the considered TMOs were taken from the Wagman and/or Kubaschewski  tables, [56,57] thus ignoring the p − V and entropic contributions, similar to previous works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [37,38,82] The  overall optimal U value for each TM was obtained by taking the average of the required U for each of the  available oxidation reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In the case of Ni oxides, oxidation of NiO to LiNiO2 by 2Li2O + 4NiO + O2  → 4LiNiO2 was considered as a proxy for the Ni2+ → Ni3+ oxidation reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [38]  3  Results  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='1  Oxidation energetics  Figure 1 displays the variation of the enthalpy of different oxidation reactions among binary TMOs, as a  function of applied U in the r2SCAN+U framework, for all TMs considered except Cr and Cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Solid lines  in each panel of Figure 1 represent DFT-calculated oxidation enthalpies, with each color corresponding to  different oxidation reactions for the TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' For instance in V oxides (Figure 1b), the solid black line corresponds  to the oxidation reaction, VO → V2O3, while the solid red and green lines indicate V2O3 → VO2 and VO2 →  V2O5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Similarly, the experimental enthalpy of each oxidation reaction is represented by dashed  horizontal line of the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' For example, the black dashed line in Figure 1b indicates the experimental  oxidation enthalpy (-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='36 eV) of VO → V2O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Also, dotted vertical line of a given color highlights the  required U value to minimize the error between DFT-calculated and experimental value for the oxidation  reaction enthalpy indicated by the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' The dotted blue line in each panel signifies the overall optimal  U for the TM that is averaged across all available oxidation reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  We report an optimal U value of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='8, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='8, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='1 eV, respectively, for Ti, V, Mn, Fe,  Co, and Ni oxides, within the r2SCAN+U framework (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Notably, the optimal U obtained with  r2SCAN is less than that reported previously for SCAN functional (Table S1) for all 3d TMs considered  (except V and Fe), which can be attributed to better accuracy of r2SCAN compared to SCAN, as observed  in non-TMOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [36] For V oxides, the required U value for VO2 → V2O5, V2O3 → VO2, VO → V2O3 is  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='7, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='2 eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Thus, the optimal U value for V is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='0 eV (average of the three required  U values), which is identical to the U correction required with SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [38] The decreasing required U with  increasing oxidation state of V in V oxides is expected due to the decrease in the strength of exchange  interactions among the d electrons as oxidation state increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In the case of Fe, FeO → Fe2O3 and FeO  → Fe3O4 reactions require a U of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='9 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='3 eV, respectively, resulting in an optimal U of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='1 eV, which  is also identical to the optimal U with SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [37] Moreover, we obtain the highest optimal U of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='1 eV  for Fe, among all TMs considered in this work, which is consistent with the fact that Fe3+ has the highest  number of unpaired d electrons resulting in the strongest exchange interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  For Ti and Ni, we observe a marginal improvement in the U -value for r2SCAN when compared to SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Specifically, we obtain an optimal U of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='3 eV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='1 eV for Ti and Ni, respectively, versus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5 eV for  both elements with SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' We find an optimal U value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='8 eV for both Mn (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='7 eV with SCAN) and  Co (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='0 eV with SCAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In Mn-oxides, the required U for the oxidation of Mn2O3 → MnO2, and MnO →  5  Mn2O3 are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='1 eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' The optimal U for Mn is transferable to other Mn oxides as well,  indicated by the robust agreement between r2SCAN+U -calculated and experimental oxidation enthalpy for  MnO → Mn3O4 (green lines in Figure 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  For Cr and Cu oxides, we obtain reasonable agreement with experimental data without a U correction  (Figure S2), similar to our observation with SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [38] In fact, for Cu, introducing U -correction worsens the  error in the calculated oxidation enthalpy for Cu2O → CuO versus experiment, similar to our observation  with SCAN(+U ) as well, which can be attributed to PAW potentials derived at the PBE-level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [38] However,  the magnitude of error (versus experiment) is smaller with r2SCAN (≈13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='1%) than with SCAN (≈25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='7%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  In case of Cr, the oxidation reaction of CrO2 → CrO3 requires U ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='9 eV, but introducing a U correction  worsens any agreement with experiment for Cr2O3 → CrO2 (where required U = 0 eV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Thus, the optimal  U for Cr oxides is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='45 eV (<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5 eV), which only provides a marginal improvement in describing oxidation  enthalpies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Hence, we recommend using only r2SCAN for calculating any Cr oxide framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  0  1  2  3  4  U(eV)  9  8  7  6  5  4  Reaction Enthalpy (eV per O2)  FeO/Fe2O3  FeO/Fe3O4  Experimental  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='9 eV  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='3 eV  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='1 eV  (d)  (e)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  3  U(eV)  7  6  5  4  3  2  1  Reaction Enthalpy (eV per O2)  CoO/Co3O4  Experimental  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='8 eV  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  U(eV)  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  Reaction Enthalpy (eV per O2)  NiO/LiNiO2  Experimental  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='1 eV  (f)  (a)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  U (eV)  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='4  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='2  8  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='8  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='6  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='4  Reaction Enthalpy (eV per O2)  Ti2O3/TiO2  Experimental  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='3 eV  (b)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  3  U (eV)  8  6  4  2  0  2  Reaction Enthalpy (eV per O2)  VO/V2O3  V2O3/VO2  VO2/V2O5  Experimental  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='0 eV  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='7 eV  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='0 eV  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='2 eV  (c)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  U(eV)  7  6  5  4  3  2  1  0  1  Reaction Enthalpy (eV per O2)  MnO/Mn2O3  Mn2O3/MnO2  Experimental  MnO/Mn3O4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5 eV  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='8 eV  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='1 eV  Figure 1: Calculated oxidation enthalpy versus the magnitude of U correction within r2SCAN+U framework  for (a) Ti, (b) V, (c) Mn, (d) Fe, (e) Co, and (f) Ni oxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Solid, dashed, and dotted lines of a given color  indicate calculated, experimental, and required U values for a given oxidation reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Optimal U for each  TM is indicated by the dotted blue line in each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='2  Lattice parameters  All r2SCAN(+U ) and SCAN(+U ) calculated lattice parameters, on-site magnetic moments, and band  gaps for each TMO are tabulated in Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Additionally, the calculated lattice volumes by the four XC  functionals are plotted against experimental data in Figure 2a for all oxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Generally, both SCAN (green  squares in Figure 2a) and r2SCAN (blue symbols) offer < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='8% deviation from the experimental lattice  parameters for all the TMOs considered, except VO, FeO, CuO, and LiNiO2, indicating robust agreement  with experiments for both functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In VO, SCAN and r2SCAN overestimate (by ∼8%) the experimental  lattice constants, while the deviation in FeO and CuO is ∼3-4% and ∼8-10%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  In LiNiO2,  6  SCAN’s β angle evaluation is ∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='1% different from experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Notably, SCAN and r2SCAN do show qualitative differences in their calculated lattice parameters (when  compared against experiments) across TMOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' For instance, both functionals overestimate the experimental  lattice constants in TiO2, Ti2O3, and VO, while they underestimate in CrO2, CrO3, MnO2, and Fe3O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  There are also examples (MnO and Mn2O3) where SCAN underestimates the experimental lattice constants  while r2SCAN overestimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Overall, there are cases where SCAN’s errors in lattice parameter estimations  are lower versus experiments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=', Cr2O3, CoO), r2SCAN’s errors are lower (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=', CrO2, CrO3, MnO2,  Fe3O4), and both functionals exhibit identical errors (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=', TiO2, Co3O4, NiO, Cu2O), signifying that both  functionals offer similar performance in terms of geometrical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Comparing r2SCAN and SCAN, we find that r2SCAN’s lattice constants are generally larger than SCAN  across TMOs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=', Ti2O3, Cr2O3, CrO3, VO2, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' As a range, r2SCAN estimates lattice constants that  are a maximum of ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5% larger than SCAN (in CrO3) and a minimum of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='1% larger than SCAN (in  Mn2O3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Having said that, there are instances where r2SCAN’s lattice constant evaluations are lower than  SCAN (VO, CoO, CuO, and LiNiO2) and cases where both functionals are identical (TiO2, Co3O4, NiO,  and Cu2O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In specific TMOs, SCAN and r2SCAN calculate an identical (individual) lattice constant, while  the other lattice constants with r2SCAN are larger than SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' For example, a and c lattice constants with  r2SCAN are higher than SCAN in V2O5 while both functionals estimate b = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='55 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  On introducing the optimal U correction, an increase in the value of calculated lattice constants is ob-  tained for both SCAN and r2SCAN functionals for all TMOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' The lattice constants computed by r2SCAN+U  (yellow symbols in Figure 2a) is up to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='3% higher than r2SCAN, except FeO (∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='2% higher).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Similar to the  comparison of r2SCAN vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' SCAN, there are systems where r2SCAN+U predicts larger, smaller, and identical  lattice constants compared to SCAN+U (red triangles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' For example, r2SCAN+U calculates larger lattice  constants than SCAN+U in VO2, V2O5, MnO, Mn2O3 and Fe3O4 (maximum of ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5% higher in V2O5),  while for Ti2O3, CoO and NiO, r2SCAN+U ’s estimations are smaller than SCAN+U (maximum deviation  of ∼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='1% in Ti2O3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Both SCAN+U and r2SCAN+U functionals evaluate identical lattice parameters for  TiO2, Co3O4 and LiNiO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Overall, lattice constants calculated by SCAN+U and r2SCAN+U deviate <∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='3% from experiments  for all TMOs, except VO and VO2 where deviations of ∼8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5% and ∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='6% are observed, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Adding  U improves the agreement with experiment for both SCAN and r2SCAN in Co3O4, while r2SCAN+U gives  the best estimate of the lattice parameters in FeO (< 1% deviation vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' experiments) compared to SCAN,  SCAN+U and r2SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Notably, all functionals break the rocksalt symmetry of VO, MnO, and FeO, while  the cubic symmetry of Fe3O4 is retained only by SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In Ti2O3, the hexagonal symmetry is broken by  SCAN but the symmetry is preserved by the other frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In summary, we find that the differences in  lattice parameter estimations to be minimal across the four functionals on average, with notable exceptions  of a few systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='3  On-site magnetic moments  On-site magnetic moments of the TMOs (Figure 2c and Table S2) computed by SCAN and r2SCAN  generally underestimate experimental values, with the exception of MnO2, Mn2O3, CrO2, and VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Note  that larger magnetic moments typically indicate stronger localization of d electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Comparing r2SCAN  and SCAN calculations, we find that r2SCAN typically estimates smaller magnetic moments than SCAN  but with several exceptions, such as MnO, MnO2, Mn2O3, Cr2O3, and VO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Thus, on average, SCAN’s  magnetic moment predictions are in better agreement with experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' However, in terms of magnitude,  7  Ti2O3  TiO2  VO  V2O3  VO2  V2O5  Cr2O3  CrO2  CrO3  MnO  Mn3O4  Mn2O3  MnO2  Fe2O3  Fe3O4  FeO  CoO  Co3O4  NiO  LiNiO2  Cu2O  CuO  SCAN  SCAN+U  r2SCAN  r2SCAN+U  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='8  On-site magnetic moment  difference (  B)  Ti2O3  TiO2  VO  V2O3  VO2  V2O5  Cr2O3  CrO2  CrO3  MnO  Mn3O4  Mn2O3  MnO2  Fe2O3  Fe3O4  FeO  CoO  Co3O4  NiO  LiNiO2  Cu2O  CuO  SCAN  SCAN+U  r2SCAN  r2SCAN+U  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='8  On-site magnetic moment  difference (  B)  SCAN  SCAN+U  r2SCAN  r2SCAN+U  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  Band gap difference (eV)  (a)  (c)  (b)  0  100  200  300  400  500  600  700  800  900  Experimental lattice volume (Å3)  0  100  200  300  400  500  600  700  800  900  Predicted lattice volume (Å3)  SCAN  SCAN+U  r2SCAN  r2SCAN+U  Figure 2: (a) Comparison of calculated and experimental lattice volume (in ˚A3) of all TMOs considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  (b) Violin plot capturing the difference between the experimental and computed band gap (in eV) across  TMO systems using the four XC frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' The empty circle and horizontal line in the inner box plot  corresponds to the mean and median of the calculated band gaps, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' (c) Heat map representation  of the differences between the experimental and calculated on-site magnetic moments (in µB) using the  four XC functionals and across all TMOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' A value of zero indicates perfect consistency, while red (blue)  colors indicate overestimation (underestimation) of magnetic moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Hatched boxes either correspond to  experimentally undetermined magnetic moments (VO) or calculations not executed with U frameworks (Cr  and Cu oxides).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  moments predicted by r2SCAN deviate by < 3% from SCAN’s estimates, except CuO (∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='8% deviation),  CrO2 (∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5%), and MnO2 (∼ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5%), highlighting that the differences in the predictions are marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Adding optimal U to both SCAN and r2SCAN increases the magnitude of the calculated on-site magnetic  moments for all TMOs (except VO2, which is predicted to be metallic by all functionals), consistent with  the expectation that the U correction facilitates d electron localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' r2SCAN+U -calculated data are  similar to the corresponding SCAN+U values (< 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='3% variation), except LiNiO2 (∼6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='3% variation), and  Ti2O3 (∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='8%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Similar to r2SCAN versus SCAN, r2SCAN+U estimates smaller magnetic moments than  SCAN+U, with notable exceptions being VO2, Mn2O3, MnO2 and FeO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Overall, we observe the accuracy  in calculated on-site magnetic moments versus experiments to follow the order SCAN+U > r2SCAN+U >  SCAN > r2SCAN for several TMOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' However, there are specific cases where specific XC frameworks offer  better accuracy in calculating magnetic moments, such as SCAN in CrO2, Mn2O3, MnO2, Fe3O4 and CuO,  8  r2SCAN in Mn3O4 and Cr2O3, and r2SCAN+U in V2O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Given the numerically marginal deviations in  calculated magnetic moments across the XC frameworks (∼10% deviation), we expect an increase/decrease  in accuracy to be marginal amongst the XC frameworks considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='4  Band gaps  The differences between calculated and experimental band gaps of all TMOs considered are visualized  as violin plots for SCAN (green violin), SCAN+U (red), r2SCAN (blue), and r2SCAN+U in Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  The top and bottom ends of the individual violins mark the highest and lowest differences in the respective  calculated data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Note that the mean values (white empty circles) are similar for SCAN and r2SCAN, and  in turn are lower than their U -corrected versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In other words, addition of the U-correction reduces  the error of calculated band gaps compared to experimental values, which is expected given that semi-local  DFT typically underestimates band gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Also, we find that SCAN+U displays the lowest mean band  gap difference among the XC functionals considered, indicating that on-average SCAN+U provides better  computed band gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  We present calculated electronic DOS of select TMOs, namely CoO (panels a and b), V2O3 (c and d),  and Mn2O3 (e and f), in Figure 3, to illustrate qualitative trends in computed band gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' The DOS for  the remaining TMOs, calculated by the four XC frameworks, are compiled in Figures S3-S19 of the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In  each DOS panel, solid orange and solid green lines correspond to the 2p-states of O and the 3d-states of the  TM, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Dashed black lines represent Fermi levels in metallic compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Dotted vertical lines  represent valence and conduction band edges in semiconducting/insulating compounds, with the band gaps  indicated by the text annotation near the conduction band minimum (CBM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' The zero of the energy scale  is set to the valence band maximum (VBM) for TMOs with a band gap and to the Fermi level in metallic  TMOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  We observe that r2SCAN generally calculates a smaller band gap than SCAN for most TMOs (maximum  of ∼66% lower in MnO2, see Table S2), as illustrated by the case of CoO in panels a and b of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Notable exceptions do exist to this observation, such as V2O5 (∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='7% larger), CrO3 (∼3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='2%), MnO (∼4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='3%),  and Fe2O3 (∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='7%), where r2SCAN calculated band gaps are marginally larger than SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Both SCAN  and r2SCAN incorrectly describe the ground state electronic configuration of narrow band gap TMOs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=',  experimental band gaps < 1 eV), including Ti2O3 (Figure S4), V2O3(Figure 3c and S3c), VO2 (Figure S7)  and Fe3O4 (Figure S15) to be metallic, with the exception of MnO2 where both SCAN and r2SCAN estimate  a narrow gap (Figures S12a and S12c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Additionally, both functionals also calculate the wrong electronic  structure in the case of a non-narrow-gap semiconductor, Mn2O3 (Figure S3), which exhibits an experimental  gap of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='2-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='3 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [83,84] However, SCAN and r2SCAN qualitatively describe the right electronic structure in  the case of wide band gap TMOs such as FeO (Figure S13), Fe2O3 (Figure S14), and NiO (Figure S17), with  a significant quantitative underestimation of the experimental gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In any case, the differences in electronic  structure predictions between SCAN and r2SCAN in TMOs are minimal, with SCAN being marginally better  in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Introducing a U correction to SCAN and r2SCAN widens or opens the band gap, especially in narrow  band gap TMOs, as illustrated by the case of V2O3 (panels c and d in Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' The opening of band  gap with U correction is expected since localization of d electrons, which form the VBM and/or CBM in  3d-TMOs, is faciliated with U addition, in turn resulting in a larger gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' However, in the case of VO2  (Figure S7), adding U does not capture the MIT that occurs at low temperatures (< 341 K [61]) with either  SCAN or r2SCAN, causing the erroneous prediction of metallic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Generally, SCAN+U calculates  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='36 eV  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='91 eV  (b)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='61 eV  (c)  (d)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='24 eV  (e)  (f)  Figure 3: DOS for CoO calculated using (a) SCAN and (b) r2SCAN, DOS for V2O3 computed using (c)  r2SCAN and (d) r2SCAN+U, and DOS for Mn2O3 estimated using (e) SCAN+U and (f) r2SCAN+U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  a larger band gap than r2SCAN+U (Table S2), as highlighted by the case of Mn2O3 (panels e and f in  Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In fact, SCAN+U is the only framework (among those considered) to estimate a band gap in  10  6  /eV  (states)  4  2  Density of states (  6  4-3 -2 -1  2  3  4  5  Energy (eV)6  /eV  (states)  4  Density of states (  6  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  4 -3-2  0  2  3  4  Energy (eV)/eV)  60  Mnd  (states)  40  20  MWN  Density of states  20  40  60  6  5  4  ¥-3  ¥-2-101  2  3  4  Energy (eV)60  /eV)  Mnd  (states)  40  20  Density of states  20  40  60  6  5  4  2  3  4  Energy (eV)/ev  10  (states)  5  Density of states  0  10  6  5  2  0  2  3  4  Energy (eV)/eV  10  (states)  Density of states  Wi  10  6  5  ¥1  2  3  4  Energy (eV)Mn2O3, which is consistent with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Moreover, SCAN+U ’s evaluations of larger band gaps results  in better (poorer) quantitative agreement with experiments in wide (narrow) gap materials, such as MnO  and FeO (V2O3 and MnO2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Note that SCAN+U and r2SCAN+U do underestimate the experimental band gaps, similar to SCAN and  r2SCAN, in wide gap TMOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' The only exception to this observation is CoO, where SCAN+U overestimates  the band gap versus experiment (Figure S3a and Table S2), as also observed in our previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [38] In select  TMOs, including Fe2O3 and V2O5, r2SCAN+U ’s band gap is larger than SCAN+U, but the magnitude of  difference (≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='2 eV) is meagre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Thus, for electronic structure predictions, we expect SCAN+U to provide the  best qualitative and quantitative band gaps across TMOs, among the functionals considered here, especially  for wide gap semiconductors/insulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' However, the qualitative trends provided by r2SCAN+U are quite  robust as well and in small gap semiconductors (< 1 eV gap), r2SCAN+U ’s quantitative accuracy is often  better than SCAN+U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5  Transferability checks  To examine the transferability of the optimal U values determined in this work (with r2SCAN), to oxide  systems not used for obtaining the values, we perform checks on systems with different oxidation state and/or  coordination environment for each TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' We compare calculated values against available experimental data,  such as structural, electronic, magnetic, and/or electrochemical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Specifically, we choose Ba2TiO4  as a check for Ti, BiVO4 for V, K3MnO4, K2MnO4, and Mn2O7 for Mn, SrFeO3 for Fe, LiCoO2-CoO2 for  Co, and LiNiO2-NiO2 for Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Data related to transferability checks are compiled in Figure 4, Table 1, and  Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  In the case of Ba2TiO4, we compare the calculated lattice parameters with experimental values (see  Table S3 and lattice voliume differences plotted in Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Ba2TiO4 crystallizes in a monoclinic structure  (space group P21/n) at low temperatures, where the unit cell is composed of four formula units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [85, 86]  Ti atoms are present in distorted tetrahedra composed of neighbouring oxygen atoms (TiO4) within the  Ba2TiO4 lattice, which is different from the octahedral environments sampled in TiO2 and Ti2O3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Upon  structure relaxation, we observe that both r22SCAN and r22SCAN+U functionals marginally overestimate  (by ∼2%) experimental lattice parameters (Figure 4 and Table S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Similar to trends observed in Table S2,  adding U to r2SCAN increases the calculated lattice parameters in Ba2TiO4 (by ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='03 ˚A), thereby marginally  reducing the agreement with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  We benchmark both structural and electronic properties of BiVO4 as a transferability check for V-based  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Note that BiVO4 transforms from tetragonal (I 41/a) to a monoclinic (I 2/b) ‘scheelite’ phase below  ∼ 528 K, [87, 88] which is a reversible second order ferroelastic transition driven by soft optical phonon  modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' The BiVO4 unit cell possesses four formula units, with tetrahedrally coordinated V ions, which is  different from the coordination environments of V in VO, V2O3, VO2, and V2O5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Importantly, monoclinic-  BiVO4 spontaneously transforms to the tetragonal structure upon structure relaxation with r22SCAN and  r22SCAN+U, similar to the observation by Liu et al [87] with GGA and hybrid functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Thus, neither  r22SCAN nor r22SCAN+U predict the correct ground state structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Additionally, BiVO4 possess a band  gap of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='4–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='48 eV [89] and is a candidate photocatalyst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [87] Both r22SCAN and r2SCAN+U provide similar  band gap predictions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='01-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='98 eV), which is in good qualitative agreement with experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Surprisingly,  r2SCAN+U evaluates a marginally lower band gap than r2SCAN (see panels a and b in Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' However,  both r22SCAN and r2SCAN+U predict similar states occupying the valence band (Op) and conduction band  (Vd) edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  11  Ba2TiO4  BiVO4  K3MnO4  K2MnO4  Mn2O7  SrFeO3  r2SCAN  r2SCAN+U  20  10  0  10  20  Lattice volume  difference (Å3)  (a)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='98 eV  (b)  (c)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='013 eV  Figure 4: DOS for BiVO4 calculated using (a) r2SCAN and (b) r2SCAN+U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' (c) Difference between experi-  mental and calculated lattice volumes (using r2SCAN and r2SCAN+U ), plotted as a heatmap, for various  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Red (blue) squares indicate overestimated (underestimated) calculated lattice volumes versus ex-  periment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  The rationale behind the choice of K3MnO4, K2MnO4, and Mn2O7 as checks for Mn-based systems is to  explore the higher, unsampled oxidation states of Mn, namely +5, +6, and +7 in K3MnO4, K2MnO4, and  Mn2O7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Also, Mn resides in tetrahedral coordination in these compounds, which is different from  the octahedral coordination observed in MnO, Mn2O3, and MnO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Although Mn2+ resides in tetrahedral  sites in spinel-Mn3O4, we had not used in the spinel structure to obtain our optimal U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' We benchmark the  calculated lattice parameters versus experiments for all Mn-based transferability checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Mn2O7 is a volatile liquid at 298 K and solidifies to a monoclinic crystal structure (P21/c) below ∼ 279 K,  with the unit cell consisting of 8 formula units of corner sharing tetrahedral MnO4 pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [90, 91] Upon  structural relaxation, both r2SCAN and r2SCAN+U underestimate the lattice constants of monoclinic-  Mn2O7 by ∼1-3% (Figure 4 and Table S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In the case of K3MnO4, the tetragonal symmetry (I 42m) [92]  is broken with r2SCAN functional resulting in an orthorhombic structure, while the symmetry is preserved  by r2SCAN+U (see Figure  4 and Table S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Nonetheless, both r2SCAN and r2SCAN+U significantly  underestimate the c parameter (by ∼ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5%) and overestimate the a or b parameter (∼ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='2%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' K2MnO4 is  an orthorhombic crystal (Pnma) with four formula units per unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [93] Here, r2SCAN and r2SCAN+U  predict identical lattice parameters, which marginally underestimate experimental values (by ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='4-1%, see  Figure 4 and Table S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  The choice of SrFeO3, a cubic perovskite, as a check for Fe is largely motivated by the 4+ oxidation  state exhibited by Fe in the structure, which is not sampled in FeO, Fe2O3, or Fe3O4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Both r2SCAN and  r2SCAN+U preserve the cubic symmetry during structure relaxation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' with r2SCAN+U ’s lattice parameters  12  (states/eV  Bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Bi  Density of states (  6  5  4 -3  ¥-2  0  2  3  4  Energy (eV)(states/eV  Bip  Bi  Density of states (  6  5  4 -3  ¥-2  0  2  3  4  Energy (eV)(states/eV  Bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Bi  Density of states (  6  5  4 -3  ¥-2  0  2  3  4  Energy (eV)(states/eV  Bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Bi  Density of states (  6  5  4 -3  ¥-2  0  2  3  4  Energy (eV)(states/eV  Bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Bi  Density of states (  6  5  4 -3  ¥-2  0  2  3  4  Energy (eV)(states/eV  Bi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Bi  Density of states (  6  5  4 -3  ¥-2  0  2  3  4  Energy (eV)identical to experiments and r2SCAN’s parameters being a slight underestimation (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5%, see Figure 4  and Table S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In terms of magnetic configuration of Fe in SrFeO3, Takeda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [94] reported a helical spin  structure via their neutron diffraction experiments, with competing FM and AFM interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' However,  Shein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [95] found a FM metallic state to be the ground state of SrFeO3, over a wide range of pressures,  based on their first principles calculations, which they attributed to stronger FM than AFM interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' We  considered a FM configuration of Fe atoms in the SrFeO3 unit cell, and the on-site magnetic moments on Fe  calculated by both r2SCAN (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='375 µB, Table 1) and r2SCAN+U (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='819 µB) overestimate the experimental  value (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='4 µB [94]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' However, our calculated magnetic moments do indicate a localization of ∼4 electrons  on the d orbitals of Fe, consistent with its +4 oxidation state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  We choose CoO2 (R3m or ‘O3‘ polymorph [96]), and NiO2 (P1m1 or ‘O1’ [97]), both layered structures,  as transferability checks for Co and Ni, respectively, owing to the unsampled 4+ oxidation states of each  TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In terms of experimental property to benchmark, we choose the average Li intercalation voltage in these  structures, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=', LiCoO2-CoO2, and LiNiO2-NiO2 pairs, since they have been measured with high precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  The reader is referred to previous works on calculating and benchmarking average ‘topotactic’ intercalation  voltages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [98,99] r2SCAN underestimates the experimental average voltage [96,99–103] in LiNiO2-NiO2 (by  ∼ 8%), while it overestimates the average voltage in LiCoO2-CoO2 (by ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='7%), similar to trends observed  with SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [99] The addition of U to r2SCAN leads to an improvement in agreement with the experimental  voltage in the Ni-system (deviation of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='8%), while it worsens the agreement in the Co-system (deviation  of ∼ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='4%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Nevertheless, r2SCAN+U does overestimate the average voltage in both Co and Ni systems,  similar to the behavior of SCAN+U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' [99]  Table 1: Voltage and magnetic moments calculated by r2SCAN, and r2SCAN+U compared against experi-  mental values (denoted by ‘Expt.’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' The U values used with r2SCAN+U are the corresponding optimal U  values obtained for each TM (from Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Composition  Source  Voltage  Magnetic moment  (space group)  (V)  (µB)  LiCoO2-CoO2  Expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='05 (R¯3m)  r2SCAN  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='12 r2SCAN+U  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='23 LiNiO2-NiO2  Expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='85 (P1m1)  r2SCAN  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='54 r2SCAN+U  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='92 SrFeO3  Expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='4  (Pm¯3m)  r2SCAN 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='375  r2SCAN+U 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='819  4  Discussion  In this work, we evaluated the performance of the r2SCAN functional among binary TMOs consisting  of 3d-TMs by calculating the oxidation enthalpies, lattice parameters, on-site magnetic moments, and band  gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Additionally, for each TM-O2 system considered, we calculated the optimal Hubbard-U corrections  to be used in a r2SCAN+U framework, based on experimental oxidation enthalpies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Although theoretical  approaches exist to derive U values, [41–47] using oxidation enthalpies nominally gives an “average” correc-  tion that is suitable across several oxidation states of a given TM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Specifically, our optimal U values are 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='3,  13  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='8, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='8, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='1 eV for Ti, V, Mn, Fe, Co, and Ni, respectively, while we don’t deem a U correction  necessary for Cr and Cu oxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Interestingly, the optimal U corrections needed with r2SCAN are lower in  magnitude compared to SCAN for Ti, Mn, Co, and Ni oxides (while the corrections are identical for V and  Fe oxides), indicating that r2SCAN exhibits lower errors with oxidation enthalpies and possibly lower SIEs  than SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' However, this is not reflected in other physical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' On an average, we find the accuracy,  versus experimental values, to be similar for r2SCAN compared to SCAN, and for r2SCAN+U compared to  SCAN+U, respectively, in lattice parameter, on-site magnetic moment, and band gap evaluations as seen in  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  The general trends in lattice parameter, magnetic moment, and band gap predictions, across the XC  frameworks considered, can be summarized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' We observe that r2SCAN generates larger lattice  constants than SCAN and on addition of the U correction to both functionals, the lattice constants further  increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Thus, in systems where SCAN underestimates experimental lattice constants (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=', CrO2, CrO3,  MnO2), shifting to r2SCAN improves agreement (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=', error in r2SCAN in CrO3 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='8% versus 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='3% with  SCAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Also, there are instances where the ground state symmetry of the TMO is not preserved by some  or all of the XC frameworks considered (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=', in VO, MnO, FeO, Fe3O4, and Ti2O3), highlighting systematic  issues in the XC treatment across the four frameworks considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' The calculated on-site magnetic moments  by r2SCAN (and r2SCAN+U ) are marginally lower than SCAN (SCAN+U ), with the U correction nom-  inally increasing the calculated moments calculated by r2SCAN and SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' However, calculated magnetic  moments across the four XC frameworks differ by < 10% (except LiNiO2), signifying marginal differences  in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Both SCAN and r2SCAN underestimate band gaps across all TMOs (except MnO2), with  band gaps calculated by r2SCAN typically being lower than SCAN, and adding the U opens/widens the  gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Thus, SCAN+U offers the best quantitative accuracy versus experimental band gaps, especially for  wide gap semiconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Note that the qualitative trends from r2SCAN+U are consistent with the trends  exhibited by SCAN+U and should be reliable in electronic structure predictions in other TM-based oxide  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  r2SCAN adopts the smooth polynomial interpolation function of rSCAN to maintain numerical stability  during SCF calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Additionally, the reformed gradient expansion for correlation introduced in r2SCAN  (partially) negates the error introduced to the slowly varying density by the non-vanishing interpolation  function, [32] which largely accounts for the observed variation in accuracy of r2SCAN versus SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Based  on our data, we observe that r2SCAN is not systematically more accurate than SCAN across all TMOs  and for all property predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' For example, we have lower optimal U values indicating lower SIEs with  r2SCAN versus SCAN, but also lower on-site magnetic moments (except Mn and Cr oxides) signifying poorer  d-electron localization with r2SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Further, the smaller band gaps with r2SCAN (versus SCAN) may be  caused by the residual SIEs, resulting in an underestimation of the CBM across TMOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Hence, usage of  r2SCAN(+U ) in TM-based systems must be done with care and efforts should be made to benchmark as  many available experimental properties as possible before performing “true” computational predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  We considered the transferability of the U values estimated in this work, with r2SCAN, by examining  systems for each TM with oxidation states and/or coordination environments not sampled while calculating  the optimal U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In general, we find that r2SCAN or its Hubbard U corrected version estimate similar lattice  parameters and hence yield similar accuracies on structural properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Analogously, the calculated on-site  magnetic moments in SrFeO3 and the band gaps in BiVO4 are similar between r2SCAN and r2SCAN+U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In  case of electrochemical properties, we do find tangible variations in the calculated average voltages of r2SCAN  and r2SCAN+U, with r2SCAN+U exhibiting overall lower errors across the Co and Ni systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Thus, we  14  SCAN+U  r2SCAN  r2SCAN+U  Ti  V  Cr  Mn  Fe  Co  Ni  Cu  0  1  2  3  4  5  6  7  8  9  Ti  V  Cr  Mn  Fe  Co  Ni  Cu  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='8  (a)  Ti  V  Cr  Mn  Fe  Co  Ni  Cu  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='8  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='2  (c)  (b)  Rela�ve overall  computa�on �me  Rela�ve computa�on �me  per ionic step  Rela�ve computa�on �me  per electronic step  Figure 5: (a) Overall computational time (electronic+ionic steps) (b) computational time per ionic step and  (c) computational time per electronic loop taken for each TM-O2 binary system with SCAN+U, r2SCAN,  and r2SCAN+U frameworks relative to SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Values greater (smaller) than 1 in each panel indicates that  a given calculation is slower (faster) than SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  15  find the optimal U values obtained in this work to be transferable across oxide frameworks not sampled a  priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Nevertheless, more benchmarking studies to compare the performance of r2SCAN+U with r2SCAN  (and experiments) will help in quantifying the reliability and errors associated with using r2SCAN+U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Given that r2SCAN(+U ) is not systematically more or less accurate than SCAN(+U ), the computational  performance and numerical stability of r2SCAN(+U ) is critical in determining its utility in property pre-  dictions across materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Thus, we have quantified the computational time of r2SCAN(+U ) and SCAN+U  relative to SCAN for each TM-O2 system considered in Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Specifically, panels a, b, and c of Fig-  ure 5 plot the overall (electronic+ionic steps), per ionic step, and per electronic step computational time,  respectively, taken by the SCAN+U (blue bars), r2SCAN (red), and r2SCAN+U (yellow) frameworks, rel-  ative to the computational time taken by the SCAN functional (dotted black lines), for each TM-based  set of oxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Details on calculating the computational times used by the functionals is described in the  ‘Computational time’ section of the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Note that our objective is not to provide a rigorous quantification of  computational resources required for each XC framework, but to provide a qualitative understanding of the  relative computational costs across the frameworks considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  For each electronic step, r2SCAN(+U ) is typically faster than SCAN (Figure 5), signifying better numer-  ical stability than SCAN, with Mn, Ni, and Cu oxides being marginal exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' In contrast, on a per-ionic  step basis, r2SCAN and r2SCAN+U is slower than SCAN, by ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='05-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='78× and ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='31×, respectively,  highlighting that r2SCAN(+U ) takes more electronic steps to converge per ionic step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Importantly, the over-  all computational time (ionic+electronic steps, Figure 5) required for structural relaxation of TMOs using  r2SCAN and r2SCAN+U is lower than SCAN, by ∼12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='1-61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='2% and ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='9-34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='5%, respectively, except in Fe  oxides, indicating that r2SCAN(+U ) takes lower number of ionic steps to converge, which possibly indicates  a better description of atomic forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' The higher overall computation time in Fe oxides with r2SCAN(+U )  than SCAN is primarily due to the difficulty in converging Fe3O4 with r2SCAN(+U ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Comparing r2SCAN  and r2SCAN+U, we find that r2SCAN+U takes a higher overall computational time to converge, except  in Fe and Ni oxides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Thus, we expect r2SCAN(+U ) to provide good utility in property predictions in  TM-containing systems given its better computational performance and reasonable accuracy compared to  SCAN(+U ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  5  Conclusion  3d-TMs and their compound phases find applications in several fields such as energy storage, solar  cells, catalysts, thermochemical water splitting, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=', and it is imperative to predict their properties such  as lattice constants, magnetic moments, reaction enthalpies, and band gaps accurately using DFT-based  techniques for designing better materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Recently, the r2SCAN metaGGA XC functional was proposed  to exhibit the accuracy of its predecessor, SCAN, and the computational performance of rSCAN in main-  group compounds, but the accuracy of r2SCAN was not rigorously tested on TM-based systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Here,  we assessed the numerical accuracy and computational performance of r2SCAN in binary 3d-TMOs, in  calculating the lattice parameters, on-site magnetic moments, binary oxidation enthalpies, and band gaps  against experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Notably, we observed that r2SCAN exhibited similar qualitative trends as that  of SCAN, with marginally larger estimations of lattice parameters than SCAN, while the on-site magnetic  moments and band gap calculations are marginally smaller than SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' While both r2SCAN and SCAN  underestimated the band gaps in wide gap TMOs, with SCAN offering slightly better accuracy, they failed  to predict the correct ground state electronic configurations of narrow band gap TMOs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=', Mn2O3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  16  On analysing the addition of Hubbard U -correction to improve the accuracy of the r2SCAN functional,  we observed that a lower optimal U value, based on experimental oxidation enthalpies, was required in  a r2SCAN+U framework for Ti, Mn, Co and Ni oxides, when compared to a SCAN+U framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' The  optimal U values were identical in both r2SCAN+U and SCAN+U frameworks for V and Fe oxides, while we  did not observe the need for a U correction in Cr and Cu oxides with r2SCAN, similar to SCAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Moreover,  introducing the U -correction to SCAN and r2SCAN increased the calculated lattice parameters, on-site  magnetic moments and the band gaps of the TMOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  r2SCAN+U and SCAN+U successfully opened a band gap for narrow gap TMOs (except VO2 and  Mn2O3 with r2SCAN+U ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Upon testing the optimal U values with r2SCAN+U on oxides with different  oxidation states and/or coordination environments, we found that the U values derived in this work are in  general transferable to other TM-containing oxides as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Furthermore, we observed that r2SCAN(+U )  took less overall computational time (ionic+electronic steps) to converge when compared to SCAN, which  indicated that r2SCAN(+U ) was computationally more efficient than SCAN(+U ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' Since r2SCAN+U offers  a reasonably accurate prediction of material properties at a lower computational expense than SCAN+U, we  observe that r2SCAN+U can be used in high-throughput materials discovery, after adequate benchmarking  tests are done in each new chemical space explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Acknowledgments  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' acknowledges the Indian Institute of Science (IISc) Seed Grant, SG/MHRD/20/0020 and SR/MHRD/20/0013  and the Science and Engineering Research Board (SERB) of the Department of Science and Technology, Gov-  ernment of India, under sanction numbers SRG/2021/000201 and IPA/2021/000007 for financial support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' thanks the Ministry of Human Resource Development, Government of India, for financial assistance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' acknowledges financial support from SERB under IPA/2021/000007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' All the authors acknowledge the  computational resources provided by the Supercomputer Education and Research Centre, IISc, for enabling  some of the density functional theory calculations showcased in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Author Contributions  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' envisioned and designed the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' performed the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content=' All authors con-  tributed in data analysis and writing the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Conflicts of Interest  The authors declare no competing financial or non-financial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Availability of data  The data that support the findings of this study are openly available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='com/sai-mat-  group/r2SCAN-U-benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
+page_content='  Supplementary Materials  Electronic Supporting Information is available online at , with details on the crystal structures used for  calculations, oxidation energetics of Cr and Cu oxides, densities of states of all systems not showcased in the  17  main text, and details on computational time calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NAyT4oBgHgl3EQfp_h6/content/2301.00535v1.pdf'}
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+Astronomy & Astrophysics manuscript no. main
+©ESO 2022
+2022-12-31
+Stellar Karaoke: Deep Blind Separation of Terrestrial Atmospheric
+Effects out of Stellar Spectra by Velocity Whitening
+Nima Sedaghat1, J. Bryce Kalmbach1, Brianna M. Smart1, and Erin L. Howard1
+DIRAC Institute and the Department of Astronomy, University of Washington, 3910 15th Avenue NE, Seattle, WA 98195, USA
+e-mail: nimaseda@uw.edu
+2022-12-31
+ABSTRACT
+We exploit the statistical independence of stellar features and atmospheric adversarial effects in stellar spectra, to remove the latter
+from observed signals using a fully unsupervised data-driven approach. Concretely, we first increase the inter-observation entropy of
+telluric absorption lines by imposing a random, virtual radial velocity to the observed spectrum. This novel “trick” results in a non-
+standard form of “whitening” in the atmospheric components of the spectrum, decorelating them across multiple observations. Then
+we use deep convolutional auto-encoders, to learn a feature-space in which the two “sources” of information, stellar and atmospheric,
+are easily separable, leading to removal of the latter. We apply the process on spectra from two different data collections: ~250,000
+HARPS spectra and ~660,000 from SDSS.
+We compare and analyze the results across datasets, as well as with existing tools, and discuss directions for utilizing the introduced
+method as a fast and more reliable tool in the future.
+Key words. stellar spectra – telluric lines – deep learning – unsupervised – whitening – decorrelation – source separation
+1. Introduction
+Throughout this paper, we introduce and elaborate a method
+based on signal processing tricks combined with a fully unsuper-
+vised deep neural network to remove the non-linear, time-variant
+effect of telluric lines out of observed spectra without any need
+for manual modeling and tuning. Concretely, we show how look-
+ing at an enormous dataset of spectra reveals new insights into
+their features, allowing for novel solutions to existing, difficult
+problems.
+In ground-based observations of stellar spectra, it is not un-
+usual for the source’s spectrum to reach our detectors in an
+altered state. The spectrum may be affected by a number of
+events in space (redshift/blueshift, emission, absorption, etc.).
+When finally reaching the Earth, the photons in the spectrum
+will undergo additional transformations as they travel through
+the Earth’s atmosphere and are collected by telescopes. Tele-
+scope effects are often well characterized and constrained (CCD
+noise, fringing, mirror defects, etc.). However, other effects are
+usually more difficult to model and correct.
+Particularly, passing through the terrestrial molecules present
+in the Earth’s atmosphere results in absorption lines in the sensed
+spectrum. These absorption lines, known as telluric lines, are the
+results of interaction of photons with molecules due to a num-
+ber of electron, rotational, and vibrational transitions, and are a
+persistent source of contamination and loss of information in the
+observed spectra. The majority of the molecules responsible for
+absorption are O2, H2O, CO2, CH4, O3, N2O, and the Chappuis
+ozone absorption bands. The many of these absorption features
+lie in the near-infrared and ultraviolet part of the electromagnetic
+spectrum, with weaker absorption features from ozone, oxygen,
+and water present in optical wavelengths. Additional emission
+features are also added to the spectrum as photons are emitted
+from the molecules. The emission features are relatively straight-
+4000
+4500
+5000
+5500
+6000
+6500
+Wavelength (Å)
+0.4
+0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Normalized Flux (-)
+Input Spectrum
+Reconstructed Spectrum
+Fig. 1. We exploit the statistical properties of stellar spectra in large
+datasets, pass them through a convolutional auto-encoder, and get tel-
+luric lines rejected with close to zero effort.
+forward to handle by observing source-free portions of the sky.
+Observed emission lines in "empty" regions provide an easily ac-
+cessible template for emission removal. The absorption features
+are more complicated, scaling non-linearly, and are affected by
+atmospheric conditions at the time of observation.
+Most of the traditional methods used for the removal of tel-
+luric lines consider each spectrum as a single independent en-
+tity, discarding the set of observed spectra as a whole – e.g. see
+Hrudková & Harmanec (2005). One group of such methods use
+standard stars to assist with atmospheric line removal. They do
+Article number, page 1 of 10
+arXiv:2301.00313v1  [astro-ph.SR]  1 Jan 2023
+
+A&A proofs: manuscript no. main
+this by observing standard stars with relatively featureless spec-
+tra and using them as atmospheric templates. This is often done
+by observing A0V or G-type stars, though this is predominantly
+used at the near-infrared and IR wavelengths (Vacca et al. 2003;
+Artigau et al. 2014). The reference star should be ideally lo-
+cated near the target star and observed close in time to measure
+the atmospheric conditions as accurately as possible. The target
+source’s spectrum is then divided by the telluric template. How-
+ever, this method is limited if there are no such standard stars
+available during an observing run or if observing conditions re-
+sult in a poor spectrum. Atmospheric absorption can also vary
+rapidly, on the order of minutes, requiring significant time and
+resources dedicated to observing the template stars. The division
+is an iterative process requiring wavelength and intensity scaling
+adjustments to match the atmospheric effects.
+Another common method is to model the Earth’s atmosphere
+and create a synthetic spectrum which precisely models the at-
+mospheric absorption. These models solve the radiative trans-
+fer equation of the Earth’s atmosphere using numerical models
+– e.g. Allart et al. (2022). These methods are dependent on at-
+mospheric conditions measurements from the night’s observa-
+tions. This is commonly done with programs such as molecfit
+(Smette et al. 2015) and telfit (Gullikson et al. 2014). The
+radiative transfer code retrieves the temperature, pressure, and
+humidity from the time of observation and uses a database of
+molecular parameters to create a fit for the telluric absorption.
+While this technique is relatively successful, it may perform
+poorly if there are a large number of intrinsic features, little or
+no continuum, low signal-to-noise ratio (S/N), or large airmass
+observations with high water vapor content. Moreover, current
+implementations of this technique suffer from rather slow perfor-
+mance to the extent that fitting a model to a single spectrum may
+take up to several minutes on today’s computers. See Ulmer-
+Moll et al. (2019) for a comparison of the above methods.
+More recent methods, however, have incorporated implicit
+modeling of the spectra by looking at a set of examples, poten-
+tially eliminating the need for manual, explicit modeling. Such
+methods, loosely dubbed data-driven approaches, have gained
+good momentum in the past couple of decades. Particularly in
+astronomy, a data-driven look at spectra has been mainly incor-
+porated in the broad context of dimensionality reduction.
+Arguably the most popular dimensionality reduction method
+used in astronomy has been the Principal Component Analy-
+sis (PCA – see Jolliffe & Cadima (2016) for a review), which
+has been used as a dimensionality reduction tool on astronom-
+ical spectra for many years now. Connolly et al. (1995) used
+it to classify galaxy spectra with only the first two principal
+components while Bailer-Jones et al. (1998) showed that PCA
+can compress stellar spectra by a factor of over 30 and classify
+anomalous and non-stellar spectra in the data set. Furthermore,
+Bailer-Jones et al. (1998) found that the compression removed
+noise as well as bogus features in the spectra such as dust ap-
+pearing on the plate during plate scanning. Some works such as
+Artigau et al. (2014), even attempt to remove telluric lines of a
+few specific objects in the HARPS dataset using PCA decom-
+position and demonstrate improved radial velocity measurement
+accuracy. However, their method does not operate solely on the
+stellar spectra and still requires observations of telluric standard
+stars. Our approach solely requires a set of input stellar spectra
+without specific telluric standard stars.
+Moreover, the very important, but often overlooked point
+about applicability of PCA for such applications, is its linear na-
+ture, making it unable to account for modeling non-linear phe-
+nomena by definition. E.g. in the case of telluric lines, a method
+simply based on PCA, may not be able to tell the difference
+between a telluric and stellar line, in crowded regions of the
+wavelength. This inability has been well observed as early as
+in Bailer-Jones et al. (1998). In this work, we show how trans-
+formation to a more sophisticated feature space is necessary for
+such a task.
+There is a closely related family of neural networks known
+as encoder-decoder networks (fig. 2). The contraction part com-
+presses the input to an often low-dimensional representation,
+also known as "latent representation" or simply "code". The ex-
+pansion part is then used to decode this low-dimensional repre-
+sentation up to the desired output. The transformation used for
+coding (and decoding) is learned during training, according to
+the task at hand. A special case of such networks, auto-encoders,
+is an unsupervised network trained to reconstruct the input with-
+out the need for labeled input data. As described in Hinton &
+Salakhutdinov (2006) these networks can be used to reduce the
+dimensionality of input data in a non-linear generalization of
+PCA.
+In Yang & Li (2015) the authors use a classical (non-
+convolutional) auto-encoder to transform 3000-dimensional
+spectra from the Sloan Digital Sky Survey (SDSS) Data Re-
+lease 7 (Abazajian et al. 2009) to lower-dimensional features,
+which are later used for estimation of atmospheric parame-
+ters. Wang et al. (2016a) also used classical (non-convolutional)
+auto-encoders for feature learning on astronomical spectra. They
+compare spectral classification based on their learned features
+with PCA and locally linear embedding (LLE), an alternate
+non-linear dimensionality reduction method. They find that their
+auto-encoder approach performs the best when classifying spec-
+tra among a data set of F, G and K-type stars.
+One problem with the mere use of auto-encoders, especially
+for applications in (astro-)physics, is the entangled mapping of
+concepts into the reduced signal – the latent space. The Varia-
+tional AutoEncoder (VAE: Kingma & Welling 2014) has shown
+to mitigate this effect to some extent. Portillo et al. (2020) used
+a VAE to reconstruct SDSS galaxy spectra resampled to an input
+spectrum with 1000 components. They found that when limiting
+the dimensionality of the latent spaces to 2, 4, 6 or 10 compo-
+nents (≥ 99% compression) the VAEs and traditional autoen-
+coders both reconstructed SDSS galaxy spectra with a lower re-
+construction error than PCA and non-negative matrix factoriza-
+tion (NMF) applied with the same number of components. The
+authors then demonstrated that different galaxy classes occupied
+different areas of the latent space enabling classification.
+Classical auto-encoders are merely composed of fully-
+connected layers and thus suffer from a lack of scalability to
+deeper networks and high-dimensional data – usually the case
+in modern astronomical spectroscopy. We address this drawback
+by incorporating convolutional (and up-convolutional) layers for
+transformation of data to the desired feature space; an idea bor-
+rowed from computer vision. In Zhang et al. (2004) the authors
+map images of people’s faces to low-dimensional manifolds, and
+reconstruct them back. They find a correlation between the face
+pose and the low-dimensional manifold, independently of the
+person in the image. Wang et al. (2016b) shows comprehensive
+experiments on dimensionality reduction using auto-encoders,
+and studies the effects of different latent dimensions using syn-
+thetic and real images. Note that in case of spectra, the networks
+need to be updated to use 1D (up-)convolutional layers, as op-
+posed to 2D in case of images.
+Perhaps the closest to our work, both in terms of the used
+network architecture, as well as the data-driven unsupervised
+concept is Sedaghat et al. (2021), where a convolutional VAE
+Article number, page 2 of 10
+
+Sedaghat et al.: Stellar Karaoke
+Fig. 2. A simple auto-encoder trained to reconstruct stellar spectra, can decompose physically meaningful components out of the input, when the
+compression is high enough and the number of convolutional kernels is limited. Blue is the input spectrum and orange is the reconstructed version.
+The region annotated by the red circle indicates a high density of such reconstruction rejections.
+is used to extract knowledge from a large number of HARPS
+spectra, in a fully unsupervised manner. We borrow and use the
+exact same neural architecture for our work. However, we addi-
+tionally utilize the conceptual decomposition of stellar features
+and atmospheric effects into two different spectra; an idea that
+turns out to have been touched as early as in Hadrava (1997),
+however without a data-driven perspective.
+Our Contributions
+• We process a huge number of very high-dimensional spec-
+tra as a whole, letting statistical properties emerge in them,
+allowing to be treated as random processes.
+• We model the telluric components in stellar spectra as inde-
+pendent stars and impose a virtual radial velocity on them to
+achieve statistical whitening/decorrelation.
+• We incorporate a convolutional auto-encoder, that automati-
+cally acts as a source separation tool, rejecting telluric lines
+in a fully unsupervised fashion, with zero explicit modeling.
+2. Problem Formulation
+We seek to clean adversarial atmospheric effects out of an ar-
+bitrary observed signal, x. The observed signal in our case is a
+stellar spectrum, and so is a function of wavelength, λ, letting us
+denote it as x(λ).
+We use the below formulation to model the various phenom-
+ena affecting a stellar spectrum, before it is captured by our sen-
+sors:
+x(λ) =
+�
+s(λ) × t(λ)
+�
+∗ h(λ) + n(λ)
+(1)
+where s is the stellar spectrum, incorporating line-of-sight ef-
+fects from relative stellar velocity and the interstellar medium. t
+is an imaginary signal representing the telluric lines affecting the
+spectrum. x is the observed signal: a single spectrum that takes
+on different values at each wavelength depending on the flux at
+that wavelength. So we use the notation, x(λ), throughout this
+article.
+We use h to denote what we call the observation transfer
+function, and models the changes the signal goes through dur-
+ing the sensing process, and includes, but is not limited to, the
+line spread function, etc. n models the additive noise which is
+not modeled in the transfer function, h.
+We work with a large ensemble of N observations, mostly
+coming from different sources. We use the subscript, i, to differ-
+entiate between various observations:
+xi(λ) =
+�
+si(λ) × ti(λ)
+�
+∗ h(λ) + ni(λ)
+i ∈ {1, 2, . . . , N}
+(2)
+Note that the spectrum, s, depends on i, as we work with various
+objects at the same time. The atmospheric conditions are also
+time-variant, an important point that is reflected in dependence
+of t on i. The same is true for noise, n. Without loss of generality,
+and for the sake of simplicity, we assume that h is constant across
+the whole set of observations.
+We seek to eliminate the effect of telluric lines from the
+observed signal, which in this model is equal to extracting
+si(λ) ∗ h(λ) for every observation – note that removal of the ob-
+servation effect, h, is not part of the objective here.
+We also model the effect of the radial velocity, v, of the target
+object on the observed spectrum, s as:
+si(λ) = V {˚si(λ), vi}
+= ˚si
+�
+λ�1 − vi
+c
+��
+(3)
+where ˚si would be the observed spectrum, if the radial velocity
+was zero – i.e. no Doppler shift. We call ˚si the static spectrum
+hereafter. The V{.} operator represents the effect of the radial
+velocity, vi, on the spectrum, as expanded in the second line of
+the above equation, and c is the speed of light. The physical units
+of vi and c can be arbitrarily chosen, as long as they are kept the
+same.
+ti on the other hand, and by definition, do not have any de-
+pendence on vi, and therefore can be modeled as:
+ti(λ) = V
+�
+˚ti(λ), 0
+�
+(4)
+2.1. Continuous vs. Discrete
+All the signals discussed above are of continuous nature up until
+the point they are sensed by the detector. Sensing by the detector
+Article number, page 3 of 10
+
+A&A proofs: manuscript no. main
+is a process that involves sampling as one of its steps, converting
+a spectrum into a series of real-valued samples. Therefore, the
+data we work with in our experiments is the discrete representa-
+tion of xi(λ), namely Xl
+i – we use l as the discrete-valued index
+for the sampled pixels.
+However, for the reasons below, keeping the formulation in
+the continuous representation is safe – and clearer. First, the
+signals modeled so far are all the constituent elements of the
+pre-sampling signal, xi(λ), and so the continuous models hold.
+Secondly, the only part of our method which explicitly modifies
+the signal along the wavelength axis, the V {, } operator (sec-
+tion 3.1), does its job by regridding the interpolated version of
+the signal; practically converting the discrete signal back to its
+continuous version, applying the transformation and sampling it
+back again to the discrete space. Therefore mathematical defini-
+tion of the operator in the continuous space is valid.
+2.2. Signals as Random Processes
+The process of sensing a signal, xi, from an arbitrarily chosen
+object in the sky, which is affected by many non-deterministic
+phenomena along the way, can be seen as one realization of a
+random process. In other words, the set of xi, or their discrete
+representation, Xi, for various i, represent an ensemble of real-
+izations of a random process, {X} 1. Note that in this particular
+application, the index set of the random process is sampled from
+wavelengths, λ – a bit counter-intuitive, as it is usually of a tem-
+poral nature in typical applications.
+Therefore each Xl represents a random variable, with out-
+comes Xl
+iii. Similarly, we can model {S }, { ˚S } and {T} as discrete-
+index random processes of the same type. {S } is then a genera-
+tor of various clean spectra, S i, while { ˚S } generates the static ˚S l
+i
+spectra. As we will see in the next section, this non-deterministic
+view on signals allows us to explain the statistical operations and
+properties of the components more clearly.
+3. Method
+The method is composed of two key steps:
+a) Increasing the entropy of telluric components across obser-
+vations and,
+b) transforming the spectra into a space where the stellar and
+telluric components are separable, then removing the non-
+dominant one.
+Median Normalization As a pre-processing step, we normalize
+each spectrum in the dataset based on its median, to mitigate the
+effect of different distances in sources, which otherwise results
+in extreme inter-sample flux range variations.
+3.1. Velocity Whitening
+In the default conditions, the spectroscopic observations are in
+the so-called topocentric reference frame where the telluric lines,
+if they exist, take on the same wavelengths – or pixels locations.
+What it means from a statistical point of view though is this:
+Let us assume, for simplicity, that all the observed objects
+have the same static spectra, ˚si(λ). The resulting spectra, si(λ)
+1 The term random in this context is not in contradiction with the struc-
+ture in stellar spectra. The structure is encoded in the basic parameters
+of the constituent set of random variables, such as the expected value
+and covariance matrix.
+would then only be different based on their radial velocities,
+vi, which, by definition, are modeled as scaling transformations
+along the wavelength axis (eq. (3)). Note that since vi are out-
+comes of a random variable, si(λ) are consequently still highly
+independent and uncorrelated.
+ti on the other hand, are composed of a set of absorption lines
+with different strengths, but all happening at pre-defined specific
+wavelengths. This feature makes them highly correlated across
+multiple observations and easy to fit for any model – fig. 3, left
+column.
+To decrease the existing correlation between realizations of
+the telluric signals, we increase the entropy across ti by random-
+izing them using an emulated radial velocity,
+t′
+i(λ) = V
+�
+˚ti(λ), v′
+i
+�
+(5)
+where each v′
+i is a random velocity and is drawn from
+V′ : R → R. The above effect can be implemented by sim-
+ply contracting or expanding the wavelength axis, according to
+eq. (3). Note though that this is a purely artificial phenomenon,
+and although it is chosen to have the same effect as the “real”
+radial velocity of stars, it has no particular meaning for telluric
+lines 2.
+In practice, however, we only have access to the observed
+signal, x, and not its forming components. Hence the proposed
+randomization cannot be applied directly on t alone, and the
+whole observed signal gets affected. But in the below, we show
+that it can still have the desired effect:
+λ −→ λ′
+i = λ × (1 − v′
+i
+c )
+(6)
+x′
+i(λ) = V �xi(λ), v′
+i
+�
+= V
+��
+si(λ) × ti(λ)
+�
+∗ h(λ) + ni(λ), v′
+i
+�
+= V
+��
+si(λ) × ti(λ)
+�
+∗ h(λ), v′
+i
+�
++ V �ni(λ), v′
+i
+�
+(7)
+which, according to the proof provided in appendix A becomes:
+x′
+i(λ) = (1 − v′
+i
+c )V �si(λ) × ti(λ), v′
+i
+� ∗ V �h(λ), v′
+i
+� + V �ni(λ), v′
+i
+�
+= (1 − v′
+i
+c )
+�
+si(λ′
+i) × ti(λ′
+i)
+�
+∗ h(λ′
+i) + ni(λ′
+i)
+(8)
+Now since
+�
+1 −
+v′
+i
+c
+�
+is, for each x′
+i, a constant coefficient and
+is normalized out before being passed to the next step of the
+method, we can see that the emulated randomized velocity has
+founds its way from xi down to ti. In other words, we have
+achieved the required velocity randomization in ti by modifying
+xi. The two main components in the tweaked signal can now be
+rewritten as:
+Telluric:
+Stellar:
+V
+�
+˚ti(λ), v′
+i
+�
+V
+�
+˚si(λ), vi + v′
+i
+�
+So, we have achieved two components which are virutally mov-
+ing independently of each other, when going over various obser-
+vations. Figure 3 illustrates this effect in practice.
+2 A similar deterministic effect occurs when taking stellar spectra to
+the barycentric frame.
+Article number, page 4 of 10
+
+Sedaghat et al.: Stellar Karaoke
+6265
+6270
+6275
+6280
+topo-centric
+6265
+6270
+6275
+6280
+ADP.2014-09-16T11:03:35.377
+randomized
+6265
+6270
+6275
+6280
+6265
+6270
+6275
+6280
+ADP.2014-09-16T11:03:36.737
+6265
+6270
+6275
+6280
+Wavelength (Å)
+6265
+6270
+6275
+6280
+Wavelength (Å)
+ADP.2014-09-16T11:03:38.307
+Fig. 3. On the left a set of exemplar spectra are depicted. Stellar lines are
+unaligned due to different radial velocities. But telluric lines are aligned,
+even though they may have different shapes due to their inherent time
+dependence (the dotted vertical line indicates the location of a specific
+telluric line in all plots). On the right the same spectra after velocity
+randomization are depicted. Telluric lines are now unaligned too, but
+with a pattern different to that of stellar lines!
+From a statistical point of view, what we achieve with the
+above velocity randomization is a degree of “whitening” of the
+tellurics random process, {T}. More precisely, we decrease the
+mutual correlation between every pair, (T l, T m), pushing the sta-
+tistical behavior of the telluric component toward white noise (Li
+& Zhang 1998; Eldar & Oppenheim 2003; Kessy et al. 2018).
+Note again that (T l, T m), the random variables we try to decor-
+relate, are each defined on a single pixel of the spectrum, and
+their outcomes vary along with different observations. A visual-
+ization of the results of the above whitening process is illustrated
+in section 5.
+3.2. Deep Feature Space
+We borrow and use the exact architecture of the 1D convolu-
+tional auto-encoder introduced by Sedaghat et al. (2021) – Fig-
+ure 2. The convolutional layers of the encoder transform the in-
+put spectrum down to a pre-defined number of “latent variables”
+at the bottleneck of the network. This low-dimensional repre-
+sentation of the input, also known as the “code”, is the most
+compressed version of the input spectrum. The dimensionality
+of this vector is chosen based on the desired compression rate in
+various experiments. Note though that in the VAE version of our
+networks, which is the case in most of the experiments of this
+work, the latent nodes are implemented probabilistically, each
+being modeled with a pair of scalars: mean and std of a normal
+distribution. On the other side of the bottleneck, the decoder re-
+Fig. 4. Visual illustration of the covariance matrix of the ensemble of
+signals, before and after whitening. On the left, the covariance ma-
+trix of the [6275, 6285]Å region for some subset of size 90 of the
+observations is visualized. On the right, the same is done after veloc-
+ity whitening, where vi were sampled from the uniform distribution:
+V ∼ U(−30km/s, 30km/s). The covariance matrix is closer to the iden-
+tity matrix now, confirming achievement of some degree of whiten-
+ing/decorrelation.
+ceives the compressed code and takes it step-by-step up to the
+same dimension as the original input (218 + 216 for HARPS). We
+keep using the same per-pixel L1 end-to-end loss function, as in-
+troduced in the original work, to achieve acceptable pixel-level
+accuracy.
+This ∼typical architecture has proven to be able to transform
+the input to a space where noise-like components of the signal
+are easily separable. As we show with our experiments, the sta-
+tistical trick developed in the previous section pushes the telluric
+components farther from the stellar features and closer to the
+noise, in the learned feature space, letting them be rejected as
+easily as noise.
+We of course need to constrain the reconstruction abilities of
+the network with a high compression rate as well as a variational
+loss at the bottleneck, not to have too strong of a network capable
+of fitting the two independent components at the same time!
+4. Data
+We use data from two publicly available colletions: HARPS and
+SDSS. Our main experiments are run on HARPS, while SDSS is
+used as a difficult test-bench.
+4.1. HARPS
+HARPS (High Accuracy Radial-velocity Planet Searcher; Mayor
+et al. 2003) is an instrument on the 3.6m La Silla Telescope.
+It is a fibre-fed high-resolution echelle spectrograph dedicated
+to the discovery of exoplanets, with a spectral resolution of
+R = 115, 000 and covers the spectral range 378–691nm 3. The
+data used has been instrument-corrected and detector-corrected,
+as well as sky-subtracted.
+Our downloaded dataset initially consisted of 272376 total
+spectra, which after automatic removal of corrupted files, NaNs
+and noise-like ones, was reduced to 267361 “stable” ones. This
+collection is mainly composed of stellar spectra. However, there
+are some random contaminant objects too, such as SUN, MOON,
+etc., which we allowed to enter our training set on purpose,
+to increase robustness of the learned features. We trimmed, re-
+gridded and homogenized all spectra before being passed to the
+3 http://archive.eso.org/wdb/wdb/adp/phase3_main/form
+Article number, page 5 of 10
+
+A&A proofs: manuscript no. main
+Fig. 5. Results Qualitative illustration of the results on an exemplar selection of HARPS spectra. Each row depicts one spectrum, with its HARPS
+ID written on the right, while columns focus on different regions of interest. Interpretation hint: reconstruction (orange) should follow the pseudo-
+truth (molecfit: green). The left-most column covers the whole spectrum, as is fed into the network, and illustrates the robustness of the network
+to different characteristics of the spectra (continuum, noise level, etc.). Major stellar lines such as Hα can be easily spotted in the less ‘busy’
+examples. In the middle column we zoom in on a potentially complex region where narrow stellar lines, when existing, can collide with telluric
+lines of similar shapes. E.g. in the second and 4th row there are examples of such cases, where the network rejects the telluric component, while
+still preserving the stellar part very well – thus rejecting the hypothesis that it might be simply rejecting narrow lines by a applying a moving
+average. In the third column, we focus on a region where similarly narrow stellar lines occur in some of the spectra, but not in the others. This way
+we reject the hypothesis that the network might have ’memorized’ the locations of the lines. Note that the network (orange) is even outperforming
+the pseudo-truth (green) in some cases, illustrating a smoother, more robust reconstruction in abrupt changes.
+network in exactly the same way as is done by Sedaghat et al.
+(2021).
+Note that the pipeline the HARPS spectra go through is set
+up such that the spectra are automatically transformed to the
+barycentric reference frame and re-gridded before being stored
+in the archive. The originally captured version, in the topocen-
+tric reference frame, is also not preserved. Therefore we had to
+transform them back to the topocentric frame for our experi-
+Article number, page 6 of 10
+
+pseudo-truth
+input
+reconstruction
+Normalized Flux (-)
+ADP.2014-09-16T11:05:10.627
+0
+4000
+5000
+6000
+6278
+6279
+6280
+6281
+6282
+5565.05567.55570.05572.5
+Normalized Flux (-)
+ADP.2014-09-16T11:03:31.413
+1
+4000
+5000
+6000
+6278
+6279
+6280
+6281
+6282
+5565.05567.55570.05572.5
+Normalized Flux (-)
+ADP.2014-09-16T11:03:31.673
+1
+0
+4000
+5000
+6000
+6278
+6279
+6280
+6281
+6282
+5565.05567.55570.05572.5
+Normalized Flux (-)
+ADP.2014-10-01T10:22:48.507
+1
+0
+4000
+5000
+6000
+6278
+6279
+6280
+6281
+6282
+5565.05567.55570.05572.5
+Wavelength (A)
+Wavelength (A)
+Wavelength (A)Sedaghat et al.: Stellar Karaoke
+ments. Although this is touching the core concept of our pre-
+sented method, it turned out to be a safe procedure: transforma-
+tion to the barycentric frame exerts added randomness on the
+radial velocity, which is perfectly compatible with our method.
+In fact, our method has been inspired by observing traces of the
+above-mentioned fact in our initial experiments.
+4.2. SDSS
+Our second dataset consisted of spectra from the SEGUE (Yanny
+et al. 2009) and SEGUE-2 spectroscopic surveys (Rockosi et al.
+2022) that were part of the larger Sloan Digital Sky Survey
+(SDSS). The SEGUE surveys both used the 2.5m Sloan Foun-
+dation Telescope (Gunn et al. 2006) located at Apache Point Ob-
+servatory with the two original Sloan Digital Sky Survey fiber
+spectrographs (Smee et al. 2013). The SDSS spectrographs have
+a resolution of R ∼ 1800 and together have 640 fibers (320 each)
+that plug into aluminum "plugplates" for each observation. In
+each plate 32 plugs are reserved for blank sky observations and
+16 for spectrophotometric standard stars in the field. Each spec-
+trograph has a red and a blue channel that collect data on separate
+CCDs with the blue wavelength range from approximately 3800
+Å to 6100 Å and the red wavelengths spanning approximately
+5900 Å to 9200 Å. Sources in the original SEGUE survey sam-
+pled Milky Way stars at a variety of distances, colors and metal-
+licities while the SEGUE-2 targets focused on stars in the Milky
+Way halo.
+In our experiments we used the ~660,000 uncalibrated spec-
+tra from the red CCD of one of the spectrographs (labeled as ’r1’
+in the SDSS data archive) provided with SDSS Data Release 17
+(Abdurro’uf et al. 2022) and accessible via the DR17 FITS web-
+site4. The uncalibrated spectra consist of multiple 10-30 minute
+exposures of each source. Since the SEGUE surveys imaged
+each source multiple times to create coadded spectra this means
+we have multiple spectra in our dataset for the same source. For
+each plate we exclude the fibers that were intentionally pointed
+at empty patches of sky and labelled "SKY" in the data.
+The uncalibrated spectra used are labelled ‘spFrame‘ in the
+SDSS data model5 and according to the details of the SDSS spec-
+troscopic pipeline (Stoughton et al. 2002) they are flat-fielded but
+not flux-calibrated. The flux calibration spectra (‘spFluxCalib‘
+in the data model) include telluric absorption calculated by the
+spectroscopic pipeline based upon the spectrophotometric stan-
+dard stars that are included in the observation set of each plate.
+These calibration are what we use as a "pseudo-truth" for com-
+parisons of our network results.
+5. Experiments and Results
+We train and evaluate multiple networks based on various com-
+binations of the below (hyper-)parameters:
+• Latent-space dimensionality
+• Velocity randomization level
+For our main experiments, that include hyper-parameter
+sweeping and controlled tests, we mainy use the HARPS dataset.
+Other datasets are used for comparative testing of the method in
+extreme conditions.
+4 https://data.sdss.org/sas/dr17/
+5 https://data.sdss.org/datamodel/
+5.1. Main Results
+Figure 5 illustrates a few examples of how the method manages
+to reject the telluric lines and preserve the stellar features. The
+example spectra and regions are specifically chosen to rule out
+potential naive hypotheses for how the network rejects telluric
+lines. In other words, the results confirm that the network has
+learned a semantic representation of the constituent components
+and is separating the sources in that feature space, as opposed
+to e.g. simply having “missed” narrower lines (simple averag-
+ing, low-pass filtering). In the caption of fig. 5 a few other such
+hypotheses are elaborated and rejected.
+5.2. Quantitative Evaluation
+Various (hyper-)parameters, such as the degree of compression
+during dimensionality reduction, or the resolution of the input
+data, may influence the capability of the network in rejecting
+the telluric lines. Precisely, there is always a trade off between
+telluric-rejection and stellar-reconstruction. Therefore, we de-
+velop and incorporate the below mutual metrics to quantify both
+aspects at the same time:
+Qt =
+� �
+Mt ��� ˆS − ˜G
+���
+�
+� Mt
+(9)
+where Qt is a proxy for the quality of rejection of telluric lines.
+ˆS represents the reconstructed spectrum (i.e. the direct output of
+the network) and ˜G is the pseudo ground truth. ˜G is a corrected
+version of the output of molecfit in case of HARPS data, and a
+modified version of the publicly available calibrated spectra in
+case of SDSS. Mt is a binary mask; a vector of the same size
+of the input, having ones at all pixels containing known telluric
+lines, and zeros everywhere else. Note that the subscript i, index-
+ing each observed spectrum, is omitted in Qt, ˆS , ˜G and Mt for
+the sake of simplicity, and the summation is calculated over all
+the pixels of each spectrum.
+We similarly define the dual metric for measuring the recon-
+struction (preservation) of the stellar features as follows:
+Qs =
+�
+�������Ms
+�������1 −
+ˆS
+˜G
+�������
+�������
+� Ms
+(10)
+where Qs is, conversely, a proxy for the quality of stellar recon-
+struction. Note that both of the above are metrics of a distance
+nature, and hence, the lower the better.
+5.3. The Effect of Compression
+It is reasonable to expect, and is supported by our experiments
+that, with a lower dimensionality at the information bottleneck
+of the network, the network’s capacity for preserving many con-
+current features during compression diminishes. This results in a
+higher rejection of details in the reconstructed spectra, which can
+be naively interpreted as an improvement in telluric line rejection
+– a decrease in Qt. Note however that stellar reconstruction qual-
+ity can get worse at the same time, and so the two metrics need
+to be considered at the same time.
+Table 1 compares results of various runs with different di-
+mensionalities at the latent space on a fixed subset of HARPS
+spectra. Note how the stellar reconstruction quality starts to de-
+cay by increasing the compression (lower latent dimensions). On
+Article number, page 7 of 10
+
+A&A proofs: manuscript no. main
+8800
+8900
+9000
+9100
+telluric
+8100
+8200
+8300
+8400
+telluric
+6400
+6500
+6600
+6700
+6800
+stellar: H-alpha
+8400
+8600
+8800
+104_3110_spFrame-r1-00055323_230
+stellar
+6000
+7000
+8000
+9000
+Normalized Flux (-)
+8800
+8900
+9000
+9100
+Wavelength (Å)
+8100
+8200
+8300
+8400
+Wavelength (Å)
+6400
+6500
+6600
+6700
+6800
+Wavelength (Å)
+8400
+8600
+8800
+Wavelength (Å)
+104_3114_spFrame-r1-00056165_144
+6000
+7000
+8000
+9000
+Wavelength (Å)
+Normalized Flux (-)
+input
+tellurics profile
+reconstruction
+input
+tellurics profile
+reconstruction
+Fig. 6. Test results on a low-resolution data collection: SDSS. On the left, the whole spectrum is depicted, while the other columns depict various
+regions of interest along the wavelength axis, so the qualitative performance of the method can be assessed on specific stellar and telluric features.
+Note that since pseudo-truth data is not easily obtainable in this case, we over-plot a "tellurics profile" in red, just for visual comparison. It should
+be interpreted differently from the pseudo-truth curves in other figures, as it does follow the telluric lines. Stellar features are preserved very well,
+even in bad signal-to-noise conditions. Some of the wider telluric lines, however, are not fully rejected in this case due to the significantly low
+resolution as compared to HARPS. Please refer to the main text for a more detailed discussion.
+Latent Dim.↓
+Telluric
+Stellar
+8
+0.053
+0.015
+32
+0.055
+0.012
+128
+0.063
+0.014
+1024
+0.176
+0.012
+Table 1. Quality of stellar line reconstruction vs. telluric line rejection
+on HARPS, for different configurations of the network. Latent Dim. is
+the number of dimensions of the code, or the latent representation. A
+lower number is better in both columns.
+the other hand, by increasing the number of latent nodes, the
+network starts to become too powerful, managing to reconstruct
+both components and losing the source separation capability.
+5.4. The Effect of Velocity-Whitening
+To demonstrate the effect of the whitening step, we compare the
+results of two experiments in controlled conditions. The first ex-
+periment is run on HARPS spectra in topocentric frame, where
+telluric lines are all aligned – zero velocity randomization. Then
+we apply velocity whitening, sampling vi from the uniform dis-
+tribution: V ∼ U(−30km/s, 30km/s). All other configurations of
+the test are kept fixed. Table 2 shows a clear difference in the
+telluric rejection performance of the two runs.
+HARPS
+Telluric
+Stellar
+Topocentric
+0.199
+0.018
+Rand. Velocity
+0.063
+0.014
+Table 2. Comparison of two experiments with zero velocity randomiza-
+tion (‘Topocentric’) and V ∼ U(−30km/s, 30km/s) (‘Rand. Velocity’).
+5.5. HARPS vs. SDSS: High- vs. Low-Resolution
+Figure 6 depicts the results of applying our method on SDSS
+spectra. The extremely low pixel resolution we used for this
+dataset (1.66Å as opposed to 0.01Å for HARPS), makes this
+in practice a stress test for our method, since many of the tel-
+luric lines get merged, appearing as wide artifacts. However, the
+method performs at an acceptable level. E.g. the middle column
+of fig. 6 shows how the method misses the very wide telluric ar-
+tifact while managing to reject the neighboring telluric lines. In
+general, it seems our method can “hunt” telluric lines up until
+the point the resolution goes so low that merging/blending of the
+lines converts them to “slow” artifacts.
+6. Conclusions and Future Directions
+We presented a method that, by incorporating a Big Data-
+inspired view at stellar spectra, exploits the statistical indepen-
+dence of the radial velocity of stars with telluric lines in their
+observed spectra, reinforces it using a novel trick, and utilizes
+a fully unsupervised convolutional neural network to reject the
+undesireable part.
+The method is superior to existing, traditional telluric line
+removal tools in terms of preparation effort, performance, and
+accuracy. The fact that it is fully unsupervised, obviates the need
+for any kind of model parameter tuning – which is the case in
+e.g. molecfit, where a list of wavelengths should be manually
+specified to initialize the model. Training the model, in practice,
+calls for merely passing a large number of spectra through the
+network – and it will do the rest.
+Training one network suffices for one whole data collection.
+Once trained, it processes each spectrum in a fraction of a second
+– depending on the size of the spectrum and, consequently, the
+size of the network. But the speed-up does not sacrifice accuracy;
+as seen in section 5, it can even detect and suppress hard-to-
+locate lines which are missed by molecfit – the opposite may
+happen too, though in rarer situations.
+Article number, page 8 of 10
+
+Sedaghat et al.: Stellar Karaoke
+Nevertheless, the current version is still a demonstration of
+a research product, showcasing the strengths of a fully unsuper-
+vised approach. But for Stellar Karaoke to become a ready-to-
+use package in every application, more work is required. No-
+tably, the decision of which component to keep and which one
+to reject (stellar vs. telluric), should not be left to the network. It
+can be enforced by a minimal supervision.
+Acknowledgements. Some of the experiments demonstrated in this work have
+been run on compute servers provided by ESO, during NS’s collaboration with
+the ESCAPE project between 2019 and 2021.
+References
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+ations: Strategies, Processes, and Systems V, Vol. 9149, International Society
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+Bailer-Jones, C. A., Irwin, M., & Von Hippel, T. 1998, Monthly Notices of the
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+Article number, page 9 of 10
+
+A&A proofs: manuscript no. main
+Appendix A: Wavelength Transformation and
+Convolution
+x(λ) = z(λ) ∗ h(λ)
+=
+�
+w
+z(w)h(λ − w)dw
+(A.1)
+x(λ′
+i) = x(aλ)
+=
+�
+w
+z(w)h(aλ − w)dw
+(A.2)
+Let w = au,
+(A.3)
+x(λ′
+i) = a
+�
+u
+z(au)h(aλ − au)du
+(A.4)
+Let z(au) = z′(u), h(au) = h′(u)
+(A.5)
+=⇒ x(λ′
+i) = a
+�
+u
+z′(u)h′(λ − u)du
+= a z′(λ) ∗ h′(λ)
+= a z(aλ) ∗ h(aλ)
+= a z(λ′
+i) ∗ h(λ′
+i)
+(A.6)
+The same could be shown in a rather shorter way using
+the Fourier transform. However, to avoid confusion between the
+terms frequency and wavelength used in two different domains,
+here we use the direct expansion of the convolution operator.
+Article number, page 10 of 10
+
diff --git a/A9AyT4oBgHgl3EQfd_iI/content/tmp_files/load_file.txt b/A9AyT4oBgHgl3EQfd_iI/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..3ae743840e2304057f376c590fa3677da5e63649
--- /dev/null
+++ b/A9AyT4oBgHgl3EQfd_iI/content/tmp_files/load_file.txt
@@ -0,0 +1,667 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf,len=666
+page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' main  ©ESO 2022  2022-12-31  Stellar Karaoke: Deep Blind Separation of Terrestrial Atmospheric  Effects out of Stellar Spectra by Velocity Whitening  Nima Sedaghat1, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Bryce Kalmbach1, Brianna M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Smart1, and Erin L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Howard1  DIRAC Institute and the Department of Astronomy, University of Washington, 3910 15th Avenue NE, Seattle, WA 98195, USA  e-mail: nimaseda@uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='edu  2022-12-31  ABSTRACT  We exploit the statistical independence of stellar features and atmospheric adversarial effects in stellar spectra, to remove the latter  from observed signals using a fully unsupervised data-driven approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Concretely, we first increase the inter-observation entropy of  telluric absorption lines by imposing a random, virtual radial velocity to the observed spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' This novel “trick” results in a non-  standard form of “whitening” in the atmospheric components of the spectrum, decorelating them across multiple observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Then  we use deep convolutional auto-encoders, to learn a feature-space in which the two “sources” of information, stellar and atmospheric,  are easily separable, leading to removal of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' We apply the process on spectra from two different data collections: ~250,000  HARPS spectra and ~660,000 from SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  We compare and analyze the results across datasets, as well as with existing tools, and discuss directions for utilizing the introduced  method as a fast and more reliable tool in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' stellar spectra – telluric lines – deep learning – unsupervised – whitening – decorrelation – source separation  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Introduction  Throughout this paper, we introduce and elaborate a method  based on signal processing tricks combined with a fully unsuper-  vised deep neural network to remove the non-linear, time-variant  effect of telluric lines out of observed spectra without any need  for manual modeling and tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Concretely, we show how look-  ing at an enormous dataset of spectra reveals new insights into  their features, allowing for novel solutions to existing, difficult  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  In ground-based observations of stellar spectra, it is not un-  usual for the source’s spectrum to reach our detectors in an  altered state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The spectrum may be affected by a number of  events in space (redshift/blueshift, emission, absorption, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  When finally reaching the Earth, the photons in the spectrum  will undergo additional transformations as they travel through  the Earth’s atmosphere and are collected by telescopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Tele-  scope effects are often well characterized and constrained (CCD  noise, fringing, mirror defects, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' However, other effects are  usually more difficult to model and correct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Particularly, passing through the terrestrial molecules present  in the Earth’s atmosphere results in absorption lines in the sensed  spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' These absorption lines, known as telluric lines, are the  results of interaction of photons with molecules due to a num-  ber of electron, rotational, and vibrational transitions, and are a  persistent source of contamination and loss of information in the  observed spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The majority of the molecules responsible for  absorption are O2, H2O, CO2, CH4, O3, N2O, and the Chappuis  ozone absorption bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The many of these absorption features  lie in the near-infrared and ultraviolet part of the electromagnetic  spectrum, with weaker absorption features from ozone, oxygen,  and water present in optical wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Additional emission  features are also added to the spectrum as photons are emitted  from the molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The emission features are relatively straight-  4000  4500  5000  5500  6000  6500  Wavelength (Å)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='2  Normalized Flux (-)  Input Spectrum  Reconstructed Spectrum  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' We exploit the statistical properties of stellar spectra in large  datasets, pass them through a convolutional auto-encoder, and get tel-  luric lines rejected with close to zero effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  forward to handle by observing source-free portions of the sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Observed emission lines in "empty" regions provide an easily ac-  cessible template for emission removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The absorption features  are more complicated, scaling non-linearly, and are affected by  atmospheric conditions at the time of observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Most of the traditional methods used for the removal of tel-  luric lines consider each spectrum as a single independent en-  tity, discarding the set of observed spectra as a whole – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' see  Hrudková & Harmanec (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' One group of such methods use  standard stars to assist with atmospheric line removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' They do  Article number, page 1 of 10  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='00313v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='SR]  1 Jan 2023  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' main  this by observing standard stars with relatively featureless spec-  tra and using them as atmospheric templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' This is often done  by observing A0V or G-type stars, though this is predominantly  used at the near-infrared and IR wavelengths (Vacca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Artigau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The reference star should be ideally lo-  cated near the target star and observed close in time to measure  the atmospheric conditions as accurately as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The target  source’s spectrum is then divided by the telluric template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' How-  ever, this method is limited if there are no such standard stars  available during an observing run or if observing conditions re-  sult in a poor spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Atmospheric absorption can also vary  rapidly, on the order of minutes, requiring significant time and  resources dedicated to observing the template stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The division  is an iterative process requiring wavelength and intensity scaling  adjustments to match the atmospheric effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Another common method is to model the Earth’s atmosphere  and create a synthetic spectrum which precisely models the at-  mospheric absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' These models solve the radiative trans-  fer equation of the Earth’s atmosphere using numerical models  – e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Allart et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' These methods are dependent on at-  mospheric conditions measurements from the night’s observa-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' This is commonly done with programs such as molecfit  (Smette et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 2015) and telfit (Gullikson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The  radiative transfer code retrieves the temperature, pressure, and  humidity from the time of observation and uses a database of  molecular parameters to create a fit for the telluric absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  While this technique is relatively successful, it may perform  poorly if there are a large number of intrinsic features, little or  no continuum, low signal-to-noise ratio (S/N), or large airmass  observations with high water vapor content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Moreover, current  implementations of this technique suffer from rather slow perfor-  mance to the extent that fitting a model to a single spectrum may  take up to several minutes on today’s computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' See Ulmer-  Moll et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' (2019) for a comparison of the above methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  More recent methods, however, have incorporated implicit  modeling of the spectra by looking at a set of examples, poten-  tially eliminating the need for manual, explicit modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Such  methods, loosely dubbed data-driven approaches, have gained  good momentum in the past couple of decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Particularly in  astronomy, a data-driven look at spectra has been mainly incor-  porated in the broad context of dimensionality reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Arguably the most popular dimensionality reduction method  used in astronomy has been the Principal Component Analy-  sis (PCA – see Jolliffe & Cadima (2016) for a review), which  has been used as a dimensionality reduction tool on astronom-  ical spectra for many years now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Connolly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' (1995) used  it to classify galaxy spectra with only the first two principal  components while Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' (1998) showed that PCA  can compress stellar spectra by a factor of over 30 and classify  anomalous and non-stellar spectra in the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Furthermore,  Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' (1998) found that the compression removed  noise as well as bogus features in the spectra such as dust ap-  pearing on the plate during plate scanning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Some works such as  Artigau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' (2014), even attempt to remove telluric lines of a  few specific objects in the HARPS dataset using PCA decom-  position and demonstrate improved radial velocity measurement  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' However, their method does not operate solely on the  stellar spectra and still requires observations of telluric standard  stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Our approach solely requires a set of input stellar spectra  without specific telluric standard stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Moreover, the very important, but often overlooked point  about applicability of PCA for such applications, is its linear na-  ture, making it unable to account for modeling non-linear phe-  nomena by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' in the case of telluric lines, a method  simply based on PCA, may not be able to tell the difference  between a telluric and stellar line, in crowded regions of the  wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' This inability has been well observed as early as  in Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' In this work, we show how trans-  formation to a more sophisticated feature space is necessary for  such a task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  There is a closely related family of neural networks known  as encoder-decoder networks (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The contraction part com-  presses the input to an often low-dimensional representation,  also known as "latent representation" or simply "code".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The ex-  pansion part is then used to decode this low-dimensional repre-  sentation up to the desired output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The transformation used for  coding (and decoding) is learned during training, according to  the task at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' A special case of such networks, auto-encoders,  is an unsupervised network trained to reconstruct the input with-  out the need for labeled input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' As described in Hinton &  Salakhutdinov (2006) these networks can be used to reduce the  dimensionality of input data in a non-linear generalization of  PCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  In Yang & Li (2015) the authors use a classical (non-  convolutional) auto-encoder to transform 3000-dimensional  spectra from the Sloan Digital Sky Survey (SDSS) Data Re-  lease 7 (Abazajian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 2009) to lower-dimensional features,  which are later used for estimation of atmospheric parame-  ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' (2016a) also used classical (non-convolutional)  auto-encoders for feature learning on astronomical spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' They  compare spectral classification based on their learned features  with PCA and locally linear embedding (LLE), an alternate  non-linear dimensionality reduction method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' They find that their  auto-encoder approach performs the best when classifying spec-  tra among a data set of F, G and K-type stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  One problem with the mere use of auto-encoders, especially  for applications in (astro-)physics, is the entangled mapping of  concepts into the reduced signal – the latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The Varia-  tional AutoEncoder (VAE: Kingma & Welling 2014) has shown  to mitigate this effect to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Portillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' (2020) used  a VAE to reconstruct SDSS galaxy spectra resampled to an input  spectrum with 1000 components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' They found that when limiting  the dimensionality of the latent spaces to 2, 4, 6 or 10 compo-  nents (≥ 99% compression) the VAEs and traditional autoen-  coders both reconstructed SDSS galaxy spectra with a lower re-  construction error than PCA and non-negative matrix factoriza-  tion (NMF) applied with the same number of components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The  authors then demonstrated that different galaxy classes occupied  different areas of the latent space enabling classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Classical auto-encoders are merely composed of fully-  connected layers and thus suffer from a lack of scalability to  deeper networks and high-dimensional data – usually the case  in modern astronomical spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' We address this drawback  by incorporating convolutional (and up-convolutional) layers for  transformation of data to the desired feature space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' an idea bor-  rowed from computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' In Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' (2004) the authors  map images of people’s faces to low-dimensional manifolds, and  reconstruct them back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' They find a correlation between the face  pose and the low-dimensional manifold, independently of the  person in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' (2016b) shows comprehensive  experiments on dimensionality reduction using auto-encoders,  and studies the effects of different latent dimensions using syn-  thetic and real images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Note that in case of spectra, the networks  need to be updated to use 1D (up-)convolutional layers, as op-  posed to 2D in case of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Perhaps the closest to our work, both in terms of the used  network architecture, as well as the data-driven unsupervised  concept is Sedaghat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' (2021), where a convolutional VAE  Article number, page 2 of 10  Sedaghat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' : Stellar Karaoke  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' A simple auto-encoder trained to reconstruct stellar spectra, can decompose physically meaningful components out of the input, when the  compression is high enough and the number of convolutional kernels is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Blue is the input spectrum and orange is the reconstructed version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  The region annotated by the red circle indicates a high density of such reconstruction rejections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  is used to extract knowledge from a large number of HARPS  spectra, in a fully unsupervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' We borrow and use the  exact same neural architecture for our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' However, we addi-  tionally utilize the conceptual decomposition of stellar features  and atmospheric effects into two different spectra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' an idea that  turns out to have been touched as early as in Hadrava (1997),  however without a data-driven perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Our Contributions  We process a huge number of very high-dimensional spec-  tra as a whole, letting statistical properties emerge in them,  allowing to be treated as random processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  We model the telluric components in stellar spectra as inde-  pendent stars and impose a virtual radial velocity on them to  achieve statistical whitening/decorrelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  We incorporate a convolutional auto-encoder, that automati-  cally acts as a source separation tool, rejecting telluric lines  in a fully unsupervised fashion, with zero explicit modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Problem Formulation  We seek to clean adversarial atmospheric effects out of an ar-  bitrary observed signal, x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The observed signal in our case is a  stellar spectrum, and so is a function of wavelength, λ, letting us  denote it as x(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  We use the below formulation to model the various phenom-  ena affecting a stellar spectrum, before it is captured by our sen-  sors:  x(λ) =  �  s(λ) × t(λ)  �  ∗ h(λ) + n(λ)  (1)  where s is the stellar spectrum, incorporating line-of-sight ef-  fects from relative stellar velocity and the interstellar medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' t  is an imaginary signal representing the telluric lines affecting the  spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' x is the observed signal: a single spectrum that takes  on different values at each wavelength depending on the flux at  that wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' So we use the notation, x(λ), throughout this  article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  We use h to denote what we call the observation transfer  function, and models the changes the signal goes through dur-  ing the sensing process, and includes, but is not limited to, the  line spread function, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' n models the additive noise which is  not modeled in the transfer function, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  We work with a large ensemble of N observations, mostly  coming from different sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' We use the subscript, i, to differ-  entiate between various observations:  xi(λ) =  �  si(λ) × ti(λ)  �  ∗ h(λ) + ni(λ)  i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' , N}  (2)  Note that the spectrum, s, depends on i, as we work with various  objects at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The atmospheric conditions are also  time-variant, an important point that is reflected in dependence  of t on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The same is true for noise, n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Without loss of generality,  and for the sake of simplicity, we assume that h is constant across  the whole set of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  We seek to eliminate the effect of telluric lines from the  observed signal, which in this model is equal to extracting  si(λ) ∗ h(λ) for every observation – note that removal of the ob-  servation effect, h, is not part of the objective here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  We also model the effect of the radial velocity, v, of the target  object on the observed spectrum, s as:  si(λ) = V {˚si(λ), vi}  = ˚si  �  λ�1 − vi  c  ��  (3)  where ˚si would be the observed spectrum, if the radial velocity  was zero – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' no Doppler shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' We call ˚si the static spectrum  hereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The V{.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='} operator represents the effect of the radial  velocity, vi, on the spectrum, as expanded in the second line of  the above equation, and c is the speed of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The physical units  of vi and c can be arbitrarily chosen, as long as they are kept the  same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  ti on the other hand, and by definition, do not have any de-  pendence on vi, and therefore can be modeled as:  ti(λ) = V  �  ˚ti(λ), 0  �  (4)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Continuous vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Discrete  All the signals discussed above are of continuous nature up until  the point they are sensed by the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Sensing by the detector  Article number, page 3 of 10  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' main  is a process that involves sampling as one of its steps, converting  a spectrum into a series of real-valued samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Therefore, the  data we work with in our experiments is the discrete representa-  tion of xi(λ), namely Xl  i – we use l as the discrete-valued index  for the sampled pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  However, for the reasons below, keeping the formulation in  the continuous representation is safe – and clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' First, the  signals modeled so far are all the constituent elements of the  pre-sampling signal, xi(λ), and so the continuous models hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Secondly, the only part of our method which explicitly modifies  the signal along the wavelength axis, the V {, } operator (sec-  tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='1), does its job by regridding the interpolated version of  the signal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' practically converting the discrete signal back to its  continuous version, applying the transformation and sampling it  back again to the discrete space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Therefore mathematical defini-  tion of the operator in the continuous space is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Signals as Random Processes  The process of sensing a signal, xi, from an arbitrarily chosen  object in the sky, which is affected by many non-deterministic  phenomena along the way, can be seen as one realization of a  random process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' In other words, the set of xi, or their discrete  representation, Xi, for various i, represent an ensemble of real-  izations of a random process, {X} 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Note that in this particular  application, the index set of the random process is sampled from  wavelengths, λ – a bit counter-intuitive, as it is usually of a tem-  poral nature in typical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Therefore each Xl represents a random variable, with out-  comes Xl  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Similarly, we can model {S }, { ˚S } and {T} as discrete-  index random processes of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' {S } is then a genera-  tor of various clean spectra, S i, while { ˚S } generates the static ˚S l  i  spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' As we will see in the next section, this non-deterministic  view on signals allows us to explain the statistical operations and  properties of the components more clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Method  The method is composed of two key steps:  a) Increasing the entropy of telluric components across obser-  vations and,  b) transforming the spectra into a space where the stellar and  telluric components are separable, then removing the non-  dominant one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Median Normalization As a pre-processing step, we normalize  each spectrum in the dataset based on its median, to mitigate the  effect of different distances in sources, which otherwise results  in extreme inter-sample flux range variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Velocity Whitening  In the default conditions, the spectroscopic observations are in  the so-called topocentric reference frame where the telluric lines,  if they exist, take on the same wavelengths – or pixels locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  What it means from a statistical point of view though is this:  Let us assume, for simplicity, that all the observed objects  have the same static spectra, ˚si(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The resulting spectra, si(λ)  1 The term random in this context is not in contradiction with the struc-  ture in stellar spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The structure is encoded in the basic parameters  of the constituent set of random variables, such as the expected value  and covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  would then only be different based on their radial velocities,  vi, which, by definition, are modeled as scaling transformations  along the wavelength axis (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Note that since vi are out-  comes of a random variable, si(λ) are consequently still highly  independent and uncorrelated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  ti on the other hand, are composed of a set of absorption lines  with different strengths, but all happening at pre-defined specific  wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' This feature makes them highly correlated across  multiple observations and easy to fit for any model – fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 3, left  column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  To decrease the existing correlation between realizations of  the telluric signals, we increase the entropy across ti by random-  izing them using an emulated radial velocity,  t′  i(λ) = V  �  ˚ti(λ), v′  i  �  (5)  where each v′  i is a random velocity and is drawn from  V′ : R → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The above effect can be implemented by sim-  ply contracting or expanding the wavelength axis, according to  eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Note though that this is a purely artificial phenomenon,  and although it is chosen to have the same effect as the “real”  radial velocity of stars, it has no particular meaning for telluric  lines 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  In practice, however, we only have access to the observed  signal, x, and not its forming components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Hence the proposed  randomization cannot be applied directly on t alone, and the  whole observed signal gets affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' But in the below,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' we show  that it can still have the desired effect:  λ −→ λ′  i = λ × (1 − v′  i  c )  (6)  x′  i(λ) = V �xi(λ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' v′  i  �  = V  ��  si(λ) × ti(λ)  �  ∗ h(λ) + ni(λ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' v′  i  �  = V  ��  si(λ) × ti(λ)  �  ∗ h(λ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' v′  i  �  + V �ni(λ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' v′  i  �  (7)  which,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' according to the proof provided in appendix A becomes:  x′  i(λ) = (1 − v′  i  c )V �si(λ) × ti(λ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' v′  i  � ∗ V �h(λ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' v′  i  � + V �ni(λ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' v′  i  �  = (1 − v′  i  c )  �  si(λ′  i) × ti(λ′  i)  �  ∗ h(λ′  i) + ni(λ′  i)  (8)  Now since  �  1 −  v′  i  c  �  is,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' for each x′  i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' a constant coefficient and  is normalized out before being passed to the next step of the  method,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' we can see that the emulated randomized velocity has  founds its way from xi down to ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' In other words, we have  achieved the required velocity randomization in ti by modifying  xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The two main components in the tweaked signal can now be  rewritten as:  Telluric:  Stellar:  V  �  ˚ti(λ), v′  i  �  V  �  ˚si(λ), vi + v′  i  �  So, we have achieved two components which are virutally mov-  ing independently of each other, when going over various obser-  vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Figure 3 illustrates this effect in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  2 A similar deterministic effect occurs when taking stellar spectra to  the barycentric frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Article number, page 4 of 10  Sedaghat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' : Stellar Karaoke  6265  6270  6275  6280  topo-centric  6265  6270  6275  6280  ADP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='2014-09-16T11:03:35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='377  randomized  6265  6270  6275  6280  6265  6270  6275  6280  ADP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='2014-09-16T11:03:36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='737  6265  6270  6275  6280  Wavelength (Å)  6265  6270  6275  6280  Wavelength (Å)  ADP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='2014-09-16T11:03:38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='307  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' On the left a set of exemplar spectra are depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Stellar lines are  unaligned due to different radial velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' But telluric lines are aligned,  even though they may have different shapes due to their inherent time  dependence (the dotted vertical line indicates the location of a specific  telluric line in all plots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' On the right the same spectra after velocity  randomization are depicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Telluric lines are now unaligned too, but  with a pattern different to that of stellar lines!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  From a statistical point of view, what we achieve with the  above velocity randomization is a degree of “whitening” of the  tellurics random process, {T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' More precisely, we decrease the  mutual correlation between every pair, (T l, T m), pushing the sta-  tistical behavior of the telluric component toward white noise (Li  & Zhang 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Eldar & Oppenheim 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Kessy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Note again that (T l, T m), the random variables we try to decor-  relate, are each defined on a single pixel of the spectrum, and  their outcomes vary along with different observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' A visual-  ization of the results of the above whitening process is illustrated  in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Deep Feature Space  We borrow and use the exact architecture of the 1D convolu-  tional auto-encoder introduced by Sedaghat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' (2021) – Fig-  ure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The convolutional layers of the encoder transform the in-  put spectrum down to a pre-defined number of “latent variables”  at the bottleneck of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' This low-dimensional repre-  sentation of the input, also known as the “code”, is the most  compressed version of the input spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The dimensionality  of this vector is chosen based on the desired compression rate in  various experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Note though that in the VAE version of our  networks, which is the case in most of the experiments of this  work, the latent nodes are implemented probabilistically, each  being modeled with a pair of scalars: mean and std of a normal  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' On the other side of the bottleneck, the decoder re-  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Visual illustration of the covariance matrix of the ensemble of  signals, before and after whitening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' On the left, the covariance ma-  trix of the [6275, 6285]Å region for some subset of size 90 of the  observations is visualized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' On the right, the same is done after veloc-  ity whitening, where vi were sampled from the uniform distribution:  V ∼ U(−30km/s, 30km/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The covariance matrix is closer to the iden-  tity matrix now, confirming achievement of some degree of whiten-  ing/decorrelation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  ceives the compressed code and takes it step-by-step up to the  same dimension as the original input (218 + 216 for HARPS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' We  keep using the same per-pixel L1 end-to-end loss function, as in-  troduced in the original work, to achieve acceptable pixel-level  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  This ∼typical architecture has proven to be able to transform  the input to a space where noise-like components of the signal  are easily separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' As we show with our experiments, the sta-  tistical trick developed in the previous section pushes the telluric  components farther from the stellar features and closer to the  noise, in the learned feature space, letting them be rejected as  easily as noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  We of course need to constrain the reconstruction abilities of  the network with a high compression rate as well as a variational  loss at the bottleneck, not to have too strong of a network capable  of fitting the two independent components at the same time!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Data  We use data from two publicly available colletions: HARPS and  SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Our main experiments are run on HARPS, while SDSS is  used as a difficult test-bench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' HARPS  HARPS (High Accuracy Radial-velocity Planet Searcher;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Mayor  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 2003) is an instrument on the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='6m La Silla Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  It is a fibre-fed high-resolution echelle spectrograph dedicated  to the discovery of exoplanets, with a spectral resolution of  R = 115, 000 and covers the spectral range 378–691nm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The  data used has been instrument-corrected and detector-corrected,  as well as sky-subtracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Our downloaded dataset initially consisted of 272376 total  spectra, which after automatic removal of corrupted files, NaNs  and noise-like ones, was reduced to 267361 “stable” ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' This  collection is mainly composed of stellar spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' However, there  are some random contaminant objects too, such as SUN, MOON,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=', which we allowed to enter our training set on purpose,  to increase robustness of the learned features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' We trimmed, re-  gridded and homogenized all spectra before being passed to the  3 http://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='eso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='org/wdb/wdb/adp/phase3_main/form  Article number, page 5 of 10  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' main  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Results Qualitative illustration of the results on an exemplar selection of HARPS spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Each row depicts one spectrum, with its HARPS  ID written on the right, while columns focus on different regions of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Interpretation hint: reconstruction (orange) should follow the pseudo-  truth (molecfit: green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The left-most column covers the whole spectrum, as is fed into the network, and illustrates the robustness of the network  to different characteristics of the spectra (continuum, noise level, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Major stellar lines such as Hα can be easily spotted in the less ‘busy’  examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' In the middle column we zoom in on a potentially complex region where narrow stellar lines, when existing, can collide with telluric  lines of similar shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' in the second and 4th row there are examples of such cases, where the network rejects the telluric component, while  still preserving the stellar part very well – thus rejecting the hypothesis that it might be simply rejecting narrow lines by a applying a moving  average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' In the third column, we focus on a region where similarly narrow stellar lines occur in some of the spectra, but not in the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' This way  we reject the hypothesis that the network might have ’memorized’ the locations of the lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Note that the network (orange) is even outperforming  the pseudo-truth (green) in some cases, illustrating a smoother, more robust reconstruction in abrupt changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  network in exactly the same way as is done by Sedaghat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Note that the pipeline the HARPS spectra go through is set  up such that the spectra are automatically transformed to the  barycentric reference frame and re-gridded before being stored  in the archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The originally captured version, in the topocen-  tric reference frame, is also not preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Therefore we had to  transform them back to the topocentric frame for our experi-  Article number, page 6 of 10  pseudo-truth  input  reconstruction  Normalized Flux (-)  ADP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='2014-09-16T11:05:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
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+page_content='5  Normalized Flux (-)  ADP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='2014-09-16T11:03:31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
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+page_content='5  Normalized Flux (-)  ADP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='2014-09-16T11:03:31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
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+page_content='05567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='55570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
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+page_content='5  Normalized Flux (-)  ADP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='2014-10-01T10:22:48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='507  1  0  4000  5000  6000  6278  6279  6280  6281  6282  5565.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='05567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='55570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='05572.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='5  Wavelength (A)  Wavelength (A)  Wavelength (A)Sedaghat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' : Stellar Karaoke  ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Although this is touching the core concept of our pre-  sented method, it turned out to be a safe procedure: transforma-  tion to the barycentric frame exerts added randomness on the  radial velocity, which is perfectly compatible with our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  In fact, our method has been inspired by observing traces of the  above-mentioned fact in our initial experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' SDSS  Our second dataset consisted of spectra from the SEGUE (Yanny  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 2009) and SEGUE-2 spectroscopic surveys (Rockosi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  2022) that were part of the larger Sloan Digital Sky Survey  (SDSS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The SEGUE surveys both used the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='5m Sloan Foun-  dation Telescope (Gunn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 2006) located at Apache Point Ob-  servatory with the two original Sloan Digital Sky Survey fiber  spectrographs (Smee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The SDSS spectrographs have  a resolution of R ∼ 1800 and together have 640 fibers (320 each)  that plug into aluminum "plugplates" for each observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' In  each plate 32 plugs are reserved for blank sky observations and  16 for spectrophotometric standard stars in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Each spec-  trograph has a red and a blue channel that collect data on separate  CCDs with the blue wavelength range from approximately 3800  Å to 6100 Å and the red wavelengths spanning approximately  5900 Å to 9200 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Sources in the original SEGUE survey sam-  pled Milky Way stars at a variety of distances, colors and metal-  licities while the SEGUE-2 targets focused on stars in the Milky  Way halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  In our experiments we used the ~660,000 uncalibrated spec-  tra from the red CCD of one of the spectrographs (labeled as ’r1’  in the SDSS data archive) provided with SDSS Data Release 17  (Abdurro’uf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 2022) and accessible via the DR17 FITS web-  site4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The uncalibrated spectra consist of multiple 10-30 minute  exposures of each source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Since the SEGUE surveys imaged  each source multiple times to create coadded spectra this means  we have multiple spectra in our dataset for the same source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' For  each plate we exclude the fibers that were intentionally pointed  at empty patches of sky and labelled "SKY" in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  The uncalibrated spectra used are labelled ‘spFrame‘ in the  SDSS data model5 and according to the details of the SDSS spec-  troscopic pipeline (Stoughton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 2002) they are flat-fielded but  not flux-calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The flux calibration spectra (‘spFluxCalib‘  in the data model) include telluric absorption calculated by the  spectroscopic pipeline based upon the spectrophotometric stan-  dard stars that are included in the observation set of each plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  These calibration are what we use as a "pseudo-truth" for com-  parisons of our network results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Experiments and Results  We train and evaluate multiple networks based on various com-  binations of the below (hyper-)parameters:  Latent-space dimensionality  Velocity randomization level  For our main experiments, that include hyper-parameter  sweeping and controlled tests, we mainy use the HARPS dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Other datasets are used for comparative testing of the method in  extreme conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  4 https://data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='org/sas/dr17/  5 https://data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='org/datamodel/  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Main Results  Figure 5 illustrates a few examples of how the method manages  to reject the telluric lines and preserve the stellar features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The  example spectra and regions are specifically chosen to rule out  potential naive hypotheses for how the network rejects telluric  lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' In other words, the results confirm that the network has  learned a semantic representation of the constituent components  and is separating the sources in that feature space, as opposed  to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' simply having “missed” narrower lines (simple averag-  ing, low-pass filtering).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' In the caption of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 5 a few other such  hypotheses are elaborated and rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Quantitative Evaluation  Various (hyper-)parameters, such as the degree of compression  during dimensionality reduction, or the resolution of the input  data, may influence the capability of the network in rejecting  the telluric lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Precisely, there is always a trade off between  telluric-rejection and stellar-reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Therefore, we de-  velop and incorporate the below mutual metrics to quantify both  aspects at the same time:  Qt =  � �  Mt ��� ˆS − ˜G  ���  �  � Mt  (9)  where Qt is a proxy for the quality of rejection of telluric lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  ˆS represents the reconstructed spectrum (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' the direct output of  the network) and ˜G is the pseudo ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' ˜G is a corrected  version of the output of molecfit in case of HARPS data, and a  modified version of the publicly available calibrated spectra in  case of SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Mt is a binary mask;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' a vector of the same size  of the input, having ones at all pixels containing known telluric  lines, and zeros everywhere else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Note that the subscript i, index-  ing each observed spectrum, is omitted in Qt, ˆS , ˜G and Mt for  the sake of simplicity, and the summation is calculated over all  the pixels of each spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  We similarly define the dual metric for measuring the recon-  struction (preservation) of the stellar features as follows:  Qs =  �  �������Ms  �������1 −  ˆS  ˜G  �������  �������  � Ms  (10)  where Qs is, conversely, a proxy for the quality of stellar recon-  struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Note that both of the above are metrics of a distance  nature, and hence, the lower the better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The Effect of Compression  It is reasonable to expect, and is supported by our experiments  that, with a lower dimensionality at the information bottleneck  of the network, the network’s capacity for preserving many con-  current features during compression diminishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' This results in a  higher rejection of details in the reconstructed spectra, which can  be naively interpreted as an improvement in telluric line rejection  – a decrease in Qt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Note however that stellar reconstruction qual-  ity can get worse at the same time, and so the two metrics need  to be considered at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Table 1 compares results of various runs with different di-  mensionalities at the latent space on a fixed subset of HARPS  spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Note how the stellar reconstruction quality starts to de-  cay by increasing the compression (lower latent dimensions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' On  Article number, page 7 of 10  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
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+page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Test results on a low-resolution data collection: SDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' On the left, the whole spectrum is depicted, while the other columns depict various  regions of interest along the wavelength axis, so the qualitative performance of the method can be assessed on specific stellar and telluric features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Note that since pseudo-truth data is not easily obtainable in this case, we over-plot a "tellurics profile" in red, just for visual comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' It should  be interpreted differently from the pseudo-truth curves in other figures, as it does follow the telluric lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Stellar features are preserved very well,  even in bad signal-to-noise conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Some of the wider telluric lines, however, are not fully rejected in this case due to the significantly low  resolution as compared to HARPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Please refer to the main text for a more detailed discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Latent Dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='↓  Telluric  Stellar  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='053  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='015  32  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='055  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='012  128  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='063  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='014  1024  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='176  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='012  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Quality of stellar line reconstruction vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' telluric line rejection  on HARPS, for different configurations of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Latent Dim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' is  the number of dimensions of the code, or the latent representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' A  lower number is better in both columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  the other hand, by increasing the number of latent nodes, the  network starts to become too powerful, managing to reconstruct  both components and losing the source separation capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The Effect of Velocity-Whitening  To demonstrate the effect of the whitening step, we compare the  results of two experiments in controlled conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The first ex-  periment is run on HARPS spectra in topocentric frame, where  telluric lines are all aligned – zero velocity randomization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Then  we apply velocity whitening, sampling vi from the uniform dis-  tribution: V ∼ U(−30km/s, 30km/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' All other configurations of  the test are kept fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Table 2 shows a clear difference in the  telluric rejection performance of the two runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  HARPS  Telluric  Stellar  Topocentric  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='199  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='018  Rand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Velocity  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='063  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='014  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Comparison of two experiments with zero velocity randomiza-  tion (‘Topocentric’) and V ∼ U(−30km/s, 30km/s) (‘Rand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Velocity’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' HARPS vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' SDSS: High- vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Low-Resolution  Figure 6 depicts the results of applying our method on SDSS  spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The extremely low pixel resolution we used for this  dataset (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='66Å as opposed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='01Å for HARPS), makes this  in practice a stress test for our method, since many of the tel-  luric lines get merged, appearing as wide artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' However, the  method performs at an acceptable level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' the middle column  of fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' 6 shows how the method misses the very wide telluric ar-  tifact while managing to reject the neighboring telluric lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' In  general, it seems our method can “hunt” telluric lines up until  the point the resolution goes so low that merging/blending of the  lines converts them to “slow” artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Conclusions and Future Directions  We presented a method that, by incorporating a Big Data-  inspired view at stellar spectra, exploits the statistical indepen-  dence of the radial velocity of stars with telluric lines in their  observed spectra, reinforces it using a novel trick, and utilizes  a fully unsupervised convolutional neural network to reject the  undesireable part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  The method is superior to existing, traditional telluric line  removal tools in terms of preparation effort, performance, and  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' The fact that it is fully unsupervised, obviates the need  for any kind of model parameter tuning – which is the case in  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' molecfit, where a list of wavelengths should be manually  specified to initialize the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Training the model, in practice,  calls for merely passing a large number of spectra through the  network – and it will do the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Training one network suffices for one whole data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Once trained, it processes each spectrum in a fraction of a second  – depending on the size of the spectrum and, consequently, the  size of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' But the speed-up does not sacrifice accuracy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  as seen in section 5, it can even detect and suppress hard-to-  locate lines which are missed by molecfit – the opposite may  happen too, though in rarer situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Article number, page 8 of 10  Sedaghat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' : Stellar Karaoke  Nevertheless, the current version is still a demonstration of  a research product, showcasing the strengths of a fully unsuper-  vised approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' But for Stellar Karaoke to become a ready-to-  use package in every application, more work is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' No-  tably, the decision of which component to keep and which one  to reject (stellar vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' telluric), should not be left to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' It  can be enforced by a minimal supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' Some of the experiments demonstrated in this work have  been run on compute servers provided by ESO, during NS’s collaboration with  the ESCAPE project between 2019 and 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
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+page_content=' main  Appendix A: Wavelength Transformation and  Convolution  x(λ) = z(λ) ∗ h(λ)  =  �  w  z(w)h(λ − w)dw  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='1)  x(λ′  i) = x(aλ)  =  �  w  z(w)h(aλ − w)dw  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='2)  Let w = au,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='3)  x(λ′  i) = a  �  u  z(au)h(aλ − au)du  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='4)  Let z(au) = z′(u), h(au) = h′(u)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='5)  =⇒ x(λ′  i) = a  �  u  z′(u)h′(λ − u)du  = a z′(λ) ∗ h′(λ)  = a z(aλ) ∗ h(aλ)  = a z(λ′  i) ∗ h(λ′  i)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='6)  The same could be shown in a rather shorter way using  the Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content=' However, to avoid confusion between the  terms frequency and wavelength used in two different domains,  here we use the direct expansion of the convolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
+page_content='  Article number, page 10 of 10' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9AyT4oBgHgl3EQfd_iI/content/2301.00313v1.pdf'}
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+Observation of anisotropic superfluid density in an artificial crystal
+J. Tao,∗ M. Zhao,∗ and I. B. Spielman
+Joint Quantum Institute, University of Maryland and National Institute
+of Standards and Technology, College Park, Maryland 20742, USA
+(Dated: January 4, 2023)
+We experimentally and theoretically investigate the anisotropic speed of sound of an atomic
+superfluid (SF) Bose-Einstein condensate in a 1D optical lattice. Because the speed of sound derives
+from the SF density, this implies that the SF density is itself anisotropic. We find that the speed
+of sound is decreased by the optical lattice, and the SF density is concomitantly reduced. This
+reduction is accompanied by the appearance of a normal fluid in the purely Bose condensed phase.
+The reduction in SF density—first predicted [A. J. Leggett, Phys.
+Rev.
+Lett.
+25 1543–1546
+(1970)] in the context of supersolidity—results from the coexistence of superfluidity and density
+modulations, but is agnostic about the origin of the modulations. We additionally measure the
+moment of inertia of the system in a scissors mode experiment, demonstrating the existence of
+rotational flow. As such we shed light on some supersolid properties using imposed, rather than
+spontaneously formed, density-order.
+Superfluidity and Bose-Einstein condensation (BEC)
+are deeply connected.
+In dilute atomic BECs, the
+superfluid (SF) and condensate densities are generally
+equal [1, 2]. By contrast, SF 4He can be a nearly pure SF,
+with only about 14 % condensate fraction [3], and infinite
+2D Berezinskii–Kosterlitz–Thouless (BKT) SFs have no
+condensate at all [4, 5]. In 1970 Tony Leggett showed
+that supersolids—systems spontaneously forming both
+SF and crystalline order (i.e., density modulations)—
+exhibit the reverse behavior: SF density far below the
+condensate density [6].
+Here we observe this effect in
+a nearly pure atomic BEC with artificial crystal order
+imprinted by an optical lattice.
+The
+complex-valued
+order
+parameter
+φ(r)
+=
+�
+ρsf exp[iϕ(r)],
+describing a SF with number den-
+sity ρsf and phase ϕ(r), gives rise to two hallmark
+SF properties: dissipationless supercurrents associated
+with spatial gradients in ϕ(r) and (Bogoliubov [2])
+sound described by traveling waves in ϕ(r).
+Because
+supercurrents arise from phase gradients, they are locally
+irrotational; in liquid 4He, the resulting non-classical
+rotational inertia appears below the SF transition
+temperature Tc.
+Supersolids are more exotic systems
+spontaneously forming crystalline order while exhibiting
+SF transport properties.
+Recent experiments with
+dipolar BECs of Dy and Er are suggestive of these
+properties [7, 8].
+Leggett argued that the modulated
+density ρ(r) of a supersolid leads to an unavoidable
+reduction in ρsf, and derived an upper bound for ρsf [6].
+This reduction results from the 3D density distribution,
+and as such is masked in tight binding descriptions such
+as the Bose-Hubbard model, which makes the unrelated
+prediction of vanishing ρsf at the superfluid to Mott
+insulator transition [9, 10].
+We created an artificial SF crystal by imprinting pe-
+riodic density modulations into an atomic BEC using a
+1D optical lattice as in Fig. 1(a).
+While these modu-
+lations do not form spontaneously, Leggett’s result still
+applies, making this an ideal system for understanding
+crystalline SFs without the added complexity of spon-
+taneously broken symmetries. We experimentally mea-
+sured an anisotropic speed of sound via Bragg spec-
+troscopy of the phonon mode. This implies the existence
+of an effective anisotropic superfluid density—which can
+be expressed as a second rank tensor ρsf
+ij—and we find
+that it saturates Leggett’s bound, in agreement with
+Gross-Pitaveskii equation (GPE) simulations.
+We also
+determined an associated anisotropic suppression of the
+moment of inertia in terms of the scissor-mode frequen-
+cies [11, 12].
+Anisotropic superfluids—Here we consider pure 3D
+BECs whose condensate mode ψ(r) = |ψ(r)| exp[iϑ(r)] is
+well described by the Gross-Pitaveskii equation (GPE).
+An optical lattice potential V (r) = (U0/2) cos(2krx) peri-
+odically modulates the condensate density ρ(r) = |ψ(r)|2
+with unit cell (UC) size a = π/kr [Fig. 1(b)-i]. By con-
+trast, the SF order parameter φ(r) is a coarse grained
+quantity describing system properties on a scale ≫ a,
+giving the nominally uniform density in Fig. 1(c)-i.
+Even disregarding potential differences in ρsf(r) and
+ρ(r), we argue that φ(r) is not simply equal to ψ(r) av-
+eraged over some scale large compared to a. The fun-
+damental origin of this effect can be understood by con-
+sidering a 1D system of size L with periodic boundary
+conditions in which both the condensate phase ϑ and SF
+phase ϕ wind by an integer multiple N of 2π [Fig. 1(b,c)-
+ii], yielding a metastable quantized supercurrent [13].
+To satisfy the steady-state continuity equation, the mi-
+croscopic current J(x) = ρ(x) [ℏ∂xϑ(x)/m] must be in-
+dependent of x [Fig. 1(b)-ii], however, the periodically
+modulated density ρ(x) > 0 implies the local velocity
+v(x) = ℏ∂xϑ(x)/m has oscillatory structure and conse-
+quently ϑ(x) follows a staircase pattern [Fig. 1(b)-iii, iv].
+From macroscopic considerations the superfluid cur-
+rent is J = ρsf [ℏ∂xϕ(x)/m] = 2πNℏρsf/(mL). Equating
+the currents obtained from considering the condensate
+arXiv:2301.01258v1  [physics.atom-ph]  3 Jan 2023
+
+2
+(a)
+−1
+0
+1
+(b) BEC
+−1
+0
+1
+(c) SF
+Position x/a
+0.0
+2.5
+ρ/¯ρ
+i.
+0
+1
+2
+3
+J (arb.)
+ii.
+0
+1
+ϑ/(2π)
+iii.
+0
+5
+v/¯v
+iv.
+ρsf/¯ρ
+i.
+J (arb.)
+ii.
+ϕ/(2π)
+iii.
+v/¯v
+iv.
+FIG. 1.
+Concept. (a) A BEC is confined in a harmonic trap
+superimposed with a 1D optical lattice (along ex, green), spa-
+tially modulating the condensate density (red). The dashed
+and dotted lines call out a region of nominally constant mean
+density and the left and right columns indicate the (b) state
+of the condensate and (c) SF in the presence of a current.
+These were computed for a 5Er deep lattice and plot: i. den-
+sity (red), ii. current (green), iii. phase (orange), and iv. local
+velocity (blue). The red dashed line plots the mean density
+¯ρ.
+mode and the SF order parameter and integrating over
+a unit cell yields Leggett’s equation [6]
+ρsf =a
+��
+UC
+dx
+ρ(x)
+�−1
+, as well as ϕ = 1
+a
+�
+UC
+ϑ(x)dx. (1)
+This implies that ρsf ≤ ¯ρ, where ¯ρ is the spatial average
+of the condensate density over a UC, and as we discuss
+below the remaining density ρn = ¯ρ − ρsf behaves as a
+pseudo-normal fluid. In the more general context where
+the GPE is inapplicable, the Leggett expression for ρsf
+is an upper bound for the SF density in systems with
+crystalline order [6].
+In a 3D system, the current Ji = ρsf
+ij [ℏ∂jϕ/m] derives
+from a SF density tensor. For systems with rectangular
+symmetry [14] ρsf
+ij is diagonal, and the analogs to Eq. (1)
+for each of the three elements use a 1D density integrated
+along the transverse directions. In our experiments this
+implies that the superfluid density is only reduced along
+the direction of the optical lattice, so ρsf
+yy = ρsf
+zz = ¯ρ.
+Experiment—We used 87Rb BECs with N ≈ 2 × 105
+atoms in the |F = 1, mF = 1⟩ hyperfine ground state. A
+1064 nm trapping laser with an elliptical cross-section,
+traveling along ex provided strong vertical confinement
+with frequency ωz/(2π) = 220 Hz; the in-plane frequen-
+cies, from ωx,y/(2π) = (34, 51) Hz to (56, 36) Hz, were
+optimized for our different experiments. We created a
+1D optical lattice using a retro-reflected λ = 532 nm
+laser traveling along ex, giving an a = 266 nm lattice pe-
+riod, comparable to the ξ = 170(20) nm minimum heal-
+ing length. The optical lattice was linearly ramped on
+in 100 ms to a final depth ≤ 10 Er, with single pho-
+ton recoil energy and momentum Er = ℏ2k2
+r /(2m), and
+ℏkr = 2πℏ/λ respectively [15]. For Bragg experiments
+the final state was measured using resonant absorption
+imaging after a 15 ms time of flight (TOF); scissors mode
+measurements were performed in-situ using partial trans-
+fer absorption imaging [16].
+Anisotropic speed of sound—The speed of sound for di-
+agonal ρsf
+ij is predicted to result from c2
+i = f sf
+ii /(κm) in
+terms of the superfluid fractions f sf
+ii = ρsf
+ij/¯ρ, the com-
+pressibility κ = ¯ρ−1 (∂¯ρ/∂µ), and the chemical potential
+µ. This reduces to the well-known value c2 = µ/m for
+an isotropic homogeneous system (See [17] for the full
+dispersion beyond the linear approximation). The sound
+speed ratio
+c2
+x
+c2y
+= ρsf
+xx
+ρsf
+yy
+= f sf
+xx,
+(2)
+provides direct access to the different components of the
+superfluid density [see [17] for a Josephson sum rule [18]
+argument]. Because the density is y-independent, Eq. (1)
+implies ρsf
+yy = ¯ρ.
+We Bragg-scattered the BEC off a weak sinusoidal po-
+tential with reciprocal lattice vector δk slowly moving
+with velocity v by patterning a laser beam with a dig-
+ital micro-mirror device (DMD [19]) and measured the
+scattered fraction p. This results from what are effec-
+tively two interfering laser beams driving two-photon
+transitions with difference-wavevector δk and angular fre-
+quency δω = δk v. We applied this potential for ≈ 5 ms.
+Bragg transitions ensued when the difference energy and
+momentum were resonant with the BEC’s Bogoliubov
+dispersion, and Fig. 2(a) shows data in the linear regime.
+The width of this spectral feature is limited by our BEC’s
+inhomogeneous density profile; the resonance (vertical
+dashed line) obtained from a Lorentzian fit (solid curve)
+therefore reflects an average speed of sound [20].
+A series of such fits lead to phonon dispersion relations
+with Bragg-lattice period from 2.25 µm to 8.5 µm. Repre-
+sentative dispersions taken along ex and ey are shown in
+Fig. 2(b), and we obtain the phonon speed of sound using
+linear fits. Figure 2(c) summarizes these data showing
+the speed of sound decreasing along the lattice direction
+ex, but slightly increasing along ey (resulting from the
+increased atomic density in the individual lattice sites).
+Finally Fig. 2(d) shows our main result: the normalized
+superfluid density obtained from these data using Eq. (2)
+decreases as a function of U0.
+We compared these data to GPE simulations in two
+ways, we:
+(1) used the Bogoliubov-de Gennes (BdG)
+equations to obtain cx and cy and (2) directly evaluated
+Eq. (1) from the GPE ground state density. The solid
+curves in Fig. 2(c) plot the sound speed obtained from
+
+3
+0
+500
+1000
+δω/2π (Hz)
+0.0
+0.1
+0.2
+0.3
+p
+(a)
+0.0
+0.2
+0.4
+δk/2π (µm−1)
+0
+200
+400
+600
+800
+δω/2π (Hz)
+(b)
+0
+2
+4
+6
+8
+10
+U0/Er
+0
+1
+2
+3
+c (mm/s)
+cx
+cy
+(c)
+0
+2
+4
+6
+8
+10
+U0/Er
+0.00
+0.25
+0.50
+0.75
+1.00
+ρsf
+xx/¯ρ
+(d)
+FIG. 2.
+Bragg spectroscopy. Black and red symbols mark excitations created along ex and ey respectively. (a) Transferred
+population fraction p as a function of frequency difference δω with wavevetor δk/2π = 0.26 µm−1 and lattice depth U0 = 5.7Er.
+The solid curve is a Lorentzian fit giving the resonance frequency marked by the vertical dashed line. (b) Phonon dispersion
+obtained from Bragg spectra. The bold symbols resulted from (a) and the linear fit (with zero intercept) gives the speed of
+sound. (c) Anisotropic speed of sound. The bold symbols are derived from (b) and the solid curves are from BdG simulations
+(no free parameters [17]). (d) SF density obtained from speed of sound measurements (blue markers, error bars mark single-
+sigma statistical uncertainties). We compare with two models: the red dashed curve plots a homogeneous gas BdG calculation,
+and the solid black curve plots the result of Eq. (1). The simulations used our calibrated experimental parameters.
+solving the 1D BdG [21], and the red dashed curve in (d)
+is the ratio of these speeds. To compare with Leggett’s
+prediction, we found the ground state of the 2D GPE
+for our experimental parameters and evaluated Eq. (1)
+throughout our inhomogeneous system. The black curve
+in Fig. 2 plots the resulting weighted average. Remark-
+ably the BdG results are in near-perfect agreement with
+Leggett’s expression.
+Scissors mode—The single-valued nature of the SF or-
+der parameter greatly affects rotational properties such
+as the moment of inertia I. For highly anisotropic traps,
+the scissors mode [11] describes a fixed density distribu-
+tion pivoting by a small angle θ about the trap center
+with frequency ωsc. Scissors mode experiments are remi-
+niscent of torsional balance experiments in 4He [22] which
+give access to the non-classical rotational inertia [6].
+It is suggestive to quantify these dynamics in terms of
+the Lagrangian L = I ˙θ2/2 − V (θ), for moment of inertia
+0
+0.25 0.5 0.75
+1
+fsf
+xx
+0.00
+0.25
+0.50
+0.75
+1.00
+ωsc/ωsc,0
+(a)
+0
+2
+4
+6
+8 10
+U0/Er
+20
+40
+60
+ωd/2π Hz
+ωx,d
+ωy,d
+0
+0.25 0.5 0.75
+1
+fsf
+xx
+−0.2
+0.0
+0.2
+0.4
+I/Ic
+(54, 36) Hz
+(36, 50) Hz
+(b)
+FIG. 3.
+Moment of inertia from scissors mode.
+(a-inset)
+Measured dipole mode frequencies (markers) along with fits
+(curves) where the frequency at U0 is the only free param-
+eter for each curve. (a) Scissors mode frequency. Blue and
+green correspond to U = 0 trap frequencies (34, 51) Hz and
+(54, 36) Hz respectively. (b) Moment of inertia in units of Ic.
+Symbols are the data computed as described in the text, and
+the solid curves are GPE predictions using I = ∂Ω⟨Lz⟩, with
+angular frequency Ω.
+I and potential energy V (θ). For small θ the potential
+can be expanded as V (θ) ≈ Iω2
+scθ2/2 with
+I
+Ic
+=
+(ω2
+x − ω2
+y)2
+ω2sc(ω2x + ω2y)
+(3)
+in terms of the classical moment of inertia Ic and in agree-
+ment with Ref. [11] for isotropic superfluids. Although
+this interpretation is highly intuitive, it does not survive
+careful consideration.
+The anisotropic superfluid den-
+sity couples radial and azimuthal flow and as a result
+a single parameter Lagrangian is insufficient to describe
+rotational dynamics.
+Instead the superfluid hydrodynamic equations pre-
+dict a moment of inertia scaled by a factor of (f sf
+xxω2
+x −
+f sf
+yyω2
+y)/(ω2
+x − ω2
+y) (see [17]) compared to Eq. (3). There-
+fore we expect ωsc, in conjunction with the superfluid
+density will give I/Ic as a function of lattice depth.
+The inset to Fig. 3(a) plots the dipole mode frequen-
+cies ωx,d and ωy,d for a trap with frequencies (54, 36) Hz.
+The frequency reduction is also related to ρsf via f sf =
+(ωx,d/ωx)2 along the lattice direction [17].
+This ratio
+can also be expressed in terms of an increased effective
+mass that converges to the predictions of single-particle
+band structure [23] when the lattice period falls below
+the healing length; in our case the value computed per-
+turbatively from the GPE differs by about 20 % from the
+band structure prediction. The result of this modeling is
+shown by the solid curves.
+We excited the scissors mode using our DMD to tilt the
+harmonic potential by 50 to 140 mrad for ≈ 1 ms (shorter
+than the trap periods) and let the BEC evolve in the orig-
+inal trap for a variable time. We measured the resulting
+dynamics in-situ and extracted the angle by fitting the re-
+sulting density profile to a rotated Gaussian. Figure 3(a)
+shows scissor mode frequency normalized to the expected
+frequency [24] of ω2
+sc,0 = f sf
+xxω2
+x + f sf
+yyω2
+y for a trap elon-
+gated either along ex [with frequencies (56, 36) Hz, blue]
+
+4
+I/Ic
+I/Ic
+−20
+0
+20
+x (µm)
+−20
+0
+20
+y (µm)
+(a)
+−20
+0
+20
+x (µm)
+−20
+0
+20
+y (µm)
+(b)
+-1.0
+-0.5
+0.0
+0.5
+1.0
+Lz(r) (arb. units)
+−0.3
+0.0
+0.3
+(c)
+0
+0.25 0.5 0.75
+1
+fsf
+xx
+0.0
+0.3
+0.6
+0
+1
+(d)
+0
+0.25 0.5 0.75
+1
+fsf
+xx
+0
+1
+FIG. 4.
+Moment of inertia in rotating systems computed
+using 2D GPE simulations. The left column indicates sim-
+ulations in which the lattice is static while in the right col-
+umn the lattice co-rotates with the confining potential. (a,b)
+Angular momentum density for trap frequencies 2π×(56,36)
+and U0 = 10Er. (c, d) Total momentum of inertia in traps
+with frequencies 2π×(56,36) (top, green) and 2π ×(36, 56) Hz
+(bottom, blue). Dashed curves plot Isf/Ic and the solid curve
+plots I/Ic.
+or along ey [with frequencies (36, 50) Hz, green]. In both
+cases ωsc is about 5 % in excess of the simple predic-
+tion, perhaps from finite temperature or anharmonicities
+in the ODT.
+We combine these observations in Fig. 3(b) to ob-
+tain I/Ic; the data (symbols) and our 2D GPE simu-
+lations (curves) are in agreement. For traps elongated
+along ex (green) I/Ic unexpectedly changes sign when
+ωx,d = ωy,d. To understand the physical origin of this
+effect we now turn our attention to rotating systems.
+Rotation—Thus far we focused exclusively on the su-
+perfluid density, while avoiding questions about any as-
+sociated normal fluid. We can deduce the existence of a
+normal fluid component by considering two thought ex-
+periments in a 1D ring geometry (with radius R) and
+quantify both in terms of the resulting angular momen-
+tum [25]. In case (i), we consider an Aharonov-Bohm ge-
+ometry and slowly thread the ring with a single quanta
+of magnetic flux (see Ref. [26] for an artificial gauge field
+proposal). The process is equivalent to imprinting a 2π
+phase winding (of the type discussed on page 1), giving
+angular velocity Ω = ℏ/(mR2) and angular momentum
+Lz/ℏ = 2πRρsf. In case (ii), we consider a complimen-
+tary experiment in which the lattice is very slowly accel-
+erated to a final angular velocity Ω; this is best under-
+stood by transforming into the frame co-rotating with
+the lattice. This leads to a lab frame angular momentum
+Lz/ℏ = 2πR(¯ρ−ρsf) which we interpret as resulting from
+the normal fluid co-moving with the lattice.
+With this insight we extended our 2D numerical sim-
+ulations to analogous cases for rotating harmonically
+trapped systems where : (i) the lattice is static in the
+lab frame (as in scissors mode experiments) or (ii) it co-
+rotates with the confining potential. In both cases we
+use the coarse graining defined in Eq. (1) to obtain the
+superfluid density and phase. In this way we compute
+the total moment of inertia I from ψ(r, t), the superfluid
+component Isf from φ(r, t), and we define the normal
+component as the difference In = I − Isf.
+Case (i): as in our 1D thought experiment only the SF
+component responds. Then although ∇ϕ is manifestly
+irrotational, because ρsf
+xx ̸= ρsf
+yy the superfluid current
+can be rotational. In this case, the relative magnitude
+of the co- and counter-rotating contributions vary with
+the lattice depth, leading to regions of negative angular
+momentum density L(r) along the BEC’s semi-minor axis
+[Fig. 4(a)]. The superfluid moment of inertia computed
+from these simulations [Fig. 4(c)] is in full agreement with
+the scissor mode simulation, and as expected for a static
+lattice Isf = I (no normal flow).
+When the lattice is along the semi-minor axis, as pic-
+tured in (a) and the green curve in (c), the counter-
+rotating contribution increases with U0, until the dipole
+mode frequencies along ex and ey invert, after which
+point, I/Ic becomes negative.
+The reverse is the case
+when the lattice is along the semi-major axis and I/Ic
+increases monotonically. This novel observation confirms
+the negative kinetic energy resulting from ˙θ.
+Case (ii): In contrast, the angular momentum density
+is strictly positive [Fig. 4(b)] for both lattice orientations
+and I/Ic increases with lattice depth [Fig. 4(d)]. In this
+case the normal fluid to co-rotate with the trap giving
+the current Jn = (−ρn
+xxy, ρn
+yyx) ˙θ. The total I/Ic is then
+the sum of the superfluid [17] and normal contribution
+I
+Ic
+=
+(f sf
+xxω2
+x − f sf
+yyω2
+y)2
+(f sf
+xxω2x + f sf
+yyω2y)(ω2x + ω2y) + f n
+xxω2
+x + f n
+yyω2
+y
+ω2x + ω2y
+. (4)
+This result, along with our 2D GPE simulations, are
+plotted in Fig. 4(d). The dashed curve plots the super-
+fluid contribution to Isf/Ic in agreement with the coarse-
+grained GPE (crosses). The solid curve and the triangles
+plot the corresponding total moment of inertia, in excess
+of the SF contribution. This implies the appearance of
+normal fluid flow.
+This agreement confirms that the superfluid contribu-
+tion derives from gradients of the coarse-grained phase
+ϕ, while the normal contribution stems from variations
+of ϑ within each lattice site.
+Discussion and outlook—Our inability to obtain I/Ic
+from scissors mode measurements without detailed mod-
+eling reinforces similar conclusions in dipolar gases [27].
+In both cases the simple argument fails because ˙θ cou-
+ples to more internal degrees of freedom than Lz alone.
+In this context Ref. [27] concluded that the scissors mode
+
+5
+does yield the moment of inertia when 1D density mod-
+ulations comove with the oscillatory motion: this is con-
+sistent with our findings comparing motion in static and
+rotating lattices. Our GPE simulations indicate that the
+analytical relations generalize to lattices with period in
+excess of the healing length.
+Although we conclude that a normal fluid exists, it is
+inseparable from the optical lattice and lacks any internal
+dynamics of its own, i.e., it is not described by a dynami-
+cal equation of motion. In contrast, both the superstripe
+phase in spin-orbit coupled BECs [28–32] and supersolid
+phases of dipolar gases [33–35], support dynamical den-
+sity modulations. Leggett’s expression applies to both
+of these systems implying a reduced superfluid density,
+which in this case could exhibit dynamics, as expected
+for a system described by a two-fluid model [11, 12].
+This leaves open questions regarding nature the nor-
+mal fluid of spin-orbit coupled systems where an interplay
+between single-particle physics and interactions govern
+supersolid-like properties.
+In addition, ρsf is expected
+to be reduced outside of the superstripe phase [31, 32]
+where the density is uniform (making Leggett’s expres-
+sion inapplicable), but the BEC’s spin vector is spatially
+periodic.
+Note: During the early stages of manuscript prepara-
+tion we become aware of a related work, using a long
+period 1D lattice applied to a homogeneously confined
+2D BEC.
+The authors thank S. Stringari for suggesting this line
+of investigation and to both S. Stringari and S. Roccuzzo
+for stimulating discussions. In addition W. D. Phillips
+and S. Mukherjee carefully read the manuscript. This
+work was partially supported by the National Institute
+of Standards and Technology, and the National Science
+Foundation through the Physics Frontier Center at the
+Joint Quantum Institute (PHY-1430094) and the Quan-
+tum Leap Challenge Institute for Robust Quantum Sim-
+ulation (OMA-2120757).
+∗ These two authors contributed equally
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+
diff --git a/DdAzT4oBgHgl3EQfT_w7/content/tmp_files/load_file.txt b/DdAzT4oBgHgl3EQfT_w7/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..fd3e62161ce6212a1cb46a7ac585ee9da41f5338
--- /dev/null
+++ b/DdAzT4oBgHgl3EQfT_w7/content/tmp_files/load_file.txt
@@ -0,0 +1,549 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf,len=548
+page_content='Observation of anisotropic superfluid density in an artificial crystal  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Tao,∗ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Zhao,∗ and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Spielman  Joint Quantum Institute, University of Maryland and National Institute  of Standards and Technology, College Park, Maryland 20742, USA  (Dated: January 4, 2023)  We experimentally and theoretically investigate the anisotropic speed of sound of an atomic  superfluid (SF) Bose-Einstein condensate in a 1D optical lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Because the speed of sound derives  from the SF density, this implies that the SF density is itself anisotropic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' We find that the speed  of sound is decreased by the optical lattice, and the SF density is concomitantly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' This  reduction is accompanied by the appearance of a normal fluid in the purely Bose condensed phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  The reduction in SF density—first predicted [A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Leggett, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  25 1543–1546  (1970)] in the context of supersolidity—results from the coexistence of superfluidity and density  modulations, but is agnostic about the origin of the modulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' We additionally measure the  moment of inertia of the system in a scissors mode experiment, demonstrating the existence of  rotational flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' As such we shed light on some supersolid properties using imposed, rather than  spontaneously formed, density-order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Superfluidity and Bose-Einstein condensation (BEC)  are deeply connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  In dilute atomic BECs, the  superfluid (SF) and condensate densities are generally  equal [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' By contrast, SF 4He can be a nearly pure SF,  with only about 14 % condensate fraction [3], and infinite  2D Berezinskii–Kosterlitz–Thouless (BKT) SFs have no  condensate at all [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' In 1970 Tony Leggett showed  that supersolids—systems spontaneously forming both  SF and crystalline order (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=', density modulations)—  exhibit the reverse behavior: SF density far below the  condensate density [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Here we observe this effect in  a nearly pure atomic BEC with artificial crystal order  imprinted by an optical lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  The  complex-valued  order  parameter  φ(r)  =  �  ρsf exp[iϕ(r)],  describing a SF with number den-  sity ρsf and phase ϕ(r), gives rise to two hallmark  SF properties: dissipationless supercurrents associated  with spatial gradients in ϕ(r) and (Bogoliubov [2])  sound described by traveling waves in ϕ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Because  supercurrents arise from phase gradients, they are locally  irrotational;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' in liquid 4He, the resulting non-classical  rotational inertia appears below the SF transition  temperature Tc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Supersolids are more exotic systems  spontaneously forming crystalline order while exhibiting  SF transport properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Recent experiments with  dipolar BECs of Dy and Er are suggestive of these  properties [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Leggett argued that the modulated  density ρ(r) of a supersolid leads to an unavoidable  reduction in ρsf, and derived an upper bound for ρsf [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  This reduction results from the 3D density distribution,  and as such is masked in tight binding descriptions such  as the Bose-Hubbard model, which makes the unrelated  prediction of vanishing ρsf at the superfluid to Mott  insulator transition [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  We created an artificial SF crystal by imprinting pe-  riodic density modulations into an atomic BEC using a  1D optical lattice as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  While these modu-  lations do not form spontaneously, Leggett’s result still  applies, making this an ideal system for understanding  crystalline SFs without the added complexity of spon-  taneously broken symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' We experimentally mea-  sured an anisotropic speed of sound via Bragg spec-  troscopy of the phonon mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' This implies the existence  of an effective anisotropic superfluid density—which can  be expressed as a second rank tensor ρsf  ij—and we find  that it saturates Leggett’s bound, in agreement with  Gross-Pitaveskii equation (GPE) simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  We also  determined an associated anisotropic suppression of the  moment of inertia in terms of the scissor-mode frequen-  cies [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Anisotropic superfluids—Here we consider pure 3D  BECs whose condensate mode ψ(r) = |ψ(r)| exp[iϑ(r)] is  well described by the Gross-Pitaveskii equation (GPE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  An optical lattice potential V (r) = (U0/2) cos(2krx) peri-  odically modulates the condensate density ρ(r) = |ψ(r)|2  with unit cell (UC) size a = π/kr [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 1(b)-i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' By con-  trast, the SF order parameter φ(r) is a coarse grained  quantity describing system properties on a scale ≫ a,  giving the nominally uniform density in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 1(c)-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Even disregarding potential differences in ρsf(r) and  ρ(r), we argue that φ(r) is not simply equal to ψ(r) av-  eraged over some scale large compared to a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' The fun-  damental origin of this effect can be understood by con-  sidering a 1D system of size L with periodic boundary  conditions in which both the condensate phase ϑ and SF  phase ϕ wind by an integer multiple N of 2π [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 1(b,c)-  ii], yielding a metastable quantized supercurrent [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  To satisfy the steady-state continuity equation, the mi-  croscopic current J(x) = ρ(x) [ℏ∂xϑ(x)/m] must be in-  dependent of x [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 1(b)-ii], however, the periodically  modulated density ρ(x) > 0 implies the local velocity  v(x) = ℏ∂xϑ(x)/m has oscillatory structure and conse-  quently ϑ(x) follows a staircase pattern [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 1(b)-iii, iv].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  From macroscopic considerations the superfluid cur-  rent is J = ρsf [ℏ∂xϕ(x)/m] = 2πNℏρsf/(mL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Equating  the currents obtained from considering the condensate  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='01258v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='atom-ph]  3 Jan 2023  2  (a)  −1  0  1  (b) BEC  −1  0  1  (c) SF  Position x/a  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='5  ρ/¯ρ  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  0  1  2  3  J (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=')  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  0  1  ϑ/(2π)  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  0  5  v/¯v  iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  ρsf/¯ρ  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  J (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=')  ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  ϕ/(2π)  iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  v/¯v  iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' (a) A BEC is confined in a harmonic trap  superimposed with a 1D optical lattice (along ex, green), spa-  tially modulating the condensate density (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' The dashed  and dotted lines call out a region of nominally constant mean  density and the left and right columns indicate the (b) state  of the condensate and (c) SF in the presence of a current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  These were computed for a 5Er deep lattice and plot: i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' den-  sity (red), ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' current (green), iii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' phase (orange), and iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' local  velocity (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' The red dashed line plots the mean density  ¯ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  mode and the SF order parameter and integrating over  a unit cell yields Leggett’s equation [6]  ρsf =a  ��  UC  dx  ρ(x)  �−1  , as well as ϕ = 1  a  �  UC  ϑ(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' (1)  This implies that ρsf ≤ ¯ρ, where ¯ρ is the spatial average  of the condensate density over a UC, and as we discuss  below the remaining density ρn = ¯ρ − ρsf behaves as a  pseudo-normal fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' In the more general context where  the GPE is inapplicable, the Leggett expression for ρsf  is an upper bound for the SF density in systems with  crystalline order [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  In a 3D system, the current Ji = ρsf  ij [ℏ∂jϕ/m] derives  from a SF density tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' For systems with rectangular  symmetry [14] ρsf  ij is diagonal, and the analogs to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' (1)  for each of the three elements use a 1D density integrated  along the transverse directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' In our experiments this  implies that the superfluid density is only reduced along  the direction of the optical lattice, so ρsf  yy = ρsf  zz = ¯ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Experiment—We used 87Rb BECs with N ≈ 2 × 105  atoms in the |F = 1, mF = 1⟩ hyperfine ground state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' A  1064 nm trapping laser with an elliptical cross-section,  traveling along ex provided strong vertical confinement  with frequency ωz/(2π) = 220 Hz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' the in-plane frequen-  cies, from ωx,y/(2π) = (34, 51) Hz to (56, 36) Hz, were  optimized for our different experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' We created a  1D optical lattice using a retro-reflected λ = 532 nm  laser traveling along ex, giving an a = 266 nm lattice pe-  riod, comparable to the ξ = 170(20) nm minimum heal-  ing length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' The optical lattice was linearly ramped on  in 100 ms to a final depth ≤ 10 Er, with single pho-  ton recoil energy and momentum Er = ℏ2k2  r /(2m), and  ℏkr = 2πℏ/λ respectively [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' For Bragg experiments  the final state was measured using resonant absorption  imaging after a 15 ms time of flight (TOF);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' scissors mode  measurements were performed in-situ using partial trans-  fer absorption imaging [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Anisotropic speed of sound—The speed of sound for di-  agonal ρsf  ij is predicted to result from c2  i = f sf  ii /(κm) in  terms of the superfluid fractions f sf  ii = ρsf  ij/¯ρ, the com-  pressibility κ = ¯ρ−1 (∂¯ρ/∂µ), and the chemical potential  µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' This reduces to the well-known value c2 = µ/m for  an isotropic homogeneous system (See [17] for the full  dispersion beyond the linear approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' The sound  speed ratio  c2  x  c2y  = ρsf  xx  ρsf  yy  = f sf  xx,  (2)  provides direct access to the different components of the  superfluid density [see [17] for a Josephson sum rule [18]  argument].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Because the density is y-independent, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' (1)  implies ρsf  yy = ¯ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  We Bragg-scattered the BEC off a weak sinusoidal po-  tential with reciprocal lattice vector δk slowly moving  with velocity v by patterning a laser beam with a dig-  ital micro-mirror device (DMD [19]) and measured the  scattered fraction p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' This results from what are effec-  tively two interfering laser beams driving two-photon  transitions with difference-wavevector δk and angular fre-  quency δω = δk v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' We applied this potential for ≈ 5 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Bragg transitions ensued when the difference energy and  momentum were resonant with the BEC’s Bogoliubov  dispersion, and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 2(a) shows data in the linear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  The width of this spectral feature is limited by our BEC’s  inhomogeneous density profile;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' the resonance (vertical  dashed line) obtained from a Lorentzian fit (solid curve)  therefore reflects an average speed of sound [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  A series of such fits lead to phonon dispersion relations  with Bragg-lattice period from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='25 µm to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Repre-  sentative dispersions taken along ex and ey are shown in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 2(b), and we obtain the phonon speed of sound using  linear fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Figure 2(c) summarizes these data showing  the speed of sound decreasing along the lattice direction  ex, but slightly increasing along ey (resulting from the  increased atomic density in the individual lattice sites).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Finally Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 2(d) shows our main result: the normalized  superfluid density obtained from these data using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' (2)  decreases as a function of U0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  We compared these data to GPE simulations in two  ways, we:  (1) used the Bogoliubov-de Gennes (BdG)  equations to obtain cx and cy and (2) directly evaluated  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' (1) from the GPE ground state density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' The solid  curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 2(c) plot the sound speed obtained from  3  0  500  1000  δω/2π (Hz)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='3  p  (a)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='4  δk/2π (µm−1)  0  200  400  600  800  δω/2π (Hz)  (b)  0  2  4  6  8  10  U0/Er  0  1  2  3  c (mm/s)  cx  cy  (c)  0  2  4  6  8  10  U0/Er  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='00  ρsf  xx/¯ρ  (d)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Bragg spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Black and red symbols mark excitations created along ex and ey respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' (a) Transferred  population fraction p as a function of frequency difference δω with wavevetor δk/2π = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='26 µm−1 and lattice depth U0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='7Er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  The solid curve is a Lorentzian fit giving the resonance frequency marked by the vertical dashed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' (b) Phonon dispersion  obtained from Bragg spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' The bold symbols resulted from (a) and the linear fit (with zero intercept) gives the speed of  sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' (c) Anisotropic speed of sound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' The bold symbols are derived from (b) and the solid curves are from BdG simulations  (no free parameters [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' (d) SF density obtained from speed of sound measurements (blue markers, error bars mark single-  sigma statistical uncertainties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' We compare with two models: the red dashed curve plots a homogeneous gas BdG calculation,  and the solid black curve plots the result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' The simulations used our calibrated experimental parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  solving the 1D BdG [21], and the red dashed curve in (d)  is the ratio of these speeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' To compare with Leggett’s  prediction, we found the ground state of the 2D GPE  for our experimental parameters and evaluated Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' (1)  throughout our inhomogeneous system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' The black curve  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 2 plots the resulting weighted average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Remark-  ably the BdG results are in near-perfect agreement with  Leggett’s expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Scissors mode—The single-valued nature of the SF or-  der parameter greatly affects rotational properties such  as the moment of inertia I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' For highly anisotropic traps,  the scissors mode [11] describes a fixed density distribu-  tion pivoting by a small angle θ about the trap center  with frequency ωsc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Scissors mode experiments are remi-  niscent of torsional balance experiments in 4He [22] which  give access to the non-classical rotational inertia [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  It is suggestive to quantify these dynamics in terms of  the Lagrangian L = I ˙θ2/2 − V (θ), for moment of inertia  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='75  1  fsf  xx  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='75  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='00  ωsc/ωsc,0  (a)  0  2  4  6  8 10  U0/Er  20  40  60  ωd/2π Hz  ωx,d  ωy,d  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='75  1  fsf  xx  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='4  I/Ic  (54, 36) Hz  (36, 50) Hz  (b)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Moment of inertia from scissors mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  (a-inset)  Measured dipole mode frequencies (markers) along with fits  (curves) where the frequency at U0 is the only free param-  eter for each curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' (a) Scissors mode frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Blue and  green correspond to U = 0 trap frequencies (34, 51) Hz and  (54, 36) Hz respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' (b) Moment of inertia in units of Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Symbols are the data computed as described in the text, and  the solid curves are GPE predictions using I = ∂Ω⟨Lz⟩, with  angular frequency Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  I and potential energy V (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' For small θ the potential  can be expanded as V (θ) ≈ Iω2  scθ2/2 with  I  Ic  =  (ω2  x − ω2  y)2  ω2sc(ω2x + ω2y)  (3)  in terms of the classical moment of inertia Ic and in agree-  ment with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' [11] for isotropic superfluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Although  this interpretation is highly intuitive, it does not survive  careful consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  The anisotropic superfluid den-  sity couples radial and azimuthal flow and as a result  a single parameter Lagrangian is insufficient to describe  rotational dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Instead the superfluid hydrodynamic equations pre-  dict a moment of inertia scaled by a factor of (f sf  xxω2  x −  f sf  yyω2  y)/(ω2  x − ω2  y) (see [17]) compared to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' There-  fore we expect ωsc, in conjunction with the superfluid  density will give I/Ic as a function of lattice depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  The inset to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 3(a) plots the dipole mode frequen-  cies ωx,d and ωy,d for a trap with frequencies (54, 36) Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  The frequency reduction is also related to ρsf via f sf =  (ωx,d/ωx)2 along the lattice direction [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  This ratio  can also be expressed in terms of an increased effective  mass that converges to the predictions of single-particle  band structure [23] when the lattice period falls below  the healing length;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' in our case the value computed per-  turbatively from the GPE differs by about 20 % from the  band structure prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' The result of this modeling is  shown by the solid curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  We excited the scissors mode using our DMD to tilt the  harmonic potential by 50 to 140 mrad for ≈ 1 ms (shorter  than the trap periods) and let the BEC evolve in the orig-  inal trap for a variable time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' We measured the resulting  dynamics in-situ and extracted the angle by fitting the re-  sulting density profile to a rotated Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Figure 3(a)  shows scissor mode frequency normalized to the expected  frequency [24] of ω2  sc,0 = f sf  xxω2  x + f sf  yyω2  y for a trap elon-  gated either along ex [with frequencies (56, 36) Hz, blue]  4  I/Ic  I/Ic  −20  0  20  x (µm)  −20  0  20  y (µm)  (a)  −20  0  20  x (µm)  −20  0  20  y (µm)  (b)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='0  Lz(r) (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' units)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='3  (c)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='75  1  fsf  xx  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='6  0  1  (d)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='75  1  fsf  xx  0  1  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Moment of inertia in rotating systems computed  using 2D GPE simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' The left column indicates sim-  ulations in which the lattice is static while in the right col-  umn the lattice co-rotates with the confining potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' (a,b)  Angular momentum density for trap frequencies 2π×(56,36)  and U0 = 10Er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' (c, d) Total momentum of inertia in traps  with frequencies 2π×(56,36) (top, green) and 2π ×(36, 56) Hz  (bottom, blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Dashed curves plot Isf/Ic and the solid curve  plots I/Ic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  or along ey [with frequencies (36, 50) Hz, green].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' In both  cases ωsc is about 5 % in excess of the simple predic-  tion, perhaps from finite temperature or anharmonicities  in the ODT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  We combine these observations in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 3(b) to ob-  tain I/Ic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' the data (symbols) and our 2D GPE simu-  lations (curves) are in agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' For traps elongated  along ex (green) I/Ic unexpectedly changes sign when  ωx,d = ωy,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' To understand the physical origin of this  effect we now turn our attention to rotating systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Rotation—Thus far we focused exclusively on the su-  perfluid density, while avoiding questions about any as-  sociated normal fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' We can deduce the existence of a  normal fluid component by considering two thought ex-  periments in a 1D ring geometry (with radius R) and  quantify both in terms of the resulting angular momen-  tum [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' In case (i), we consider an Aharonov-Bohm ge-  ometry and slowly thread the ring with a single quanta  of magnetic flux (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' [26] for an artificial gauge field  proposal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' The process is equivalent to imprinting a 2π  phase winding (of the type discussed on page 1), giving  angular velocity Ω = ℏ/(mR2) and angular momentum  Lz/ℏ = 2πRρsf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' In case (ii), we consider a complimen-  tary experiment in which the lattice is very slowly accel-  erated to a final angular velocity Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' this is best under-  stood by transforming into the frame co-rotating with  the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' This leads to a lab frame angular momentum  Lz/ℏ = 2πR(¯ρ−ρsf) which we interpret as resulting from  the normal fluid co-moving with the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  With this insight we extended our 2D numerical sim-  ulations to analogous cases for rotating harmonically  trapped systems where : (i) the lattice is static in the  lab frame (as in scissors mode experiments) or (ii) it co-  rotates with the confining potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' In both cases we  use the coarse graining defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' (1) to obtain the  superfluid density and phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' In this way we compute  the total moment of inertia I from ψ(r, t), the superfluid  component Isf from φ(r, t), and we define the normal  component as the difference In = I − Isf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Case (i): as in our 1D thought experiment only the SF  component responds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Then although ∇ϕ is manifestly  irrotational, because ρsf  xx ̸= ρsf  yy the superfluid current  can be rotational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' In this case, the relative magnitude  of the co- and counter-rotating contributions vary with  the lattice depth, leading to regions of negative angular  momentum density L(r) along the BEC’s semi-minor axis  [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 4(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' The superfluid moment of inertia computed  from these simulations [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 4(c)] is in full agreement with  the scissor mode simulation, and as expected for a static  lattice Isf = I (no normal flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  When the lattice is along the semi-minor axis, as pic-  tured in (a) and the green curve in (c), the counter-  rotating contribution increases with U0, until the dipole  mode frequencies along ex and ey invert, after which  point, I/Ic becomes negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  The reverse is the case  when the lattice is along the semi-major axis and I/Ic  increases monotonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' This novel observation confirms  the negative kinetic energy resulting from ˙θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Case (ii): In contrast, the angular momentum density  is strictly positive [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 4(b)] for both lattice orientations  and I/Ic increases with lattice depth [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 4(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' In this  case the normal fluid to co-rotate with the trap giving  the current Jn = (−ρn  xxy, ρn  yyx) ˙θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' The total I/Ic is then  the sum of the superfluid [17] and normal contribution  I  Ic  =  (f sf  xxω2  x − f sf  yyω2  y)2  (f sf  xxω2x + f sf  yyω2y)(ω2x + ω2y) + f n  xxω2  x + f n  yyω2  y  ω2x + ω2y  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' (4)  This result, along with our 2D GPE simulations, are  plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' 4(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' The dashed curve plots the super-  fluid contribution to Isf/Ic in agreement with the coarse-  grained GPE (crosses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' The solid curve and the triangles  plot the corresponding total moment of inertia, in excess  of the SF contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' This implies the appearance of  normal fluid flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  This agreement confirms that the superfluid contribu-  tion derives from gradients of the coarse-grained phase  ϕ, while the normal contribution stems from variations  of ϑ within each lattice site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Discussion and outlook—Our inability to obtain I/Ic  from scissors mode measurements without detailed mod-  eling reinforces similar conclusions in dipolar gases [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  In both cases the simple argument fails because ˙θ cou-  ples to more internal degrees of freedom than Lz alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  In this context Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' [27] concluded that the scissors mode  5  does yield the moment of inertia when 1D density mod-  ulations comove with the oscillatory motion: this is con-  sistent with our findings comparing motion in static and  rotating lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Our GPE simulations indicate that the  analytical relations generalize to lattices with period in  excess of the healing length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Although we conclude that a normal fluid exists, it is  inseparable from the optical lattice and lacks any internal  dynamics of its own, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=', it is not described by a dynami-  cal equation of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' In contrast, both the superstripe  phase in spin-orbit coupled BECs [28–32] and supersolid  phases of dipolar gases [33–35], support dynamical den-  sity modulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Leggett’s expression applies to both  of these systems implying a reduced superfluid density,  which in this case could exhibit dynamics, as expected  for a system described by a two-fluid model [11, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  This leaves open questions regarding nature the nor-  mal fluid of spin-orbit coupled systems where an interplay  between single-particle physics and interactions govern  supersolid-like properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  In addition, ρsf is expected  to be reduced outside of the superstripe phase [31, 32]  where the density is uniform (making Leggett’s expres-  sion inapplicable), but the BEC’s spin vector is spatially  periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  Note: During the early stages of manuscript prepara-  tion we become aware of a related work, using a long  period 1D lattice applied to a homogeneously confined  2D BEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content='  The authors thank S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Stringari for suggesting this line  of investigation and to both S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Stringari and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Roccuzzo  for stimulating discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' In addition W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Phillips  and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' Mukherjee carefully read the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
+page_content=' This  work was partially supported by the National Institute  of Standards and Technology, and the National Science  Foundation through the Physics Frontier Center at the  Joint Quantum Institute (PHY-1430094) and the Quan-  tum Leap Challenge Institute for Robust Quantum Sim-  ulation (OMA-2120757).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdAzT4oBgHgl3EQfT_w7/content/2301.01258v1.pdf'}
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diff --git a/DdE4T4oBgHgl3EQfew2P/content/tmp_files/2301.05102v1.pdf.txt b/DdE4T4oBgHgl3EQfew2P/content/tmp_files/2301.05102v1.pdf.txt
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@@ -0,0 +1,1804 @@
+Improvement of Computational Performance of Evolutionary AutoML in a Heterogeneous
+Environment
+Nikolay O. Nikitin, Sergey Teryoshkin, Valerii Pokrovskii, Sergey Pakulin, Denis Nasonov
+ITMO University, Saint-Petersburg, Russia
+Abstract
+Resource-intensive computations are a major factor that limits the effectiveness of automated machine learning solutions. In the
+paper, we propose a modular approach that can be used to increase the quality of evolutionary optimization for modelling pipelines
+with a graph-based structure. It consists of several stages - parallelization, caching and evaluation. Heterogeneous and remote
+resources can be involved in the evaluation stage. The conducted experiments confirm the correctness and effectiveness of the
+proposed approach. The implemented algorithms are available as a part of the open-source framework FEDOT.
+Keywords: AutoML, heterogeneous infrastructure, evolutionary optimization, caching
+1. Introduction
+Nowadays,
+automated machine learning (AutoML) is
+widely used in science, and industry [16, 33]. The major prob-
+lem of solving real-world tasks with AutoML is the high com-
+putational cost of the search for an optimal modelling pipeline.
+During the evaluation of the candidate pipelines’ quality, many
+machine learning models are trained. This task is very resource-
+intensive, so it can take a considerable amount of time to
+achieve the appropriate result. It can be considered a bottle-
+neck for any existing AutoML solution. This issue raises vari-
+ous problems from different fields: from integration of AutoML
+to business processes [31] to carbon emission and sustainability
+concerns[35].
+There are many approaches for improving computational
+performance that are used in state-of-the-art (SOTA) AutoML
+solutions [8]. First of all, almost all solutions support parallel
+execution. Some of them also support caching of evaluated can-
+didates [22]. Also, the graphics processing unit can be used to
+reduce the training time [15].
+There is a variety of open-source tools that could improve
+the efficiency of certain steps of machine learning pipelines.
+For instance, involving various MLOps tools like MLFlow [38],
+task-specific databases [37] and scaling tools like Ray [18] al-
+lows the effectiveness of ML applications to be notably in-
+creased.
+However, the optimal design of the computational strategy
+depends on the infrastructure and the underlying AutoML al-
+gorithm. The SOTA AutoML solutions are based on differ-
+ent optimization methods: random search, Bayesian optimiza-
+tion, genetic algorithms, and meta-learning [8, 2]. The struc-
+tural patterns used in modelling pipelines can also be different:
+linear pipelines or ensembling techniques (stacking, blending,
+Email address: nnikitin@itmo.ru (Nikolay O. Nikitin)
+and boosting) [39]. The most complicated case is a composite
+pipeline represented as a directed acyclic graph [21]. At the
+same time, there is no ready-to-use solution for improving the
+computational performance of an automated open-ended search
+for pipelines in the composite AI field.
+In the paper, we want to propose a adaptive approach
+to reduce the computational cost of AutoML for compos-
+ite pipelines.
+Several techniques are implemented: pipeline
+caching, parallelization of the fitness function evaluation, com-
+putation with hybrid (GPU and CPU) systems, and integration
+with remote distributed systems.
+This approach differs from existing solutions since it can be
+configured for automated machine learning in various computa-
+tional environments (including distributed and heterogeneous).
+Also, the caching procedure can be effectively used for vari-
+ous pipeline designs (linear, weighted ensembles, multi-layer
+ensembles, etc).
+To confirm the effectiveness of the proposed approach in
+empirical way, we conducted a set of numerical experiments us-
+ing set of open datasets of various sizes (described in Table 1).
+The results presented in Section 6 allow us to conclude that a
+larger number of pipelines can be evaluated and better qual-
+ity metrics can be achieved by AutoML using this approach.
+The software implementation is available in the open-source
+AutoML framework FEDOT.
+The paper is organized as follows: Section 2 describes the
+computational strategies used in state-of-the-art AutoML tools.
+Section 3 provides the problem statement for AutoML perfor-
+mance improvement. Section 4 proposes a set of novel im-
+provements for the composite evolutionary AutoML. Section 5
+describes the software implementation of these techniques in an
+open-source framework. Section 6 provides the experimental
+evaluation of the proposed techniques for different case studies.
+Finally, Sec. 7 provides an analysis of the obtained results and
+possible extensions of the research.
+Preprint submitted to Algorithms
+January 13, 2023
+arXiv:2301.05102v1  [cs.LG]  12 Jan 2023
+
+2. Related Works
+There are dozens of open-source AutoML solutions that can
+be used for designing modelling pipelines. The first frame-
+works that became well-known are H2O [15], TPOT [14] and
+Auto-sklearn [4].
+As more novel AutoML solutions, Auto-
+Gluon [3] and LAMA [36] can be noted. Also, there are a lot
+of other AutoML tools with various specific features [17].
+There are different strategies for performance improve-
+ment used in the noted frameworks. In the TPOT framework,
+pipeline caching is implemented [14].
+TPOT-SH [26] uses
+the concept of Successive Halving to explore the search space
+faster, especially for larger datasets.
+Various techniques are
+used to evaluate the pipelines on different subsets of training
+data (e.g. layering [6]).
+A widely-used parallelization tool is the joblib library im-
+plemented in Python. However, there are more advanced frame-
+works for parallelization that can be noted. For example, Ray
+[18] can be used to scale AI and Python applications in dis-
+tributed environments. It provides various instruments for dis-
+tributed data preprocessing, distributed training of ML models
+and scalable hyperparameter tuning.
+Improving the computational performance for evolutionary
+algorithms outside AutoML is also discussed in the literature.
+As an example, parallel GPU-based evaluation of the fitness
+function can be used [24] to solve the expensive problems re-
+lated to big data [23]. There are various techniques that im-
+prove the performance of evolutionary algorithms in concurrent
+mode [7]. The tensor-based computational model can be used
+to achieve cross-platform hardware acceleration [12].
+Also,
+platform-specific open-source solutions are presented in this
+field (e.g. scikit-learn-intelex 1).
+One of the widely used techniques to avoid fitness evalua-
+tion bottlenecks in evolutionary algorithms is caching [29]. Fi-
+nal values of the fitness evaluation can be cached [13] as well
+as partial results [30, 11, 10].
+Moreover, a number of solutions exist that can perform
+remote/distributed training (e.g.
+Auto-sklearn, H2O, TPOT,
+LAMA). These AutoML frameworks use different frameworks
+for distributed computing. Autosklearn and TPOT use Dask2,
+LAMA uses Apache Spark3. H2O uses its own Apache Spark
+modification called Sparkling Water4. Distributed computing
+frameworks allow the processing of large datasets spread over
+the nodes of a cluster system.
+We can conclude that there is a large number of techniques
+and solutions that can reduce the resource consumption for Au-
+toML and EA. However, there is still no well-developed ap-
+proach that can be used to identify graph-based pipelines in
+the heterogeneous computational environment in composite AI
+problems. For this reason, we decided to formulate the problem
+statement specific to composite AutoML and propose possible
+solutions.
+1https://github.com/intel/scikit-learn-intelex
+2https://dask.org
+3https://spark.apache.org
+4https://h2o.ai/products/h2o-sparkling-water
+3. Problem Statement
+We want to design multi-task and multi-modal pipelines for
+various tasks using a single flexible instrument. Consequently,
+it becomes necessary to implement the framework’s architec-
+ture more abstractly to separate the pipeline search process
+from the top-level API. The modelling pipeline is represented
+as a directed acyclic graph in this case. Each node (modelling
+or data transformation operation) is described by the operation’s
+name and set of hyper-parameters. If necessary, different data
+sources (tables, time series, images, texts) can be involved in
+the pipeline. Also, metadata is attached to the data flow, making
+it possible to change the task several times during the pipeline
+evaluation (e.g., solve a classification task and then - a regres-
+sion task).
+The drawback of this approach is the increased search space
+that should be explored during the optimization.
+In auto-
+mated modelling, we want to control the balance between open-
+endedness [25] and local search. The simplest way is to apply
+the of direct constraints (e.g. limit to the pipeline size). Also, it
+can be more effective to apply the regularization and sensitivity
+analysis procedures [21] and adaptive optimisation strategies
+to control the convergence of optimisation. At the same time,
+avoiding over-complicated pipelines and reckless spending of a
+limited time budget is also essential. It makes the effectiveness
+of the computational part even more critical for open-ended Au-
+toML.
+It pushes us to compromise between pipeline complexity
+and training time. However, if we can improve computing ef-
+ficiency, the framework will probably be able to build more
+complicated models with higher quality while consuming the
+same training time.
+There are many approaches to improv-
+ing evolutionary algorithms’ computing performance, such as
+parallelization, caching, etc. These approaches can be divided
+into single-machine optimizations and horizontal scaling tech-
+niques. Single-machine optimizations aim to improve comput-
+ing performance only on the machines performing the compu-
+tations. Horizontal scaling allows involving additional servers
+to speed up computing. Both techniques can be used separately
+or combined.
+Evolutionary algorithms’ computational time mainly de-
+pends on the population size and the number of generations.
+Increasing population size leads to an increased probability of
+getting better individuals. A more significant number of gener-
+ations means more attempts to grow better individuals based on
+the best previous generation.
+From a computational point of view, we have several iter-
+ations, each of which requires the results from the previous it-
+eration. So, it is complicated to scale computations over the
+iterations, and the total computation time is Equation 1.
+Ttotal =
+n
+�
+i=1
+Ti
+(1)
+where n is the number of iterations and Ti - the computa-
+tion time of generation i. Inside one generation, all individuals
+are processed independently from one another, allowing us to
+2
+
+scale these computations according to the available computa-
+tional resources. The population training time can be estimated
+with Equation 2.
+Ttotal =
+n
+�
+i=1
+argmax
+di
+j∈Di
+(argmin(τ(di
+j,
+rj∈Ri/{rj−1,..,r1}
+ω(Di−1,..,D1), rj))
+(2)
+where Di is the set of individuals in the population i and di
+j is
+individual j, Ri is the set of available resources on iteration i,
+ω is a cache function with pre-calculated elements on iterations
+i − 1, i − 2, ..., 1, and τ is a function that returns the calculation
+time considering caching and evaluation of individuals Di.
+Due to the Equations 1 and 2, we should increase the popu-
+lation size as much as possible. It allows to speed up the algo-
+rithm convergence and improve its result using horizontal scal-
+ing. For this purpose, we can use both remote computing on
+production servers and distributed computing using homoge-
+neous and heterogeneous computing clusters.
+Remote computing allows model training to be delegated to
+a remote infrastructure. This approach is justified if the local
+machine computational resources are insufficient to train a set
+of models in a reasonable amount of time. Remote computing
+may take place on a dedicated computing server or a cluster of
+servers that accepts tasks to train models using REST API, RPC
+or message queues.
+The main challenge in the investigated problem is to pro-
+pose the performance improvement strategy for AutoML that
+is adaptive to various types of computational infrastructures.
+In Figure 1, five classes are noted: shared memory system,
+multi-node cluster with distributed memory, complex homo-
+geneous and heterogeneous supercomputer environments, and
+hybrid systems. The system with structure (a) can execute par-
+allel tasks in a straightforward way. In the systems (b)-(e), the
+remote nodes are involved (homogeneous and heterogeneous).
+For the system (e), the structure is hybrid since various remote
+nodes have different computational performance and connec-
+tions overheads. For this reason, the adaptability of the compu-
+tational strategy is especially important.
+There are various ways can be uses to adapt the compu-
+tational strategy to specific infrastructure.
+For example, the
+empirical performance models [9] can be used to choose the
+optimal infrastructure for evaluating specific pipelines. Sim-
+ple pipelines with low fitting time can be assigned to low-
+performance computational nodes.
+Otherwise, complicated
+pipelines with high fitting time can be assigned to high-
+performance nodes. It makes it necessary to develop a modular
+approach that effectively utilizes all available resources.
+Our main motivation is to develop an approach that can be
+used at the computational layer of AutoML. It should be pos-
+sible to adapt this layer to the specified infrastructure (local or
+remote) in a frame of the same AutoML approach. This solution
+should be high-level, modular and flexible to allow integrating
+it with different AutoML tools. Also, it should support the dif-
+ferent types of pipelines (from simplest linear pipeline to the
+multi-level ensembles).
+4. Proposed Improvements
+This section is devoted to various aspects of the proposed
+approach for improving the computational performance of evo-
+lutionary AutoML. The high-level scheme of the approach is
+presented in Figure 2. Four main aspects are considered: (1)
+parallelization of the fitness function evaluation; (2) partial
+caching of evaluated individuals; (3) combining CPU and GPU
+to accelerate the processing of individuals (4) integration with
+remote infrastructure for a complex task. Algorithmic-based
+improvements (e.g.
+surrogate-assisted optimization) are not
+considered here.
+The detailed implementation of the proposed approach is
+described in Alg. 1. In this notation, graph represents the struc-
+ture of the composite pipeline. The details of the evolutionary
+optimisation are hidden to make the proposed improvements
+more clear.
+Algorithm 1 High-level pseudocode of the evaluation dispatch-
+ing algorithm implemented in the proposed approach. Paral-
+lelization, caching and evaluation stages are demonstrated for
+processing one generation of the evolutionary algorithm.
+1: procedure ProcessPopulation
+2:
+Input:
+inds (set of non-evaluated individuals),
+objective (objective function that calculates the fitness
+of an individual),
+n (number of parallel jobs)
+timer (timer-like object)
+infrastructure (description of setup)
+3:
+Output: evaluated inds
+4:
+do in parallel(n)
+5:
+if timer.enough time( ) then
+6:
+graph ← inds[i].graph ▷ get structure of each ind.
+7:
+if infrastructure.is remote( ) then
+8:
+cache ← DistributedCache()
+9:
+sync cache
+▷ sync cache database
+10:
+task id ← create task(graph)
+11:
+wait task id
+12:
+inds[i].fitness ← request result(graph)
+13:
+else
+14:
+prepare graph
+▷ assign CPU and GPU to
+nodes
+15:
+cache ← LocalCache()
+16:
+load cache
+▷ init cache database
+17:
+if cache.exists(graph)( ) then
+18:
+fit from cache(graph)
+19:
+inds[i].fitness ← obj(graph, cache)
+20:
+fit(graph)
+▷ Fit nodes that are not in cache
+21:
+save cache
+▷ preserve updated cache
+22:
+if not inds[i].is valid then
+23:
+delete inds[i]
+▷ for unsuccessful evaluation
+24:
+else
+25:
+delete inds[i]
+▷ not enough time, skipping
+26:
+return inds
+▷ candidates for selection
+3
+
+Figure 1: Different types of computational infrastructures that can be used in AutoML: (a) shared memory (SM) system (b) multi-node cluster with distributed
+memory (c) complex homogeneous supercomputer system with spatially distributed infrastructure (d) supercomputer system with heterogeneous distributed infras-
+tructure.
+Figure 2: Workflow of the proposed approach for the improvement of compu-
+tational performance for composite AutoML
+4.1. Parallelization
+Parallelizing evolutionary algorithms is not a novel idea.
+There are a lot of papers and open-source solutions devoted
+to this problem. However, parallelization in AutoML has its
+specifics. For example, various computationally efficient strate-
+gies of parallel evolution can be used [9].
+We are considering an evolutionary algorithm for search-
+ing for the best solution in the space of pipelines that can be
+represented as directed acyclic graphs. The classic approach
+to parallelizing evolutionary optimization is evaluating all indi-
+viduals in the population concurrently [9]. It works because of
+the nature of the evolutionary algorithm. There are no depen-
+dencies between individuals in a generation. Other approaches
+suggest dividing populations into isolated parts [32] or using
+co-evolutionary algorithms to divide tasks into subtasks [5].
+The proposed algorithm considers the maximum evaluation
+time length for each pipeline evaluation to resolve the possi-
+ble evaluation time anomalies caused by the stochastic nature
+of data-driven model training. If the training process does not
+converge at least in one cross-validation fold, the time required
+for corresponding fitness evaluation can be increased signifi-
+cantly. So, the individuals that spend excess time on evaluation
+are skipped to preserve the overall performance of the evolu-
+tionary optimizer.
+4.2. Caching
+The existing caching approaches are aimed at preserving
+and reusing fitted pipelines [22]. However, separate nodes of
+composite pipelines can be cached individually [21]. It makes
+it possible to reuse the fitted models and reduce the fitness func-
+tion’s evaluation time. The optimizer can share the in-memory
+cache across the populations, and individuals [9]. However, it
+raises the problems of memory consumption.
+After analyzing existing solutions, we focused on the rela-
+tional database approach for pipeline caching. More specifi-
+cally, the sqlite3 library was used to implement it. First of all,
+it provides only one output file, which is not guaranteed for
+non-relational databases - e.g. shelve. Secondly, all concurrent
+save-load operations can be fully processed during the parallel
+evaluation of the fitness functions without direct usage of syn-
+chronization primitives, atomic variables and other instruments
+necessary for simultaneous access to data.
+Finally, this approach allows extracting several operations
+simultaneously, which helps to improve the overall perfor-
+mance of caching. Also, the set of operations can be saved
+to the database taking into account the existence of cache items
+with the same primary key.
+The caching procedure for the multi-layer ensemble
+pipelines should take into account that the cached model/-
+operations are suitable only for the specific configuration of
+previous nodes and edges in the modelling pipelines.
+So,
+the key contains the recursive description of the structure
+4
+
+a)Shared memory
+b) Cluster with
+distributed memory
+system (SM)
+c)Homogeneoussupercomputer
+environment
+Core
+Core
+Core
+Main
+1
+N
+Node
+Sheduler
+Node
+Node
+Node
+1
+N2
+Node
+Node
+Node
+1
+..
+N
+Node
+Node
+Node
+Node
+Node
+Node
+1
+N1
+d) Heterogeneous
+1
+N3
+supercomputerenvironment
+e) Hybrid environment
+Sheduler
+SM
+System
+Low-speedchannel
+High-speed channel
+Low-speed channel Mid speed channel High-speed channel
+SM
+Node
+Node
+Node
+SM
+Clust.
+Clust.
+Embed.
+Embed.
+Syst.
+1
+N
+Syst.
+system
+systemCandidate
+ML pipeline
+Population K
+0000
+N
+Individuals
+不
+Parallelization
+Evaluation
+Caching
+Local
+Remote
+CPU
+GPUof previous nodes and edges.
+Also, the identifier of cross-
+validation fold is specified. The following notation is used:
+(/[node name] [hparams];)/[node name] [hparams]...”, where
+/ denotes the beginning of the node name and round brackets
+represent the nested edges. The caching details are presented in
+Figure 3.
+Figure 3: Interaction between operations’ cache and the modelling pipeline that
+should be fitted
+4.3. GPU
+Evaluating ML models with GPUs is a well-developed fea-
+ture in many solutions. For example, the RAPIDS library [34]
+contains the CuML module that allows training classification,
+regression and clustering models with GPUs. To adapt this so-
+lution to composite pipelines, we should consider a setup in
+which only a part of the nodes can be evaluated with GPUs. In
+this situation, the pipeline should be fitted in a heterogeneous
+way.
+The proposed approach makes it possible to use both CPUs
+and GPUs for fitting by separating the ML model type and its
+implementation. The same model (e.g. random forest) can have
+several implementations (CPU-based and GPU-based).
+Figure 4 shows an example of a computationally hetero-
+geneous composite model structure. Data transfer between the
+GPU-based nodes (yellow) is performed within the video mem-
+ory, and the models themselves in the nodes are trained on
+graphics processing units (GPUs). Other nodes are executed
+on CPUs.
+Due to the multiple software limitations set by RAPIDS li-
+braries (CUDA-compatible GPU driver, restricted set of sup-
+ported operation systems), it is practical to conduct the compu-
+tations within Docker-based containers.
+4.4. Remote evaluation
+Remote evaluation can be integrated into the evolutionary
+optimiser in various ways. Both dataset folds and population
+parts can be distributed across several computational nodes to
+satisfy time or memory limits. Since evolutionary algorithms
+do not always require processing large datasets, we have fo-
+cused on the parallelism aspect of remote computing. The pro-
+posed implementation relies on Kubernetes. The REST API
+Figure 4: The structure of a pipeline that can be evaluated in a heterogeneous
+(CPU - blue and GPU - yellow) way
+service inside the Kubernetes cluster is used to run computa-
+tions via HTTP requests. The client implements a wrapper for
+requests.
+During the population training, the evaluator uses the
+client’s methods to process individuals on the Kubernetes clus-
+ter. Then, after starting processing all individuals, the evaluator
+waits for computations to be completed via client methods. The
+run request contains the container image, resources limit for the
+container, mount paths and model parameters. The REST API
+service creates the requested container and keeps monitoring it.
+The client uses requests to the REST API service to get actual
+containers’ statuses to see if it is still running, completed or
+failed.
+Finally, the client downloads the fitted pipeline when the
+training is completed. We wrap the result into a compressed
+archive to reduce the amount of data transferred over the net-
+work. Then, the files are sent to the client. This process scheme
+is presented in Figure 5.
+Figure 5: Communication between AutoML and remote cluster
+This way, we can divide the population training process into
+three stages: (1) requests to evaluate individuals, (2) computing
+and waiting for the completion and (3) fetching the results.
+5. Software Implementation
+The proposed approach can be used as a part of the archi-
+tecture that includes:
+5
+
+Preprocessor cache
+Nodes cache
+data_description 1: fitted preprocessor 1
+node uid 1: fitted node 1
+data description 2: fitted preprocessor 2
+node uid 2: fitted node 2
+Data fold 1
+preprocessor
+Node 1
+uid=node 4/
+evaluation
+(node1;node2;node3)
+Fitness
+Data
+Data fold 2
+Node 2
+Node 4
+Node 3
+Data fold 3
+Pipeline
+8
+Evo. opt
+88
+N
+N+1
+pop.
+pop.Decision
+Tree
+(GPU)
+SVC
+(GPU)
+Random
+Scaling
+Forest
+(GPU)
+Logit
+(CPU)
+GPU memory
+(CPU)
+XGBoost
+(CPU)LocalAutoML
+Kubernetes
+RemoteEvaluatol
+Client
+AutoML
+Create individual
+Individual-1
+create task
+RESTAPI
+AutoML
+Wait until ready
+get task status
+Service
+Individual-2
+AutoML
+Fetch individual
+download result
+Individual-N• The model repository block, which provides storage and
+selection of various implementations of predictive mod-
+els and data processing blocks. One model can contain
+several implementations (e.g., for CPU and GPU);
+• The block of the generative design of composite mod-
+els, which implements the creation of models with spec-
+ified properties by evolutionary algorithms. The proper-
+ties of the models are determined by the target function
+passed to the optimizer. If there is more than one tar-
+get function specified (as an example, the training time
+and modelling error can be used together as objectives
+for AutoML), then the multi-criteria formulation of the
+optimization problem is implemented, where the result
+of the model design is a Pareto front containing various
+compromising solutions. The genotype is represented in
+graph form, and the crossing and mutation operators are
+implemented accordingly.
+• The pipeline execution block on a given computational
+infrastructure. It allows individual pipeline execution on
+the given computational nodes.
+This architecture is implemented in the core of the open-
+source FEDOT framework. Different aspects of its implemen-
+tation are already detailed in a series of papers: [19] describes
+the main schemes and the implementation of the evolutionary
+operators, [28] is devoted to the multi-objective modification of
+this approach, and [20] provides an extended description of the
+various aspects of the evolutionary design for composite mod-
+elling pipelines. The tuning strategy of the pipeline hyperpa-
+rameters is based on Bayesian optimization.
+Custom models can be put inside this node. Search space
+for hyperparameters and initial approximations for the models
+should be specified manually if necessary. It makes it possi-
+ble to involve the infrastructure-specific implementation of the
+model in AutoML.
+The example below demonstrates the AutoML workflow
+from input data processing to obtaining prediction.
+api = Fedot(problem=’classification ’,
+seed =42, timeout =30, preset=’gpu’)
+api.fit(features=x_train , target=
+y_train)
+predictions = api.predict(features=
+x_test)
+Figure 6 provides the UML class diagram for the imple-
+mentation of various evaluation strategies that allow combining
+CPU- and GPU-based nodes in a single modelling pipeline. A
+high-level modelling method (e.g. Support Vector Classifica-
+tion) can be implemented using different algorithms: a CPU-
+optimised implementation of SVC can be obtained from the
+scikit-learn library [27]. In contrast, a GPU-optimised imple-
+mentation is available in the CUML library. The proposed ar-
+chitecture makes it possible to hide these details inside the spe-
+cific modelling pipeline and use the same optimisation logic for
+different implementations of the algorithms.
+Figure 6: The class diagram for the implementation of a modelling pipeline that
+consists of several operations. The Operation class represents a high-level mod-
+elling strategy that is used inside the operation. EvaluationStrategy is a base
+class for the algorithmic implementation of this strategy. SklearnEvaluation-
+Strategy represents the implementation obtained from the scikit-learn library
+and CumlEvaluationStrategy represents the implementation from the CUML
+library.
+The optimizer operates on individual models as a black box
+with input, output and fit/predict methods. The following build-
+ing blocks can be used for pipelines: models (Bernoulli Naive
+Bayes classifier, logistic regression, multilayer perceptron, ran-
+dom forest, gradient boosting, k-nearest classifier, QDA, LDA,
+decision tree) and data transformation operations (scaling, nor-
+malization, polynomial features transformation, principal com-
+ponent analysis, independent component analysis, isolation for-
+est, resampling).
+6. Experimental Studies
+We conducted a series of experiments to confirm the cor-
+rectness and effectiveness of the proposed approach.
+It can
+be divided into experiments with local and remote infrastruc-
+ture. As benchmarks, various classification datasets from the
+OpenML base [1] and synthetic datasets were used (the full list
+is presented in Table 1). A description of the computational
+infrastructure is provided for each experiment.
+The following methodology was used for experimental
+studies: each experiment started with dividing samples into two
+groups: ‘learning’ and ‘validation’ samples in the ratio 70% to
+30% to avoid data leaks. Then, the learning sample was trans-
+ferred to the evolutionary optimizer. During the optimisation,
+the 5-fold cross-validation procedure was applied to estimate
+the values of the fitness function.
+The experiment is repeated three times for each dataset to
+take the stochasticity of the optimizer into account. The quality
+metrics are averaged over these iterations.
+6.1. Local infrastructure
+For experiments with the local infrastructure, we configured
+a server based on Xeon Cascadelake (2900MHz) with 12 cores
+6
+
+EvaluationStrategy
+operation_type
+fito
+Operation
+predictO
+operation_type
+A
+strategy
+define_strategy0
+exectute_strategy0
+SklearnEvaluationStrategy
+operation_implementation
+fito
+predicto
+CUMLEvaluationStrategy
+Pipeline
+operation_implementation
+fito
+predict0Table 1: The properties of OpenML datasets that were used during the exper-
+iments. The random forest model is used as a baseline for the training time
+estimation.
+Dataset
+name
+Rows,
+10ˆ3
+Feat.
+Total
+elem.,
+10ˆ3
+Base.
+train.
+time,
+sec
+Num.
+of
+clas
+ses
+adult
+49
+14
+684
+12.5
+2
+amazon
+employee
+access
+33
+9
+295
+1.5
+2
+australian
+0.69
+15
+10
+0.2
+2
+bank-
+marketing
+45
+17
+769
+6.9
+2
+blood-
+transfusion
+0.75
+5
+4
+0.1
+2
+car
+1,8
+7
+12
+0.5
+4
+cnae-9
+1,1
+857
+926
+14.7
+9
+jungle chess
+2pcs
+45
+7
+314
+48.0
+3
+numerai28
+96
+22
+2119
+8.4
+2
+phoneme
+54
+6
+32
+0.12
+2
+sylvine
+51
+21
+108
+0.5
+2
+volkert
+58
+181
+10554
+128.4
+10
+synthetic
+blobs
+100
+10
+1000
+6.2
+2
+synthetic
+moons
+1
+2
+2
+0.12
+2
+and 24Gb memory.
+As our approach claims to increase the number of evaluated
+pipelines during fitting due to caching, it will be correct to com-
+pare this metric with and without the caching option. For that
+reason, we created the benchmark considering different compu-
+tational setups for AutoML. It utilizes a dataset for classifica-
+tion present using the FEDOT framework as a test bench.
+In Figure 7 the comparison of cache-based and cache-free
+configurations is provided. For the first one, both the pipelines
+cache and data preprocessing cache are activated. The number
+of parallel jobs used during optimization is one.
+During evolutionary optimization, a lot of candidate solu-
+tions (pipelines) are evaluated. We repeated the experiment for
+different timeouts that limit the execution time for the entire Au-
+toML run since they affect the number of evaluated pipelines.
+Also, an additional time limit is applied to the entire pipeline (to
+process the fit time anomalies for large pipelines). It is specified
+as 1/4 of the total timeout.
+Because of the stochastic nature of the optimization-based
+experiments, each run was repeated three times, and the ob-
+tained metrics were averaged.
+The results presented in Figure 8 are obtained with the
+n jobs hyperparameter value equal to 12.
+Table 2 summarises the averaged metrics of the experiments
+with single-process and multi-process caching. The average
+performance was increased by 14 %, which empirically con-
+Figure 7: The dependence between the number of pipelines and the usage of
+cache (averaged for ten runs). Single-processing is used.
+Figure 8: The dependence between the number of pipelines and the usage of
+cache (averaged for ten runs). 8 parallel jobs are used.
+firms the effectiveness of the proposed approach.
+The next stage of the experiment is devoted to the analysis
+of the evolutionary algorithm’s performance in multiprocessing
+mode. We compared algorithm performance with the number
+of processes equal to 1 and 8 with a timeout set to 10 minutes.
+The optimization of the pipeline structure was repeated three
+times with no seed and with five cross validation folds to take
+stochasticity into account.
+The dependency of correctly evaluated pipelines on a speci-
+fied number of jobs for a single dataset is presented in Figure 9.
+It can be seen that near-linear improvement in parallel speedup
+is achieved. Figure 10 demonstrates the dependency of the best
+fitness calculated using cross-validation on the timestamp from
+the configuration. Launches with 8 processes find a better solu-
+tion faster than launches with one process.
+Table 3 summarises the averaged results of experiments in
+single-process and multiprocessing modes. The fitness score
+calculated using cross-validation increases linearly with the
+number of evaluated pipelines. It confirms the effectiveness of
+the local parallelization of evolutionary AutoML.
+7
+
+350
+without cache
+with cache
+5
+actual time for optimization in minutes
+300
+correctly evaluated pipelines
+4
+250
+200
+3
+150
+2
+100
+1
+50
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+4.5
+5.0
+timeout in minuteswithout cache
+with cache
+5
+1200
+actual time for optimization in minutes
+correctly evaluated pipelines
+4
+1000
+800
+3
+600
+2
+400
+1
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+4.5
+5.0
+timeout in minutesTable 2: The results of experiments with caching of pipeline nodes and data preprocessing operations. The first column indicates whether the cache has been used,
+and the second column represents the number of parallel processes. The next three columns represent different metric values (the number of evaluated pipelines,
+ROC AUC for validation sample, and ROC AUC for cross-validation of training sample).
+Dataset
+Configuration
+Pipelines count
+ROC-AUC final
+ROC-AUC cross-validation
+Cache
+Number of processes
+adult
+on
+1
+27
+0,92
+0,9117
+off
+23
+0,9213
+0,913
+on
+8
+190
+0,921
+0,9131
+off
+170
+0,922
+0,9137
+amazon employee access
+on
+1
+85
+0,8447
+0,8346
+off
+78
+0,8497
+0,8376
+on
+8
+416
+0,8507
+0,8356
+off
+369
+0,849
+0,8398
+australian
+on
+1
+879
+0,9313
+0,9432
+off
+838
+0,9283
+0,9401
+on
+8
+6354
+0,928
+0,9411
+off
+6199
+0,934
+0,9442
+bank-marketing
+on
+1
+38
+0,93
+0,931
+off
+30
+0,9313
+0,93
+on
+8
+205
+0,931
+0,932
+off
+211
+0,932
+0,931
+blood-transfusion
+-service-center
+on
+1
+2175
+0,748
+0,75
+off
+2064
+0,7383
+0,759
+on
+8
+13943
+0,745
+0,761
+off
+13834
+0,749
+0,7659
+car
+on
+1
+812
+0,921
+0,933
+off
+728
+0,9233
+0,9319
+on
+8
+4856
+0,922
+0,935
+off
+4608
+0,92
+0,934
+cnae-9
+on
+1
+214
+0,995
+0,9939
+off
+195
+0,995
+0,9939
+on
+8
+1100
+0,995
+0,9942
+off
+1161
+0,995
+0,9953
+jungle chess 2pcs raw
+endgame complete
+on
+1
+30
+0,9671
+0,9637
+off
+44
+0,9667
+0,9627
+on
+8
+89
+0,969
+0,9631
+off
+99
+0,9713
+0,9649
+numerai28
+on
+1
+3
+0.508
+0,51
+off
+6
+0,511
+0,5182
+on
+8
+20
+0,527
+0,528
+off
+23
+0,5273
+0,528
+phoneme
+on
+1
+354
+0,9599
+0,951
+off
+325
+0,9597
+0,9515
+on
+8
+1954
+0,9631
+0,955
+off
+1885
+0,963
+0,9547
+sylvine
+on
+1
+221
+0,9852
+0,9809
+off
+206
+0,9853
+0,9806
+on
+8
+932
+0,9878
+0,981
+off
+827
+0,9877
+0,9829
+volkert
+on
+1
+4
+0,932
+0,9298
+off
+6
+0,9393
+0,9344
+on
+8
+20
+0,9313
+0,934
+off
+21
+0,9317
+0,9273
+synthetic blobs
+on
+1
+31
+1
+1
+off
+27
+1
+1
+on
+8
+235
+1
+1
+off
+224
+1
+1
+synthetic moons
+on
+1
+1124
+1
+1
+off
+1026
+1
+1
+on
+8
+12356
+1
+1
+off
+12227
+1
+1
+8
+
+Table 3: The results of experiments with parallelization of evolution. The first two rows for each dataset (1 and 8 jobs) represent the results obtained with a fit time
+limit for pipelines, while the ”without limit” row contain the results obtained without limits with 8 jobs.
+Dataset
+Number of processes
+Pipelines
+count
+ROC-AUC
+final
+ROC-AUC
+cross-validation
+adult
+1 (with limit)
+23
+0,9213
+0,913
+8 (with limit)
+170
+0,922
+0,9137
+8 (without limit)
+126
+0,9217
+0,9141
+amazon employee access
+1 (with limit)
+78
+0,8497
+0,8376
+8 (with limit)
+369
+0,849
+0,8398
+8 (without limit)
+329
+0,849
+0,8387
+australian
+1 (with limit)
+838
+0,9283
+0,9401
+8 (with limit)
+6199
+0,934
+0,9442
+8 (without limit)
+10009
+0,9307
+0,9444
+bank-marketing
+1 (with limit)
+30
+0,9313
+0,93
+8 (with limit)
+211
+0,932
+0,931
+8 (without limit)
+184
+0,9313
+0,93
+blood-transfusion-service-center
+1 (with limit)
+2064
+0,7383
+0,759
+8 (with limit)
+13834
+0,749
+0,7659
+8 (without limit)
+8329
+0,716
+0,7658
+car
+1 (with limit)
+728
+0,9233
+0,9319
+8 (with limit)
+4608
+0,92
+0,934
+8 (without limit)
+4612
+0,925
+0,9345
+cnae-9
+1 (with limit)
+195
+0,995
+0,9939
+8 (with limit)
+1161
+0,995
+0,9953
+8 (without limit)
+1168
+0,995
+0,9949
+jungle chess 2pcs raw endgame complete
+1 (with limit)
+44
+0,9667
+0,9627
+8 (with limit)
+99
+0,9713
+0,9649
+8 (without limit)
+144
+0,9723
+0,9661
+numerai28
+1 (with limit)
+6
+0,511
+0,5182
+8 (with limit)
+23
+0,5273
+0,528
+8 (without limit)
+22
+0,5253
+0,5266
+phoneme
+1 (with limit)
+325
+0,9597
+0,9515
+8 (with limit)
+1885
+0,963
+0,9547
+8 (without limit)
+1917
+0,963
+0,9536
+sylvine
+1 (with limit)
+206
+0,9853
+0,9806
+8 (with limit)
+827
+0,9877
+0,9829
+8 (without limit)
+816
+0,986
+0,9821
+volkert
+1 (with limit)
+6
+0,9373
+0,9329
+8 (with limit)
+21
+0,9393
+0,9344
+8 (without limit)
+21
+0,9317
+0,9273
+synthetic blobs
+1 (with limit)
+8
+1
+1
+8 (with limit)
+37
+1
+1
+8 (without limit)
+30
+1
+1
+synthetic moons
+1 (with limit)
+1026
+1
+1
+8 (with limit)
+12227
+1
+1
+8 (without limit)
+12569
+1
+1
+9
+
+Figure 9: The detailed analysis of dependency of number of pipelines from the
+number of jobs (dataset blood-transfusion-service-center)
+Figure 10: The dependency of the best fitness values on the optimization time.
+The intervals represent the stochasticity of the optimization runs.
+6.2. Heterogeneous infrastructure
+In the next series of experiments, we aim to estimate the
+efficiency of heterogeneous infrastructure for large datasets.
+Computation experiments were performed in a supercomputer
+environment configured based on two DGX-1 clusters. Each
+cluster contains eight Tesla V100 graphics cards and 128 GB of
+video memory. The number of graphics cores is 40960.
+The first experiment compares AutoML performance for the
+CPU-only and the hybrid infrastructures for various tasks. The
+aim of the experiment is to estimate the decreasing of fitting
+time after involvement of GPU-based nodes.
+To reduce the
+computational complexity of experiments, we decided not to
+use the full set of datasets from Table 1. In this experiment, four
+synthetic binary classification datasets with 10 features and dif-
+ferent number of rows (10000, 100000, 200000 and and 300000
+rows).
+Both single-model (SVC) and multi-model pipelines
+(consisting of SVC, Logistic Regression, and Random Forest)
+are considered to estimate the overhead for data flow transfer
+between models in the pipeline. The results are presented in
+Table 4: The training time of the pipelines on synthetic data under different
+conditions (single SVC classifier and composite pipeline with several models
+are considered). Averaged efficiency estimations are presented for homoge-
+neous (one server with multi-core CPU) and heterogeneous (CPU and GPU)
+computing environments.
+Rows, 103
+Fitting time, sec
+Improvement,
+%
+Single
+model
+Comp.
+pipeline
+Single
+model
+Comp.
+pipeline
+CPU
+CPU+
+GPU
+CPU
+CPU+
+GPU
+10
+0.3
+2.2
+0.7
+2.4
+-
+-
+100
+11.4
+1.9
+8.3
+1.6
+91
+88
+200
+39.3
+2.8
+21
+3.2
+94
+85
+300
+76.0
+4.2
+37.8
+5.5
+95
+86
+Table 4.
+The results confirmed that the overhead could exceed the
+performance gain for a small amount of data. However, the
+proposed hybrid approach to pipeline evaluation is reasonably
+practical for large datasets.
+6.3. Remote infrastructure
+The next series of experiments uses a homogeneous cluster
+of 20 nodes under Kubernetes control. Each node has 40 CPU
+cores and 256 Gb RAM. We have trained populations of 50,
+100 and 200 individuals. Each population has been trained four
+times, then the estimated time values were averaged. The aim
+of this experiment is to analyze the structure of computing time
+for remote evaluation and confirm that remote evaluation can
+be viable for large datasets regardless of existing overheads. To
+make the results of the experiment more compact, we focused
+on an analysis of a single synthetic binary classification dataset
+with 300000 rows and 10 features.
+Figure 11a presents training time depending on population
+size without results fetching. The evaluation of each individual
+has no CPU and system memory limits. Also, we have drawn
+a linear fit time as reference line. The initial point for this line
+is the time for the population size of 50. We found that training
+time increases almost linearly with the increase in population.
+To explain the near-linear time growth, we can consider the
+operations that took the most time during the computing and
+the overheads they have. Since requests overheads are nearly 0
+seconds, they are not presented in Figure 12.
+Figure 12 shows that computing time and overheads for
+each individual are the same and are independent of popula-
+tion size. It means that the performance bottleneck is not on the
+cluster side. The fit stages timeline (including results fetching)
+is presented in Figure 13.
+This timeline shows that the most time is spent on results
+fetching. Result fetching consists of zip-file downloading over
+the network, unpacking and then deserializing the model. Since
+the results are fetched concurrently, a large number of individu-
+als lead to a high network and drive load on the local machine.
+Moreover, even though the overhead for the request to run
+one individual is less than 1 sec, a large number of requests also
+10
+
+8 process
+0.855
+1process
+0.850
+0.845
+score
+ROC-AUC
+0.840
+0.835
+0.830
+0.825
+0
+500
+1000
+1500
+2000
+2500
+3000
+Time in secondsDependency ofthe bestfitness from numberof jobs
+14000
+8
+minutes
+12000
+7
+correctly evaluated pipelines
+6
+actual time for optimization in 
+10000
+8000
+4
+6000
+m
+4000
+2
+1
+2000
+1
+2
+3
+4
+5
+6
+7
+8
+numberof jobs(a) Without limits (fast calculations)
+(b) CPU limit = 0.2 core (heavy calculations emulation)
+Figure 11: The dependence of the total fit time on individual numbers in different computational setups. The orange line represents linear acceleration; the blue line
+represents the observed values of fit time.
+Figure 12: Overheads and computing time during experiments with remote infrastructure
+Figure 13: The explanation of remote training timeline with remote infrastruc-
+ture.
+consume a significant amount of time. The time range between
+the last request and the last completed individual is less than 20
+seconds, and it falls within the 75-percentile of computing time.
+It means that the computing cluster is underutilized because it
+is not running the individuals in parallel as well as expected
+because it is waiting for requests from the client.
+To sum up, it is not reasonable to use remote training if we
+have lightweight and fast computations. Overheads in the form
+of requests and results fetching will be significantly larger than
+the payload. To prove this assumption, we have repeated the
+same experiment, but we have artificially introduced CPU limit-
+ing for each individual (up to 0.2 CPU core) to emulate ”heavy”
+computing. The results for heavy-weight tasks are presented in
+Figure 11b.
+We can conclude that remote computing provides a signif-
+icant speedup for expensive computations. However, the over-
+head for small datasets should be taken into account.
+7. Conclusions and Discussions
+In the paper, we propose a modular approach that improves
+the efficiency of evolutionary AutoML in a heterogeneous en-
+11
+
+remote fit time
+linear fit timeremote fit time
+linear fit time主vironment. The proposed approach differs from existing solu-
+tions since it can be configured for automated machine learning
+in various computational environments. It makes it possible
+to parallelize and distribute the computational tasks across hy-
+brid and/or remote computational systems. Also, caching al-
+gorithms are implemented to increase the optimization perfor-
+mance for composite pipelines.
+The AutoML-based experimental setup consisted of (1) the
+estimation of parallel speedup for a different number of pro-
+cesses; (2) an analysis of the efficiency of the cache; (3) an anal-
+ysis of the GPU computations efficiency; (4) optimisation runs
+with remote infrastructure involved. The experiments confirm
+the proposed approach’s efficiency. It allows achieving signifi-
+cant improvements in the number of evaluated individuals and
+in the fitness function.
+There are several ways to improve remote computing per-
+formance aimed at different bottlenecks that can be used sepa-
+rately or combined:
+1. Efficient cluster resources utilization requires a custom
+scheduler and additional plugins for batch workload such
+as Volcano5.
+2. Refuse to request to run each individual. Better to use
+one request to run a batch of individuals. This way, the
+number of requests will be reduced to one independent
+request for the all population. Also, we can apply specu-
+lative computing mode when the number of rest individ-
+uals is small;
+3. Provide a cluster file system mount on the local machine.
+This will reduce the number of requests for downloading
+results, and the client will also skip zip file unpacking.
+Instead, the client will read the results from the mounted
+file system. If it is impossible, then we have to implement
+not only batch run requests but also batch download re-
+quests;
+4. Perform model validation using remote infrastructure
+too.
+This way, we also have to provide a validation
+dataset to the remote system. Remote computing will
+validate individuals and save the score. It will make it
+unnecessary to fetch the trained models, and the calcu-
+lated score will be enough for further decisions;
+5. Heterogeneous environment.
+We can use a heteroge-
+neous environment not only on the cluster layer but on
+the client-server layer. For example, the client can per-
+form lightweight calculations locally, heavy calculations
+at the same time will be sent to the cluster, and the heavi-
+est calculations may be sent to the most powerful cluster
+nodes (e.g. special GPU nodes).
+Another direction of improvement is the support of large
+dataset processing. It can be based on the implementation of
+the distributed evaluation of different folds of the data set. The
+caching system can also be implemented in a distributed way.
+5https://volcano.sh
+8. Code and Data Availability
+The software implementation of all described methods and
+algorithms is available in the open repository https://gith
+ub.com/ITMO-NSS-team/fedot-performance-improve
+ment-benchmark.
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+
diff --git a/DdE4T4oBgHgl3EQfew2P/content/tmp_files/load_file.txt b/DdE4T4oBgHgl3EQfew2P/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf,len=1027
+page_content='Improvement of Computational Performance of Evolutionary AutoML in a Heterogeneous  Environment  Nikolay O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Nikitin, Sergey Teryoshkin, Valerii Pokrovskii, Sergey Pakulin, Denis Nasonov  ITMO University, Saint-Petersburg, Russia  Abstract  Resource-intensive computations are a major factor that limits the effectiveness of automated machine learning solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' In the  paper, we propose a modular approach that can be used to increase the quality of evolutionary optimization for modelling pipelines  with a graph-based structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' It consists of several stages - parallelization, caching and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Heterogeneous and remote  resources can be involved in the evaluation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The conducted experiments confirm the correctness and effectiveness of the  proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The implemented algorithms are available as a part of the open-source framework FEDOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Keywords: AutoML, heterogeneous infrastructure, evolutionary optimization, caching  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Introduction  Nowadays,  automated machine learning (AutoML) is  widely used in science, and industry [16, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The major prob-  lem of solving real-world tasks with AutoML is the high com-  putational cost of the search for an optimal modelling pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  During the evaluation of the candidate pipelines’ quality, many  machine learning models are trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' This task is very resource-  intensive, so it can take a considerable amount of time to  achieve the appropriate result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' It can be considered a bottle-  neck for any existing AutoML solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' This issue raises vari-  ous problems from different fields: from integration of AutoML  to business processes [31] to carbon emission and sustainability  concerns[35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  There are many approaches for improving computational  performance that are used in state-of-the-art (SOTA) AutoML  solutions [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' First of all, almost all solutions support parallel  execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Some of them also support caching of evaluated can-  didates [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Also, the graphics processing unit can be used to  reduce the training time [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  There is a variety of open-source tools that could improve  the efficiency of certain steps of machine learning pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  For instance, involving various MLOps tools like MLFlow [38],  task-specific databases [37] and scaling tools like Ray [18] al-  lows the effectiveness of ML applications to be notably in-  creased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  However, the optimal design of the computational strategy  depends on the infrastructure and the underlying AutoML al-  gorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The SOTA AutoML solutions are based on differ-  ent optimization methods: random search, Bayesian optimiza-  tion, genetic algorithms, and meta-learning [8, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The struc-  tural patterns used in modelling pipelines can also be different:  linear pipelines or ensembling techniques (stacking, blending,  Email address: nnikitin@itmo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='ru (Nikolay O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Nikitin)  and boosting) [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The most complicated case is a composite  pipeline represented as a directed acyclic graph [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' At the  same time, there is no ready-to-use solution for improving the  computational performance of an automated open-ended search  for pipelines in the composite AI field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  In the paper, we want to propose a adaptive approach  to reduce the computational cost of AutoML for compos-  ite pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Several techniques are implemented: pipeline  caching, parallelization of the fitness function evaluation, com-  putation with hybrid (GPU and CPU) systems, and integration  with remote distributed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  This approach differs from existing solutions since it can be  configured for automated machine learning in various computa-  tional environments (including distributed and heterogeneous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Also, the caching procedure can be effectively used for vari-  ous pipeline designs (linear, weighted ensembles, multi-layer  ensembles, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  To confirm the effectiveness of the proposed approach in  empirical way, we conducted a set of numerical experiments us-  ing set of open datasets of various sizes (described in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The results presented in Section 6 allow us to conclude that a  larger number of pipelines can be evaluated and better qual-  ity metrics can be achieved by AutoML using this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The software implementation is available in the open-source  AutoML framework FEDOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The paper is organized as follows: Section 2 describes the  computational strategies used in state-of-the-art AutoML tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Section 3 provides the problem statement for AutoML perfor-  mance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Section 4 proposes a set of novel im-  provements for the composite evolutionary AutoML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Section 5  describes the software implementation of these techniques in an  open-source framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Section 6 provides the experimental  evaluation of the proposed techniques for different case studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Finally, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' 7 provides an analysis of the obtained results and  possible extensions of the research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Preprint submitted to Algorithms  January 13, 2023  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='05102v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='LG]  12 Jan 2023  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Related Works  There are dozens of open-source AutoML solutions that can  be used for designing modelling pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The first frame-  works that became well-known are H2O [15], TPOT [14] and  Auto-sklearn [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  As more novel AutoML solutions, Auto-  Gluon [3] and LAMA [36] can be noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Also, there are a lot  of other AutoML tools with various specific features [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  There are different strategies for performance improve-  ment used in the noted frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' In the TPOT framework,  pipeline caching is implemented [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  TPOT-SH [26] uses  the concept of Successive Halving to explore the search space  faster, especially for larger datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Various techniques are  used to evaluate the pipelines on different subsets of training  data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' layering [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  A widely-used parallelization tool is the joblib library im-  plemented in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' However, there are more advanced frame-  works for parallelization that can be noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' For example, Ray  [18] can be used to scale AI and Python applications in dis-  tributed environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' It provides various instruments for dis-  tributed data preprocessing, distributed training of ML models  and scalable hyperparameter tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Improving the computational performance for evolutionary  algorithms outside AutoML is also discussed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  As an example, parallel GPU-based evaluation of the fitness  function can be used [24] to solve the expensive problems re-  lated to big data [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' There are various techniques that im-  prove the performance of evolutionary algorithms in concurrent  mode [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The tensor-based computational model can be used  to achieve cross-platform hardware acceleration [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Also,  platform-specific open-source solutions are presented in this  field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' scikit-learn-intelex 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  One of the widely used techniques to avoid fitness evalua-  tion bottlenecks in evolutionary algorithms is caching [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Fi-  nal values of the fitness evaluation can be cached [13] as well  as partial results [30, 11, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Moreover, a number of solutions exist that can perform  remote/distributed training (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Auto-sklearn, H2O, TPOT,  LAMA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' These AutoML frameworks use different frameworks  for distributed computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Autosklearn and TPOT use Dask2,  LAMA uses Apache Spark3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' H2O uses its own Apache Spark  modification called Sparkling Water4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Distributed computing  frameworks allow the processing of large datasets spread over  the nodes of a cluster system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  We can conclude that there is a large number of techniques  and solutions that can reduce the resource consumption for Au-  toML and EA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' However, there is still no well-developed ap-  proach that can be used to identify graph-based pipelines in  the heterogeneous computational environment in composite AI  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' For this reason, we decided to formulate the problem  statement specific to composite AutoML and propose possible  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='com/intel/scikit-learn-intelex  2https://dask.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='org  3https://spark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='apache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='org  4https://h2o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='ai/products/h2o-sparkling-water  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Problem Statement  We want to design multi-task and multi-modal pipelines for  various tasks using a single flexible instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Consequently,  it becomes necessary to implement the framework’s architec-  ture more abstractly to separate the pipeline search process  from the top-level API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The modelling pipeline is represented  as a directed acyclic graph in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Each node (modelling  or data transformation operation) is described by the operation’s  name and set of hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' If necessary, different data  sources (tables, time series, images, texts) can be involved in  the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Also, metadata is attached to the data flow, making  it possible to change the task several times during the pipeline  evaluation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=', solve a classification task and then - a regres-  sion task).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The drawback of this approach is the increased search space  that should be explored during the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  In auto-  mated modelling, we want to control the balance between open-  endedness [25] and local search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The simplest way is to apply  the of direct constraints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' limit to the pipeline size).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Also, it  can be more effective to apply the regularization and sensitivity  analysis procedures [21] and adaptive optimisation strategies  to control the convergence of optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' At the same time,  avoiding over-complicated pipelines and reckless spending of a  limited time budget is also essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' It makes the effectiveness  of the computational part even more critical for open-ended Au-  toML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  It pushes us to compromise between pipeline complexity  and training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' However, if we can improve computing ef-  ficiency, the framework will probably be able to build more  complicated models with higher quality while consuming the  same training time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  There are many approaches to improv-  ing evolutionary algorithms’ computing performance, such as  parallelization, caching, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' These approaches can be divided  into single-machine optimizations and horizontal scaling tech-  niques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Single-machine optimizations aim to improve comput-  ing performance only on the machines performing the compu-  tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Horizontal scaling allows involving additional servers  to speed up computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Both techniques can be used separately  or combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Evolutionary algorithms’ computational time mainly de-  pends on the population size and the number of generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Increasing population size leads to an increased probability of  getting better individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' A more significant number of gener-  ations means more attempts to grow better individuals based on  the best previous generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  From a computational point of view, we have several iter-  ations, each of which requires the results from the previous it-  eration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' So, it is complicated to scale computations over the  iterations, and the total computation time is Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Ttotal =  n  �  i=1  Ti  (1)  where n is the number of iterations and Ti - the computa-  tion time of generation i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Inside one generation, all individuals  are processed independently from one another, allowing us to  2  scale these computations according to the available computa-  tional resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The population training time can be estimated  with Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Ttotal =  n  �  i=1  argmax  di  j∈Di  (argmin(τ(di  j,  rj∈Ri/{rj−1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='.,r1}  ω(Di−1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='.,D1), rj))  (2)  where Di is the set of individuals in the population i and di  j is  individual j, Ri is the set of available resources on iteration i,  ω is a cache function with pre-calculated elements on iterations  i − 1, i − 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=', 1, and τ is a function that returns the calculation  time considering caching and evaluation of individuals Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Due to the Equations 1 and 2, we should increase the popu-  lation size as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' It allows to speed up the algo-  rithm convergence and improve its result using horizontal scal-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' For this purpose, we can use both remote computing on  production servers and distributed computing using homoge-  neous and heterogeneous computing clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Remote computing allows model training to be delegated to  a remote infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' This approach is justified if the local  machine computational resources are insufficient to train a set  of models in a reasonable amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Remote computing  may take place on a dedicated computing server or a cluster of  servers that accepts tasks to train models using REST API, RPC  or message queues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The main challenge in the investigated problem is to pro-  pose the performance improvement strategy for AutoML that  is adaptive to various types of computational infrastructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  In Figure 1, five classes are noted: shared memory system,  multi-node cluster with distributed memory, complex homo-  geneous and heterogeneous supercomputer environments, and  hybrid systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The system with structure (a) can execute par-  allel tasks in a straightforward way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' In the systems (b)-(e), the  remote nodes are involved (homogeneous and heterogeneous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  For the system (e), the structure is hybrid since various remote  nodes have different computational performance and connec-  tions overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' For this reason, the adaptability of the compu-  tational strategy is especially important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  There are various ways can be uses to adapt the compu-  tational strategy to specific infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  For example, the  empirical performance models [9] can be used to choose the  optimal infrastructure for evaluating specific pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Sim-  ple pipelines with low fitting time can be assigned to low-  performance computational nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Otherwise, complicated  pipelines with high fitting time can be assigned to high-  performance nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' It makes it necessary to develop a modular  approach that effectively utilizes all available resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Our main motivation is to develop an approach that can be  used at the computational layer of AutoML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' It should be pos-  sible to adapt this layer to the specified infrastructure (local or  remote) in a frame of the same AutoML approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' This solution  should be high-level, modular and flexible to allow integrating  it with different AutoML tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Also, it should support the dif-  ferent types of pipelines (from simplest linear pipeline to the  multi-level ensembles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Proposed Improvements  This section is devoted to various aspects of the proposed  approach for improving the computational performance of evo-  lutionary AutoML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The high-level scheme of the approach is  presented in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Four main aspects are considered: (1)  parallelization of the fitness function evaluation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' (2) partial  caching of evaluated individuals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' (3) combining CPU and GPU  to accelerate the processing of individuals (4) integration with  remote infrastructure for a complex task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Algorithmic-based  improvements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  surrogate-assisted optimization) are not  considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The detailed implementation of the proposed approach is  described in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' In this notation, graph represents the struc-  ture of the composite pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The details of the evolutionary  optimisation are hidden to make the proposed improvements  more clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Algorithm 1 High-level pseudocode of the evaluation dispatch-  ing algorithm implemented in the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Paral-  lelization, caching and evaluation stages are demonstrated for  processing one generation of the evolutionary algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  1: procedure ProcessPopulation  2:  Input:  inds (set of non-evaluated individuals),  objective (objective function that calculates the fitness  of an individual),  n (number of parallel jobs)  timer (timer-like object)  infrastructure (description of setup)  3:  Output: evaluated inds  4:  do in parallel(n)  5:  if timer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='enough time( ) then  6:  graph ← inds[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='graph ▷ get structure of each ind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  7:  if infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='is remote( ) then  8:  cache ← DistributedCache()  9:  sync cache  ▷ sync cache database  10:  task id ← create task(graph)  11:  wait task id  12:  inds[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='fitness ← request result(graph)  13:  else  14:  prepare graph  ▷ assign CPU and GPU to  nodes  15:  cache ← LocalCache()  16:  load cache  ▷ init cache database  17:  if cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='exists(graph)( ) then  18:  fit from cache(graph)  19:  inds[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='fitness ← obj(graph, cache)  20:  fit(graph)  ▷ Fit nodes that are not in cache  21:  save cache  ▷ preserve updated cache  22:  if not inds[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='is valid then  23:  delete inds[i]  ▷ for unsuccessful evaluation  24:  else  25:  delete inds[i]  ▷ not enough time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' skipping  26:  return inds  ▷ candidates for selection  3  Figure 1: Different types of computational infrastructures that can be used in AutoML: (a) shared memory (SM) system (b) multi-node cluster with distributed  memory (c) complex homogeneous supercomputer system with spatially distributed infrastructure (d) supercomputer system with heterogeneous distributed infras-  tructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Figure 2: Workflow of the proposed approach for the improvement of compu-  tational performance for composite AutoML  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Parallelization  Parallelizing evolutionary algorithms is not a novel idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  There are a lot of papers and open-source solutions devoted  to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' However, parallelization in AutoML has its  specifics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' For example, various computationally efficient strate-  gies of parallel evolution can be used [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  We are considering an evolutionary algorithm for search-  ing for the best solution in the space of pipelines that can be  represented as directed acyclic graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The classic approach  to parallelizing evolutionary optimization is evaluating all indi-  viduals in the population concurrently [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' It works because of  the nature of the evolutionary algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' There are no depen-  dencies between individuals in a generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Other approaches  suggest dividing populations into isolated parts [32] or using  co-evolutionary algorithms to divide tasks into subtasks [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The proposed algorithm considers the maximum evaluation  time length for each pipeline evaluation to resolve the possi-  ble evaluation time anomalies caused by the stochastic nature  of data-driven model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' If the training process does not  converge at least in one cross-validation fold, the time required  for corresponding fitness evaluation can be increased signifi-  cantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' So, the individuals that spend excess time on evaluation  are skipped to preserve the overall performance of the evolu-  tionary optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Caching  The existing caching approaches are aimed at preserving  and reusing fitted pipelines [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' However, separate nodes of  composite pipelines can be cached individually [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' It makes  it possible to reuse the fitted models and reduce the fitness func-  tion’s evaluation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The optimizer can share the in-memory  cache across the populations, and individuals [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' However, it  raises the problems of memory consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  After analyzing existing solutions, we focused on the rela-  tional database approach for pipeline caching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' More specifi-  cally, the sqlite3 library was used to implement it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' First of all,  it provides only one output file, which is not guaranteed for  non-relational databases - e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' shelve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Secondly, all concurrent  save-load operations can be fully processed during the parallel  evaluation of the fitness functions without direct usage of syn-  chronization primitives, atomic variables and other instruments  necessary for simultaneous access to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Finally, this approach allows extracting several operations  simultaneously, which helps to improve the overall perfor-  mance of caching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Also, the set of operations can be saved  to the database taking into account the existence of cache items  with the same primary key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The caching procedure for the multi-layer ensemble  pipelines should take into account that the cached model/-  operations are suitable only for the specific configuration of  previous nodes and edges in the modelling pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  So,  the key contains the recursive description of the structure  4  a)Shared memory  b) Cluster with  distributed memory  system (SM)  c)Homogeneoussupercomputer  environment  Core  Core  Core  Main  1  N  Node  Sheduler  Node  Node  Node  1  N2  Node  Node  Node  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='.  N  Node  Node  Node  Node  Node  Node  1  N1  d) Heterogeneous  1  N3  supercomputerenvironment  e) Hybrid environment  Sheduler  SM  System  Low-speedchannel  High-speed channel  Low-speed channel Mid speed channel High-speed channel  SM  Node  Node  Node  SM  Clust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Clust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Embed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Embed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  1  N  Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  system  systemCandidate  ML pipeline  Population K  0000  N  Individuals  不  Parallelization  Evaluation  Caching  Local  Remote  CPU  GPUof previous nodes and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Also, the identifier of cross-  validation fold is specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The following notation is used:  (/[node name] [hparams];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=')/[node name] [hparams].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='..”, where  / denotes the beginning of the node name and round brackets  represent the nested edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The caching details are presented in  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Figure 3: Interaction between operations’ cache and the modelling pipeline that  should be fitted  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' GPU  Evaluating ML models with GPUs is a well-developed fea-  ture in many solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' For example, the RAPIDS library [34]  contains the CuML module that allows training classification,  regression and clustering models with GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' To adapt this so-  lution to composite pipelines, we should consider a setup in  which only a part of the nodes can be evaluated with GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' In  this situation, the pipeline should be fitted in a heterogeneous  way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The proposed approach makes it possible to use both CPUs  and GPUs for fitting by separating the ML model type and its  implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The same model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' random forest) can have  several implementations (CPU-based and GPU-based).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Figure 4 shows an example of a computationally hetero-  geneous composite model structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Data transfer between the  GPU-based nodes (yellow) is performed within the video mem-  ory, and the models themselves in the nodes are trained on  graphics processing units (GPUs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Other nodes are executed  on CPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Due to the multiple software limitations set by RAPIDS li-  braries (CUDA-compatible GPU driver, restricted set of sup-  ported operation systems), it is practical to conduct the compu-  tations within Docker-based containers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Remote evaluation  Remote evaluation can be integrated into the evolutionary  optimiser in various ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Both dataset folds and population  parts can be distributed across several computational nodes to  satisfy time or memory limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Since evolutionary algorithms  do not always require processing large datasets, we have fo-  cused on the parallelism aspect of remote computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The pro-  posed implementation relies on Kubernetes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The REST API  Figure 4: The structure of a pipeline that can be evaluated in a heterogeneous  (CPU - blue and GPU - yellow) way  service inside the Kubernetes cluster is used to run computa-  tions via HTTP requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The client implements a wrapper for  requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  During the population training, the evaluator uses the  client’s methods to process individuals on the Kubernetes clus-  ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Then, after starting processing all individuals, the evaluator  waits for computations to be completed via client methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The  run request contains the container image, resources limit for the  container, mount paths and model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The REST API  service creates the requested container and keeps monitoring it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The client uses requests to the REST API service to get actual  containers’ statuses to see if it is still running, completed or  failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Finally, the client downloads the fitted pipeline when the  training is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' We wrap the result into a compressed  archive to reduce the amount of data transferred over the net-  work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Then, the files are sent to the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' This process scheme  is presented in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Figure 5: Communication between AutoML and remote cluster  This way, we can divide the population training process into  three stages: (1) requests to evaluate individuals, (2) computing  and waiting for the completion and (3) fetching the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Software Implementation  The proposed approach can be used as a part of the archi-  tecture that includes:  5  Preprocessor cache  Nodes cache  data_description 1: fitted preprocessor 1  node uid 1: fitted node 1  data description 2: fitted preprocessor 2  node uid 2: fitted node 2  Data fold 1  preprocessor  Node 1  uid=node 4/  evaluation  (node1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='node2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='node3)  Fitness  Data  Data fold 2  Node 2  Node 4  Node 3  Data fold 3  Pipeline  8  Evo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' opt  88  N  N+1  pop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  pop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='Decision  Tree  (GPU)  SVC  (GPU)  Random  Scaling  Forest  (GPU)  Logit  (CPU)  GPU memory  (CPU)  XGBoost  (CPU)LocalAutoML  Kubernetes  RemoteEvaluatol  Client  AutoML  Create individual  Individual-1  create task  RESTAPI  AutoML  Wait until ready  get task status  Service  Individual-2  AutoML  Fetch individual  download result  Individual-N• The model repository block, which provides storage and  selection of various implementations of predictive mod-  els and data processing blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' One model can contain  several implementations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=', for CPU and GPU);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The block of the generative design of composite mod-  els, which implements the creation of models with spec-  ified properties by evolutionary algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The proper-  ties of the models are determined by the target function  passed to the optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' If there is more than one tar-  get function specified (as an example, the training time  and modelling error can be used together as objectives  for AutoML), then the multi-criteria formulation of the  optimization problem is implemented, where the result  of the model design is a Pareto front containing various  compromising solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The genotype is represented in  graph form, and the crossing and mutation operators are  implemented accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The pipeline execution block on a given computational  infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' It allows individual pipeline execution on  the given computational nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  This architecture is implemented in the core of the open-  source FEDOT framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Different aspects of its implemen-  tation are already detailed in a series of papers: [19] describes  the main schemes and the implementation of the evolutionary  operators, [28] is devoted to the multi-objective modification of  this approach, and [20] provides an extended description of the  various aspects of the evolutionary design for composite mod-  elling pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The tuning strategy of the pipeline hyperpa-  rameters is based on Bayesian optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Custom models can be put inside this node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Search space  for hyperparameters and initial approximations for the models  should be specified manually if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' It makes it possi-  ble to involve the infrastructure-specific implementation of the  model in AutoML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The example below demonstrates the AutoML workflow  from input data processing to obtaining prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  api = Fedot(problem=’classification ’,  seed =42, timeout =30, preset=’gpu’)  api.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='fit(features=x_train , target=  y_train)  predictions = api.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='predict(features=  x_test)  Figure 6 provides the UML class diagram for the imple-  mentation of various evaluation strategies that allow combining  CPU- and GPU-based nodes in a single modelling pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' A  high-level modelling method (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Support Vector Classifica-  tion) can be implemented using different algorithms: a CPU-  optimised implementation of SVC can be obtained from the  scikit-learn library [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' In contrast, a GPU-optimised imple-  mentation is available in the CUML library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The proposed ar-  chitecture makes it possible to hide these details inside the spe-  cific modelling pipeline and use the same optimisation logic for  different implementations of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Figure 6: The class diagram for the implementation of a modelling pipeline that  consists of several operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The Operation class represents a high-level mod-  elling strategy that is used inside the operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' EvaluationStrategy is a base  class for the algorithmic implementation of this strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' SklearnEvaluation-  Strategy represents the implementation obtained from the scikit-learn library  and CumlEvaluationStrategy represents the implementation from the CUML  library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The optimizer operates on individual models as a black box  with input, output and fit/predict methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The following build-  ing blocks can be used for pipelines: models (Bernoulli Naive  Bayes classifier, logistic regression, multilayer perceptron, ran-  dom forest, gradient boosting, k-nearest classifier, QDA, LDA,  decision tree) and data transformation operations (scaling, nor-  malization, polynomial features transformation, principal com-  ponent analysis, independent component analysis, isolation for-  est, resampling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Experimental Studies  We conducted a series of experiments to confirm the cor-  rectness and effectiveness of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  It can  be divided into experiments with local and remote infrastruc-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' As benchmarks, various classification datasets from the  OpenML base [1] and synthetic datasets were used (the full list  is presented in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' A description of the computational  infrastructure is provided for each experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The following methodology was used for experimental  studies: each experiment started with dividing samples into two  groups: ‘learning’ and ‘validation’ samples in the ratio 70% to  30% to avoid data leaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Then, the learning sample was trans-  ferred to the evolutionary optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' During the optimisation,  the 5-fold cross-validation procedure was applied to estimate  the values of the fitness function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The experiment is repeated three times for each dataset to  take the stochasticity of the optimizer into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The quality  metrics are averaged over these iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Local infrastructure  For experiments with the local infrastructure,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' we configured  a server based on Xeon Cascadelake (2900MHz) with 12 cores  6  EvaluationStrategy  operation_type  fito  Operation  predictO  operation_type  A  strategy  define_strategy0  exectute_strategy0  SklearnEvaluationStrategy  operation_implementation  fito  predicto  CUMLEvaluationStrategy  Pipeline  operation_implementation  fito  predict0Table 1: The properties of OpenML datasets that were used during the exper-  iments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
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+page_content='2  2  synthetic  moons  1  2  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='12  2  and 24Gb memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  As our approach claims to increase the number of evaluated  pipelines during fitting due to caching, it will be correct to com-  pare this metric with and without the caching option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' For that  reason, we created the benchmark considering different compu-  tational setups for AutoML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' It utilizes a dataset for classifica-  tion present using the FEDOT framework as a test bench.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  In Figure 7 the comparison of cache-based and cache-free  configurations is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' For the first one, both the pipelines  cache and data preprocessing cache are activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The number  of parallel jobs used during optimization is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  During evolutionary optimization, a lot of candidate solu-  tions (pipelines) are evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' We repeated the experiment for  different timeouts that limit the execution time for the entire Au-  toML run since they affect the number of evaluated pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Also, an additional time limit is applied to the entire pipeline (to  process the fit time anomalies for large pipelines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' It is specified  as 1/4 of the total timeout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Because of the stochastic nature of the optimization-based  experiments, each run was repeated three times, and the ob-  tained metrics were averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The results presented in Figure 8 are obtained with the  n jobs hyperparameter value equal to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Table 2 summarises the averaged metrics of the experiments  with single-process and multi-process caching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The average  performance was increased by 14 %, which empirically con-  Figure 7: The dependence between the number of pipelines and the usage of  cache (averaged for ten runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Single-processing is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Figure 8: The dependence between the number of pipelines and the usage of  cache (averaged for ten runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' 8 parallel jobs are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  firms the effectiveness of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The next stage of the experiment is devoted to the analysis  of the evolutionary algorithm’s performance in multiprocessing  mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' We compared algorithm performance with the number  of processes equal to 1 and 8 with a timeout set to 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The optimization of the pipeline structure was repeated three  times with no seed and with five cross validation folds to take  stochasticity into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The dependency of correctly evaluated pipelines on a speci-  fied number of jobs for a single dataset is presented in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  It can be seen that near-linear improvement in parallel speedup  is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Figure 10 demonstrates the dependency of the best  fitness calculated using cross-validation on the timestamp from  the configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Launches with 8 processes find a better solu-  tion faster than launches with one process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Table 3 summarises the averaged results of experiments in  single-process and multiprocessing modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The fitness score  calculated using cross-validation increases linearly with the  number of evaluated pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' It confirms the effectiveness of  the local parallelization of evolutionary AutoML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  7  350  without cache  with cache  5  actual time for optimization in minutes  300  correctly evaluated pipelines  4  250  200  3  150  2  100  1  50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
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+page_content='0  timeout in minuteswithout cache  with cache  5  1200  actual time for optimization in minutes  correctly evaluated pipelines  4  1000  800  3  600  2  400  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
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+page_content='0  timeout in minutesTable 2: The results of experiments with caching of pipeline nodes and data preprocessing operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The first column indicates whether the cache has been used,  and the second column represents the number of parallel processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The next three columns represent different metric values (the number of evaluated pipelines,  ROC AUC for validation sample, and ROC AUC for cross-validation of training sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
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+page_content='9317  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='9273  synthetic blobs  1 (with limit)  8  1  1  8 (with limit)  37  1  1  8 (without limit)  30  1  1  synthetic moons  1 (with limit)  1026  1  1  8 (with limit)  12227  1  1  8 (without limit)  12569  1  1  9  Figure 9: The detailed analysis of dependency of number of pipelines from the  number of jobs (dataset blood-transfusion-service-center)  Figure 10: The dependency of the best fitness values on the optimization time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The intervals represent the stochasticity of the optimization runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Heterogeneous infrastructure  In the next series of experiments, we aim to estimate the  efficiency of heterogeneous infrastructure for large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Computation experiments were performed in a supercomputer  environment configured based on two DGX-1 clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Each  cluster contains eight Tesla V100 graphics cards and 128 GB of  video memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The number of graphics cores is 40960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The first experiment compares AutoML performance for the  CPU-only and the hybrid infrastructures for various tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The  aim of the experiment is to estimate the decreasing of fitting  time after involvement of GPU-based nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  To reduce the  computational complexity of experiments, we decided not to  use the full set of datasets from Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' In this experiment, four  synthetic binary classification datasets with 10 features and dif-  ferent number of rows (10000, 100000, 200000 and and 300000  rows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Both single-model (SVC) and multi-model pipelines  (consisting of SVC, Logistic Regression, and Random Forest)  are considered to estimate the overhead for data flow transfer  between models in the pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The results are presented in  Table 4: The training time of the pipelines on synthetic data under different  conditions (single SVC classifier and composite pipeline with several models  are considered).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Averaged efficiency estimations are presented for homoge-  neous (one server with multi-core CPU) and heterogeneous (CPU and GPU)  computing environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Rows, 103  Fitting time, sec  Improvement,  %  Single  model  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  pipeline  Single  model  Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  pipeline  CPU  CPU+  GPU  CPU  CPU+  GPU  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='7  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='4 100  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='9  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='6  91  88  200  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='8  21  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='2  94  85  300  76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='2  37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='8  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='5  95  86  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The results confirmed that the overhead could exceed the  performance gain for a small amount of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' However, the  proposed hybrid approach to pipeline evaluation is reasonably  practical for large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Remote infrastructure  The next series of experiments uses a homogeneous cluster  of 20 nodes under Kubernetes control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Each node has 40 CPU  cores and 256 Gb RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' We have trained populations of 50,  100 and 200 individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Each population has been trained four  times, then the estimated time values were averaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The aim  of this experiment is to analyze the structure of computing time  for remote evaluation and confirm that remote evaluation can  be viable for large datasets regardless of existing overheads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' To  make the results of the experiment more compact, we focused  on an analysis of a single synthetic binary classification dataset  with 300000 rows and 10 features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Figure 11a presents training time depending on population  size without results fetching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The evaluation of each individual  has no CPU and system memory limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Also, we have drawn  a linear fit time as reference line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The initial point for this line  is the time for the population size of 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' We found that training  time increases almost linearly with the increase in population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  To explain the near-linear time growth, we can consider the  operations that took the most time during the computing and  the overheads they have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Since requests overheads are nearly 0  seconds, they are not presented in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Figure 12 shows that computing time and overheads for  each individual are the same and are independent of popula-  tion size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' It means that the performance bottleneck is not on the  cluster side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The fit stages timeline (including results fetching)  is presented in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  This timeline shows that the most time is spent on results  fetching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Result fetching consists of zip-file downloading over  the network, unpacking and then deserializing the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Since  the results are fetched concurrently, a large number of individu-  als lead to a high network and drive load on the local machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Moreover, even though the overhead for the request to run  one individual is less than 1 sec, a large number of requests also  10  8 process  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='855  1process  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='850  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='845  score  ROC-AUC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='840  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='835  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='830  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='825  0  500  1000  1500  2000  2500  3000  Time in secondsDependency ofthe bestfitness from numberof jobs  14000  8  minutes  12000  7  correctly evaluated pipelines  6  actual time for optimization in  10000  8000  4  6000  m  4000  2  1  2000  1  2  3  4  5  6  7  8  numberof jobs(a) Without limits (fast calculations)  (b) CPU limit = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='2 core (heavy calculations emulation)  Figure 11: The dependence of the total fit time on individual numbers in different computational setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The orange line represents linear acceleration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' the blue line  represents the observed values of fit time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Figure 12: Overheads and computing time during experiments with remote infrastructure  Figure 13: The explanation of remote training timeline with remote infrastruc-  ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  consume a significant amount of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The time range between  the last request and the last completed individual is less than 20  seconds, and it falls within the 75-percentile of computing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  It means that the computing cluster is underutilized because it  is not running the individuals in parallel as well as expected  because it is waiting for requests from the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  To sum up, it is not reasonable to use remote training if we  have lightweight and fast computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Overheads in the form  of requests and results fetching will be significantly larger than  the payload.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' To prove this assumption, we have repeated the  same experiment, but we have artificially introduced CPU limit-  ing for each individual (up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='2 CPU core) to emulate ”heavy”  computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The results for heavy-weight tasks are presented in  Figure 11b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  We can conclude that remote computing provides a signif-  icant speedup for expensive computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' However, the over-  head for small datasets should be taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Conclusions and Discussions  In the paper, we propose a modular approach that improves  the efficiency of evolutionary AutoML in a heterogeneous en-  11  remote fit time  linear fit timeremote fit time  linear fit time主vironment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The proposed approach differs from existing solu-  tions since it can be configured for automated machine learning  in various computational environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' It makes it possible  to parallelize and distribute the computational tasks across hy-  brid and/or remote computational systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Also, caching al-  gorithms are implemented to increase the optimization perfor-  mance for composite pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  The AutoML-based experimental setup consisted of (1) the  estimation of parallel speedup for a different number of pro-  cesses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' (2) an analysis of the efficiency of the cache;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' (3) an anal-  ysis of the GPU computations efficiency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' (4) optimisation runs  with remote infrastructure involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The experiments confirm  the proposed approach’s efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' It allows achieving signifi-  cant improvements in the number of evaluated individuals and  in the fitness function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  There are several ways to improve remote computing per-  formance aimed at different bottlenecks that can be used sepa-  rately or combined:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Efficient cluster resources utilization requires a custom  scheduler and additional plugins for batch workload such  as Volcano5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Refuse to request to run each individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Better to use  one request to run a batch of individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' This way, the  number of requests will be reduced to one independent  request for the all population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Also, we can apply specu-  lative computing mode when the number of rest individ-  uals is small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Provide a cluster file system mount on the local machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  This will reduce the number of requests for downloading  results, and the client will also skip zip file unpacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Instead, the client will read the results from the mounted  file system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' If it is impossible, then we have to implement  not only batch run requests but also batch download re-  quests;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Perform model validation using remote infrastructure  too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  This way, we also have to provide a validation  dataset to the remote system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Remote computing will  validate individuals and save the score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' It will make it  unnecessary to fetch the trained models, and the calcu-  lated score will be enough for further decisions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Heterogeneous environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  We can use a heteroge-  neous environment not only on the cluster layer but on  the client-server layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' For example, the client can per-  form lightweight calculations locally, heavy calculations  at the same time will be sent to the cluster, and the heavi-  est calculations may be sent to the most powerful cluster  nodes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' special GPU nodes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  Another direction of improvement is the support of large  dataset processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' It can be based on the implementation of  the distributed evaluation of different folds of the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' The  caching system can also be implemented in a distributed way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  5https://volcano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='sh  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content=' Code and Data Availability  The software implementation of all described methods and  algorithms is available in the open repository https://gith  ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='com/ITMO-NSS-team/fedot-performance-improve  ment-benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
+page_content='  References  [1] Bischl, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DdE4T4oBgHgl3EQfew2P/content/2301.05102v1.pdf'}
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diff --git a/GNE2T4oBgHgl3EQfTQeG/content/tmp_files/2301.03801v1.pdf.txt b/GNE2T4oBgHgl3EQfTQeG/content/tmp_files/2301.03801v1.pdf.txt
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+UNIFYSPEECH: A UNIFIED FRAMEWORK FOR ZERO-
+SHOT TEXT-TO-SPEECH AND VOICE CONVERSION
+Haogeng Liu1, Tao Wang1, Ruibo Fu2, Jiangyan Yi2, Zhengqi Wen2
+Chinese Academy of Science
+Beijing, China
+Jianhua Tao1
+Department of Automation, Tsinghua University
+School of Artificial Intelligence, University of Chinese Academy of Sciences, China
+Beijing, China
+ABSTRACT
+Text-to-speech (TTS) and voice conversion (VC) are two different tasks both aim-
+ing at generating high quality speaking voice according to different input modality.
+Due to their similarity, this paper proposes UnifySpeech, which brings TTS and
+VC into a unified framework for the first time. The model is based on the assump-
+tion that speech can be decoupled into three independent components: content
+information, speaker information, prosody information. Both TTS and VC can be
+regarded as mining these three parts of information from the input and completing
+the reconstruction of speech. For TTS, the speech content information is derived
+from the text, while in VC it’s derived from the source speech, so all the remaining
+units are shared except for the speech content extraction module in the two tasks.
+We applied vector quantization and domain constrain to bridge the gap between
+the content domains of TTS and VC. Objective and subjective evaluation shows
+that by combining the two task, TTS obtains better speaker modeling ability while
+VC gets hold of impressive speech content decoupling capability.
+Index Terms: decoupling, zero-shot learning, text-to-speech, voice conversion,
+vector quantization
+1
+INTRODUCTION
+Cloning the voice of the target speaker is an attractive technology, which can be applied to various
+scenes (Sisman et al., 2020), such as entertainment creation, personalized mobile assistants, security
+field, etc. The most ideal voice cloning operation is to give only one relatively short speech of the
+unseen target speaker as a reference and then any speech of the target speaker can be synthesized,
+which is called zero-shot voice clone. In the speech research community, text-to-speech (TTS) and
+voice conversion (VC) are two mainstream ways to realize voice clone (Sisman et al., 2020). There-
+fore, a variety of techniques for zero-shot TTS and VC have been proposed recently (Gorodetskii &
+Ozhiganov, 2022; Tang et al., 2021).
+However, although TTS and VC techniques are two important ways of voice clone with same output
+form, the research of TTS and VC is more or less independent. There isn’t much interaction between
+them. But they are both speech synthesis task. In terms of speech generation, we categorize the
+information of the target speaker’s speech into three kinds of information: (1) speech content, the
+characters of phonemes or phonetic posteriorgram (PPG) in voice conversion, represents the content
+of the speech. (2) speaker information, which represents the characteristics of speakers, is related
+to the speaker’s articulation organ. (3) prosody information, which covers the intonation, stress,
+and rhythm of speech. According to FastSpeech2 (Ren et al., 2020), pitch, energy and duration
+information can reflected them certainly. As TTS extracts speech content directly from phonemes,
+it is easier to obtain content information irrelevant to speaker than VC. As VC encounters more
+speakers, it’s possible to obtain more robust speaker modeling ability. So, by integrating TTS and
+1
+arXiv:2301.03801v1  [cs.SD]  10 Jan 2023
+
+VC into a unified framework and combining their training data, it can help the model learn these
+three kinds of information better.
+Unfortunately, investigating TTS and VC in the same framework is challenging, as the speech
+content are extracted from different modality. Specifically, the speech content in TTS is obtained
+through phoneme information while the phonemes and speech in TTS are unequal sequences, need-
+ing attention mechanism to align them. However, the attention mechanism is often affected by the
+speaker’s information, so it is impossible to learn the speech content representation completely ir-
+relevant to the speaker. While in VC, the source speech and target speech are aligned in speech
+content, so the speech content can be extracted directly from the source speech, which is different
+from the TTS. In contrast to speech content information, speaker information and prosody infor-
+mation can be modeled using the same network in TTS and VC. Therefore, the difficulty here is
+to keep speech content for TTS and VC consistent. With the development of speech synthesis, the
+recently proposed Adaspeech2 (Yan et al., 2021) can combine text information with speech infor-
+mation, which can effectively decouple the speech content from the input. The unified framework
+becomes possible.
+Overall, the main contributions of this paper are:
+• We propose UnifySpeech, a unified framework for TTS and VC. VC enables unlabeled data
+to join training process, making TTS encounters more speakers. TTS enhance the ability
+for voice content decoupling in VC. Thus, both pipeline benefits from the other one.
+• We apply vector quantization and domain constrain to bridge the gap between the content
+domains of TTS and VC. Ablation experiment shows this method’s effectiveness.
+• We perform extensive experiments: zero-shot TTS, zero-shot VC. Results proves that
+jointly trained TTS outperformes StyleSpeech (Min et al., 2021)and jointly trained VC
+gains better speech decoupling ability.
+Demos for this paper are available at https://liuhaogeng.github.io/UnifySpeech/.
+2
+BACKGROUND
+In this section, we will briefly review the background of this work, including neural TTS and VC
+models, and the zero-shot learning for TTS and VC tasks.
+2.1
+TEXT-TO-SPEECH TASK
+TTS task is to model the mapping between text and speech, which is a modeling problem between
+two unequal length sequences. According to the alignment mechanism (Battenberg et al., 2020)
+between text and speech in the model, the end-to-end TTS model can be divided into two categories:
+1) Using a neural network to learn the alignment information between text and speech, such as
+local sensitive attention in Tacotron (Wang et al., 2017b). Various improvements to the attention
+mechanism have been proposed. In addition, inspired by CTC (Kim et al., 2017) in ASR, glow-
+TTS (Kim et al., 2020), VITS (Kim et al., 2021) can automatically learn the alignment information.
+2) By introducing the duration information (Ren et al., 2019) of phonemes as prior knowledge, the
+text is upsampled to achieve alignment (McAuliffe et al., 2017). Since the upsampled information
+based on text is independent of the speaker, the speech content can be well separated from the
+speech. Therefore, we introduce the duration information to build the TTS model in this paper.
+2.2
+VOICE CONVERSION TASK
+Voice conversion can be seen as two steps (Sisman et al., 2020).
+Firstly, extract the speaker-
+independent speech content information from the source speech, and then embed the target speaker
+information to the speech content to reconstruct the speech of the target speaker. According to the
+way of extracting speech content, the VC model can be divided into two categories: (1) text-based
+approach. (2) Information bottleneck approach. The first approach requires an additional pre-trained
+ASR model. Since the ASR is trained in a supervised manner, it demands a lot of paired text and
+speech. Additionally, pipeline modeling is easy to accumulate errors and affects the performance of
+2
+
+the system. Therefore, a lot of research work is focused on the latter. By adding some restrictions
+to the information bottleneck, different kinds of information can be decoupled. However, if the in-
+formation bottleneck can not be decoupled well, the performance of the model will be significantly
+reduced.
+2.3
+ZERO SHOT LEARNING FOR TTS AND VC
+The research of zero-shot learning based on TTS and VC focused on how to extract effective speaker
+information and then embed it into TTS or VC model for joint training or segmented training.
+Typical speaker features include i-vector (Wang et al., 2017a), d-vector (Variani et al., 2014), x-
+vector (Snyder et al., 2018) and so on. In addition, the modules that extract speaker-style informa-
+tion through the specially designed network structure, such as GST (Wang et al., 2018), VAE (Van
+Den Oord et al., 2017), can also achieve good results.
+3
+UNIFYSPEECH
+In this section, we introduce the details of UnifySpeech for zero-shot TTS and VC tasks. We first
+give the key idea of UnifySpeech: speech factorization, and then introduce the formulation of Uni-
+fySpeech. Finally, we will describe the model structure of UnifySpeech.
+Figure 1: Structure of UnifySpeech.
+3.1
+SPEECH REPRESENTATION DISENTANGLEMENT
+The core of the controllable and migratable speech generation task is to decouple the components of
+the generated speech first, and then control and transfer each component. Although some end-to-end
+models can directly model the relationship between text and speech (TTS) or speech-to-speech task
+(VC), due to the mutual coupling of various components of the end-to-end model, it brings great
+difficulties to the transfer learning of the model. Therefore, we first decouple the speech generation
+task into three independent components and then input them into the decoder to synthesize speech.
+This idea is also the main architecture of UnifySpeech. The three components and their sources will
+be described in detail below.
+Speech Content: To generate intelligible speech signals, it is important to model accurate speech
+content information. Speech content is linguistic information, which is irrelevant to the speaker.
+Due to the different types of input signals of TTS and VC, the ways of extracting speech content are
+different. For TTS, the source of speech content is text. Firstly, to learn the context information of
+the text, a text encoder is used to encode the text to obtain the context representation. The context
+representation is up-sampled according to the phoneme’s duration information to obtain the speech
+content information. For VC, since the source speech is aligned with the target speech, we directly
+use a content encoder to extract speech content information from the source speech.
+Speaker: The speaker information includes the speaker’s characteristics, such as the timbre, volume,
+etc. We can extract the speaker information from the given speech of the target speaker. This is
+3
+
+Decoder
+Decoder
+Pitch
+Pitch
+ Pitch predictor
+Pitch predictor
+Prosody information Speaker embedding
+Speech content
+Speech content
+Speaker embedding Prosody information
+VQ
+VQ
+Prosody encoder
+Speaker encoder
+Speaker encoder
+Prosody encoder
+Text encoder
+ Content encoder
+Pitch
+Reference mel
+Phonemes
+Speech
+Reference mel
+Pitchcommon for TTS and VC tasks, so the speaker extraction network can be shared in the two tasks.
+Since the speaker extraction network can directly extract information from speech without text, so a
+large number of data without text annotation in the VC task can be used for training, which can help
+to improve the transfer learning ability of the model.
+Prosody: The prosody information represents how the speaker says the content information. It
+is independent of the speaker information and related to the way of expression. Since the pitch
+information can reflect the rhythm of speech, therefore, the pitch information is used to extract
+prosody information. The prosody information, like speaker information, can be shared by both TTS
+and VC. In addition, in the training process, we can obtain pitch information from the ground truth
+speech, but there is no ground truth in the inference stage. Therefore, a pitch prediction module (Ren
+et al., 2020) is proposed in the training stage, which takes the speech content information and speaker
+information as the input to predict the pitch information.
+3.2
+SPEECH CONTENT WITH VECTOR QUANTIZATION
+Since there are different ways to obtain the speech content in TTS and VC, it is very easy to de-
+viate between the two domains. If this deviation occurs, it will cause devastating damage to some
+downstream shared modules (such as pitch predictor and decoder). Therefore, to ensure that the
+consistency of the two speech content domains as close as possible, we reconstruct them from two
+aspects:
+• First, we use the shared codebook to quantify the continuous feature space.
+• Second, we use the labeled data to narrow two discrete feature spaces.
+The detailed process is described below. Suppose that the vector obtained by the text encoder in
+TTS pipeline is Cp = (C1
+p, C2
+p, · · · , CT
+p ) with length T, the vector obtained by the content encoder
+in VC pipeline is Cs = (C1
+s, C2
+s, · · · , CT
+′
+s ) with length T
+′. It should be noted that we add a length
+regulator module in the text encoder to solve the problem of length mismatch between the text and
+speech sequence, which is introduced in FastSpeech (Ren et al., 2019). Therefore, if text and speech
+are paired, T
+′ = T. The vector Cp and Cs is a sequence of continuous vector in Eq. Due to the
+large representation range of continuous features, Cp and Cs are difficult to match. We borrow
+the discretization method for latent variables from Vector Quantized Variational AutoEncoder (Van
+Den Oord et al., 2017). Specifically, for each time step t, the continuous latent representations Ct
+p in
+Cp can be mapped into C
+t
+p by finding the nearest pre-defined discretized embedding in the dictionary
+as:
+C
+t
+p = ek,
+k = argminj
+��Ct
+p − ej
+��
+2
+(1)
+where ej is the j-th embedding in the codebook dictionary, and j ∈ 1, 2, · · · , V . Since selecting
+the entry with the minimum distance will cause the operation to be non-differentiable, the straight-
+through gradient estimator can be used to approximate the gradient, which can be expressed as:
+¯ht = ht + ev − sg (ht) ,
+v = arg min
+k
+∥ht − ek∥2
+(2)
+where sg(·) is the stop-gradient (Van Den Oord et al., 2017) operation that treats its input as constant
+during back-propagation.
+After vector quantization, the quantized sequence Cp
+=
+(C
+1
+p, C
+2
+p, · · · , C
+T
+p ) and Cs
+=
+(C
+1
+s, C
+2
+s, · · · , C
+T
+s ) can be obtained. It should be noted that when Cp and Cs are quantized into
+Cp and Cs, they share the same codebook e. The advantage of this is that since the speech content
+features Cp in the TTS pipeline are independent of the speaker, sharing the same codebook can
+help learn the speech content features Cs independent of the speaker in the VC pipeline, which is
+essential for the VC task.
+Although both pipeline use the same codebook for coding, there is no guarantee that there is no
+deviation between the two fields after coding. Therefore, to further eliminate the deviation between
+the two domains, we use the labeled speech in the TTS pipeline to supervise the training of the two
+domains. Specifically, for paired text and speech data, we constrain the feature distance between the
+quantized sequence Cp and Cs:
+4
+
+Lpair =
+��Cp − Cs
+��2
+2
+(3)
+In this way, we can efficiently minimize the domain discrepancy by using limited labeled data.
+Figure 2: Structure of vector quantized operation.
+3.3
+UNIFYSPEECH PIPELINE
+An overview of our proposed UnifySpeech architecture is illustrated in Fig. 2. It consists of a
+sequence-to-sequence TTS, and a sequence-to-sequence VC. The key idea is to share most of the
+module parameters (speaker encoder, prosody encoder, decoder and pitch predictor) and map the
+speech content in TTS and VC to the same space. As mentioned above, the UnifySpeech allows
+us to train the model on the concatenation of both the labeled and unlabeled data. For supervised
+training with labeled data, both models can be trained independently by minimizing the loss between
+their predicted speech and the ground truth. For unsupervised training with unlabeled data, the VC
+pipeline can be trained, and the parameters are shared with TTS.
+To further clarify the training process, we unrolled the framework as follows.
+3.3.1
+TTS PIPELINE
+Denote the text and speech sequence pair (x, y, F0) ∈ D, where D is the paired text and speech
+corpus. Each element in the text sequence x represents a phoneme or character, while each element
+in the speech sequence y represents a frame of speech. F0 is the pitch information of y. The
+representation obtained after three encoders are speech content C, speaker S and prosody P.
+Then, the three parts are added and input into a decoder to obtain the predicted speech y′. In addition,
+to obtain the pitch information in the interference stage, we use the content information and speaker
+information to predict the F0. These processes can be expressed as:
+y′ = decoder(C + S + P)
+(4)
+F0′ = pitch predictor(C + S)
+(5)
+Therefore, the reconstruction loss in TTS process includes two parts:
+LV C
+rec = MSE(y, y′) + MSE(F0, F0′)
+(6)
+5
+
+C
+Look up min
+Look up min
+distance
+distance
+vector
+vector
+1
+V
+L2
+L2
+distance
+ev
+distance
+Ct
+Cs
+Text encoder
+Content encoderIn addition, we use the content encoder in the VC pipeline to extract the content representation Cs
+for the training speech in TTS, and close the distance between the two domains by calculating the
+distance loss of Cs and Cp, which is explained in Sec. 3.2.
+Lpair =
+��Cp − Cs
+��2
+2
+(7)
+The loss of TTS pipeline can be expressed as:
+LT T S = LT T S
+rec
++ Lpair
+(8)
+where MSE denotes the mean squared errors.
+3.3.2
+VC PIPELINE
+Denote all the unlabeled or labeled speech (y, F0) ∈ Y . We first extract three information from
+(y, F0), which are speech content Cs, speaker Ss and prosody Ps.
+Then, similar to the TTS pipeline, the shared decoder and pitch predictor module is used to predict
+the speech signal y′ and F0′, which is similar to the Eq.1 and Eq.2.
+The loss of VC pipeline only includes reconstruction loss, which can be expressed as:
+LV C = MSE(Y, Y ′) + MSE(F0, F0′)
+(9)
+where MSE denotes the mean squared errors.
+3.3.3
+TRAINING PROCESS
+With such a unified framework, TTS and VC can learn from each other through joint training. The
+details of the algorithm can be found below.
+Procedure 1 UnifySpeech training algorithm
+1: Input: Paired speech and text dataset (x, y), speech only dataset y
+′
+2: repeat
+3:
+# A. TTS pipeline with speech-text data pairs
+1.
+Sample paired speech and text (x, y)
+2.
+Extract speech content information from x for domain loss
+3.
+Generate the predict speech y, pitch F0 and speech content from text
+4.
+Calculate the loss for TTS LT T S
+rec
+4:
+# B. VC pipeline with speech-only data
+1.
+Sample paired speech and text (x, y)
+2.
+Extract speech content information from y for domain loss
+3.
+Calculate the domain loss for the two pipeline Lpair
+4.
+Sample speech y
+′ in speech only dataset
+5.
+Generate the predict speech y, pitch F0 and speech content from speech y
+′
+6.
+Calculate the loss for VC LV C
+5:
+# C. Loss combination:
+1.
+Combine all loss (LT T S
+rec , Lpair, LV C) into a single loss variable
+2.
+Calculate TTS and VC parameters gradient
+3.
+Update TTS and VC parameters with gradient descent optimization
+6: until convergence
+4
+EXPERIMENTS AND RESULTS ANALYSIS
+In this section, we conduct experiments to evaluate our proposed methods. The experiments are
+carried out from two aspects: zero-shot TTS, zero-shot VC.
+6
+
+4.1
+DATASETS
+Two datasets are used to simulate labeled data and unlabeled data, respectively. VCTK dataset, an
+English language dataset containing 44 hours of speech and 109 speakers is used as labeled data.
+Each speaker has approximately 400 sentences. LibriTTS (Zen et al., 2019) are used as unlabeled
+data, which consists of 585 hours of speech data from 2484 speakers. We only use the speech data in
+LibriTTS and discard the text for unsupervised training. In this way, it can simulate the scene where
+a large number of speech that we can obtain are unlabeled. We use a 16-bit, 22050 Hz sampling rate
+for all experiments. The 80-dim Mel spectrogram is extracted with Hann windowing, frame shift
+of 12.5-ms, frame length of 50-ms, and 1024-point Fourier transform. In this experiment, we use
+hifigan (Kong et al., 2020) as vocoder.
+4.2
+MODEL DETAILS
+Figure 3: Structure of each module in UnifySpeech. FFT (Ren et al., 2019) means feed-forward
+Transformer.
+The detail of each module in our proposed method is shown in the Fig. 3. Specifically, to make
+the sequence of speech content extracted from the text encoder and content encoder equal, a length
+regulator is added to the text encoder, which is inspired by the FastSpeech (Ren et al., 2019). The
+structure of the duration predictor is the same as that in FastSpeech. The structure of the decoder
+and content encoder is similar, but the dimensions of input and output are opposite. For the prosody
+encoder, we first quantize F0 of each frame to 32 possible values and encode them into a learnable
+embedding vector according to the value. And we change the output of the pitch predictor into a
+distribution. By this way, pitch prediction becomes a classification task, reducing the difficulty of
+frame-level pitch prediction. We use the speaker encoder in StyleSpeech and fuse features with Style
+Adaptive Layer Norm(SALN) (Min et al., 2021) method to make the comparision fairly.
+The number of feed-forward Transformer (Vaswani et al., 2017) (FFT) blocks in the text encoder is
+4 and it is 6 in the decoder module. In each FFT block, the dimension of hidden states is 256. The
+kernel sizes of all the 1D-convolution are set to 3. The dropout rate is set to 0.5. The dimension of the
+last linear layer in the decoder is 256. The dimension of last linear layer in encoders (text encoder,
+pitch encoder, content encoder) is 256. An Adam optimizer (Kingma & Ba, 2014) is used to update
+the parameters. The initial learning rate is 0.001 and the learning rate decreased exponentially.
+7
+
+Linear layer
+Linear layer
+Duration
+Length regulator
+predictor
+m block FFT
+不
+不
+LN + Dropout
+n block FFT
+不
+Speaker
+Speaker
+Embedding
+ConvlD + ReLU
+Phoneme
+Embedding
+Embedding
+Linear layer
+Linear layer
+Pooling
+个
+LN + Dropout
+LN + Dropout
+FFT
+ConvlD + ReLU
+ConvlD + ReLU
+LN + Dropout
+LN + Dropout
+不
+Pitch Embedding
+ConvlD + ReLU
+ConvlD + ReLU
+Randomly select a clip
+LSTM
+Quantify to [0, 32]
+from target speaker4.3
+ZERO-SHOT TTS
+We first carried out zero-shot TTS task. We choose four speakers from VCTK that are not used
+during training process as target speakers. For each speaker, we randomly select about 20 sentences
+to be our target. Then we caculate F0 RMSE (root of mean square errors of F0), MCD (Kubichek,
+1993) (Mel-cepstrum distortion), V/UV (the error rate of voicing/unvoicing flags) and F0 CORR
+(correlation factor of F0) between synthesized speech and ground truth speech as the objective met-
+rics.
+Table 1: Objective evaluation results for zero-shot TTS.
+Model
+F0 RMSE (Hz)
+MCD (db)
+V/UV
+F0 CORR
+UnifySpeech
+17.84
+2.51
+16.9%
+0.93
+StyleSpeech
+19.02
+2.63
+18.06%
+0.92
+Subjective evaluation was also conducted to compare the speech’s quality and similarity. We choose
+mean opinion scores(MOS) for naturalness and similarity mean opinion scores(SMOS) for similar-
+ity. Both metrics are rated in 1-to-5 scale and reported with the 95% confidence intervals (CI).
+Table 2: Mean opinion score (MOS) of the models. With VC means the model is jointly trained.
+Model
+MOS
+SMOS
+GT
+4.32 ± 0.15
+−
+GT mel + Vocoder
+4.09 ± 0.15
+−
+StyleSpeech
+3.52 ± 0.13
+3.82 ± 0.13
+UnifySpeech-TTS (with VC)
+3.76 ± 0.12
+3.95 ± 0.13
+To better show our method’s effectiveness, we demonstrate the t-SNE projection of the speaker
+embedding vectors from speakers in both VCTK and LibriTTS. (Van der Maaten & Hinton, 2008)
+Fig. 4 shows the speaker visualization. For the seen speakers (x) and unseen speakers (o), the
+corresponding speaker embedding form a cluster and distinct from others. The boundary between
+different speakers is clear. This shows that the speaker encoder performs well.
+Figure 4: Speaker visualization of generated speeches,where the circle and triangle indicate unseen
+speaker,while x and square indicate seen speaker.
+4.4
+ZERO-SHOT VC
+We carried out zero-shot VC task, using unseen speakers voice to be the reference speech. As we
+are lack of parallel corpus, we only conduct subjective evaluation. But VC and TTS shares the same
+8
+
+30
+p225
+p226
+p227
+20
+p228
+p279
+X
+p274
+10
+p307
+p311
+8012
+0
+￥5181
+3885
+2961
+-10
+1089
+7021
+20
+KX
+-20
+15
+-10
+-5
+0
+5
+10
+15speaker encoder, Fig. 4 in zero-shot TTS can also be a reference.
+Just as in zero-shot TTS task, we choose mean opinion scores(MOS) for naturalness and similarity
+mean opinion scores(SMOS) for similarity. Both metrics are rated in 1-to-5 scale and reported with
+the 95% confidence intervals (CI).
+Table 3: Mean opinion score (MOS) of the VC models. With TTS means the model is jointly
+trained.
+Model
+MOS
+SMOS
+GT
+4.32 ± 0.15
+−
+GT mel + Vocoder
+4.09 ± 0.15
+−
+UnifySpeech-VC(without TTS)
+3.63 ± 0.13
+1.31 ± 0.06
+UnifySpeech-VC(with TTS)
+3.58 ± 0.12
+3.31 ± 0.13
+It can be found that when VC pipeline is trained alone, its performance is poor. In other words, it
+doesn’t have the ability to discriminate. But jointly training with TTS improves its speech decou-
+pling ability, indirectly improving the speaker modeling ability, which we will analysis later.
+4.5
+ABLATION EXPERIMENT
+To figure out whether jointly training is effective, we separately train the TTS and VC parts, carrying
+out objective evaluation on them. We also remove the VQ parts and test the model. For TTS,
+we caculate F0 RMSE (root of mean square errors of F0), MCD (Kubichek, 1993) (Mel-cepstrum
+distortion), V/UV (the error rate of voicing/unvoicing flags) and F0 CORR (correlation factor of F0)
+between synthesized speech and ground truth speech as the objective metrics.
+Table 4: Objective evaluation results for zero-shot TTS.
+Model
+F0 RMSE (Hz)
+MCD (db)
+V/UV
+F0 CORR
+UnifySpeech-TTS(without VC)
+19.31
+2.55
+16.9%
+0.91
+UnifySpeech-TTS-novq(with VC)
+19.41
+2.58
+17.2%
+0.92
+UnifySpeech-TTS(with VC)
+17.84
+2.51
+16.9%
+0.93
+For VC, we separately calculate the Average Cosine Similarity (ACS) (Lei et al., 2022) for the
+embedding from same (S-ACS) and different speakers (D-ACS). And then their ratio is used as an
+evaluation metric.
+Table 5: Ratio of ACS from same and different speaker. Unseen or seen means whether the speaker
+is from test set. With or without TTS means whether the VC pipeline is trained with TTS pipeline.
+Model
+S−ACS
+D−ACS (unseen)
+S−ACS
+D−ACS (seen)
+UnifySpeech-VC (with TTS)
+2.8
+6.0
+UnifySpeech-VC (without TTS)
+1.0
+1.0
+To validate whether VQ can bridge the gap between the speech content parts in TTS and VC, we
+caculate the L2 distance between the phoneme representation from TTS and VC. As in VC there
+are many frames represent same phoneme, so we choose their clustering center as the correspond-
+ing phoneme representation. We randomly select 4 sentences from validation set to carry out the
+experiment.
+Table 6: L2 distance between the phoneme representation in TTS and VC. S1 means sentence1.
+Model
+S1
+S2
+S3
+S4
+average
+UnifySpeech-VC(without VQ)
+0.464
+0.476
+0.474
+0.553
+0.492
+UnifySpeech-VC(with VQ)
+0.152
+0.205
+0.208
+0.205
+0.193
+9
+
+4.6
+RESULT ANALYSIS
+Above results show that jointly training actually improves TTS’s speaker modeling ability and VC’s
+speech decoupling ability. And VQ enhances the consistency of representation of the same content
+from phonemes and speeches, ensuring the model’s correctly working.
+For TTS, sharing modules with VC enables unlabeled data to participate in its training process.
+Along with the richer speaker style pattern, the speaker style modeling capability has been enhanced.
+For VC, when trained alone, it doesn’t have the speaker modeling ability. As the self-supervised
+training process aims at reducing the reconstruction loss of source audio, the reference audio of the
+target speaker isn’t crucial in the process. Only the content encoder and the decoder are enough for
+the process, that’s possibly why the speaker embeddings are all similar, though they are from differ-
+ent speakers. When jointly trained, the text loss plays a role of regularization factor, resulting that the
+content encoder just extracting the content information from the source speech (speaker information
+is reserved and others are discarded). This makes the reference speech with rich speaker information
+become indispensable for the reconstruction process. Thus the model’s speaker modeling ability has
+been improved.
+5
+CONCLUSIONS
+In this paper, we propose UnifySpeech, a unified framework for TTS and VC. Both task benefits
+from the other one. Due to training with large amounts of unlabeled data, their few-shot modeling
+ability makes progress as well as the synthesized speech’s quality. In the future, further improving
+the synthesized speech’s quality and making the generated speech’s style more similar to target
+speaker will be our endeavor.
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+
diff --git a/GNE2T4oBgHgl3EQfTQeG/content/tmp_files/load_file.txt b/GNE2T4oBgHgl3EQfTQeG/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf,len=465
+page_content='UNIFYSPEECH: A UNIFIED FRAMEWORK FOR ZERO-  SHOT TEXT-TO-SPEECH AND VOICE CONVERSION  Haogeng Liu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Tao Wang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Ruibo Fu2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Jiangyan Yi2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Zhengqi Wen2  Chinese Academy of Science  Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' China  Jianhua Tao1  Department of Automation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Tsinghua University  School of Artificial Intelligence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' China  Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' China  ABSTRACT  Text-to-speech (TTS) and voice conversion (VC) are two different tasks both aim-  ing at generating high quality speaking voice according to different input modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Due to their similarity, this paper proposes UnifySpeech, which brings TTS and  VC into a unified framework for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The model is based on the assump-  tion that speech can be decoupled into three independent components: content  information, speaker information, prosody information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Both TTS and VC can be  regarded as mining these three parts of information from the input and completing  the reconstruction of speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' For TTS, the speech content information is derived  from the text, while in VC it’s derived from the source speech, so all the remaining  units are shared except for the speech content extraction module in the two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  We applied vector quantization and domain constrain to bridge the gap between  the content domains of TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Objective and subjective evaluation shows  that by combining the two task, TTS obtains better speaker modeling ability while  VC gets hold of impressive speech content decoupling capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Index Terms: decoupling, zero-shot learning, text-to-speech, voice conversion,  vector quantization  1  INTRODUCTION  Cloning the voice of the target speaker is an attractive technology, which can be applied to various  scenes (Sisman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2020), such as entertainment creation, personalized mobile assistants, security  field, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The most ideal voice cloning operation is to give only one relatively short speech of the  unseen target speaker as a reference and then any speech of the target speaker can be synthesized,  which is called zero-shot voice clone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In the speech research community, text-to-speech (TTS) and  voice conversion (VC) are two mainstream ways to realize voice clone (Sisman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' There-  fore, a variety of techniques for zero-shot TTS and VC have been proposed recently (Gorodetskii &  Ozhiganov, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  However, although TTS and VC techniques are two important ways of voice clone with same output  form, the research of TTS and VC is more or less independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' There isn’t much interaction between  them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' But they are both speech synthesis task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In terms of speech generation, we categorize the  information of the target speaker’s speech into three kinds of information: (1) speech content, the  characters of phonemes or phonetic posteriorgram (PPG) in voice conversion, represents the content  of the speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' (2) speaker information, which represents the characteristics of speakers, is related  to the speaker’s articulation organ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' (3) prosody information, which covers the intonation, stress,  and rhythm of speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' According to FastSpeech2 (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2020), pitch, energy and duration  information can reflected them certainly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' As TTS extracts speech content directly from phonemes,  it is easier to obtain content information irrelevant to speaker than VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' As VC encounters more  speakers, it’s possible to obtain more robust speaker modeling ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' So, by integrating TTS and  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='03801v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='SD]  10 Jan 2023  VC into a unified framework and combining their training data, it can help the model learn these  three kinds of information better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Unfortunately, investigating TTS and VC in the same framework is challenging, as the speech  content are extracted from different modality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Specifically, the speech content in TTS is obtained  through phoneme information while the phonemes and speech in TTS are unequal sequences, need-  ing attention mechanism to align them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' However, the attention mechanism is often affected by the  speaker’s information, so it is impossible to learn the speech content representation completely ir-  relevant to the speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' While in VC, the source speech and target speech are aligned in speech  content, so the speech content can be extracted directly from the source speech, which is different  from the TTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In contrast to speech content information, speaker information and prosody infor-  mation can be modeled using the same network in TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Therefore, the difficulty here is  to keep speech content for TTS and VC consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' With the development of speech synthesis, the  recently proposed Adaspeech2 (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2021) can combine text information with speech infor-  mation, which can effectively decouple the speech content from the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The unified framework  becomes possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Overall, the main contributions of this paper are:  We propose UnifySpeech, a unified framework for TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' VC enables unlabeled data  to join training process, making TTS encounters more speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' TTS enhance the ability  for voice content decoupling in VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Thus, both pipeline benefits from the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  We apply vector quantization and domain constrain to bridge the gap between the content  domains of TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Ablation experiment shows this method’s effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  We perform extensive experiments: zero-shot TTS, zero-shot VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Results proves that  jointly trained TTS outperformes StyleSpeech (Min et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2021)and jointly trained VC  gains better speech decoupling ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Demos for this paper are available at https://liuhaogeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='io/UnifySpeech/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  2  BACKGROUND  In this section, we will briefly review the background of this work, including neural TTS and VC  models, and the zero-shot learning for TTS and VC tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='1  TEXT-TO-SPEECH TASK  TTS task is to model the mapping between text and speech, which is a modeling problem between  two unequal length sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' According to the alignment mechanism (Battenberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2020)  between text and speech in the model, the end-to-end TTS model can be divided into two categories:  1) Using a neural network to learn the alignment information between text and speech, such as  local sensitive attention in Tacotron (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Various improvements to the attention  mechanism have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In addition, inspired by CTC (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2017) in ASR, glow-  TTS (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2020), VITS (Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2021) can automatically learn the alignment information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  2) By introducing the duration information (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2019) of phonemes as prior knowledge, the  text is upsampled to achieve alignment (McAuliffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Since the upsampled information  based on text is independent of the speaker, the speech content can be well separated from the  speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Therefore, we introduce the duration information to build the TTS model in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='2  VOICE CONVERSION TASK  Voice conversion can be seen as two steps (Sisman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Firstly, extract the speaker-  independent speech content information from the source speech, and then embed the target speaker  information to the speech content to reconstruct the speech of the target speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' According to the  way of extracting speech content, the VC model can be divided into two categories: (1) text-based  approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' (2) Information bottleneck approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The first approach requires an additional pre-trained  ASR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Since the ASR is trained in a supervised manner, it demands a lot of paired text and  speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Additionally, pipeline modeling is easy to accumulate errors and affects the performance of  2  the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Therefore, a lot of research work is focused on the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' By adding some restrictions  to the information bottleneck, different kinds of information can be decoupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' However, if the in-  formation bottleneck can not be decoupled well, the performance of the model will be significantly  reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='3  ZERO SHOT LEARNING FOR TTS AND VC  The research of zero-shot learning based on TTS and VC focused on how to extract effective speaker  information and then embed it into TTS or VC model for joint training or segmented training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Typical speaker features include i-vector (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2017a), d-vector (Variani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2014), x-  vector (Snyder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2018) and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In addition, the modules that extract speaker-style informa-  tion through the specially designed network structure, such as GST (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2018), VAE (Van  Den Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2017), can also achieve good results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  3  UNIFYSPEECH  In this section, we introduce the details of UnifySpeech for zero-shot TTS and VC tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We first  give the key idea of UnifySpeech: speech factorization, and then introduce the formulation of Uni-  fySpeech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Finally, we will describe the model structure of UnifySpeech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Figure 1: Structure of UnifySpeech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='1  SPEECH REPRESENTATION DISENTANGLEMENT  The core of the controllable and migratable speech generation task is to decouple the components of  the generated speech first, and then control and transfer each component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Although some end-to-end  models can directly model the relationship between text and speech (TTS) or speech-to-speech task  (VC), due to the mutual coupling of various components of the end-to-end model, it brings great  difficulties to the transfer learning of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Therefore, we first decouple the speech generation  task into three independent components and then input them into the decoder to synthesize speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  This idea is also the main architecture of UnifySpeech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The three components and their sources will  be described in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Speech Content: To generate intelligible speech signals, it is important to model accurate speech  content information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Speech content is linguistic information, which is irrelevant to the speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Due to the different types of input signals of TTS and VC, the ways of extracting speech content are  different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' For TTS, the source of speech content is text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Firstly, to learn the context information of  the text, a text encoder is used to encode the text to obtain the context representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The context  representation is up-sampled according to the phoneme’s duration information to obtain the speech  content information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' For VC, since the source speech is aligned with the target speech, we directly  use a content encoder to extract speech content information from the source speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Speaker: The speaker information includes the speaker’s characteristics, such as the timbre, volume,  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We can extract the speaker information from the given speech of the target speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' This is  3  Decoder  Decoder  Pitch  Pitch  Pitch predictor  Pitch predictor  Prosody information Speaker embedding  Speech content  Speech content  Speaker embedding Prosody information  VQ  VQ  Prosody encoder  Speaker encoder  Speaker encoder  Prosody encoder  Text encoder  Content encoder  Pitch  Reference mel  Phonemes  Speech  Reference mel  Pitchcommon for TTS and VC tasks, so the speaker extraction network can be shared in the two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Since the speaker extraction network can directly extract information from speech without text, so a  large number of data without text annotation in the VC task can be used for training, which can help  to improve the transfer learning ability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Prosody: The prosody information represents how the speaker says the content information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' It  is independent of the speaker information and related to the way of expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Since the pitch  information can reflect the rhythm of speech, therefore, the pitch information is used to extract  prosody information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The prosody information, like speaker information, can be shared by both TTS  and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In addition, in the training process, we can obtain pitch information from the ground truth  speech, but there is no ground truth in the inference stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Therefore, a pitch prediction module (Ren  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2020) is proposed in the training stage, which takes the speech content information and speaker  information as the input to predict the pitch information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='2  SPEECH CONTENT WITH VECTOR QUANTIZATION  Since there are different ways to obtain the speech content in TTS and VC, it is very easy to de-  viate between the two domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' If this deviation occurs, it will cause devastating damage to some  downstream shared modules (such as pitch predictor and decoder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Therefore, to ensure that the  consistency of the two speech content domains as close as possible, we reconstruct them from two  aspects:  First, we use the shared codebook to quantify the continuous feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Second, we use the labeled data to narrow two discrete feature spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  The detailed process is described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Suppose that the vector obtained by the text encoder in  TTS pipeline is Cp = (C1  p, C2  p, · · · , CT  p ) with length T, the vector obtained by the content encoder  in VC pipeline is Cs = (C1  s, C2  s, · · · , CT  ′  s ) with length T  ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' It should be noted that we add a length  regulator module in the text encoder to solve the problem of length mismatch between the text and  speech sequence, which is introduced in FastSpeech (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Therefore, if text and speech  are paired, T  ′ = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The vector Cp and Cs is a sequence of continuous vector in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Due to the  large representation range of continuous features, Cp and Cs are difficult to match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We borrow  the discretization method for latent variables from Vector Quantized Variational AutoEncoder (Van  Den Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Specifically, for each time step t, the continuous latent representations Ct  p in  Cp can be mapped into C  t  p by finding the nearest pre-defined discretized embedding in the dictionary  as:  C  t  p = ek,  k = argminj  ��Ct  p − ej  ��  2  (1)  where ej is the j-th embedding in the codebook dictionary, and j ∈ 1, 2, · · · , V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Since selecting  the entry with the minimum distance will cause the operation to be non-differentiable, the straight-  through gradient estimator can be used to approximate the gradient, which can be expressed as:  ¯ht = ht + ev − sg (ht) ,  v = arg min  k  ∥ht − ek∥2  (2)  where sg(·) is the stop-gradient (Van Den Oord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2017) operation that treats its input as constant  during back-propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  After vector quantization, the quantized sequence Cp  =  (C  1  p, C  2  p, · · · , C  T  p ) and Cs  =  (C  1  s, C  2  s, · · · , C  T  s ) can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' It should be noted that when Cp and Cs are quantized into  Cp and Cs, they share the same codebook e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The advantage of this is that since the speech content  features Cp in the TTS pipeline are independent of the speaker, sharing the same codebook can  help learn the speech content features Cs independent of the speaker in the VC pipeline, which is  essential for the VC task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Although both pipeline use the same codebook for coding, there is no guarantee that there is no  deviation between the two fields after coding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Therefore, to further eliminate the deviation between  the two domains, we use the labeled speech in the TTS pipeline to supervise the training of the two  domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Specifically, for paired text and speech data, we constrain the feature distance between the  quantized sequence Cp and Cs:  4  Lpair =  ��Cp − Cs  ��2  2  (3)  In this way, we can efficiently minimize the domain discrepancy by using limited labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Figure 2: Structure of vector quantized operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='3  UNIFYSPEECH PIPELINE  An overview of our proposed UnifySpeech architecture is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' It consists of a  sequence-to-sequence TTS, and a sequence-to-sequence VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The key idea is to share most of the  module parameters (speaker encoder, prosody encoder, decoder and pitch predictor) and map the  speech content in TTS and VC to the same space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' As mentioned above, the UnifySpeech allows  us to train the model on the concatenation of both the labeled and unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' For supervised  training with labeled data, both models can be trained independently by minimizing the loss between  their predicted speech and the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' For unsupervised training with unlabeled data, the VC  pipeline can be trained, and the parameters are shared with TTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  To further clarify the training process, we unrolled the framework as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='1  TTS PIPELINE  Denote the text and speech sequence pair (x, y, F0) ∈ D, where D is the paired text and speech  corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Each element in the text sequence x represents a phoneme or character, while each element  in the speech sequence y represents a frame of speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' F0 is the pitch information of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The  representation obtained after three encoders are speech content C, speaker S and prosody P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Then, the three parts are added and input into a decoder to obtain the predicted speech y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In addition,  to obtain the pitch information in the interference stage, we use the content information and speaker  information to predict the F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' These processes can be expressed as:  y′ = decoder(C + S + P)  (4)  F0′ = pitch predictor(C + S)  (5)  Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' the reconstruction loss in TTS process includes two parts:  LV C  rec = MSE(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' y′) + MSE(F0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' F0′)  (6)  5  C  Look up min  Look up min  distance  distance  vector  vector  1  V  L2  L2  distance  ev  distance  Ct  Cs  Text encoder  Content encoderIn addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' we use the content encoder in the VC pipeline to extract the content representation Cs  for the training speech in TTS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' and close the distance between the two domains by calculating the  distance loss of Cs and Cp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' which is explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Lpair =  ��Cp − Cs  ��2  2  (7)  The loss of TTS pipeline can be expressed as:  LT T S = LT T S  rec  + Lpair  (8)  where MSE denotes the mean squared errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='2  VC PIPELINE  Denote all the unlabeled or labeled speech (y, F0) ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We first extract three information from  (y, F0), which are speech content Cs, speaker Ss and prosody Ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Then, similar to the TTS pipeline, the shared decoder and pitch predictor module is used to predict  the speech signal y′ and F0′, which is similar to the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  The loss of VC pipeline only includes reconstruction loss, which can be expressed as:  LV C = MSE(Y, Y ′) + MSE(F0, F0′)  (9)  where MSE denotes the mean squared errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='3  TRAINING PROCESS  With such a unified framework, TTS and VC can learn from each other through joint training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The  details of the algorithm can be found below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Procedure 1 UnifySpeech training algorithm  1: Input: Paired speech and text dataset (x, y), speech only dataset y  ′  2: repeat  3:  # A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' TTS pipeline with speech-text data pairs  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Sample paired speech and text (x, y)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Extract speech content information from x for domain loss  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Generate the predict speech y, pitch F0 and speech content from text  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Calculate the loss for TTS LT T S  rec  4:  # B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' VC pipeline with speech-only data  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Sample paired speech and text (x, y)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Extract speech content information from y for domain loss  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Calculate the domain loss for the two pipeline Lpair  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Sample speech y  ′ in speech only dataset  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Generate the predict speech y, pitch F0 and speech content from speech y  ′  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Calculate the loss for VC LV C  5:  # C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Loss combination:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Combine all loss (LT T S  rec , Lpair, LV C) into a single loss variable  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Calculate TTS and VC parameters gradient  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Update TTS and VC parameters with gradient descent optimization  6: until convergence  4  EXPERIMENTS AND RESULTS ANALYSIS  In this section, we conduct experiments to evaluate our proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The experiments are  carried out from two aspects: zero-shot TTS, zero-shot VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  6  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='1  DATASETS  Two datasets are used to simulate labeled data and unlabeled data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' VCTK dataset, an  English language dataset containing 44 hours of speech and 109 speakers is used as labeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Each speaker has approximately 400 sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' LibriTTS (Zen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2019) are used as unlabeled  data, which consists of 585 hours of speech data from 2484 speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We only use the speech data in  LibriTTS and discard the text for unsupervised training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In this way, it can simulate the scene where  a large number of speech that we can obtain are unlabeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We use a 16-bit, 22050 Hz sampling rate  for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The 80-dim Mel spectrogram is extracted with Hann windowing, frame shift  of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='5-ms, frame length of 50-ms, and 1024-point Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In this experiment, we use  hifigan (Kong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2020) as vocoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='2  MODEL DETAILS  Figure 3: Structure of each module in UnifySpeech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' FFT (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2019) means feed-forward  Transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  The detail of each module in our proposed method is shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Specifically, to make  the sequence of speech content extracted from the text encoder and content encoder equal, a length  regulator is added to the text encoder, which is inspired by the FastSpeech (Ren et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The  structure of the duration predictor is the same as that in FastSpeech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The structure of the decoder  and content encoder is similar, but the dimensions of input and output are opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' For the prosody  encoder, we first quantize F0 of each frame to 32 possible values and encode them into a learnable  embedding vector according to the value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' And we change the output of the pitch predictor into a  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' By this way, pitch prediction becomes a classification task, reducing the difficulty of  frame-level pitch prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We use the speaker encoder in StyleSpeech and fuse features with Style  Adaptive Layer Norm(SALN) (Min et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2021) method to make the comparision fairly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  The number of feed-forward Transformer (Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2017) (FFT) blocks in the text encoder is  4 and it is 6 in the decoder module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In each FFT block, the dimension of hidden states is 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The  kernel sizes of all the 1D-convolution are set to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The dropout rate is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The dimension of the  last linear layer in the decoder is 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The dimension of last linear layer in encoders (text encoder,  pitch encoder, content encoder) is 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' An Adam optimizer (Kingma & Ba, 2014) is used to update  the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The initial learning rate is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='001 and the learning rate decreased exponentially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  7  Linear layer  Linear layer  Duration  Length regulator  predictor  m block FFT  不  不  LN + Dropout  n block FFT  不  Speaker  Speaker  Embedding  ConvlD + ReLU  Phoneme  Embedding  Embedding  Linear layer  Linear layer  Pooling  个  LN + Dropout  LN + Dropout  FFT  ConvlD + ReLU  ConvlD + ReLU  LN + Dropout  LN + Dropout  不  Pitch Embedding  ConvlD + ReLU  ConvlD + ReLU  Randomly select a clip  LSTM  Quantify to [0, 32]  from target speaker4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='3  ZERO-SHOT TTS  We first carried out zero-shot TTS task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We choose four speakers from VCTK that are not used  during training process as target speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' For each speaker, we randomly select about 20 sentences  to be our target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Then we caculate F0 RMSE (root of mean square errors of F0), MCD (Kubichek,  1993) (Mel-cepstrum distortion), V/UV (the error rate of voicing/unvoicing flags) and F0 CORR  (correlation factor of F0) between synthesized speech and ground truth speech as the objective met-  rics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Table 1: Objective evaluation results for zero-shot TTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Model  F0 RMSE (Hz)  MCD (db)  V/UV  F0 CORR  UnifySpeech  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='84  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='51  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='9%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='93  StyleSpeech  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='02  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='63  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='06%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='92  Subjective evaluation was also conducted to compare the speech’s quality and similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We choose  mean opinion scores(MOS) for naturalness and similarity mean opinion scores(SMOS) for similar-  ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Both metrics are rated in 1-to-5 scale and reported with the 95% confidence intervals (CI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Table 2: Mean opinion score (MOS) of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' With VC means the model is jointly trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Model  MOS  SMOS  GT  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='15  −  GT mel + Vocoder  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='15  −  StyleSpeech  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='82 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='13  UnifySpeech-TTS (with VC)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='13  To better show our method’s effectiveness, we demonstrate the t-SNE projection of the speaker  embedding vectors from speakers in both VCTK and LibriTTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' (Van der Maaten & Hinton, 2008)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' 4 shows the speaker visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' For the seen speakers (x) and unseen speakers (o), the  corresponding speaker embedding form a cluster and distinct from others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' The boundary between  different speakers is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' This shows that the speaker encoder performs well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Figure 4: Speaker visualization of generated speeches,where the circle and triangle indicate unseen  speaker,while x and square indicate seen speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='4  ZERO-SHOT VC  We carried out zero-shot VC task, using unseen speakers voice to be the reference speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' As we  are lack of parallel corpus, we only conduct subjective evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' But VC and TTS shares the same  8  30  p225  p226  p227  20  p228  p279  X  p274  10  p307  p311  8012  0  ￥5181  3885  2961  10  1089  7021  20  KX  20  15  10  5  0  5  10  15speaker encoder, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' 4 in zero-shot TTS can also be a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Just as in zero-shot TTS task, we choose mean opinion scores(MOS) for naturalness and similarity  mean opinion scores(SMOS) for similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Both metrics are rated in 1-to-5 scale and reported with  the 95% confidence intervals (CI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Table 3: Mean opinion score (MOS) of the VC models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' With TTS means the model is jointly  trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Model  MOS  SMOS  GT  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='15  −  GT mel + Vocoder  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='15  −  UnifySpeech-VC(without TTS)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='13  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='06  UnifySpeech-VC(with TTS)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='58 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='31 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='13  It can be found that when VC pipeline is trained alone, its performance is poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In other words, it  doesn’t have the ability to discriminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' But jointly training with TTS improves its speech decou-  pling ability, indirectly improving the speaker modeling ability, which we will analysis later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='5  ABLATION EXPERIMENT  To figure out whether jointly training is effective, we separately train the TTS and VC parts, carrying  out objective evaluation on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We also remove the VQ parts and test the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' For TTS,  we caculate F0 RMSE (root of mean square errors of F0), MCD (Kubichek, 1993) (Mel-cepstrum  distortion), V/UV (the error rate of voicing/unvoicing flags) and F0 CORR (correlation factor of F0)  between synthesized speech and ground truth speech as the objective metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Table 4: Objective evaluation results for zero-shot TTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Model  F0 RMSE (Hz)  MCD (db)  V/UV  F0 CORR  UnifySpeech-TTS(without VC)  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='31  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='55  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='9%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='91  UnifySpeech-TTS-novq(with VC)  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='41  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='58  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='2%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='92  UnifySpeech-TTS(with VC)  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='84  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='51  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='9%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='93  For VC, we separately calculate the Average Cosine Similarity (ACS) (Lei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=', 2022) for the  embedding from same (S-ACS) and different speakers (D-ACS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' And then their ratio is used as an  evaluation metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Table 5: Ratio of ACS from same and different speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Unseen or seen means whether the speaker  is from test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' With or without TTS means whether the VC pipeline is trained with TTS pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Model  S−ACS  D−ACS (unseen)  S−ACS  D−ACS (seen)  UnifySpeech-VC (with TTS)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='8  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='0  UnifySpeech-VC (without TTS)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='0  To validate whether VQ can bridge the gap between the speech content parts in TTS and VC, we  caculate the L2 distance between the phoneme representation from TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' As in VC there  are many frames represent same phoneme, so we choose their clustering center as the correspond-  ing phoneme representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' We randomly select 4 sentences from validation set to carry out the  experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Table 6: L2 distance between the phoneme representation in TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' S1 means sentence1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Model  S1  S2  S3  S4  average  UnifySpeech-VC(without VQ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='464  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='476  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='474  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='553  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='492  UnifySpeech-VC(with VQ)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='152  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='205  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='208  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='205  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='193  9  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='6  RESULT ANALYSIS  Above results show that jointly training actually improves TTS’s speaker modeling ability and VC’s  speech decoupling ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' And VQ enhances the consistency of representation of the same content  from phonemes and speeches, ensuring the model’s correctly working.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  For TTS, sharing modules with VC enables unlabeled data to participate in its training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  Along with the richer speaker style pattern, the speaker style modeling capability has been enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  For VC, when trained alone, it doesn’t have the speaker modeling ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' As the self-supervised  training process aims at reducing the reconstruction loss of source audio, the reference audio of the  target speaker isn’t crucial in the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Only the content encoder and the decoder are enough for  the process, that’s possibly why the speaker embeddings are all similar, though they are from differ-  ent speakers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' When jointly trained, the text loss plays a role of regularization factor, resulting that the  content encoder just extracting the content information from the source speech (speaker information  is reserved and others are discarded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' This makes the reference speech with rich speaker information  become indispensable for the reconstruction process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Thus the model’s speaker modeling ability has  been improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  5  CONCLUSIONS  In this paper, we propose UnifySpeech, a unified framework for TTS and VC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Both task benefits  from the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' Due to training with large amounts of unlabeled data, their few-shot modeling  ability makes progress as well as the synthesized speech’s quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content=' In the future, further improving  the synthesized speech’s quality and making the generated speech’s style more similar to target  speaker will be our endeavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
+page_content='  REFERENCES  Eric Battenberg, RJ Skerry-Ryan, Soroosh Mariooryad, Daisy Stanton, David Kao, Matt Shannon,  and Tom Bagby.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GNE2T4oBgHgl3EQfTQeG/content/2301.03801v1.pdf'}
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+Non-Lorentzian Kaˇc-Moody Algebras
+Arjun Bagchi,a,b Ritankar Chatterjee,a Rishabh Kaushik,a,c Amartya Saha,a and
+Debmalya Sarkar.a,c
+aIndian Institute of Technology Kanpur, Kanpur 208016, India.
+bCentre de Physique Theorique, Ecole Polytechnique de Paris, 91128 Palaiseau Cedex, France.
+cInternational Centre for Theoretical Sciences (ICTS-TIFR), Bengaluru 560089, India.
+E-mail: (abagchi, ritankar, amartyas)@iit.ac.in,
+rishabh.kaushik@icts.res.in, sarkardebmalya01@gmail.com
+Abstract: We investigate two dimensional (2d) quantum field theories which exhibit Non-
+Lorentzian Kaˇc-Moody (NLKM) algebras as their underlying symmetry. Our investigations
+encompass both 2d Galilean (speed of light c → ∞) and Carrollian (c → 0) CFTs with ad-
+ditional number of infinite non-Abelian currents, stemming from an isomorphism between
+the two algebras. We alternate between an intrinsic and a limiting analysis. Our NLKM
+algebra is constructed first through a contraction and then derived from an intrinsically
+Carrollian perspective. We then go on to use the symmetries to derive a Non-Lorentzian
+(NL) Sugawara construction and ultimately write down the NL equivalent of the Knizhnik
+Zamolodchikov equations. All of these are also derived from contractions, thus providing
+a robust cross-check of our analyses.
+arXiv:2301.04686v1  [hep-th]  11 Jan 2023
+
+Contents
+1
+Introduction
+2
+2
+Non-Lorentzian Kaˇc-Moody algebra in 2d
+5
+2.1
+Carrollian and Galilean CFTs
+5
+2.2
+NL Affine Lie algebras
+7
+3
+An intrinsic Carrollian derivation
+10
+3.1
+An Infinity of Conserved Quantities
+10
+3.2
+Current Ward Identities
+11
+3.3
+Current-primary fields
+13
+3.4
+Current-Current OPEs
+15
+3.5
+Global internal symmetry
+18
+3.6
+EM tensor-current OPEs
+19
+3.7
+The Algebra of Modes
+22
+4
+Sugawara Construction
+24
+4.1
+Intrinsic construction from algebra
+24
+4.2
+Consistency with OPEs
+27
+5
+Tensionless String as NLKM
+29
+6
+Non-Lorentzian Knizhnik Zamolodchikov Equations
+31
+7
+NLKM from Contractions
+32
+7.1
+A brief detour to representations of BMS
+32
+7.2
+Contraction of the affine parameters
+33
+7.3
+Contracting the Sugawara construction
+35
+7.4
+NLKZ equations from Contraction
+36
+8
+Conclusions
+38
+8.1
+Summary
+38
+8.2
+Discussions and future directions
+39
+A Carroll Multiplets
+40
+B Calculation of Sugawara Construction Commutators
+42
+C Modified Sugawara Construction
+46
+D Details of OPE calculations
+47
+E K-Z equation in field theory approach
+49
+– 1 –
+
+F NL KZ equation as a limit
+50
+1
+Introduction
+Relativistic conformal field theory (CFT) is one of the most potent tools of modern theo-
+retical physics, with applications ranging from statistical mechanics of phase transitions to
+quantum gravity through holography and string theory. Especially powerful are methods
+of two dimensional (2d) CFTs [1] where symmetries enhance to two copies of the infinite
+dimensional Virasoro algebra. The ideas and methods of 2d CFTs are of particular impor-
+tance to the success of string theory, where this arises as residual symmetry on the string
+worldsheet after the fixing of conformal gauge [2].
+Non-abelian current algebras arise on the string worldsheet when one considers strings
+moving on arbitrary group manifolds [3], generalising the abelian versions which arise
+for strings propagating on flat backgrounds. These Kaˇc-Moody algebras give rise to the
+worldsheet 2d CFT by the Sugawara construction. The construction of strings on arbitrary
+backgrounds is thus intimately linked to these Kaˇc-Moody algebras.
+Kaˇc-Moody (KM) algebras also arise when we think of 2d CFTs augmented by additional
+symmetries [4]. For example, a CFT with additional U(1) global symmetry arises in a
+number of places, including the study of black holes in AdS3 with charged U(1) hair that
+are solutions to Einstein-Chern Simons theory [5].
+Virasoro with U(1) KM symmetry
+forms the chiral algebra of a large number of theories including N = 2 superconformal field
+theories and theories with W1+∞ symmetry.
+In this paper, we will be interested in the construction of Non-Lorentzian (NL) versions
+of Kaˇc-Moody algebras. Specifically, we are concerned with Galilean and Carrollian CFTs
+in 2d with additional symmetry. Galilean and Carrollian CFTs are obtained from their
+relativistic counterparts by a process of contraction where the speed of light is taken to
+infinity (Galilean theory) or zero (Carrollian theory). In two dimensions, the symmetry
+algebras turn out to be isomorphic and it is in this 2d case we will focus our attention. We
+will obtain the algebras of interest by a contraction and then construct various properties
+of these algebras by methods that have no connections with the limiting procedure and can
+be thought of as completely intrinsic analyses. We also show that suitable singular limits
+also reproduce our intrinsic answers.
+Possible applications
+We have a variety of applications in mind for our algebraic explorations in this work.
+Holography of flat spacetimes
+Following closely related observation in [6], it was shown in [7] that d-dimensional Con-
+formal Carrollian algebras are isomorphic to asymptotic symmetry algebras of (d + 1)
+– 2 –
+
+dimensional flat spacetimes discovered first by Bondi, van der Burg, Metzner [8] and Sachs
+[9] and called BMS algebras. The Carroll CFTs can thus act as putative duals to asypm-
+totically flat spacetimes [6, 10, 11]. Some important evidence for this duality has been
+provided in e.g. [12–18]. Of particular importance is the recent result [19] that links 3d
+Carroll CFTs and scattering amplitudes in 4d asymptotically flat spacetimes.
+Our explorations in this paper are of direct importance in the context of 3d flatspace.
+Here, the U(1) version of NLKM symmetries are of interest for the study of Flat Space
+Cosmological solutions [20] with U(1) hair, which are solutions of Einstein-Chern-Simons
+theory, like the AdS case. This was addressed recently in [21, 22].
+The above approach to holography in asymptotically flat spacetimes goes under the name
+of Carrollian holography. There is an alternate formulation called Celestial holography
+which posits that there is a 2d (relativistic) CFT that computes S-matrix elements in 4d
+asymptotically flat spacetimes. This has been instrumental in the uncovering of many new
+results in asymptotic symmetries and scattering amplitudes in 4d. The interested reader is
+pointed to the excellent reviews [23–25]. For connections between Celestial and Carrollian
+holography, we point to [19, 26, 27].
+Interestingly, of late there have been studies of tree-level massless scattering amplitudes
+which suggest that the asymptotic symmetries are far richer than the extended BMS group.
+In [28, 29], it was shown that there is an SL(2) current algebra at level zero underlying the
+symmetries of tree-level graviton amplitudes. More recently, massless scattering amplitudes
+have revealed an infinite dimensional w1+∞ algebra [30].
+If we assume that the field theory duals with these additional symmetries would be related
+to a co-dimension one holographic description of 4d flatspace, and that these theories should
+live on the whole of the null boundary and not only on the celestial sphere, the structure
+emerging from the above discussions should only be a part of the whole symmetries, very
+much like the relation between the two copies of the Virasoro algebra that make up the
+Celestial CFT and the whole (extended) BMS4 algebra. It is very likely that the algebras
+of interest would then be the 3d versions of the Carrollian Kac-Moody algebras that we
+discuss at length in this work.
+Tensionless strings
+The tensionless limit of strings, which is analogous to the massless limit of point particles,
+is an important sector of string theory that remains relatively less explored. In this limit,
+the string worldsheet becomes null [31, 32] and the 2d relativistic conformal symmetry that
+arises on the tensile worldsheet is replaced by 2d Carrollian Conformal symmetry in the
+tensionless theory [33–35]. The study of tensionless strings on arbitrary group manifolds
+would naturally incorporate the NLKM algebras we study in this paper. We would, in
+future, attempt a construction of a Wess-Zumino-Witten model with the NLKM algebras
+we discuss in this work.
+It is of interest to mention that in [36], it was argued that tensionless strings appeared when
+the level of the affine algebra corresponding to the WZW model of the strings propagating
+in a group manifold. We believe that the intrinsic formulation of such strings should involve
+– 3 –
+
+the NLKM algebras we are studying here. A direction of future work is the connection of
+these two ideas and the aim would be to show that the NLKM algebras appear when the
+level of the relativistic affine algebras are dialled to their critical value.
+Other applications
+There are, of course, other natural applications. The Galilean version of our story is of
+relevance for understanding 2d Galilean CFTs with additional symmetry which may appear
+in real life non-relativistic systems. These would be of interest also in understanding non-
+relativistic strings in curved Newton-Cartan backgrounds [37].
+Interesting enough, Carrollian structures also arise condensed matter systems, e.g. in the
+physics of flat bands that is of relevance in the context of “magic” superconductivity in
+bi-layer graphene and also in fractional quantum Hall systems [38]. It is easy to envision
+condensed matter systems with additional symmetry and hence our methods in this paper
+which lay the foundation for systems with Carrollian (and Galilean) affine Lie algebras,
+should have applicability in a wide number of condensed matter systems. Finally, a U(1)
+affine algebra was also found to emerge in studies of the BMS scalar field theory in 2d [39].
+It would be of interest to figure out if this is a feature of all free BMS field theories. Our
+methods outlined in this paper should then be useful even for free BMS field theories.
+Outline of the paper
+In section 2, we will start with a brief review of Carrollian and Galilean CFTs in gen-
+eral dimensions, subsequently specializing the discussion to two dimensions followed by
+a brief introduction to Non Lorentzian Kac Moody (NLKM) algebras via contraction of
+relativistic Affine Lie Algebras. In section 3, we present an intrinsic carrollian derivation
+to the NLKM Algebra. Then, we formulate the non-Lorentzian version of the Sugawara
+construction for these algebras in section 4 and verify its validity by showing its consis-
+tency with the OPEs we obtained in section 3. In section 5, we work out the example of
+tensionless strings on a flat background geometry which exhibits the U(1) NLKM algebra
+as the current algebra. Section 6 contains the derivation of the Non-Lorentzian analog
+of Knizhnik Zamolodchikov(KZ) equations using the OPE definition of the primaries in
+section 3. And finally in section 7, we derive NLKM Algebra, Sugawara Construction and
+the Non Lorentzian KZ equations through contractions in detail. There are six appendices
+which collect the details of various calculations that have been skipped in the main text
+for the ease of readability.
+– 4 –
+
+2
+Non-Lorentzian Kaˇc-Moody algebra in 2d
+In this section, we will begin our analysis by building the algebra we wish to study in the
+remainder of the paper. The NLKM algebra will be defined as a limit from the relativistic
+Virasoro KM algebra. We will see later how the current part of the algebra can be used
+to generate the entire algebra by a NL version of the Suwagara construction. We begin by
+reminding the reader of the Galilean (c → ∞) and the Carrollian (c → 0) contractions of
+relativistic CFTs in generic dimensions and the fact that this leads to isomorphic algebras
+in d = 2
+2.1
+Carrollian and Galilean CFTs
+The Galilean (c → ∞) and Carrollian (c → 0) limits of the Poincare algebra leads to two
+different contractions of the parent relativistic algebra and two different algebras in the
+limit in general dimensions. The Poincare algebra in D dimension ISO(D − 1, 1) is given
+by
+[Pµ, Pν] = 0,
+[Mµν, Pρ] = −2ηρ[µPν],
+[Mµν, Mρσ] = 4η[µρMσ]ν],
+(2.1)
+where µ = 0, 1, 2..., D − 1, Pµ = −∂µ are translation generators and Mµν = xµ∂ν − xν∂µ
+are Lorentz generators. Galilean limit is achieved by taking c → ∞ limit, alternatively by
+taking the contraction t → t, xi → ϵxi, ϵ → 0 limit where i ∈ {1, 2, ..., D − 1} [40, 41].
+Under this limit we see that
+M0i = t∂i + xi∂t → 1
+ϵ t∂i + ϵxi∂t
+=⇒ Bi = lim
+ϵ→0 ϵM0i = t∂i.
+(2.2)
+The spatial rotation generators Mij remains same under this contraction.
+Doing this
+contraction, we end up with the Galilei algebra, where all the non-zero commutators are
+given by
+[Mij, Mkl] = 4δi[kMl]j], [Mij, Pk] = −2δk[iPj], [Mij, Bk] = 2δk[iBj], [Bi, H] = −Pi.
+(2.3)
+Carrollian limit is achieved by taking c → 0 limit, alternatively by taking the contraction
+t → ϵt, xi → xi, ϵ → 0. Under this limit, we have
+M0i = ϵt∂i + xi∂t → ϵt∂i + 1
+ϵ xi∂t
+=⇒ Bi = lim
+ϵ→0 ϵM0i = xi∂t,
+(2.4)
+with Mij intact again. This gives us the Carroll algebra where the non-zero commutators
+are given by
+[Mij, Mkl] = 4δi[kMl]j], [Mij, Pk] = −2δk[iPj], [Mij, Bk] = −2δk[iBj] [Pi, Bj] = δijH, (2.5)
+where H = −∂t is the Hamiltonian which has now become a central element.
+The tale of the two contractions is also true for the relativistic conformal algebra.
+In
+relativistic Conformal symmetry group there are two additional generators
+D = −xµ∂µ
+Kµ = −(2xµxν∂ν − x2∂µ),
+(2.6)
+– 5 –
+
+giving us the following additional commutators along with (2.1)
+[D, Pµ] = Pµ, [D, Kµ] = −Kµ, [Kµ, Pν] = 2(ηµνD − Mµν), [Kρ, Lµν] = 2ηρ[µKν].
+(2.7)
+Taking the non-relativistic limit this time, we will end up with the following additional
+generators along with the generators of Galilei algebra
+D = −(xi∂i + t∂t)
+K = K0 = −(2txi∂i + t2∂t)
+Ki = t2∂i.
+(2.8)
+These generators, along with the generators of the Galilei algebra gives us the Galilean
+Conformal Algebra (GCA) in D dimensions [40] where non-zero commutators apart from
+(2.3) are given by
+[K, Bi] = Ki,
+[K, Pi] = 2Bi
+[Mij, Kr] = −2K[iδj]r,
+[H, Ki] = −2Bi
+[D, Ki] = −2Ki,
+[D, Pi] = Pi
+[D, H] = H,
+[H, K] = −2D,
+[D, K] = −K.
+(2.9)
+When we take the Carrollian limit of the relativistic Conformal algebra, we get the following
+new generators apart form the Carroll generators previously encountered:
+D = −(xi∂i + t∂t)
+K = K0 = −xixi∂t
+Kj = −2xj(xi∂i + t∂t) + xixi∂j.
+(2.10)
+These additional generators give us the following non-vanishing commutators along with
+those given in (2.5)
+[Mij, Kk] = δk[jKi]
+[Bi, Kj] = δijK,
+[D, K] = −K
+[K, Pi] = −2Bi
+[Ki, Pj] = −2Dδij − 2Mij,
+[H, Ki] = 2Bi
+[D, H] = −H,
+[D, Pi] = −Pi
+[D, Ki] = Ki.
+(2.11)
+(2.5) and (2.11) together form Carrollian Conformal Algebra (CCA). We see that in general
+dimensions the two different contractions give us different algebras that are not isomorphic
+to each other.
+Let us now move to the interesting case of d = 2. In 2d, the relativistic conformal algebra
+becomes infinite dimensional and is given by two copies of the Virasoro algebra:
+[Lm, Ln] = (m − n)Lm+n + c
+12(m3 − m)δm+n,0,
+[ ¯Lm, ¯Ln] = (m − n) ¯Lm+n + ¯c
+12(m3 − m)δm+n,0,
+(2.12)
+[Lm, ¯Ln] = 0.
+Here c, ¯c are central charges of the Virasoro algebra (not to be confused with the speed
+of light which we also had called c earlier). The Galilean contraction [42] of the above is
+given by
+Ln = Ln + ¯Ln,
+Mn = ϵ( ¯Ln − Ln),
+(2.13)
+– 6 –
+
+This contraction of the Virasoro algebra leads to
+[Lm, Ln] = (m − n)Lm+n + cL
+12(m3 − m)δn+m,0,
+(2.14a)
+[Lm, Mn] = (m − n)Mm+n + cM
+12 (m3 − m)δn+m,0,
+(2.14b)
+[Mm, Mn] = 0.
+(2.14c)
+The way to see that this combination yields the non-relativistic limit, it is instructive to
+write the generators of the Virasoro algebra in cylindrical cooridinates
+Ln = einω∂ω,
+¯Ln = ein¯ω∂¯ω,
+ω, ¯ω = τ ± σ,
+(2.15)
+The limit (2.13) then translates to
+σ → ϵσ, τ → τ, ϵ → 0.
+(2.16)
+which essentially means scaling velocities to be very small compared to 1 and since we are
+doing all of this in units of speed of light c = 1, this is indeed the non-relativistic limit
+c → ∞.
+On the other hand, the Carrollian contraction of the Virasoro algebra is given by
+Ln = Ln − ¯L−n,
+Mn = ϵ(Ln − ¯L−n).
+(2.17)
+Again if we go back to the cylindrical coordinates, this limit translates to
+σ → σ, τ → ϵτ, ϵ → 0.
+(2.18)
+The velocities are very large compared to 1 now and this means v/c → ∞, which equiv-
+alently translates to c → 0. This is thus the Carrollian limit. The surprising thing is
+that even this contraction yields the same algebra (2.14). In order to avoid confusion with
+Galilean or Carrollian notions, we will exclusively call this the BMS3 algebra. Galilei and
+Carroll contractions in d = 2 yield isomorphic algebras and this is down to the fact that
+there is only one contracted and one uncontracted direction in each case. The algebra does
+not differentiate between a contracted spatial and a contracted temporal direction.
+2.2
+NL Affine Lie algebras
+We start with the two copies of Virasoro Kac-Moody algebra whose holomorphic part is,
+[Lm, Ln] = (m − n)Lm+n + c
+12(m3 − m)δm+n,0,
+[Lm, ja
+n] = −nja
+m+n
+[ja
+m, jb
+n] = i
+dim(g)
+�
+c=1
+fabcjc
+m+n + mkδm+n,0δab.
+(2.19)
+There is an equivalent anti-holomorphic part with ¯fabc, ¯c and ¯k in place of fabc, c and k
+respectively which are not necessarily equal to their holomorphic counterparts. We are
+take fabc ̸= ¯fabc for generality, but the dimensions of the two Lie groups are the same.
+– 7 –
+
+We will take a contraction of the algebra as follows. We will work with the following linear
+combinations of the relativistic KM generators:
+Ln = Ln + ¯Ln,
+Mn = ϵ( ¯Ln − Ln),
+Ja
+m = ja
+m + ¯ja
+m,
+Ka
+m = ϵ(¯ja
+m − ja
+m).
+(2.20)
+We will the consider the limit ϵ → 0. The contracted algebra is given by:
+[Lm, Ln] = (m − n)Lm+n + cL
+12(m3 − m)δn+m,0,
+(2.21a)
+[Lm, Mn] = (m − n)Mm+n + cM
+12 (m3 − m)δn+m,0,
+(2.21b)
+[Lm, Ja
+n] = −nJa
+m+n , [Lm, Ka
+n] = −nKa
+m+n , [Mm, Ja
+n] = −nKa
+m+n
+(2.21c)
+[Ja
+m, Jb
+n] = i
+dim(g)
+�
+c=1
+F abcJc
+m+n + i
+dim(g)
+�
+c=1
+GabcKc
+m+n + mk1δabδm+n,0,
+(2.21d)
+[Ja
+m, Kb
+n] = i
+dim(g)
+�
+c=1
+F abcKc
+m+n + mk2δabδm+n,0,
+(2.21e)
+with rest of the commutators vanishing. We recognise the first two lines as the familiar
+BMS3 algebra, equivalently the 2d Galilean or 2d Carrollian Conformal Algebra. Through-
+out the paper, we will call this sub-algebra the BMS algebra. In the above, the structure
+constants are related to their relativistic counterparts by
+F abc = 1
+2
+�
+fabc + ¯fabc�
+, Gabc = 1
+2ϵ
+�
+¯fabc − fabc�
+.
+(2.22)
+while the central terms are given by
+cL = c + ¯c,
+cM = ϵ(¯c − c), k1 = ¯k + k, k2 = ϵ(¯k − k).
+(2.23)
+We will call the algebra (2.21) the 2d Non-Lorentzian Kaˇc-Moody algebra. We can also carry
+out the contraction ultrarelativistically for which we take the following linear combinations
+of generators,
+Ln = Ln − ¯L−n,
+Mn = ϵ(Ln + ¯L−n),
+Ja
+n = (ja
+n + ¯ja
+−n),
+Ka
+n = ϵ(ja
+n − ¯ja
+−n).
+(2.24)
+Contracted algebra will be same as (2.21) with relations analogous to (2.22) and(2.23)
+taking the following form:
+F abc = 1
+2
+�
+fabc + ¯fabc
+�
+, Gabc = 1
+2ϵ
+�
+fabc − ¯fabc
+�
+(2.25)
+cL = c − ¯c,
+cM = ϵ(c + ¯c),
+k1 = k − ¯k,
+k2 = ϵ(k + ¯k).
+(2.26)
+This (2.21) will be the algebra of interest for the rest of our paper.
+Note that if we start with 2 identical Kac-Moody algebras for the holomorphic and anti-
+holomorphic sections, then after contraction we get
+fabc = ¯fabc ⇒ F abc = fabc, Gabc = 0.
+(2.27)
+– 8 –
+
+For most of our analysis, we will use this simplified algebra, but the results can be easily
+generalised for the general algebra.
+We should also clarify something about the new Lie algebra structure and our notation.
+We started with (the affine version of) two copies of a Lie algebra g of dimension
+n = dim(g),
+constructed from the generators {ja, a ∈ {1, 2, . . . , n}} and {¯ja, a ∈ {1, 2, . . . , n}}. Via
+contraction we obtain a (non-semisimple) Lie algebra ˜g of dimension
+˜n = dim(˜g) = 2 dim(g)
+consisting of generators {Ja, Ka, a ∈ {1, 2, . . . , n}}.
+So in all our notation, the indices
+a, b, c run from 1 to n = dim(g) (not ˜n), and whenever we write dim(g) (like in expressions
+of central charge cL later in the paper), we mean the dimension of the parent algebra g,
+which is actually half of the dimension of the new algebra ˜g. This is done for the tidiness
+of expressions.
+Before we conclude this section, we would like to comment on the choice of contraction of
+the currents to get to the NLKM algebra. Note that the chosen linear combinations of the
+Kac Moody generators are motivated by the Galilean/Carrollian contractions analogous to
+(2.16) and (2.18). The generators in cylindrical coordinates look like,
+ja
+n = ja ⊗ einω,
+¯ja
+n = ¯ja ⊗ ein¯ω,
+ω, ¯ω = τ ± σ
+(2.28)
+Galilean limit will correspond to (2.16) and Carrollian limit corresponds to (2.18).
+In
+Galilean case, we have (upto linear order in ϵ),
+ja
+n + ¯ja
+n = (ja + ¯ja) ⊗ einτ + inσϵ(ja − ¯ja) ⊗ einτ,
+(2.29a)
+ja
+n − ¯ja
+n = (ja − ¯ja) ⊗ einτ + inσϵ(ja + ¯ja) ⊗ einτ.
+(2.29b)
+We now have two choices, keeping the BMS contraction the same:
+• Ja = ja + ¯ja, Ka = ϵ(¯ja − ja) ⇒ (2.20).
+• Ja = ¯ja − ja, Ka = ϵ(¯ja + ja) ⇒ Ja
+m = ¯ja
+m − ja
+m, Ka
+m = ϵ(¯ja
+m + ja
+m).
+Both of these choices lead to relations (2.23) but (2.22) need to be altered for the second
+choice to:
+F abc = 1
+2( ¯fabc − fabc), Gabc = 1
+2ϵ(fabc + ¯fabc).
+(2.30)
+Similarly, in Carrollian case, we have two choices: (2.24) and
+Ja
+n = ja
+n − ja
+−n,
+Ka
+n = ϵ(ja
+n + ja
+−n).
+in which case (2.26) remains same but we have
+F abc = 1
+2(fabc − ¯fabc),
+Gabc = 1
+2ϵ(fabc + ¯fabc)
+(2.31)
+– 9 –
+
+instead of (2.25). Since we will be considering the special case when fabc = ¯fabc, we will
+stick to the first choice in this paper. For the intrinsic analysis, which only uses the algebra
+(2.21), of course these choices do not matter. But when we derive results from the limit,
+in Sec. 7, it is important to state everything works for the other contraction as well. E.g.
+Sugawara construction through contraction (Sec 7.3) can be carried out for the second
+choice if we specialize to the case ¯fabc = −fabc in case of Galilean contraction.
+3
+An intrinsic Carrollian derivation
+Having derived the non-Lorentzian current algebra through a contraction, we now go on to
+present an intrinsically Carrollian derivation of the same, where we would not be alluding
+to a limiting procedure at all. This section heavily borrows from the machinery detailed in
+[43], some of the important features of which are described in Appendix A. Although we
+will try and be self consistent so that the section (with the help of the related appendix A)
+stands on its own, for any details that we may have inadvertently skipped in what follows,
+the reader is referred back to [43].
+3.1
+An Infinity of Conserved Quantities
+A 2D Carrollian CFT on the flat Carrollian background (t, x) is invariant under an infinite
+number of 2D Carrollian conformal (CC) transformations whose infinitesimal versions are
+given as:
+x′ = x + ϵxf(x) ,
+t′ = t + ϵxtf′(x) + ϵtg(x)
+(3.1)
+As a consequence, the EM tensor components classically satisfy the following conditions
+[43]:
+∂µT µ
+ν = 0 ;
+T x
+t = 0 ;
+T µ
+µ = 0
+=⇒ ∂tT t
+t = 0 ;
+∂tT t
+x = ∂xT t
+t.
+(3.2)
+This allows for an infinite number of Noether currents. The two conserved currents corre-
+sponding to the symmetry transformation (3.1) are noted below:
+jµ
+t =
+�
+jt
+t , jx
+t
+�
+=
+�
+g(x)T t
+t , 0
+�
+,
+jµ
+x =
+�
+jt
+x , jx
+x
+�
+=
+�
+f(x)T t
+x + tf′(x)T t
+t , −f(x)T t
+t
+�
+.
+(3.3)
+Now, we suppose that there are some other pairs of fields {J a
+x , J a
+t } in the theory that obey
+the following conditions analogous to (3.2):
+∂tJ a
+t (t, x) = 0 ;
+∂tJ a
+x (t, x) = ∂xJ a
+t (t, x)
+(3.4)
+where a is to be thought of as a ‘flavor’ index (but t and x in subscript are not tensor
+indices). Using these fields, we can construct an infinite number of conserved quantities:
+kaµ =
+�
+kat , kax�
+= (g(x)J a
+t , 0) ,
+(3.5a)
+jaµ =
+�
+jat , jax�
+=
+�
+f(x)J a
+x + tf′(x)J a
+t , −f(x)J a
+t
+�
+.
+(3.5b)
+We shall regard these conserved quantities as the Noether currents associated to some
+‘internal’ symmetries of the action.
+– 10 –
+
+3.2
+Current Ward Identities
+We now consider an infinitesimal internal transformation of a (possibly multi-component)
+field Φ(t, x):
+Φ(t, x) → ˜Φ(t, x) = Φ(t, x) + ϵa (Fa · Φ) (t, x)
+(3.6)
+where Fa · Φ denotes the functional changes of the (multi-component) field Φ under in-
+finitesimal transformations labeled by ϵa. So, the generator GaΦ of this transformation is
+given by:
+−iϵaGaΦ(t, x) := ˜Φ(t, x) − Φ(t, x) = ϵa (Fa · Φ) (t, x)
+(3.7)
+We shall now find the Ward identity corresponding to this internal transformation which
+is assumed to be a symmetry of the 2D Carrollian CFT. For this purpose, we analytically
+continue the real space variable x ∈ R∪{∞} to the complex plane; thus, the Ward identity
+reads [43]:
+∂µ⟨jµ
+a(t, x)X⟩ ∼ −i
+n
+�
+i=1
+δ(t − ti)
+�
+�−(Fa)i · ⟨X⟩
+x − xi
++
+�
+k≥2
+⟨Y (k)
+a
+⟩i(x1, ...xn)
+(x − xi)k
+�
+�
+(3.8)
+where the yet unknown correlators ⟨Y (k)
+a
+⟩i depend on the transformation properties of the
+fields in the string-of-fields X and the transformation itself and ∼ denotes ‘modulo terms
+holomorphic in x inside [. . .]’. We also use the shorthand x1 = (t1, x1). All the correlators
+are time-ordered.
+Let us now assume that the symmetry transformation (3.6) is associated to the following
+conserved current operator:
+kaµ = (J a
+t , 0) .
+(3.9)
+The corresponding Ward identity then is:
+∂t⟨J a
+t (t, x)X⟩ ∼ −i
+n
+�
+i=1
+δ(t − ti)
+�
+�−(Fa)i · ⟨X⟩
+x − xi
++
+�
+k≥2
+⟨Y (k)
+a
+⟩i(x1, ...xn)
+(x − xi)k
+�
+�
+⇒ ⟨J a
+t (t, x)X⟩ = −i
+n
+�
+i=1
+θ(t − ti)
+�
+�−(Fa)i · ⟨X⟩
+x − xi
++
+�
+k≥2
+⟨Y (k)
+a
+⟩i(x1, ...xn)
+(x − xi)k
+�
+�
+(3.10)
+where, following [43], the initial condition has been taken to be:
+lim
+t→−∞⟨J a
+t (t, x)X⟩ = 0
+(3.11)
+and as in 2D relativistic CFT [4], ⟨J a
+t (t, x)X⟩ is assumed to be finite whenever x ̸= {xi};
+this condition makes the holomorphic terms inside [. . .] vanish in this Ward identity.
+– 11 –
+
+Similarly, if the conserved current:
+jaµ = (J a
+x , −J a
+t )
+(3.12)
+is associated to another internal symmetry transformation:
+Φ(t, x) → ˜Φ(t, x) = Φ(t, x) + ϵa (Ga · Φ) (t, x)
+(3.13)
+the corresponding Ward identity is:
+⟨(∂tJ a
+x (t, x) − ∂xJ a
+t (t, x)) X⟩ ∼ −i
+n
+�
+i=1
+δ(t − ti)
+�
+�−(Ga)i · ⟨X⟩
+x − xi
++
+�
+k≥2
+⟨Y (k)
+a
+⟩i(x1, ...xn)
+(x − xi)k
+�
+�
+(3.14)
+Assuming the initial condition:
+lim
+t→−∞⟨J a
+x (t, x)X⟩ = 0
+(3.15)
+and the finite-ness property of ⟨J a
+x (t, x)X⟩ for x ̸= {xi}, this Ward identity together with
+(3.10) lead us to:
+⟨J a
+x (t, x)X⟩ = −i
+n
+�
+i=1
+θ(t − ti)
+�
+�−(Ga)i · ⟨X⟩
+x − xi
++
+�
+k≥2
+⟨Z(k)
+a ⟩i(x1, ...xn)
+(x − xi)k
+−(t − ti)
+�
+�−(Fa)i · ⟨X⟩
+(x − xi)2
++
+�
+k≥2
+k⟨Y (k)
+a
+⟩i(x1, ...xn)
+(x − xi)k+1
+�
+�
+�
+�
+(3.16)
+Again, the correlators ⟨Z(k)
+a ⟩i can not be determined without knowing the explicit internal
+transformation properties of the fields in X.
+For future references, we note the iϵ-form [43] of the Ward identities (3.10) and (3.16)
+below with ∆˜xp := x − xp − iϵ(t − tp) 1 :
+i⟨J a
+t (t, x)X⟩ = lim
+ϵ→0+
+n
+�
+i=1
+�
+�−(Fa)i · ⟨X⟩
+∆˜xi
++
+�
+k≥2
+⟨Y (k)
+a
+⟩i
+(∆˜xi)k
+�
+�
+(3.17)
+i⟨J a
+x (t, x)X⟩ = lim
+ϵ→0+
+n
+�
+i=1
+�
+�−(Ga)i · ⟨X⟩
+∆˜xi
++
+�
+k≥2
+⟨Z(k)
+a ⟩i
+(∆˜xi)k
+−(t − ti)
+�
+�−(Fa)i · ⟨X⟩
+(∆˜xi)2
++
+�
+k≥2
+k ⟨Y (k)
+a
+⟩i
+(∆˜xi)k+1
+�
+�
+�
+� (3.18)
+1We hope the reader does not confuse the delta appearing in ∆˜xp with the conformal weight ∆. The ∆
+appearing in the difference of coordinates would always appear with a coordinate.
+– 12 –
+
+Thus, a general (possibly multi-component) 2D Carrollian conformal field Φ(t, x) has the
+following OPEs (in the iϵ-form) with the current-vector components (∆˜x′ := x′−x−iϵ(t′−
+t)):
+iJ a
+t (t′, x′)Φ(t, x) ∼ lim
+ϵ→0+
+�
+. . . + −(Fa · Φ)(t, x)
+∆˜x′
+�
+(3.19)
+iJ a
+x (t′, x′)Φ(t, x) ∼ lim
+ϵ→0+
+�
+. . . + −(Ga · Φ)(t, x)
+∆˜x′
+− (t′ − t)
+�
+. . . + −(Fa · Φ)(t, x)
+(∆˜x′)2
+��
+(3.20)
+where . . . represents higher order poles at x′ = x and ∼ denotes ‘modulo terms holomorphic
+in x′ that have vanishing VEVs’.
+In this work, we shall only consider currents with scaling dimension ∆ = 1. Comparing
+the behavior of both sides of the OPEs (3.19) and (3.20) under dilatation, we then infer
+that the scaling dimensions of the local fields Fa · Φ and Ga · Φ must be same as that of Φ.
+Since this is true for any arbitrary local field Φ, we conclude that Fa · Φ and Ga · Φ must
+be linear combinations of the components of the multi-component field Φ that ‘internally’
+transforms under a bi-matrix representation of the global current symmetry algebra; all of
+these components must have an equal scaling dimension. Our goal is to find this global
+algebra and its infinite extension.
+We now express Fa · Φ and Ga · Φ as explicit linear combinations:
+(Fa · Φ)ii′ = (ta
+K)i
+j Φji′ ;
+(Ga · Φ)ii′ = Φij′ (ta
+J) i′
+j′
+(3.21)
+where ta
+J and ta
+K are just two matrices as of now.
+Later, we shall relate them to the
+generators of the internal symmetry algebra.
+3.3
+Current-primary fields
+In the operator formalism of a QFT, the conserved charge QA is the generator of an
+infinitesimal symmetry transformation on the space of the quantum fields:
+˜Φ(x) − Φ(x) = −iϵA[QA , Φ(x)]
+(3.22)
+In 2D Carroll CFT, the above generator equation for any conserved charge operator QA is
+related to the following contour integral prescription involving an OPE [43]2:
+QA =
+1
+2πi
+�
+Cu
+dx jt
+A(t, x)
+generates
+[QA , Φ(t, x)] =
+1
+2πi
+�
+x
+dx′ jt
+A(t+, x′)Φ(t, x)
+(3.23)
+where t+ > t and the counter-clockwise contour Cu encloses the upper half-plane along
+with the real line. The contour around x must not enclose any possible singularities of the
+vector field.
+2Section 5 of this reference contains a derivation.
+– 13 –
+
+The conserved quantum charges Qa
+t [g] and Qa
+x[f] of the respective currents (3.5a) and
+(3.5b) are thus given by:
+Qa
+t [g] =
+1
+2πi
+�
+Cu
+dx g(x)J a
+t (t, x)
+(3.24)
+Qa
+x[f] =
+1
+2πi
+�
+Cu
+dx
+�
+f(x)J a
+x (t, x) + tf′(x)J a
+t (t, x)
+�
+(3.25)
+The conserved charges of all flavors collectively induce the following infinitesimal changes
+to a generic quantum field, as deduced from the OPEs (3.19) and (3.20):
+−i
+�
+a
+ϵa [Qa
+t [ga] , Φ(x)] = − 1
+2π
+�
+a
+ϵa
+�
+x
+dx′ ga(x′)J a
+t (t+, x′)Φ(t, x)
+=
+�
+a
+ϵa [ga(x) (ta
+K · Φ) (t, x) + (h.d.t.)]
+(3.26)
+−i
+�
+a
+ϵa [Qa
+x[fa] , Φ(x)] = − 1
+2π
+�
+a
+ϵa
+�
+x
+dx′ �
+fa(x′)J a
+x (t+, x′) + t+fa′(x′)J a
+t (t+, x′)
+�
+Φ(t, x)
+=
+�
+a
+ϵa �
+fa(x) (Φ · ta
+J) (t, x) + tfa′(x) (ta
+K · Φ) (t, x) + (h.d.t.)
+�
+(3.27)
+where h.d.t. denotes terms necessarily containing derivatives (of order at least 1) of ga(x)
+and fa(x).
+For the currents (3.9) and (3.12), we simply have f(x) = 1 = g(x). For any arbitrary field
+Φ(t, x) this immediately leads to:
+− i
+�
+a
+ϵa [Qa
+t [1] , Φ(x)] =
+�
+a
+ϵa (ta
+K · Φ) (t, x)
+(3.28a)
+− i
+�
+a
+ϵa [Qa
+x[1] , Φ(x)] =
+�
+a
+ϵa (Φ · ta
+J) (t, x)
+(3.28b)
+Thus, the finite internal transformation that is generated by the charges {Qa
+k[1]} is:
+Φ(t, x) → ˜Φ(t, x) = e−i �
+a ϵaQa
+t [1]Φ(t, x)ei �
+a ϵaQa
+t [1] = e
+�
+a ϵata
+K · Φ(t, x)
+(3.29)
+which is obtained by using the BCH lemma. Similarly, the charges {Qa
+x[1]} generate the
+following finite transformation:
+Φ(t, x) → ˜Φ(t, x) = Φ(t, x) · e
+�
+a ϵata
+J
+(3.30)
+On the other hand, as derived from (3.26), an arbitrary field finitely transforms under the
+action of the charges {Qa
+t [ga]} as:
+Φ(t, x) → ˜Φ(t, x) = e
+�
+a ϵaga(x)ta
+K · Φ(t, x) + extra terms
+(3.31)
+– 14 –
+
+while (3.27) leads to the following finite action of the charges {Qa
+x[fa]}:
+Φ(t, x) → ˜Φ(t, x) = e
+�
+a ϵatfa′(x)ta
+K · Φ(t, x) · e
+�
+a ϵafa(x)ta
+J + extra terms
+(3.32)
+In view of the above discussion, we emphasize that while a generic field transforms co-
+variantly under the action of the charges associated to the conserved currents (3.9) and
+(3.12), that is not the case for the generic conserved currents (3.5a) and (3.5b). A field
+that transforms covariantly (i.e. for which the extra terms in (3.31) and (3.32) vanish)
+even under the action of the charges of any currents of the form (3.5a) and (3.5b) is called
+a current-primary field. Consequently, there is no h.d.t. in (3.26) and (3.27) appropriate
+for a current-primary field. This enables us to completely specify the pole structures of the
+current-primary OPEs for a primary field Φ(t, x):
+J a
+t (t′, x′)Φ(t, x) ∼ lim
+ϵ→0+ ita
+K · Φ(t, x)
+∆˜x′
+(3.33)
+J a
+x (t′, x′)Φ(t, x) ∼ lim
+ϵ→0+ i
+�Φ(t, x) · ta
+J
+∆˜x′
+− (t′ − t)ta
+K · Φ(t, x)
+(∆˜x′)2
+�
+(3.34)
+that immediately imply the following Ward identities for a string X of primary fields:
+⟨J a
+t (t, x)X⟩ = lim
+ϵ→0+ i
+n
+�
+i=1
+(ta
+K)i · ⟨X⟩
+∆˜xi
+(3.35)
+⟨J a
+x (t, x)X⟩ = lim
+ϵ→0+ i
+n
+�
+i=1
+�⟨X⟩ · (ta
+J)i
+∆˜xi
+− (t − ti)(ta
+K)i · ⟨X⟩
+(∆˜xi)2
+�
+(3.36)
+where (ta
+J)i and (ta
+K)i denotes transformation-matrices appropriate for the i-th primary
+field in X.
+3.4
+Current-Current OPEs
+To derive the current-current OPEs using the machinery just developed, we shall assume
+that no field in the theory has negative scaling dimension with the identity being the only
+field with ∆ = 0.
+Under these assumptions, the OPEs between the EM tensor components were derived using
+only symmetry arguments in [43]; the results are:
+T t
+t(t′, x′)T t
+t(t, x) ∼ 0
+T t
+x(t′, x′)T t
+t(t, x) ∼ lim
+ϵ→0+ −i
+� −i cM
+2
+(∆˜x′)4 + 2T t
+t(t, x)
+(∆˜x′)2
++ ∂xT t
+t(t, x)
+∆˜x′
+�
+T t
+t(t′, x′)T t
+x(t, x) ∼ lim
+ϵ→0+ −i
+� −i cM
+2
+(∆˜x′)4 + 2T t
+t(t, x)
+(∆˜x′)2
++ ∂tT t
+x(t, x)
+∆˜x′
+�
+(3.37)
+T t
+x(t′, x′)T t
+x(t, x) ∼ lim
+ϵ→0+ −i
+� −i cL
+2
+(∆˜x′)4 + 2T t
+x(t, x)
+(∆˜x′)2
++ ∂xT t
+x(t, x)
+∆˜x′
+−(t′ − t)
+�−2icM
+(∆˜x′)5 + 4T t
+t(t, x)
+(∆˜x′)3
++ ∂tT t
+x(t, x)
+(∆˜x′)2
+��
+.
+– 15 –
+
+The constants cL and cM are the central charges of the 2D Carrollian conformal QFT.
+We shall use the same technique to find the current-current OPEs below. Keeping in mind
+that here we are dealing with currents with scaling dimension ∆ = 1, below we note the
+most general allowed form of the Jt − Jt OPE compatible with the general OPE (3.19):
+J a
+t (t′, x′)J b
+t (t, x) ∼ lim
+ϵ→0+ i
+�
+Aab
+(∆˜x′)2 + (ta
+K · Jt)b (t, x)
+∆˜x′
+�
+(3.38)
+where Aab is a field proportional to identity so that it has vanishing scaling dimension.
+Clearly, (ta
+K · Jt)b (t, x) must have scaling dimension ∆ = 1. So, in a generic 2D CCFT,
+(ta
+K · Jt)b must be a linear combination of {J a
+x , J a
+t }.
+Correspondingly to the classical conservation equation ∂tJ a
+t (t, x) = 0 , in the QFT we
+should have:
+J a
+t (t′, x′)∂tJ b
+t (t, x) ∼ 0
+⇒
+∂t (ta
+K · Jt)b (t, x) = 0
+(3.39)
+which means that {J a
+x } can not contribute to the linear combination (ta
+K · Jt)b.
+Since the currents have scaling dimension ∆ = 1, they must satisfy the bosonic3 exchange
+property. For the J a
+t (t, x) field, it is:
+J a
+t (t′, x′)J b
+t (t, x) = J b
+t (t, x)J a
+t (t′, x′)
+(3.40)
+This condition implies the following restrictions:
+Aab = Aba
+and
+(ta
+K · Jt)b = −
+�
+tb
+K · Jt
+�a
+(3.41)
+Looking at (3.20), we write the allowed form of a Jx − Jt OPE:
+J a
+x (t′, x′)J b
+t (t, x) ∼ lim
+ϵ→0+ i
+�
+Bab
+(∆˜x′)2 + (Jt · ta
+J)b (t, x)
+∆˜x′
+− (t′ − t)
+�
+2Aab
+(∆˜x′)3 + (ta
+K · Jt)b (t, x)
+(∆˜x′)2
+��
+(3.42)
+where Bab are some constants. Now, we have the following restrictions:
+J a
+x (t′, x′)∂tJ b
+t (t, x) ∼ 0
+=⇒
+Aab = 0
+and
+(ta
+K · Jt)b = 0
+and
+∂t (Jt · ta
+J)b (t, x) = 0
+(3.43)
+Thus, again {J a
+x } do not contribute to the linear combination (Jt · ta
+J)b.
+On the other hand, from (3.19), we get the following Jt − Jx OPE:
+J b
+t (t′, x′)J a
+x (t, x) ∼ lim
+ϵ→0+ i
+�
+Cba
+(∆˜x′)2 +
+�
+tb
+K · Jx
+�a (t, x)
+∆˜x′
+�
+(3.44)
+3Since, as will be shown later, the currents are 2D CC primary fields with integer scaling dimension,
+this statement is justified [43].
+– 16 –
+
+with Cab being constants. Using the following bosonic exchange property:
+J b
+t (t′, x′)J a
+x (t, x) = J a
+x (t, x)J b
+t (t′, x′)
+to compare the OPE (3.42) with (3.44), we get the following conditions:
+Bab = Cba
+and
+(Jt · ta
+J)b = −
+�
+tb
+K · Jx
+�a
+(3.45)
+which implies that {J a
+x } do not appear also in the linear combination
+�
+tb
+K · Jx
+�a.
+Finally, we write the Jx − Jx OPE in accordance with the general form (3.20):
+J a
+x (t′, x′)J b
+x(t, x) ∼ lim
+ϵ→0+ i
+�
+Dab
+(∆˜x′)2 + (Jx · ta
+J)b (t, x)
+∆˜x′
+− (t′ − t)
+�
+2Cab
+(∆˜x′)3 + (ta
+K · Jx)b (t, x)
+(∆˜x′)2
+��
+(3.46)
+from which, the bosonic exchange property:
+J b
+x(t′, x′)J a
+x (t, x) = J a
+x (t, x)J b
+x(t′, x′)
+leads to the following conditions:
+Dab = Dba ;
+Cab = Cba
+(3.47)
+(Jx · ta
+J)b = −
+�
+Jx · tb
+J
+�a
+;
+(ta
+K · Jx)b = −
+�
+tb
+K · Jx
+�a
+;
+∂t (Jx · ta
+J)b = ∂x (ta
+K · Jx)b
+(3.48)
+It can be readily checked that these conditions are compatible with the quantum versions
+(in the OPE language) of the classical conservation laws (3.4).
+We now explicitly write the allowed forms of the linear combinations appearing in the
+above OPEs:
+(Jt · ta
+J)b = (ta
+K · Jx)b = F abcJ c
+t
+with
+F abc = −F bac,
+(Jx · ta
+J)b = F abcJ c
+x + GabcJ c
+t
+with
+Gabc = −Gbac.
+(3.49)
+Thus, the final forms of the current-current OPEs are:
+J a
+t (t′, x′)J b
+t (t, x) ∼ 0
+J a
+x (t′, x′)J b
+t (t, x) ∼ lim
+ϵ→0+ i
+� Cab
+(∆˜x′)2 + F abcJ c
+t (t, x)
+∆˜x′
+�
+J a
+t (t′, x′)J b
+x(t, x) ∼ lim
+ϵ→0+ i
+� Cab
+(∆˜x′)2 + F abcJ c
+t (t, x)
+∆˜x′
+�
+(3.50)
+J a
+x (t′, x′)J b
+x(t, x) ∼ lim
+ϵ→0+ i
+�
+Dab
+(∆˜x′)2 +
+�
+F abcJ c
+x + GabcJ c
+t
+�
+(t, x)
+∆˜x′
+− (t′ − t)
+� 2Cab
+(∆˜x′)3 + F abcJ c
+t (t, x)
+(∆˜x′)2
+��
+These OPEs imply that the currents themselves are not current-primary fields in general.
+– 17 –
+
+3.5
+Global internal symmetry
+All the correlation functions in the theory must be invariant under the global internal
+transformations associated to which are the conserved currents (3.9) and (3.12). This fact
+will put constraints on Cab and Dab, as we will now see.
+We begin by noting the following 2-point correlators between the currents, from the above
+OPEs:
+�
+J a
+t (t′, x′)J b
+t (t, x)
+�
+= 0
+�
+J a
+t (t′, x′)J b
+x(t, x)
+�
+=
+�
+J a
+x (t′, x′)J b
+t (t, x)
+�
+= lim
+ϵ→0+ i Cab
+(∆˜x′)2
+(3.51)
+�
+J a
+x (t′, x′)J b
+x(t, x)
+�
+= lim
+ϵ→0+ i
+� Dab
+(∆˜x′)2 − (t′ − t) 2Cab
+(∆˜x′)3
+�
+since the currents, with ∆ = 1, must have vanishing VEVs.
+Next, due to the global internal symmetry, an arbitrary n-point correlator in the theory
+must satisfy:
+n
+�
+i=1
+⟨Φ1(t1, x1) . . . (Φi · ta
+J) (ti, xi) . . . Φn(tn, xn)⟩ = 0
+n
+�
+i=1
+⟨Φ1(t1, x1) . . . (ta
+K · Φi) (ti, xi) . . . Φn(tn, xn)⟩ = 0
+Thus the 2-point current correlators explicitly satisfy:
+�
+(Jx · ta
+J)b (t1, x1)J c
+x(t2, x2)
+�
++
+�
+J b
+x(t1, x1) (Jx · ta
+J)c (t2, x2)
+�
+= 0
+=⇒ F abdDdc + F acdDdb + GabdCdc + GacdCdb = 0
+and
+F abdCdc + F acdCdb = 0
+(3.52)
+The analogues relations obtained from the invariance of the 3-point current correlators are:
+F abeF ecfCfd + F caeF ebfCfd + F adeF bcfCfe = 0
+F abeF ecfDfd + F caeF ebfDfd + F adeF bcfDfe +
+�
+F abeGecf + GabeF ecf�
+Cfd
++
+�
+F caeGebf + GcaeF ebf�
+Cfd +
+�
+F adeGbcf + GadeF bcf�
+Cfe = 0
+(3.53)
+In what follows, we shall see that {F abc} and {Gabc} must also obey the following constraints
+arising as the Jacobi identity of the infinite-dimensional Lie algebra of the current-modes,
+which we had previously described in our initial algebraic description from the contraction
+in (2.21) and also will derive independently later in this section (3.80):
+F abeF ecd + F caeF ebd + F bceF ead = 0
+– 18 –
+
+F abeGecd + GabeF ecd + F caeGebd + GcaeF ebd + F bceGead + GbceF ead = 0
+(3.54)
+No new constraint for {F abc} and {Gabc} arises from the global internal invariance of
+n-point current correlators for n ≥ 4 .
+Upon comparison, we notice that (3.53) reduces to (3.54) if we choose:
+Dab = −ik1δab
+and
+Cab = −ik2δab with k2 ̸= 0
+(3.55)
+In that case, F abc and Gabc are anti-symmetric in all indices, as seen from (3.52).
+3.6
+EM tensor-current OPEs
+We now show under the assumption that no field in the theory has negative scaling dimen-
+sion with the identity field being the only one with ∆ = 0 , that the currents J a
+x (t, x) and
+J a
+t (t, x) must transform as a rank-1
+2 primary multiplet under 2D CC transformations.
+From [43], we recall the OPEs of a general 2D CC (multi-component) field Φ(l)(t, x) having
+scaling dimension ∆, Carrollian boost-charge ξ and boost rank l with the EM tensor
+components:
+T t
+x(t′, x′)Φ(l)(t, x) ∼ lim
+ϵ→0+ −i
+�
+. . . + ∆Φ(l)(t, x)
+(∆˜x′)2
++ ∂xΦ(l)(t, x)
+∆˜x′
+−(t′ − t)
+�
+. . . + 2
+�
+ξ · Φ(l)
+�
+(t, x)
+(∆˜x′)3
++ ∂tΦ(l)(t, x)
+(∆˜x′)2
+��
+T t
+t(t′, x′)Φ(l)(t, x) ∼ lim
+ϵ→0+ −i
+�
+. . . +
+�
+ξ · Φ(l)
+�
+(t, x)
+(∆˜x′)2
++ ∂tΦ(l)(t, x)
+∆˜x′
+�
+(3.56)
+The defining feature of 2D CC primary fields is the vanishing of the higher order poles in
+the above OPEs that leads to:
+T t
+x(t′, x′)Φ(l)(t, x) ∼ lim
+ϵ→0+ −i
+�∆Φ(l)(t, x)
+(∆˜x′)2
++ ∂xΦ(l)(t, x)
+∆˜x′
+−(t′ − t)
+�
+2
+�
+ξ · Φ(l)
+�
+(t, x)
+(∆˜x′)3
++ ∂tΦ(l)(t, x)
+(∆˜x′)2
+��
+T t
+t(t′, x′)Φ(l)(t, x) ∼ lim
+ϵ→0+ −i
+��
+ξ · Φ(l)
+�
+(t, x)
+(∆˜x′)2
++ ∂tΦ(l)(t, x)
+∆˜x′
+�
+(3.57)
+Now, along with the above assumption, the classical time-independence of the field J a
+t
+restricts the general OPE (3.56) to the following form:
+T t
+t(t′, x′)J a
+t (t, x) ∼ lim
+ϵ→0+ −i
+�
+Aa
+1
+(∆˜x′)3 + (ξ · J a
+t ) (t, x)
+(∆˜x′)2
+�
+(3.58)
+where the field Aa
+1 is proportional to the identity field. Similarly, we write the most general
+allowed form of the following OPE from (3.56):
+T t
+x(t′, x′)J a
+t (t, x) ∼ lim
+ϵ→0+ −i
+�
+Aa
+2
+(∆˜x′)3 + J a
+t (t, x)
+(∆˜x′)2 + ∂xJ a
+t (t, x)
+∆˜x′
+− (t′ − t)
+� 3Aa
+1
+(∆˜x′)4 + 2 (ξ · J a
+t ) (t, x)
+(∆˜x′)3
+��
+(3.59)
+– 19 –
+
+with Aa
+2 being another constant. But, the quantum counterpart of the conservation law
+(3.4) leads to:
+T t
+t(t′, x′)∂tJ a
+t (t, x) ∼ 0 =⇒ ∂t (ξ · J a
+t ) (t, x) = 0,
+T t
+x(t′, x′)∂tJ a
+t (t, x) ∼ 0 =⇒ Aa
+1 = 0
+and
+ξ · J a
+t = 0.
+(3.60)
+The OPEs for J a
+x (t, x) may have the most general form given below:
+T t
+t(t′, x′)J a
+x (t, x) ∼ lim
+ϵ→0+ −i
+�
+Aa
+3
+(∆˜x′)3 + (ξ · J a
+x ) (t, x)
+(∆˜x′)2
++ ∂tJ a
+x (t, x)
+∆˜x′
+�
+(3.61)
+with the conservation law (3.4) forcing:
+∂t (ξ · J a
+x ) (t, x) = 0.
+(3.62)
+The remaining one OPE is given below:
+T t
+x(t′, x′)J a
+x (t, x) ∼ lim
+ϵ→0+ −i
+�
+Aa
+4
+(∆˜x′)3 + J a
+x (t, x)
+(∆˜x′)2 + ∂xJ a
+x (t, x)
+∆˜x′
+(3.63)
+−(t′ − t)
+� 3Aa
+3
+(∆˜x′)4 + 2 (ξ · J a
+x ) (t, x)
+(∆˜x′)3
++ ∂tJ a
+x (t, x)
+(∆˜x′)2
+��
+where Aa
+3 and Aa
+4 are constants. From the quantum version of (3.4), one then obtains:
+T t
+x(t′, x′) [∂tJ a
+x (t, x) − ∂xJ a
+t (t, x)] ∼ 0
+=⇒
+ξ · J a
+x = J a
+t
+and
+Aa
+2 = Aa
+3
+(3.64)
+i.e. the currents J a
+x (t, x) and J a
+t (t, x) transform under Carrollian boost as a rank-1
+2 mul-
+tiplet with boost charge ξ = 1.
+The OPEs for the currents then are:
+T t
+t(t′, x′)J a
+t (t, x) ∼ 0
+T t
+x(t′, x′)J a
+t (t, x) ∼ lim
+ϵ→0+ −i
+�
+Aa
+3
+(∆˜x′)3 + J a
+t (t, x)
+(∆˜x′)2 + ∂xJ a
+t (t, x)
+∆˜x′
+�
+T t
+t(t′, x′)J a
+x (t, x) ∼ lim
+ϵ→0+ −i
+�
+Aa
+3
+(∆˜x′)3 + J a
+t (t, x)
+(∆˜x′)2 + ∂tJ a
+x (t, x)
+∆˜x′
+�
+(3.65)
+T t
+x(t′, x′)J a
+x (t, x) ∼ lim
+ϵ→0+ −i
+�
+Aa
+4
+(∆˜x′)3 + J a
+x (t, x)
+(∆˜x′)2 + ∂xJ a
+x (t, x)
+∆˜x′
+−(t′ − t)
+� 3Aa
+3
+(∆˜x′)4 + 2J a
+t (t, x)
+(∆˜x′)3
++ ∂tJ a
+x (t, x)
+(∆˜x′)2
+��
+On the other hand, applying the bosonic exchange property between the currents and the
+EM tensor components, we obtain:
+J a
+t (t′, x′)T t
+t(t, x) ∼ 0
+– 20 –
+
+J a
+t (t′, x′)T t
+x(t, x) ∼ lim
+ϵ→0+ −i
+�
+−
+Aa
+3
+(∆˜x′)3 + J a
+t (t, x)
+(∆˜x′)2
+�
+J a
+x (t′, x′)T t
+t(t, x) ∼ lim
+ϵ→0+ −i
+�
+−
+Aa
+3
+(∆˜x′)3 + J a
+t (t, x)
+(∆˜x′)2
+�
+(3.66)
+J a
+x (t′, x′)T t
+x(t, x) ∼ lim
+ϵ→0+ −i
+�
+−
+Aa
+4
+(∆˜x′)3 + J a
+x (t, x)
+(∆˜x′)2 − (t′ − t)
+�
+− 3Aa
+3
+(∆˜x′)4 + 2J a
+t (t, x)
+(∆˜x′)3
+��
+Comparing these with (3.19) and (3.20), we immediately note the following:
+�
+ta
+K · T t
+t
+�
+(t, x) =
+�
+ta
+K · T t
+x
+�
+(t, x) =
+�
+T t
+t · ta
+J
+�
+(t, x) =
+�
+T t
+x · ta
+J
+�
+(t, x) = 0
+(3.67)
+which implies that the EM tensor components actually transform under the singlet repre-
+sentation of the global internal symmetry algebra.
+Next we look at the consequences of the global internal symmetry on the 2-point correlators
+between the currents and the EM tensor components to find possible constraints on Aa
+3
+and Aa
+4. It suffices to consider the following:
+�
+(Jx · ta
+J)b (t1, x1)T t
+x(t2, x2)
+�
++
+�
+J b
+x(t1, x1)
+�
+T t
+x · ta
+J
+�
+(t2, x2)
+�
+= 0
+=⇒
+Aa
+3 = Aa
+4 = 0
+(3.68)
+Causing the vanishing of the poles of appropriate orders in the OPEs (3.65), we finally
+conclude, comparing with (3.57), that:
+the currents tranform as a 2D CC primary multiplet of rank-1
+2 with ∆ = ξ = 1 .
+Thus, the currents have the following infinitesimal 2D CC transformation property under
+(3.1), as is obtained using the prescription (3.23) for the corresponding conserved currents
+(3.3):
+− iGxJ a
+t (t, x) = −[f(x)∂x + f′(x)]J a
+t (t, x)
+;
+−iGtJ a
+t (t, x) = 0
+− iGtJ a
+x (t, x) = −g(x)∂tJ a
+x (t, x) − g′(x)J a
+t (t, x)
+(3.69)
+− iGxJ a
+x (t, x) = −[f(x)∂x + tf′(x)∂t + f′(x)]J a
+x (t, x) − tf′′(x)J a
+t (t, x)
+As an aside, from (3.66) we note down below the infinitesimal internal transformation
+properties of the EM tensor components, generated by the conserved charges of the currents
+(3.5a) and (3.5b):
+− iGJT t
+t(t, x) = −fa′(x)J a
+t (t, x)
+;
+−iGKT t
+t(t, x) = 0
+(3.70)
+− iGKT t
+x(t, x) = −ga′(x)J a
+t (t, x)
+;
+−iGJT t
+x(t, x) = −fa′(x)J a
+x (t, x) − tfa′′(x)J a
+t (t, x)
+– 21 –
+
+3.7
+The Algebra of Modes
+The EM tensor components have the following mode-expansions:
+T t
+t(t, x) = −i
+�
+n∈Z
+x−n−2Mn
+;
+T t
+x(t, x) = −i
+�
+n∈Z
+x−n−2 [Ln − (n + 2) t
+xMn]
+(3.71)
+=⇒ Mn = i
+�
+0
+dx
+2πi xn+1 T t
+t(t, x)
+;
+Ln = i
+�
+0
+dx
+2πi
+�
+xn+1T t
+x(t, x) + (n + 1)xnt T t
+t(t, x)
+�
+(3.72)
+In [43], it was shown from the EM tensor OPEs (3.37) that the EM tensor modes indeed
+generate the centrally extended BMS3 algebra:
+[Mn , Mm] = 0
+[Ln , Mm] = (n − m)Mn+m + cM
+12 (n3 − n)δn+m,0
+(3.73)
+[Ln , Lm] = (n − m)Ln+m + cL
+12(n3 − n)δn+m,0
+The infinitesimal 2D CC transformation properties of the currents are expressed in the
+operator language [43] using the prescription (3.23) for the charges in (3.72):
+[Ln , J a
+x (t, x)] = [xn+1∂x + t(n + 1)xn∂t + (n + 1)xn]J a
+x (t, x) + t(n + 1)nxn−1J a
+t (t, x)
+[Mn , J a
+x (t, x)] = xn+1∂tJ a
+x (t, x) + (n + 1)xnJ a
+t (t, x)
+(3.74)
+[Ln , J a
+t (t, x)] = xn+1∂xJ a
+t (t, x) + (n + 1)xnJ a
+t (t, x)
+;
+[Mn , J a
+t (t, x)] = 0
+Next, the classical conservation laws (3.4) imply the following space-time dependence of
+the fields {J a
+t } and {J a
+x }:
+∂tJ a
+t (t, x) = 0
+=⇒
+J a
+t (t, x) = J a
+t (x)
+∂tJ a
+x (t, x) = ∂xJ a
+t (x)
+=⇒
+J a
+x (t, x) = t∂xJ a
+t (x) + Ra(x)
+with Ra(x) being arbitrary functions.
+Guided by the above functional dependence and using the fact that the pair of the fields
+J a
+x and J a
+t forms a 2D CC primary rank-1
+2 multiplet with scaling dimension ∆ = 1, we
+write down the mode-expansions following [43]:
+J a
+t (x) =
+�
+n∈Z
+x−n−1Ka
+n
+;
+J a
+x (t, x) =
+�
+n∈Z
+x−n−1
+�
+Ja
+n − (n + 1) t
+xKa
+n
+�
+(3.75)
+Ka
+n =
+�
+Cu
+dx
+2πi xnJ a
+t (x)
+;
+Ja
+n =
+�
+Cu
+dx
+2πi
+�
+xnJ a
+x (t, x) + nxn−1tJ a
+t (x)
+�
+(3.76)
+where the counter-clockwise contour Cu encloses the upper half-plane and the real line.
+– 22 –
+
+Comparing (3.76) with (3.24) and (3.25), we immediately see that:
+Ja
+n is the conserved charge of the current jaµ =
+�
+xnJ a
+x + tnxn−1J a
+t , −xnJ a
+t
+�
+Ka
+n is the conserved charge of the current kaµ = (xnJ a
+t , 0)
+Thus, using the prescription (3.23) on the OPEs (3.33) and (3.34) we reach the definition
+of a current primary field Φ(t, x) in the operator formalism (for any n ∈ Z):
+[Ka
+n , Φ(t, x)] = ixn (ta
+K · Φ) (t, x),
+(3.77a)
+[Ja
+n , Φ(t, x)] = i
+�
+xn (Φ · ta
+J) + tnxn−1 (ta
+K · Φ)
+�
+(t, x)
+(3.77b)
+For the EM tensor components, the analogues commutation relations are found directly
+from (3.70):
+[Ka
+n , T t
+t(t, x)] = 0, [Ja
+n , T t
+t(t, x)] = −inxn−1J a
+t (t, x), [Ka
+n , T t
+x(t, x)] = −inxn−1J a
+t (t, x),
+[Ja
+n , T t
+x(t, x)] = −i
+�
+nxn−1J a
+x + tn(n − 1)xn−2J a
+t
+�
+(t, x)
+(3.78)
+Substituting the current mode-expansion (3.75) and EM tensor mode expansion (3.71) in
+these operator relations, we obtain the cross-commutation relations between the modes of
+the EM tensor and the currents:
+[Ln , Ja
+m] = −mJa
+m+n
+;
+[Mn , Ja
+m] = −mKa
+m+n = [Ln , Ka
+m] ;
+[Mn , Ka
+m] = 0
+(3.79)
+Similarly, from the current-current OPEs (3.50), using the condition (3.55) and the current
+mode-expansion (3.75), we reach the Lie algebra of the current modes:
+[Ja
+n , Jb
+m] = iF abcJc
+n+m + iGabcKc
+n+m + nk1δabδn+m,0
+[Ja
+n , Kb
+m] = iF abcKc
+n+m + nk2δabδn+m,0
+;
+[Ka
+n , Kb
+m] = 0
+(3.80)
+The global internal symmetry is governed by the subalgebra of the zero-modes:
+[Ja
+0 , Jb
+0] = iF abcJc
+0 + iGabcKc
+0
+;
+[Ja
+0 , Kb
+0] = iF abcKc
+0
+;
+[Ka
+0 , Kb
+0] = 0
+(3.81)
+This is precisely the algebra we have obtained for the NL currents in (2.21).
+– 23 –
+
+4
+Sugawara Construction
+As is well known, in 2d CFT, the Sugawara construction is employed to construct (the
+modes of) the energy momentum tensor in terms of (the modes of) the currents. In this
+section, here we will attempt to construct a NL version of the Sugawara construction and
+express Lm and Mm in terms of Jm and Km.
+We shall see that the Lm’s and Mm’s
+constructed in such manner indeed satisfy the BMS algebra. Similar constructions have
+been studied earlier in [44, 45].
+4.1
+Intrinsic construction from algebra
+We begin here by assuming we have only the algebra of the NL currents, i.e.
+[Ja
+m, Jb
+n] = ifabcJc
+m+n + mk1δabδm+n,0,
+(4.1a)
+[Ja
+m, Kb
+n] = ifabcKc
+m+n + mk2δabδm+n,0,
+[Ka
+m, Kb
+n] = 0.
+(4.1b)
+In the above, the sum over the group indices e.g. fabcJc
+m+n = �dim(¯g)
+c=1
+fabcJc
+m+n is implied.
+Comparing with (2.21), as stressed before, we work with the case where F abc = fabc, Gabc =
+0. Let us first consider the zero modes of J and K. Putting n = m = 0 in (4.1), the algebra
+for the zero modes become
+[Ja
+0 , Jb
+0] = ifabcJc
+0 ;
+[Ja
+0 , Kb
+0] = ifabcKc
+0 ;
+[Ka
+0, Kb
+0] = 0
+(4.2)
+Now let us look for quadratic Casimir operators for the above algebra.
+The possible
+combinations are
+�
+a
+Ja
+0 Ja
+0 ,
+�
+a
+Ja
+0 Ka
+0 ,
+�
+a
+Ka
+0Ja
+0 ,
+�
+a
+Ka
+0Ka
+0.
+(4.3)
+Keeping in mind that Casimir operators must commute with all the generators, we can
+exclude �
+a Ja
+0 Ja
+0 since it does not commute with Ka
+0
+� �
+a
+Ja
+0 Ja
+0 , Kb
+0
+�
+= i
+�
+a,c
+fabc(JaKc + KcJa) ̸= 0
+(4.4)
+All other combinations in (4.3) commute with all Ja
+0 ’s and Ka
+0’s. Hence a generic Casimir
+operator constructed from the zero modes of J and K will be a linear combination of the
+above three combinations of Ja
+0 ’s and Ka
+0’s. We therefore want to construct the zero level
+generator L0, M0 from these combinations.
+Since L0 and M0 are expected to be quadratic in terms of Ja
+n’s and Ka
+n’s, we can write
+down the following generic expression for them (The term � JJ is excluded since we have
+already seen that the term �
+a Ja
+0 Ja
+0 does not contribute to the zero mode part of L0 and
+M0)
+L0 = α
+�
+a,l,n
+Ja
+l Ka
+n + β
+�
+a,l,n
+Ka
+l Ja
+n + ρ
+�
+a,l,n
+Ka
+l Ka
+n
+(4.5a)
+M0 = µ
+�
+a,l,n
+Ja
+l Ka
+n + ν
+�
+a,l,n
+Ka
+l Ja
+n + η
+�
+a,l,n
+Ka
+l Ka
+n
+(4.5b)
+– 24 –
+
+Now putting the conditions that
+[L0, Ja
+n] = −nJa
+n, [M0, Ja
+n] = −nKa
+n
+and looking at just the level of current generators on the RHS, we can see that only (l = −n)
+terms should contribute, so we obtain
+L0 = α
+�
+a,n
+Ja
+nKa
+−n + β
+�
+a,n
+Ka
+nJa
+−n + ρ
+�
+a,n
+Ka
+nKa
+−n = αX0 + βY0 + ρZ0
+(4.6a)
+M0 = µ
+�
+a,l
+Ja
+l Ka
+−l + ν
+�
+a,l
+Ka
+l Ja
+−l + η
+�
+a,l
+Ka
+l Ka
+−l = µX0 + νY0 + ηZ0
+(4.6b)
+Now take
+Y0 =
+�
+a,l
+Ka
+l Ja
+−l =
+�
+a,l
+Ja
+−lKa
+l + k2
+dimg
+2
+∞
+�
+l=−∞
+l = X0 + c,
+(4.7)
+where c is a (possibly infinite) constant. Since we always have the independence of redefin-
+ing BMS generators by constant shifts, we can define level 0 BMS generators as
+L0 ≡ L′
+0 = α + β
+2
+(X0 + Y0) + ρZ0,
+M0 ≡ M′
+0 = µ + ν
+2
+(X0 + Y0) + ηZ0.
+(4.8)
+The normal ordering of the operators L0, M0 is achieved by individual normal ordering
+of X0, Y0, Z0. We generalise the definition to the other BMS generators as (with normal
+ordering)
+Lm =
+dimg
+�
+a=1
+�
+�α + β
+2
+�
+�
+�
+�
+l≤−1
+(Ja
+l Ka
+m−l + Ka
+l Ja
+m−l) +
+�
+l>−1
+(Ja
+m−lKa
+l + Ka
+m−lJa
+l )
+�
+�
+� + ρ
+�
+l
+Ka
+l Ka
+m−l
+�
+�
+Mm =
+dimg
+�
+a=1
+�
+�µ + ν
+2
+�
+�
+�
+�
+l≤−1
+(Ja
+l Ka
+m−l + Ka
+l Ja
+m−l) +
+�
+l>−1
+(Ja
+m−lKa
+l + Ka
+m−lJa
+l )
+�
+�
+� + η
+�
+l
+Ka
+l Ka
+m−l
+�
+�
+(4.9)
+Using the form of the BMS generators in (4.9), and substituting in the BMS-current cross
+commutators, i.e.
+[Lm, Ja
+n] = −nJa
+m+n ; [Lm, Ka
+n] = −nKa
+m+n ; [Mm, Ja
+n] = −nKa
+m+n,
+(4.10)
+we obtain the following values for the coefficients
+α + β = 1
+k2
+; ρ = −k1 + 2Cg
+2k2
+2
+;
+µ + ν = 0 ; η =
+1
+2k2
+.
+(4.11)
+So we obtain the final form for our NL Sugawara construction
+Lm =
+1
+2k2
+dim(g)
+�
+a=1
+�
+�
+�
+�
+l≤−1
+(Ja
+l Ka
+m−l + Ka
+l Ja
+m−l) +
+�
+l>−1
+(Ja
+m−lKa
+l + Ka
+m−lJa
+l ) − (k1 + 2Cg)
+k2
+�
+l
+Ka
+l Ka
+m−l
+�
+�
+�
+– 25 –
+
+Mm =
+1
+2k2
+dim(g)
+�
+a=1
+�
+l
+Ka
+l Ka
+m−l
+(4.12)
+As a check of the validity of the analysis, we compute the algebra of the L and M generators.
+These satisfy the following algebra (see Appendix B for detailed calculations)
+[Lm, Ln] = (m − n)Lm+n + dim(g)
+6
+(m3 − m)δm+n,0 ; [Lm, Mn] = (m − n)Mm+n
+(4.13)
+Hence, we see that by doing the NL Sugawara construction of NL currents, we can find
+BMS algebra with
+cL = 2dim(g),
+cM = 0.
+(4.14)
+Non-zero cM:
+We can obtain a non-zero cM with a slight modification to the Sugawara
+construction presented above. In case of Virasoro algebra with additional symmetries, we
+can define new Virasoro generators [46] as
+˜Ln = LS
+n + inθaja
+n + 1
+2kθ2δn,0
+(4.15)
+where LS
+n is the Virasoro generators obtained from Sugawara construction and θ = θata is a
+vector belonging to the Lie algebra with generators ta. It can be showed that (see Appendix
+C) if Ln satisfy Virasoro algebra with central charge c then ˜Ln will satisfy Virasoro algebra
+with shifted central charge ˜c where
+˜c = c + 12kθ2.
+(4.16)
+In case of NL Kac-Moody algebra, inspired by this, we introduce the following redefinitions
+˜Ln = LS
+n + inθaJa
+n + 1
+2k1θ2δn,0
+˜
+Mn = MS
+n + inθaKa
+n + 1
+2k2θ2δn,0
+(4.17)
+where LS
+n and MS
+n are given by (4.9). This redefinition will give us the following algebra
+(see Appendix C)
+[˜Lm, ˜Ln] = (m − n)˜Lm+n + cL
+12(m3 − m)δn+m,0
+[˜Lm, ˜
+Mn] = (m − n) ˜
+Mm+n + cM
+12 (m3 − m)δn+m,0
+(4.18)
+where the central charges are given by
+cL = 2dim(g) + 12k1θ2
+cM = 12k2θ2
+(4.19)
+Hence we can obtain the fully centrally extended BMS algebra.
+– 26 –
+
+4.2
+Consistency with OPEs
+We will recast the NL Sugawara construction in terms of the NL EM tensor that we
+introduced in Sec. 3 instead of the generators of the BMS: {Ln, Mn}. {Ln, Mn} are of
+course modes of the NL EM tensor. Hence the calculations in this subsection provide a
+sanity check for our analysis making sure the various formulations are consistent with each
+other.
+The NL Sugawara construction gave us (4.12).
+We now normal order the products to
+rewrite this as
+Ln =
+1
+2k2
+�
+l,a
+�
+: Ja
+l Ka
+n−l : + : Ka
+l Ja
+n−l : −k1 + 2Cg
+k2
+: Ka
+l Ka
+n−l :
+�
+Mn =
+1
+2k2
+�
+l,a
+: Ka
+l Ka
+n−l :
+(4.20)
+Here, : Al : is a shorthand for normal ordered products. Now substituting (4.20) in (3.71)
+and rearranging the terms, we get uv to xt
+Tu(u, v) =
+1
+2k2
+�
+a
+��
+(J a
+u J a
+v )(u, v) + (J a
+v J a
+u )(u, v)
+�
+− k1 + 2Cg
+k2
+(J a
+v J a
+v )(v)
+�
+Tv(v) =
+1
+2k2
+�
+a
+(J a
+v J a
+v )(v)
+(4.21)
+Here (. . . ) means normal ordered products of fields, which can be expressed in terms of
+contour integrals.
+Here we are using u, v coordinates instead of x, t of Sec. 3 because
+this construction applies to both Galilean and Carrollian CFTs, because of the similarity
+between the 2 theories mentioned in the introduction. So by substituting u → x, v → t or
+u → t, v → x, we can obtain the Galilean or Carrollian theory expressions, respectively.
+So (4.21) gives the field expression for the NL Sugawara construction. For the modified
+Sugawara construction as defined in (4.17), the EM tensor fields turn out to be (by doing
+the mode expansion using the new generators)
+˜Tu(u, v) = Tu(u, v) − iθa∂vJ a
+u (u, v) − iθa J a
+u (u, v)
+v
+− iθa uJ a
+v (v)
+v2
++ 1
+2
+k1θ2
+v2
++ uk2θ2
+v3
+˜Tv(v) = Tv(v) − iθa∂vJ a
+v (v) − iθa J a
+v (v)
+v
++ 1
+2
+k2θ2
+v2
+(4.22)
+where, Tu(u, v), Tv(u, v) refers to the quantities in (4.21), i.e. the normal NL Sugawara
+construction expressions.
+T − J OPE:
+Now, for a consistency check of our analysis so far, we will rederive the
+OPEs from the above definitions of the NL Sugawara construction. We begin with the T-J
+OPEs. Taking the definitions as in (4.21) and contracting with the currents, we get the
+– 27 –
+
+following expressions (See Appendix D for detailed calculation):
+Tv(u1, v1)J b
+v (u2, v2) ∼ regular
+Tu(u1, v1)J b
+v (u2, v2) ∼ J b
+v (u2, v2)
+v2
+12
++ ∂vJ b
+v (u2, v2)
+v12
++ . . .
+Tv(u1, v1)J a
+u (u2, v2) ∼ J a
+v (u2, v2)
+v2
+12
++ ∂vJ a
+v (u2, v2)
+v12
++ . . .
+Tu(u1, v1)J a
+u (u2, v2) ∼ J a
+u (u2, v2)
+v2
+12
++ ∂vJ a
+u (u2, v2)
+v12
++ u12
+v2
+12
+∂uJ a
+u (u2, v2)
++ 2u12
+v12
+�J a
+v (u2, v2)
+v2
+12
++ ∂vJ a
+v (u2, v2)
+v12
+�
++ . . .
+(4.23)
+Which are equivalent to the [Lm, Ja
+n] type commutation relations given in (2.21). Clearly
+these relation satisfy the OPE relations (3.66) that the currents and the EM tensors of a
+theory are supposed to satisfy.
+T − T OPE:
+Next we use the definition of the Sugawara construction to compute the
+T-T type OPEs. Doing the calculations (see Appendix D for detailed Calculations), we
+get,
+Tu(u1, v1)Tu(u2, v2) ∼2Tu(u2, v2)
+v2
+12
++ ∂vTu(u2, v2)
+v12
++ u12
+v2
+12
+∂uTu(u2, v2)
++ 2u12
+v12
+�2Tv(u2, v2)
+v2
+12
++ ∂vTv(u2, v2)
+v12
+�
++ dim(g)
+v4
+12
++ . . .
+Tu(u1, v1)Tv(u2, v2) ∼2Tv(u2, v2)
+v2
+12
++ ∂vTv(u2, v2)
+v12
++ . . .
+(4.24)
+Tv(u1, v1)Tv(u2, v2) ∼ regular
+These results (4.24) match with the OPEs of the energy momentum tensors of a 2d Carrol-
+lian or Galilean CFT, as given in (3.37). So, the NL Sugawara construction really gives the
+EM tensors of a 2d NL CFT (proved both at algebra level and now at field level). Also from
+the OPE expressions, we can verify the earlier results of cL = 2dim(g), cM = 0 obtained ear-
+lier. Again if we start with the modified Sugawara construction (4.22), we can obtain same
+OPE relations as above, but with modified central charges cL = 2dim(g) + 12k1θ2, cM =
+12k2θ2. To see this, let’s look at, for example, the ˜Tu(u1, v1) ˜Tv(v2) OPE and focus our
+attention to a specific term
+˜Tu(u1, v1) ˜Tv(v2) ∼ −θaθb∂vJ a
+u (u1, v1)∂vJ a
+v (v2) + · · · ∼ −θ2∂v1∂v2
+k2
+v2
+12
++ · · · ∼ 6k2θ2
+v4
+12
++ . . .
+Matching the above expression with (3.37), we can see cM = 12k2θ2. Similarly we can
+obtain the other shifted central charge.
+– 28 –
+
+5
+Tensionless String as NLKM
+The tensionless limit of string theory is the limit that is diametrically opposite to the
+usual point particle limit where known supergravity appears from strings. This limit that
+explores the very strong gravity regime and, when quantized, the very highly quantum and
+highly stringy regime of string theory. In this section, we explore how this is connected
+to the NLKM structures we have discussed in this paper. We will see that the currents
+satisfying the U(1) NLKM algebra come up intrinsically when we are looking at tensionless
+strings propagating in flat spacetimes.
+As is very well known, the action for a tensile relativistic string propagating in a flat
+d-dimensional spacetime is given by the Polyakov action:
+SP = T
+�
+d2ξ √γγαβ∂αXµ∂βXνηµν.
+(5.1)
+In order to take the tensionless limit, it is helpful to work with the phase space action of
+string. One can then systematically take the limit [] and the resulting action can be cast
+in a Polyakov like form, which we will call the ILST action after the authors:
+SILST =
+�
+d2ξ V αV β∂αXµ∂βXνηµν.
+(5.2)
+Here Xµ are the coordinates in the background flat space ηµν which are scalar fields on
+the worldsheet parametrized by ξa = σ, τ. The worldsheet metric γαβ degenerates in the
+limit and the following replacement is made:
+T√γγαβ → V αV β,
+(5.3)
+where V α is a vector density. The action is invariant under worldsheet diffeomorphisms
+and hence one needs to fix a gauge. It is helpful to go to the equivalent of the conformal
+gauge
+V α = (v, 0).
+(5.4)
+There is some symmetry left over even after this gauge fixing. In the usual tensile string,
+the residual symmetry gives two copies of the Virasoro algebra and the appearance of
+a 2d CFT on the worldsheet is the central reason why we understand string theory as
+well as we do. Now in the tensionless case, the residual gauge symmetry that appears on
+the worldsheet is the BMS3 algebra. This now dictates the theory of tensionless strings.
+The reason behind the appearance of the BMS algebra is that the tensionless string in
+flat spacetimes is actually a null string. This is the string equivalent of a massless point
+particle which is constrained to travel on a null geodesic of the ambient spacetime. The
+string sweeps out a worldsheet which is null and since there are no mass terms, the resulting
+action has to have conformal Carrollian symmetry in d = 2 or equivalently BMS3.
+Now, let us connect to our discussion in this paper. In the favourable conformal gauge
+(5.4), the equations of motion and constraints arising from (5.2) take the form
+EOM:
+¨Xµ = 0 ;
+Constraints:
+˙X2 = 0 ,
+˙X.X′ = 0.
+(5.5)
+– 29 –
+
+A convenient form of the solution is given by
+Xµ(σ, τ) = xµ +
+√
+2c′�
+Jµ
+0 σ + Kµ
+0 τ + i
+�
+n̸=0
+1
+n(Jµ
+n − inτKµ
+n)e−inσ�
+(5.6)
+Here σ, τ are the coordinates on the cylinder, which are related to the planar Non-Lorentzian
+coordinates as
+u = e−iσ ; v = −τe−iσ
+(5.7)
+The periodic condition Xµ(σ + 2π, τ) = Xµ(σ, τ) forces us to have Jµ
+0 = 0. From (5.6), we
+obtain
+Jµ
+τ = ∂σXµ =
+√
+2c′ �
+n
+(Jµ
+n − inτKµ
+n)e−inσ,
+Jµ
+σ = −∂τXµ = −
+√
+2c′ �
+n
+Kµ
+ne−inσ
+(5.8)
+Clearly the currents in (5.8) satisfy similar relations as (3.4), i.e.
+∂τJµ
+σ = 0 ; ∂τJµ
+τ + ∂σJµ
+σ = 0.
+(5.9)
+If we canonically quantise the system by demanding [Xµ(τ, σ), Πν(τ, σ′)] = iδ(σ − σ′)ηµν,
+we obtain the following relations for the current modes
+[Jµ
+m, Jν
+n] = [Kµ
+m, Kν
+n] = 0 ; [Jµ
+m, Kν
+n] = 2mδm+n,0ηµν
+(5.10)
+Clearly, the current modes for the same spacetime index µ follow a special case of U(1) NL
+Kac-Moody algebra (look at (2.21) for comparison), with k(µν)
+1
+= 0, k(µν)
+2
+= 2ηµν.
+Next if we look into the (classical) energy momentum tensor of this theory, their mode
+expansion coefficients are given by
+Ln = 1
+2
+�
+m
+Jµ
+mKν
+n−mηµν ; Mn = 1
+4
+�
+m
+Kµ
+mKν
+n−mηµν.
+(5.11)
+This matches classically with the relation (4.12) for the NL Sugawara construction with
+k1 = 0,
+k2 = 2.
+(5.12)
+Note that Cg = 0 for U(1) algebra. The generators in (5.11) satisfy the (centreless) BMS3
+algebra, which is expected as the action (5.2) has BMS symmetry as a residual gauge
+symmetry.
+Thus we can see that the Non-Lorentzian U(1) Kac-Moody algebra and the associated NL
+Sugawara construction come up in the theory of tensionless or null strings propagating on
+a flat background geometry. An outstanding question is what happens when we look at the
+propagation of null strings on arbitrary curved manifolds. These can be viewed as group
+manifolds and there will be the NL equivalent of a Wess-Zumino-Witten model appearing
+for tensionless strings in this context. We expect that the non-abelian NLKM algebras
+explored in this work would naturally appear on these NL WZW models. This is work in
+progress and we hope to report on this in the near future.
+– 30 –
+
+6
+Non-Lorentzian Knizhnik Zamolodchikov Equations
+In usual Lorentzian Kac-Moody algebras, the Knzhnik-Zamolodchikov equations are the
+linear differential equations that are satisfied by the correlation functions of these CFTs
+endowed with additional affine Lie symmetry. In this section, we write down the non-
+Lorentzian analogues of these KZ equations based on our construction of the NLKM algebra
+and its representations in this paper.
+In what follows, we will outline the steps to get to the NL Knizhnik Zamolodchikov equa-
+tions. The details of the calculation are presented in a separate appendix E.
+We begin with the OPE definition of BMS primary field (3.57) and obtain
+∂uΦ(u′, v′) = −
+�
+v′
+dv
+2πi
+�
+u′
+du
+2πi(u − u′)−1Tv(u, v)Φ(u′, v′) = −
+�
+v′
+dv
+2πiTv(v)Φ(u′, v′)
+∂vΦ(u′, v′) =
+�
+v′
+dv
+2πi
+�
+u′
+du
+2πi(u − u′)−1Tu(u, v)Φ(u′, v′) =
+�
+v′
+dv
+2πiTu(u′, v)Φ(u′, v′)
+(6.1)
+Next, we take the correlation function of a string of primary fields as shown below
+⟨∂uΦ(u, v)Φ1(u1, v1) . . . Φn(un, vn)⟩ = ⟨∂uΦ(u, v)X({ui, vi})⟩ =
+�
+v
+dv′
+2πi⟨Tv(u′, v′)Φ(u, v)X({ui, vi})⟩
+=
+�
+v
+dv′
+2πi⟨ 1
+2k2
+�
+(J a
+v J a
+v )(u′, v′)
+�
+Φ(u, v)X({ui, vi})⟩
+and use relations (3.57, 6.1) to finally get the following expression (details in Ap. E)
+⟨∂uΦ(u, v)X({ui, vi})⟩ = − 1
+k2
+��
+j
+ta
+R,K ⊗ ta
+Rj,K
+(v − vj)
+�
+⟨Φ(u, v)X({ui, vi})⟩
+(6.2)
+which can be written as
+�
+∂ui + 1
+k2
+�
+j̸=i
+ta
+Ri,K ⊗ ta
+Rj,K
+(vi − vj)
+�
+⟨Φ1(u1, v1) . . . Φn(un, vn)⟩ = 0.
+(6.3)
+This is one of the Non-Lorentzian Knizhnik Zamolodchikov equations.
+Similarly we can start with
+⟨∂vΦ(u, v)Φ1(u1, v1) . . . Φn(un, vn)⟩ = ⟨∂vΦ(u, v)X({ui, vi})⟩ =
+�
+v
+dv′
+2πi⟨Tu(u′, v′)Φ(u, v)X({ui, vi})⟩
+=
+�
+v
+dv′
+2πi⟨ 1
+2k2
+�
+(J a
+v J a
+u )(u′, v′) + (J a
+u J a
+v )(u′, v′) − k1 + 2Cg
+k2
+(J a
+v J a
+v )(u′, v′)
+�
+Φ(u, v)X({ui, vi})⟩
+(6.4)
+From here, we can proceed in the same way as Appendix E to obtain the other equation
+�
+∂vi − 1
+k2
+�
+j̸=i
+1
+(vi − vj)
+dimg
+�
+a=1
+��
+ta
+Ri,J ⊗ ta
+Rj,K + ta
+Ri,K ⊗ ta
+Rj,J
+�
++
+�ui − uj
+vi − vj
+− k1 + 2Cg
+k2
+�
+ta
+Ri,K ⊗ ta
+Rj,K
+��
+⟨ΦR1(u1, v1)ΦR2(u2, v2)...ΦRn(un, vn)⟩ = 0
+(6.5)
+– 31 –
+
+This above equation (6.5) is the second of the Non-Lorentzian Knizhnik Zamolodchikov
+equations. The solutions to these two equations (6.3), (6.5) would give the correlation
+functions of the underlying NLKM theory. For usual relativistic theories, the KZ equations
+are difficult to solve in general, but one can do it for the four-point functions yielding a
+closed solution in terms of hypergeometric functions.
+It would be instructive to check
+whether something similar happens here. If one can find the solutions to the NLKM four
+point functions, it would also be a nice exercise to check whether these can be arrived
+at as a limit of the relativistic answers. This process of attempting to generate the non-
+Lorentzian answers from the relativistic ones is something we outline in detail in the section
+that follows.
+7
+NLKM from Contractions
+In this section, we rederive various results we have obtained earlier in the paper through
+a systematic limit on the algebraic structures obtained in the relativistic set-up. Before
+moving to recovering the answers, we spend a bit of time understanding which limit is
+appropriate for our purposes.
+7.1
+A brief detour to representations of BMS
+We said in the introduction that there were two distinct contractions that land us up on
+the BMS3 algebra starting from two copies of the Virasoro algebra.
+One of them was
+a Carrollian or ultra-relativistic limit (2.17), where there was a mixing of positive and
+negative Virasoro modes creating the BMS generators, while the other was a Galilean or
+non-relativistic (2.13), where no mixing took place. This mixing of modes is critical for the
+understanding of the representations in the limit.
+We begin with the Galilean contraction.
+In the parent relativistic CFT, the theory is
+best described in terms of the highest weight representation. The states of the theory are
+labelled by the zero modes:
+L0|h, ¯h⟩ = h|h, ¯h⟩,
+¯L0|h, ¯h⟩ = ¯h|h, ¯h⟩
+(7.1)
+There is a class of states called primary states which are annihilated by all positive modes:
+Ln|h, ¯h⟩p = 0,
+¯Ln|h, ¯h⟩p = 0,
+∀n > 0.
+(7.2)
+The Virasoro modules are built on these primary states by acting with raising operators
+L−n. Now, looking back at the Galilean contraction (2.13), we see that
+Ln = 1
+2
+�
+Ln + 1
+ϵ Mn
+�
+,
+¯Ln = 1
+2
+�
+Ln − 1
+ϵ Mn
+�
+(7.3)
+The 2d CFT primary conditions then boil down to:
+L0|∆, ξ⟩ = ∆|∆, ξ⟩, M0|∆, ξ⟩ = ξ|∆, ξ⟩;
+Ln|∆, ξ⟩p = 0, Mn|∆, ξ⟩p = 0, ∀n > 0.
+(7.4)
+– 32 –
+
+In the above, the assumption is that the state |h, ¯h⟩ goes to the state |∆, ξ⟩ in the limit. So
+we see that highest weight states map to highest weight states in the Galilean contraction.
+This analysis holds in a similar way when we consider the full NLKM algebra. We will be
+using this for the analysis in the rest of the section.
+But before we get there, let us point out why we are not using the Carroll limit (2.17) for
+this purpose. In the Carroll contraction, we can read off
+Ln = 1
+2
+�
+Ln + 1
+ϵ Mn
+�
+,
+¯Ln = 1
+2
+�
+−L−n + 1
+ϵ M−n
+�
+(7.5)
+The 2d CFT primary conditions in the Carroll limit become:
+M0|M, s⟩ = M|M, s⟩,
+L0|M, s⟩ = s|M, s⟩,
+Mn|M, s⟩ = 0,
+∀n ̸= 0.
+(7.6)
+In the above, the state |h, ¯h⟩ in the Carroll limit becomes the state |M, s⟩. This is clearly
+not a highest weight state. The set of these states form what is called the induced repre-
+sentation. The story is again similar for the full NLKM. In the analysis that follows, we
+will not focus on the induced representations. It would be of interest to consider them in
+future work.
+We should stress however, that the answers we have obtained in the intrinsic Carrollian
+form earlier are for highest weight representations. The Galilean limit would be an effective
+way of reproducing these answers with a flip of space and time directions at the end of the
+analysis.
+7.2
+Contraction of the affine parameters
+In this section we will establish the mode expansion of the NLKM currents (3.76) from
+another approach, by doing a contraction of the original loop extended algebra. Starting
+from the finite Lie algebra
+[ja, jb] = ifabcjc ; [¯ja, ¯jb] = ifabc¯jc
+(7.7)
+we can define loop extended generators
+ja
+n = ja ⊗ zn ; ¯ja
+n = ¯ja ⊗ ¯zn,
+(7.8)
+which satisfy
+[ja
+n, jb
+m] = ifabcjc ⊗ zn+m = ifabcjc
+n+m.
+(7.9)
+Loop extended algebra obtained above admits a central extension to give us the current
+algebra in (2.12). Now, we can do a contraction of the affine parameter as z = t + ϵx and
+¯z = t − ϵx in the Galilean limit. We then obtain (keeping upto first order in ϵ),
+ja
+n = ja ⊗ (tn + nϵxtn−1 + ...) ; ¯ja
+n = ¯ja ⊗ (tn − nϵxtn−1 + ...)
+(7.10)
+Now we can define new contracted generators of the finite algebra as follows:
+Ja = ja + ¯ja ; Ka = ϵ(¯ja − ja)
+(7.11)
+– 33 –
+
+Above definition can be extended to general modes of J and K using (7.10),
+Ja
+n = lim
+ϵ→0(ja
+n + ¯ja
+n) = Ja ⊗ tn − nKa ⊗ xtn−1
+(7.12a)
+Ka
+n = lim
+ϵ→0 ϵ(¯ja
+n − ja
+n) = Ka ⊗ tn
+(7.12b)
+In this way, we get the structure of the power series expansion of J and K in terms of x
+and t, which is the Galilean analog of (3.76), or identical upto the flip of temporal and
+spatial directions.
+If we define primary field φ(x, t) as a representation of the finite algebra, i.e. in terms of
+the action of zero modes of the generators J and K,
+[Ja
+0 , φ(x, t)] ≡ [Ja, φ(x, t)] = ta
+Jφ(x, t)
+[Ka
+0, φ(x, t)] ≡ [Ka, φ(x, t)] = ta
+Kφ(x, t)
+(7.13)
+We can get the action of general modes using (7.12),
+[Ja
+n, φ(x, t)] = ta
+Jφ(x, t) ⊗ tn − nta
+Kφ(x, t) ⊗ xtn−1 ≡ tnta
+Jφ(x, t) − nxtn−1ta
+Kφ(x, t)
+[Ka
+n, φ(x, t)] = ta
+Kφ(x, t) ⊗ tn ≡ tnta
+Kφ(x, t)
+(7.14)
+This definition of a primary field agrees with our earlier result (3.77).
+This action of Ja
+n and Ka
+n on primary fields also appears when we draw motivation from
+[41]. We can define the primary field as,
+[Ja
+0 , φ(0, 0)] = ta
+Jφ(0, 0) ; [Ka
+0, φ(0, 0)] = ta
+kφ(0, 0),
+[Ja
+n, φ(0, 0)] = 0 ; [Ka
+n, φ(0, 0)] = 0
+∀ n > 0.
+(7.15)
+For n ≥ 0 and U = etL−1−xM−1,
+[Ja
+n, φ(x, t)] = [Ja
+n, Uφ(0)U −1] = U[U −1Ja
+nU, φ(0)]U −1
+(7.16)
+Consider,
+U −1Ja
+nU = e−tL−1+xM−1Ja
+netL−1−xM−1
+=
+n
+�
+k=0
+tk
+k!
+n!
+(n − k)!Ja
+n−k − nx
+n−1
+�
+k=0
+tk
+k!
+(n − 1)!
+(n − k − 1)!Ka
+n−k−1
+(7.17)
+where we have used the Baker-Campbell-Hausdorff(BCH) formula twice. Putting this back
+in (7.16), we finally get,
+[Ja
+n, φ(x, t)] = U
+� n
+�
+k=0
+tk
+k!
+n!
+(n − k)![Ja
+n−k, φ(0)] − nx
+n−1
+�
+k=0
+tk
+k!
+(n − 1)!
+(n − k − 1)![Ka
+n−k−1, φ(0)]
+�
+U −1
+= (tnta
+J − nxtn−1ta
+K)φ(x, t)
+(7.18)
+where only the terms corresponding to k = n and k = n − 1 survived respectively in the
+first and the second sum because of (7.15). Similarly for Ka
+n, we get,
+[Ka
+n, φ(x, t)] = U
+� n
+�
+k=0
+tk
+k!
+n!
+(n − k)![Ka
+n−k, φ(0)]
+�
+U −1 = tnta
+Kφ(x, t)
+(7.19)
+These results verify (7.14) again from a different perspective.
+– 34 –
+
+7.3
+Contracting the Sugawara construction
+In section 4.1, we constructed the BMS generators by taking quadratic products of the
+NL currents and ended up with (4.12). In this subsection, we shall reproduce the same
+from the Galilean limit of Sugawara construction for CFT. We start from the following
+expression for virasoro modes Lm and ¯Lm in relativistic sugawara construction.
+Lm = γ
+dim(g)
+�
+a=1
+(
+�
+l≤−1
+ja
+l ja
+m−l +
+�
+l>−1
+ja
+m−lja
+l )
+(7.20a)
+¯Lm = ¯γ
+dim(g)
+�
+a=1
+(
+�
+l≤−1
+¯ja
+l ¯ja
+m−l +
+�
+l>−1
+¯ja
+m−l¯ja
+l )
+(7.20b)
+where γ =
+1
+2(k + Cg),¯γ =
+1
+2(¯k + Cg) and Cg = −
+1
+2dim(g)
+�
+b,c
+fbacfbcd
+(7.20c)
+We can now take the Galilean limit by using the inverted version of relations (2.20) and
+get the following form of Virasoro Generators by collecting the terms with same order in ϵ,
+Lm = γ
+4(Am − Bm
+ϵ
++ Cm
+ϵ2 ) ; ¯Lm = ¯γ
+4(Am + Bm
+ϵ
++ Cm
+ϵ2 )
+(7.21)
+where,
+Am =
+dim(g)
+�
+a=1
+�
+l≤−1
+Ja
+l Ja
+m−l +
+�
+l>−1
+Ja
+m−lJa
+l
+(7.22a)
+Bm =
+dim(g)
+�
+a=1
+{
+�
+l≤−1
+(Ja
+l Ka
+m−l + Ka
+l Ja
+m−l) +
+�
+l>−1
+(Ja
+m−lKa
+l + Ka
+m−lJa
+l )}
+(7.22b)
+Cm =
+dim(g)
+�
+a=1
+�
+l
+Ka
+l Ka
+m−l
+(7.22c)
+where we have employed the commutativity of K’s in order to write 1
+ϵ2 term(Cm) as a sum
+from l = −∞ to ∞. We can use (2.13) and the above relations to obtain:
+Lm = 1
+4
+�
+(γ + ¯γ)Am − (γ − ¯γ)
+ϵ
+Bm + (γ + ¯γ)
+ϵ2
+Cm
+�
+(7.23a)
+Mm = −1
+4
+�
+ϵ(γ − ¯γ)Am − (γ + ¯γ)Bm + (γ − ¯γ)
+ϵ
+Cm
+�
+(7.23b)
+Inverting the relations (2.20), we can write k and ¯k in terms of k1 and k2,
+k = 1
+2(k1 − k2
+ϵ ) ; ¯k = 1
+2(k1 + k2
+ϵ )
+(7.24)
+Using the definitions of γ and ¯γ in (7.20) and using (7.24), we can write,
+lim
+ϵ→0
+γ + ¯γ
+ϵ2
+= −2(k1 + 2Cg)
+k2
+2
+and lim
+ϵ→0
+γ − ¯γ
+ϵ
+= − 2
+k2
+(7.25)
+– 35 –
+
+Hence, in limit ϵ → 0 (7.23) can be written as,
+Lm =
+1
+2k2
+�
+Bm − (k1 + 2Cg)
+k2
+Cm
+�
+and Mm =
+1
+2k2
+Cm
+(7.26)
+which agrees with (4.12).
+As we have already seen in the previous section that (4.12) satisfies BMS algebra with
+central charges cL = 2dim(g) and cM = 0. This fact can also be verified using the following
+definitions of central charges in the relativistic Sugawara construction,
+c = kdim(g)
+k + Cg
+; ¯c =
+¯kdim(g)
+¯k + Cg
+(7.27)
+Using (7.27) and following similar steps as before using the (7.24), we can get the following,
+cL = lim
+ϵ→0(c + ¯c) = lim
+ϵ→0
+�
+dim(g)
+�
+1
+2(k1 − k2
+ϵ )
+1
+2(k1 − k2
+ϵ ) + Cg
++
+1
+2(k1 + k2
+ϵ )
+1
+2(k1 + k2
+ϵ ) + Cg
+��
+⇒ cL = 2dim(g)
+(7.28)
+Similarly, we can get,
+cM = lim
+ϵ→0 ϵ(¯c − c) = 0
+(7.29)
+These are the same values of central charges appeared in the L, M commutation relations
+as we got in (4.13).
+The modification we have done in order to get non-zero cM too can be retrieved from
+contraction. For this, we can start from (4.15) and its conjugate. Now if we take the limit
+in (2.20), we shall again retrieve the redefined Lns and Mns as we have seen in (4.17), and
+the commutators will be same as (4.18) with central charges (4.19).
+7.4
+NLKZ equations from Contraction
+Finally, we show how to obtain the non-Lorentzian Knizhnik Zamolodchikov equations
+from a limit of the ones for a 2d CFT with additional symmetry. Some of the details of the
+analysis are contained in Appendix F. We start with the original Knizhnik Zamolodchikov
+equations:
+�
+�∂wi − 2γ
+�
+j̸=i
+�
+a(ta
+Ri)ri
+si(ta
+Rj)rj
+sj
+wi − wj
+�
+� ⟨...φsi
+Ri(wj)...φsj
+Rj(wj)...⟩ = 0
+(7.30a)
+�
+�∂ ¯wi − 2¯γ
+�
+j̸=i
+�
+a(¯ta¯Ri)¯ri¯si(¯ta¯Rj)¯rj
+¯sj
+¯wi − ¯wj
+�
+� ⟨...¯φ¯si¯Ri( ¯wj)...¯φ¯sj
+¯Rj( ¯wj)...⟩ = 0
+(7.30b)
+where
+(ta
+Ri)ri
+si(ta
+Rj)rj
+sj⟨...φsi
+Ri(wj)...φsj
+Rj(wj)...⟩ = ((ta
+Ri ⊗ ta
+Rj)⟨...φRi(wj)...φRj(wj)...⟩)ri,rj
+(7.31)
+– 36 –
+
+Using the separability of the primary fields, Φr,¯r
+R, ¯R(w, ¯w) = φr
+R(w) ⊗′ ¯φ¯r¯R( ¯w), we can write,
+⟨Φr1,¯r1
+R1, ¯R1(w1, ¯w1)...ΦrN,¯rN
+RN, ¯RN (wN, ¯wN)⟩ = ⟨φr1
+R1(w1)...φrN
+RN (wN)⟩⟨¯φ¯r1¯R1( ¯w1)...¯φ¯rN
+¯RN ( ¯wN)⟩ (7.32)
+where the primed tensor product(⊗′) ensures independent action of operators on chiral and
+anti-chiral sectors whereas the unprimed tensor product(⊗) ensures independent action on
+ith and jth insertion in the n-point function.
+Using the following linear combinations in limit ϵ → 0,
+(7.30a) × ⟨¯φ¯r1¯R1( ¯w1)...¯φ¯rN
+¯RN ( ¯wN)⟩ + (7.30b) × ⟨φr1
+R1(w1)...φrN
+RN (wN)⟩ = 0
+ϵ{(7.30a) × ⟨¯φ¯r1¯R1( ¯w1)...¯φ¯rN
+¯RN ( ¯wN)⟩ − (7.30b) × ⟨φr1
+R1(w1)...φrN
+RN (wN)⟩} = 0
+(7.33)
+We get (shown in details in Appendix F),
+�
+∂ti − 1
+k2
+�
+j̸=i
+��
+a(ta
+Ri,J ⊗ ta
+Rj,K + ta
+Ri,K ⊗ ta
+Rj,J)
+tij
++
+�xij
+t2
+ij
+− (k1 + 2Cg)
+k2tij
+� �
+a
+(ta
+Ri,K ⊗ ta
+Rj,K)
+��
+⟨...ΦRi, ¯Ri(xi, ti)...ΦRj, ¯
+Rj(xj, tj)...⟩ = 0
+�
+∂xi + 1
+k2
+�
+j̸=i
+�
+a(ta
+Ri,K ⊗ ta
+Rj,K)
+tij
+�
+⟨...ΦRi, ¯Ri(xi, ti)...ΦRj, ¯Rj(xj, tj)...⟩ = 0
+(7.34)
+where, ta
+Ri,J = ta
+Ri ⊗′ ¯I +I ⊗′ ¯ta¯Ri and ta
+Ri,J = ϵ(I ⊗′ ¯ta¯Ri −ta
+Ri ⊗′ ¯I) in limit ϵ → 0. The above
+equations are same as what we got form intrinsic analysis i.e. (6.3) and (6.5), with t → v
+and x → u as its supposed to for a Galilean contracted result.
+– 37 –
+
+8
+Conclusions
+8.1
+Summary
+In this paper we have explored aspects of Non-Lorentzian CFTs with additional Lie Al-
+gebraic symmetries. First we have reproduced the Non-Lorentzian Kaˇc-Moody algebra by
+taking singular limit from the Virasoro Kaˇc-moody algebra.
+After this we attempt to construct the same from intrinsic viewpoint without any knowledge
+of the parent algebra. We see that 2d Carrollian Conformal symmetry allows an infinite
+number of Noether currents. We then take a few pairs of conserved currents (introducing
+flavour indices to distinguish them) satisfying the conditions for EM tensor components in
+2d Carrollian Conformal symmetry. After this we derive the Ward identities associated with
+those conserved currents. We also derive OPEs of a general 2d Carrollian Conformal field
+with the current. After this we find the conserved charge operators associated with these
+currents. The transformation generated by these charges on a generic field is also derived.
+Here we first encounter the transformation matrices which we encounter in relativistic
+CFT with Kac-Moody algebra. We introduce current primary fields which turn out to
+be analogous to the Virasoro Kaˇc-Moody primary fields. After this the current current
+OPEs are derived. While doing so, the structure constants emerge from the action of the
+transformation matrices on the currents. After this, global internal symmetry is applied on
+two point and three point correlation and it turns out that the structure constants we have
+encountered while calculating the current current OPEs satisfy Jacobi identity. After this
+we derive the OPE between the Energy Momentum Tensor components and the current
+components. Using all these OPEs we derive the algebra of the current modes and the
+Energy Momentum tensor modes. The algebra transpire to be identical to the algebra
+obtained from limits of Virasoro Kaˇc-Moody algebra.
+Later in the paper, we attempt to construct the EM tensor modes (which forms the BMS
+algebra) from the current modes through Sugawara construction. Here we see that the
+Sugawara construction only takes us to the BMS algebra with one of the central charges
+to be zero. We needed another modification to the Sugawara construction in order to get
+a fully centrally extended BMS algebra. Using the expression of the EM tensor modes in
+terms of the current modes, we calculate the OPEs of EM tensor fields with themselves and
+with currents and see that the OPEs thus derived matches with the OPEs in the earlier
+section.
+After this we have a brief look at the tensionless string.
+When we look at the mode
+expansion of the coordinates (treating them as scalar fields), we see that the modes satisfy
+a special case of Non Lorentzian U(1) Kac-Moody algebra. We also look at the expression
+of modes of the classical energy momentum tensor in terms of these U(1) modes and see
+that classically this matches with the expression we have earlier derived for Non-Lorentzian
+Sugawara construction. Hence we see that U(1) NLKM currents are intrinsically present
+in the tensionless string.
+In section 6, we take a correlation function of a string of BMS current primary fields. Using
+the OPE definition of the BMS primary field, we arrive at the Non-Lorentzian version of the
+– 38 –
+
+Knizhnik Zamolodchikov equations. Finally, in section 7, we derive all the earlier results
+by taking limits from the parent algebra.
+8.2
+Discussions and future directions
+We mentioned in the introduction that our results in this paper lay the groundwork for a
+large number of applications, most importantly to the construction of a holographic dictio-
+nary for asymptotic flat spacetimes and also the understanding of tensionless strings. We
+have built the underlying algebraic structures in this paper which would be of importance
+to the quantum field theories that are at the heart of these problems.
+One of the most important and immediate next steps is to construct a Non-Lorentzian
+Wess-Zumino-Witten model that realises these symmetries. This would be central to the
+understanding of tensionless null strings moving on arbitrarily curved manifolds. The work
+on this is currently underway.
+The tensionless limit of string theory on arbitrary backgrounds is intimately related to this
+and as mentioned in the introduction, we wish to revisit the construction of [36] in the
+light of our findings in this paper. We believe that this limit would lead to null tensionless
+strings in AdS. This is to be contrasted with tensionless strings in AdS considered in e.g.
+[47] and subsequent work in this direction, which are tensionless but not null. A better
+understanding of the differences and perhaps similarities between the two approaches would
+be important.
+A generalisation of the methods outlined in this paper would be carried out for 3d Carrollian
+and 3d Galilean theories. The structures are expected to remain similar for the 3d Galilean
+theories, as the infinite dimensional structure of the algebra without the extra currents
+remains intact when generalised to higher dimensions. But for Carrollian theories, owing
+to the fundamental difference between BMS3 and BMS4, where one copy of the Virasoro
+in the 3d case gets enhanced to two Virasoros in the 4d case and supertranslations develop
+two legs instead of one, the construction of the quantum field theories with additional
+non-abelian currents would be more involved.
+Acknowledgements
+We thank Aritra Banerjee, Rudranil Basu and Niels Obers for interesting discussions and
+comments on an initial version of the manuscript.
+The work of AB is partially supported by a Swarnajayanti fellowship (SB/SJF/2019-
+20/08) from the Science and Engineering Research Board (SERB) India, the SERB grant
+(CRG/2020/002035), and a visiting professorship at ´Ecole Polytechnique Paris. AB also
+acknowledges the warm hospitality of the Niels Bohr Institute, Copenhagen during later
+stages of this work. RC is supported by the CSIR grant File No: 09/092(0991)/2018-EMR-
+I. AS is financially supported by a PMRF fellowship, MHRD, India. RK acknowledges the
+support of the Department of Atomic Energy, Government of India, under project number
+RTI4001. DS would like to thank ICTS, Bengaluru for hospitality during the course of this
+project.
+– 39 –
+
+APPENDICES
+A
+Carroll Multiplets
+In this appendix, we review the construction of Carrollian boost multiplets in two dimen-
+sions.
+In two space-time dimensions, the Carrollian boost (CB) transformation is defined as:
+x → x′ = x , t → t′ = t + vx ; or equivalently, as:
+�
+x
+t
+�
+−→
+�
+x′
+t′
+�
+=
+�
+exp
+��
+0 0
+v 0
+��� �
+x
+t
+�
+⇐⇒ xµ → x′µ =
+�
+evB(2)�µ
+ν xν
+(A.1)
+with
+B(2) :=
+�
+0 0
+1 0
+�
+(A.2)
+being the 2D representation of the CB generator B that is clearly not diagonalizable.
+Taking a cue from the Lorentz covariance of Lorentz tensors, it was postulated in [] that
+a rank-n Carrollian Cartesian tensor field Φ with ‘boost-charge’ ξ transforms under the
+Carrollian boost as:
+Φµ1...µn(t, x) −→ ˜Φµ1...µn(t′, x′) =
+�
+e−ξvB(2)
+�µ1
+ν1...
+�
+e−ξvB(2)
+�µn
+νnΦν1...νn(t, x)
+⇐⇒
+Φ(t, x) −→ ˜Φ(t′, x′) =
+� n
+�
+i=1
+e−ξvB(2)
+�
+Φ(t, x) = e
+−ξv
+n
+�
+i=1
+B(2)Φ(t, x)
+(A.3)
+where µi, νi are Carrollian space-time indices and for matrices, the left index denotes row
+while the right one denotes column; repeated indices are summed over and, in (A.3), indices
+are suppressed. It is to be noted that the up/down appearance of a tensor-index does not
+matter; only the left/right ordering is important.
+Clearly, the Carrollian Cartesian tensors defined above are decomposible.
+So, we now
+construct indecomposible Carrollian multiplets from these tensors. We begin by recognizing
+that:
+B(2) = J−
+(l= 1
+2 )
+(A.4)
+which is the lowering ladder operator in the SU(2) spin-1
+2 representation. Thus,
+n�
+i=1
+B(2) in
+(A.3) can be decomposed into indecomposable representations of J− using the technique
+of ‘addition of n spin-1
+2 angular momenta’ in quantum mechanics, such that:
+B(d) ≡ J−
+(l= d−1
+2 )
+(A.5)
+It is evident that the representations B(d) of the classical CB generator are indecomposable
+since their only generalized eigenvalue is 0 and it has geometric multiplicity 1. But, these
+representations are reducible for d ≥ 2.
+– 40 –
+
+A multi-component field transforming under the d-dimensional representation of CB, B(d),
+will be called a Carrollian multiplet of rank d−1
+2
+with d components, denoted by
+Φm
+(l= d−1
+2 )
+with
+m = 1 − d
+2
+, 3 − d
+2
+, ..., d − 1
+2
+By treating the µ = t index as spin-1
+2 up-state and the µ = x index as spin-1
+2 down-state,
+components Φm
+(l) of a Carrollian multiplet arise precisely as such linear combinations (with
+proper Clebsch-Gordon coefficients) of the components of a Cartesian tensor of an allowed
+rank n that would appear while expanding the |l, m⟩ states in an allowed |s1, s2, ..., sn⟩
+basis (where |si⟩ are Jz
+( 1
+2 ) eigenstates). So, as a linear combination of the components of
+a rank-n Cartesian tensor, one can obtain multipltes of ranks: 0, 1, 2, ..., n
+2 for even n and
+1
+2, 3
+2, 5
+2, ..., n
+2 for odd n. As an example, we see how Carrollian multiplets of ranks 1
+2 and 3
+2
+are constructed from a rank-3 Cartesian tensor:
+Φ
+3
+2
+( 3
+2 )(t, x) := Φttt(t, x)
+Φ
+1
+2
+( 3
+2 )(t, x) :=
+1
+√
+3
+�
+Φttx + Φtxt + Φxtt�
+(t, x)
+Φ
+− 1
+2
+( 3
+2 )(t, x) :=
+1
+√
+3
+�
+Φtxx + Φxtx + Φxxt�
+(t, x)
+Φ
+− 3
+2
+( 3
+2 )(t, x) := Φxxx(t, x)
+Φ
+1
+2
+( 1
+2 )(t, x) :=
+1
+√
+a2 + b2 + c2
+�
+aΦttx + bΦtxt + cΦxtt�
+(t, x)
+with a + b + c = 0
+Φ
+− 1
+2
+( 1
+2 )(t, x) :=
+1
+√
+a2 + b2 + c2
+�
+aΦxxt + bΦxtx + cΦtxx�
+(t, x)
+(As the tuple (a, b, c) in R3 lies on the plane a + b + c = 0 which is spanned by two basis
+vectors, two linearly independent rank-1
+2 multiplets arise.)
+A rank-l Carrollian multiplet with boost-charge ξ thus transforms under the 2l + 1 dimen-
+sional representation of the CB as:
+Φm
+(l)(t, x) −→ ˜Φm
+(l)(t′, x′) =
+�
+e−ξvJ−
+(l)
+�m
+m′Φm′
+(l)(t, x)
+(A.6)
+After constructing the Carrollian multiplets from the Cartesian tensors as demonstrated
+above, the components of the multiplets can always be redefined such that:
+in (A.6), J−
+(l) is replaced by M(l) := sub-diag (1, 1, ..., 1)2l+1 .
+Hence, instead of the actual J−
+(l) matrix, only the indecomposable Jordan-block structure
+is important for defining the CB transformation property of the Carrollian multiplets.
+We conclude this appendix with the following observation. Since the finite dimensional
+indecomposable representations of B are not symmetric (or Hermitian), one can start
+with:
+�
+t
+x
+�
+−→
+�
+t′
+x′
+�
+=
+�
+exp
+��
+0 v
+0 0
+��� �
+t
+x
+�
+⇐⇒ xµ → x′µ =
+�
+evB′
+(2)
+�µ
+ν xν
+– 41 –
+
+where
+B′
+(2) :=
+�
+0 1
+0 0
+�
+and follow the preceding argument to construct
+B′
+(d) ≡ J+
+(l= d−1
+2 )
+which is the SU(2) raising ladder operator. But, as J+
+(l) = (J−
+(l))T, the raising and lowering
+operators’ representation matrices are related to each other by the similarity transforma-
+tion:
+S = anti-diag (1, 1, ..., 1)2l+1
+and consequently, B and B′ furnish two equivalent representations of the CB generator.
+B
+Calculation of Sugawara Construction Commutators
+In this appendix, we provide the details of the calculation of commutators of the NLKM
+algebra from the Sugawara construction.
+Calculating [Mm, Jb
+n]
+[Mm, Jb
+n] =
+1
+2k2
+�
+a
+�
+l
+[Ka
+l Ka
+m−l, Jb
+n]
+(Using (4.12))
+=
+1
+2k2
+�
+a
+�
+l
+�
+[Ka
+l , Jb
+n]Ka
+m−l + Ka
+l [Ka
+m−l, Jb
+n]
+�
+=
+1
+2k2
+�
+a
+�
+l
+�
+(−i
+�
+c
+fbacKc
+n+l − nk2δn+l,0δab)Ka
+m−l
++ Ka
+l (−i
+�
+c
+fbacKc
+m+n−l − nk2δm+n−l,0δab)
+�
+(Using (2.21))
+=
+i
+2k2
+�
+a,c
+�
+l
+fabc�
+Kc
+n+lKa
+m−l + Ka
+l Kc
+m+n−l
+�
+− nKb
+m+n
+(∵ −fbac = fabc)
+where we have omitted the limits in summation over a and c and it is understood that
+it runs over 1 to dim(g). Now, �
+l Kc
+n+lKa
+m−l = �
+l Kc
+m+n−lKa
+l simply by translating l.
+Hence,
+[Mm, Jb
+n] = i
+k2
+�
+a,c
+�
+l
+fabcKa
+l Kc
+m+n−l − nKb
+m+n
+(B.1)
+Now, again using antisymmetry property of fabc
+�
+a,c
+�
+l
+fabcKa
+l Kc
+m+n−l =
+�
+a,c
+�
+l
+fabcKa
+m+n−lKc
+l
+– 42 –
+
+=
+�
+a,c
+�
+l
+fcbaKc
+m+n−lKa
+l
+= −
+�
+a,c
+�
+l
+fabcKa
+l Kc
+m+n−l
+⇒
+�
+a,c
+�
+l
+fabcKa
+l Kc
+m+n−l = 0
+(B.2)
+Hence, (B.1) can be written as,
+[Mm, Jb
+n] = −nKb
+m+n
+(B.3)
+Calculating [Lm, Kb
+n]
+Again, using (4.12) and (2.21), we can get the following,
+[Lm, Kb
+n] =
+1
+2k2
+�
+a
+� �
+l≤−1
+([Ja
+l , Kb
+n]Ka
+m−l + Ka
+l [Ja
+m−l, Kb
+n])
++
+�
+l>−1
+([Ja
+m−l, Kb
+n]Ka
+l + Ka
+m−l[Ja
+l , Kb
+n])
+�
+=
+1
+2k2
+�
+a
+� �
+l≤−1
+�
+i
+�
+c
+fabc(Kc
+l+nKa
+m−l + Ka
+l Kc
+m+n−l) + k2lKa
+m−lδl+n,0δab
++ k2(m − l)Ka
+l δm+n−l,0δab
+�
++
+�
+l>−1
+�
+i
+�
+c
+fabc(Kc
+m+n−lKa
+l + Ka
+m−lKc
+l+n)
++ k2(m − l)Ka
+l δm+n−l,0δab + k2lKa
+m−lδl+n,0δab
+��
+= i
+k2
+�
+a,c
+�
+l
+fabc(Kc
+m+n−lKa
+l + Ka
+m−lKc
+l+n)
++
+1
+2k2
+�
+a
+�
+l
+�
+k2(m − l)Ka
+l δm+n−l,0δab + k2lKa
+m−lδl+n,0δab
+�
+(B.4)
+First term vanishes again because of the antisymmetry of f and commutativity of K’s.
+Therefore,
+[Lm, Kb
+n] = −nKb
+m+n
+(B.5)
+Calculating [Lm, Jb
+n]
+Following similar steps as before, we get,
+[Lm, Jb
+n] = 1
+2k2
+�
+a
+� �
+l≤−1
+([Ja
+l Ka
+m−l, Jb
+n] + [Ka
+l Ja
+m−l, Jb
+n]) +
+�
+l>−1
+([Ja
+m−lKa
+l , Jb
+n]
++ [Ka
+m−lJa
+l , Jb
+n])
+�
+− (k1 + 2Cg)
+k2
+[Mm, Jb
+n]
+(B.6)
+– 43 –
+
+First term can be simplified as,
+�
+a
+�
+l≤−1
+([Ja
+l Ka
+m−l, Jb
+n] + [Ka
+l Ja
+m−l, Jb
+n])
+=
+�
+a
+�
+l≤−1
+�
+Ja
+l [Ka
+m−l, Jb
+n] + [Ja
+l , Jb
+n]Ka
+m−l + Ka
+l [Ja
+m−l, Jb
+n] + [Ka
+l , Jb
+n]Ja
+m−l
+�
+=
+�
+a
+�
+l≤−1
+�
+i
+�
+c
+fabcJa
+l Kc
+m+n−l + i
+�
+c
+fabcJc
+l+nKa
+m−l + i
+�
+c
+fabcKa
+l Jc
+m+n−l
++ i
+�
+c
+fabcKc
+l+nJa
+m−l − k2nJa
+l δm+n−l,0δab + k1lKa
+m−lδl+n,0δab
++ k1(m − l)Ka
+l δm+n−l,0δab − k2nJa
+m−lδl+n,0δab
+�
+=
+�
+l≤−1
+�
+i
+�
+a,c
+fabcJa
+l Kc
+m+n−l + i
+�
+a,c
+fabcJc
+l+nKa
+m−l + i
+�
+a,c
+fabcKa
+l Jc
+m+n−l
++ i
+�
+a,c
+fabcKc
+l+nJa
+m−l
+�
+− n
+�
+l≤−1
+(k2Jb
+m+n + k1Kb
+m+n)(δm+n−l,0 + δl+n,0) (B.7)
+Similarly, the second term looks like,
+�
+a
+�
+l>−1
+([Ka
+m−lJa
+l , Jb
+n] + [Ja
+m−lKa
+l , Jb
+n])
+=
+�
+l>−1
+�
+i
+�
+a,c
+fabcKa
+m−lJc
+l+n + i
+�
+a,c
+fabcKc
+m+n−lJa
+l + i
+�
+a,c
+fabcJa
+m−lKc
+l+n
++ i
+�
+a,c
+fabcJc
+m+n−lKa
+l
+�
+− n
+�
+l>−1
+(k2Jb
+m+n + k1Kb
+m+n)(δm+n−l,0 + δl+n,0) (B.8)
+Hence, we have,
+[Lm, Jb
+n] =
+1
+2k2
+�
+i
+�
+a,c
+fabc
+�
+�
+0≤l≤n−1
+Jc
+l Ka
+m+n−l −
+�
+0≤l≤n−1
+Ka
+l Jc
+m+n−l −
+�
+0≤l≤n−1
+Ka
+m+n−lJc
+l
++
+�
+0≤l≤n−1
+Jc
+m+n−lKa
+l
+��
+− nJb
+m+n + 2Cg
+k2
+nKb
+m+n
+=
+1
+2k2
+�
+i
+�
+a,c
+fabc
+�
+�
+0≤l≤n−1
+[Jc
+l , Ka
+m+n−l] +
+�
+0≤l≤n−1
+[Jc
+m+n−l, Ka
+l ]
+��
+− nJb
+m+n + 2Cg
+k2
+nKb
+m+n
+=
+1
+2k2
+�
+− 2n
+�
+a,c,d
+fabcfcadKd
+m+n + 2ik2
+�
+a,c
+�
+0≤l≤n−1
+fabcδm+n,0δab
+�
+− nJb
+m+n + 2Cg
+k2
+nKb
+m+n
+= −nJb
+m+n + n
+k2
+(2CgKb
+m+n −
+�
+a,c,d
+fabcfcadKd
+m+n)
+– 44 –
+
+= −nJb
+m+n
+Hence, we have,
+[Lm, Jb
+n] = −nJb
+m+n
+(B.9)
+Calculating [Lm, Mn]
+[Lm, Mn] =
+1
+2k2
+�
+a
+�
+l
+�
+[Lm, Ka
+l ]Ka
+n−l + Ka
+l [Lm, Ka
+n−l]
+�
+=
+1
+2k2
+�
+a
+�
+l
+�
+− lKa
+m+lKa
+n−l − (n − l)Ka
+l Ka
+m+n−l
+�
+(Using (B.5))
+=
+1
+2k2
+�
+a
+�
+l
+{−(l − m)Ka
+l Ka
+n+m−l − (n − l)Ka
+l Ka
+m+n−l}
+= (m − n) 1
+2k2
+�
+a
+�
+l
+Ka
+l Ka
+n+m−l
+where , we have done the re-labelling l → l − m in the second step. Hence, we get,
+[Lm, Mn] = (m − n)Mm+n
+(B.10)
+Calculating [Lm, Ln]
+It can be carried out in similar fashion by writing one of the L’s in terms of J’s and K’s
+using (4.12) and then using (B.5), (B.9) and (B.10), we can do the following,
+[Lm, Ln] = 1
+2k2
+�
+a
+� �
+l≤−1
+([Lm, Ja
+l ]Ka
+n−l + Ja
+l [Lm, Ka
+n−l] + [Lm, Ka
+l ]Ja
+n−l + Ka
+l [Lm, Ja
+n−l])
++
+�
+l>−1
+([Lm, Ja
+n−l]Ka
+l + Ja
+n−l[Lm, Ka
+l ] + [Lm, Ka
+n−l]Ja
+l + Ka
+n−l[Lm, Ja
+l ])
+�
+− (k1 + 2Cg)
+k2
+[Lm, 1
+2k2
+�
+l
+Ka
+l Ka
+n−l]
+=
+1
+2k2
+�
+a
+� �
+l≤−1
+(−lJa
+m+lKa
+n−l + lJa
+l Ka
+m+n−l − lKa
+m+lJa
+n−l + lKa
+l Ja
+m+n−l)
++
+�
+l>−1
+(lJa
+m+n−lKa
+l − lJa
+n−lKa
+m+l + lKa
+m+n−lJa
+l − lKa
+n−lJa
+m+l)
+�
+− m(k1 + 2Cg)
+k2
+1
+2k2
+�
+a
+�
+l
+Ka
+l Ka
+m+n−l − nLm+n
+(B.11)
+Changing the index l → (l − m) in the negative terms in the curly brackets and then
+simplifying, we can get,
+[Lm, Ln] =
+1
+2k2
+�
+a
+� �
+l≤−1
+m(Ja
+l Ka
+n+m−l + Ka
+l Ja
+m+n−l) +
+�
+l>−1
+m(Ja
+n+m−lKa
+l + Ka
+n+m−lJa
+l )
+– 45 –
+
++
+m−1
+�
+l=0
+(m − l)([Ja
+l , Ka
+n+m−l] − [Ja
+n+m−l, Ka
+l ])
+�
+− m(k1 + 2Cg)
+k2
+1
+2k2
+�
+a
+�
+l
+Ka
+l Ka
+m+n−l − nLm+n
+=
+1
+2k2
+�
+a
+� m−1
+�
+l=0
+(m − l)(k2lδm+n,0 − k2(n + m − l)δm+n,0)
+�
++ (m − n)Lm+n
+(B.12)
+which upon further simplification gives,
+[Lm, Ln] = (m − n)Lm+n + dim(g)
+6
+m(m2 − 1)δm+n,0
+(B.13)
+Other commutation relations of type [Mm, Kb
+n] and [Mm, Mn] vanish trivially because of
+the vanishing [Ka
+m, Kb
+n] commutator.
+C
+Modified Sugawara Construction
+In this appendix, we give details of the modified Sugawara construction. First we start
+with the modified construction in the relativistic case and then explain the construction in
+the non-Lorentzian case.
+We begin by calculating [ ˜Lm, ˜Ln] with ˜Lm given in (4.15)
+[ ˜Lm, ˜Ln] = [LS
+m + imθaja
+m + 1
+2kθ2δm,0, LS
+n + inθbjb
+n + 1
+2kθ2δn,0]
+= [LS
+m, LS
+n] + inθb[LS
+m, jb
+n] + imθa[ja
+m, LS
+n] − mnθaθb[ja
+m, jb
+n]
+= (m − n)LS
+m+n − in2θbjb
+m+n + im2θaja
+m+n − mnθaθb�
+ifabcjc
+m+n + mkδm+nδab
+�
+= (m − n) ˜Lm+n − i(m2 − n2)θaja
+m+n − 1
+2kθ2(m − n)δm+n,0 + i(m2 − n2)θaja
+m+n
+− imnθaθbfabcjc
+m+n + m3kθ2δm+n,0 + c
+12(m3 − m)δm+n,0
+= (m − n) ˜Lm+n +
+� c
+12 + kθ2�
+(m3 − m)δm+n,0
+= (m − n) ˜Lm+n + ˜c
+12(m3 − m)δm+n,0
+(C.1)
+Here ˜c = c + 12kθ2. In the fifth line we have used the fact that θaθbfabc vanishes due to
+antisymmetry of fabc over indices a and b, also the fact that nδm+n,0 = −mδm+n,0.
+Now defining ˜L and ˜
+M as in (4.17) we would like to calculate [˜Lm, ˜Ln] and [˜Lm, ˜
+Mn]. The
+calculation of [˜Lm, ˜Ln] will be exactly similar to that of [ ˜Lm, ˜Ln], while that of [˜Lm, ˜
+Mn]
+will be as following
+[˜Lm, ˜
+Mn] = [LS
+m + imθaJa
+m + 1
+2k2θ2δn,0, MS
+n + inθbKb
+n + 1
+2k2θ2δn,0]
+= [LS
+m, MS
+n ] + imθa[LS
+m, Ka
+n] + inθb[Ja
+m, MS
+n ] − mnθaθb[Jb
+m, Ka
+n]
+– 46 –
+
+= (m − n)MS
+m+n − in2θbKb
+m+n + im2θaKa
+m+n − mnθaθb�
+ifabcKc
+m+n + mk2δm+nδab
+�
+= (m − n)MS
+m+n − i(m2 − n2)θaKa
+m+n − 1
+2k2θ2(m − n)δm+n,0 + i(m2 − n2)θaKa
+m+n
+− imnθaθbfabcKc
+m+n + m3k2θ2δm+n,0
+= (m − n)Mm+n + k2θ2(m3 − m)δm+n,0
+(C.2)
+Hence, just by doing a slight modification to the Sugawara construction we end up with
+fully centrally extended BMS algebra with central charges given in (4.19).
+D
+Details of OPE calculations
+Calculation of T-J OPE
+First consider
+J a
+v (u1, v1)
+�
+b
+�
+(J b
+uJ b
+v )(u2, v2)
+�
+=
+�
+v2
+dv′
+v′ − v2
+�
+u2
+du′
+u′ − u2
+�
+b
+�
+J a
+v (u1, v1)J b
+u(u′, v′)J b
+v (u2, v2) + J b
+u(u′, v′)J a
+v (u1, v1)J b
+v (u2, v2)
+�
+=
+�
+v2
+dv′
+v′ − v2
+�
+u2
+du′
+u′ − u2
+�
+b
+�
+ifabc J c
+v (u′, v′)J b
+v (u2, v2)
+(v1 − v′)
++ k2δab J b
+v (u2, v2)
+(v1 − v′)2 + J b
+u(u′, v′) × 0
+�
+=
+�
+v2
+dv′
+v′ − v2
+�
+u2
+du′
+u′ − u2
+��
+b
+ifabc (J c
+v J b
+v )(u2, v2)
+(v1 − v′)
++ k2
+J a
+v (u2, v2)
+(v1 − v′)2
+�
+=
+�
+b
+ifabc (J c
+v J b
+v )(u2, v2)
+v12
++ k2
+J a
+v (u2, v2)
+v2
+12
+(D.1)
+Similarly we get
+J a
+v (u1, v1)
+�
+b
+�
+(J b
+v J b
+u)(u2, v2)
+�
+=
+�
+v2
+dv′
+v′ − v2
+�
+u2
+du′
+u′ − u2
+��
+b
+ifabc (J b
+v J c
+v )(u2, v2)
+v12
++ k2
+J a
+v (u′, v′)
+v2
+12
+�
+=
+�
+b
+iF abc (J b
+v J c
+v )(u2, v2)
+v12
++ k2
+J a
+v (u2, v2)
+v2
+12
+(D.2)
+Summing up (3.66) and (D.2), we see that the first terms in the expressions cancel each
+other due to the antisymmetry of the structure constant, so we get
+J a
+v (u1, v1)
+�
+b
+�
+(J b
+uJ b
+v )(u2, v2) + (J b
+v J b
+u)(u2, v2)
+�
+= 2k2
+J a
+v (u2, v2)
+v2
+12
+= 2k2
+�J a
+v (u1, v1)
+v2
+21
++ ∂vJ a
+v (u1, v1)
+v21
+�
+(D.3)
+– 47 –
+
+Also it can be trivially shown that
+J a
+v (u1, v1)
+�
+b
+�
+(J a
+v J a
+v )(u2, v2)
+�
+= 0
+(D.4)
+From (D.3) and (D.4), we can determine
+Tv(u1, v1)J b
+v (u2, v2) ∼ regular
+Tu(u1, v1)J b
+v (u2, v2) ∼ J b
+v (u2, v2)
+v2
+12
++ ∂vJ b
+v (u2, v2)
+v12
++ . . .
+(D.5)
+Similarly we get
+Tv(u1, v1)J a
+u (u2, v2) ∼ J a
+v (u2, v2)
+v2
+12
++ ∂vJ a
+v (u2, v2)
+v12
++ . . .
+Tu(u1, v1)J a
+u (u2, v2) ∼ J a
+u (u2, v2)
+v2
+12
++ ∂vJ a
+u (u2, v2)
+v12
++ u12
+v2
+12
+∂uJ a
+u (u2, v2)
++ 2u12
+v12
+�J a
+v (u1, v1)
+v2
+12
++ ∂vJ a
+v (u2, v2)
+v2
+12
+�
++ . . .
+(D.6)
+Calculation of T-T OPE
+First consider
+Tu(u1, v1)(J a
+v J a
+v )(u2, v2)
+=
+�
+v2
+dv′
+v′ − v2
+�
+u2
+du′
+u′ − u2
+�
+Tu(u1, v1)J a
+v (u, v)J a
+v (u2, v2) + J a
+v (u, v)Tu(u1, v1)v2, u2
+�
+=
+�
+v2
+dv′
+v′ − v2
+�
+u2
+du′
+u′ − u2
+�J a
+v (u, v)J a
+v (u2, v2)
+(v1 − v′)2
++ ∂v′J a
+v (u, v)J a
+v (u2, v2)
+(v1 − v′)
++ J a
+v (u, v)J a
+v (u2, v2)
+v2
+12
++ J a
+v (u, v)∂v2J a
+v (u2, v2)
+v12
+�
+= 2(J a
+v J a
+v )(u2, v2)
+v2
+12
++
+�
+v2
+dv′
+v′ − v2
+�
+u2
+du′
+u′ − u2
+�∂v′[(J a
+v J a
+v )(u2, v2) + (∂vJ a
+v J a
+v )(u2, v2) + . . . ]
+(v1 − v′)
++ ∂v2[(J a
+v J a
+v )(u2, v2) + (∂vJ a
+v J a
+v )(u2, v2) + . . . ]
+v12
+�
+= 2(J a
+v J a
+v )(u2, v2)
+v2
+12
++ ∂v(J a
+v J a
+v )(u2, v2)
+v12
+(D.7)
+So we obtain
+Tu(u1, v1)Tv(u2, v2) ∼ 2(J a
+v J a
+v )(u2, v2)
+v2
+12
++ ∂v(J a
+v J a
+v )(u2, v2)
+v12
++ . . .
+(D.8)
+Similarly, we can check
+Tv(u1, v1)Tv(u2, v2) ∼ regular,
+(D.9)
+– 48 –
+
+and
+Tu(u1, v1)Tu(u2, v2) ∼ 2Tu(u2, v2)
+v2
+12
++∂vTu(u2, v2)
+v12
++ u12
+v2
+12
+∂uTu(u2, v2)
++ 2u12
+v12
+�
+2Tv(u2, v2)
+v2
+12
++ ∂vTv(u2, v2)
+v12
+�
++ dim(g)
+v4
+12
++ . . .
+(D.10)
+E
+K-Z equation in field theory approach
+We start with
+⟨∂uΦ(u, v)Φ1(u1, v1) . . . Φn(un, vn)⟩ = ⟨∂uΦ(u, v)X⟩
+= −
+�
+v
+dv′
+2πi⟨Tv(u′, v′)Φ(u, v)X⟩
+= −
+�
+v
+dv′
+2πi
+1
+2k2
+⟨(J a
+v J a
+v )(u′, v′)Φ(u, v)X⟩
+= −
+�
+v
+dv′
+2πi
+1
+2k2
+�
+v′
+dv′′
+2πi
+1
+(v′′ − v′)⟨
+�
+J a
+v (u′′, v′′)Φ(u, v)J a
+v (u′, v′) + J a
+v (u′′, v′′)J a
+v (u′, v′)Φ(u, v)
+�
+X⟩
+−
+�
+v
+dv′
+2πi
+1
+2k2
+⟨(J a
+v J a
+v Φ)(u, v)X⟩ + . . .
+= −
+�
+v
+dv′
+2πi
+1
+2k2
+�
+v′
+dv′′
+2πi
+1
+(v′′ − v′)⟨
+�
+ta
+k
+(v′′ − v)Φ(u, v)J a
+v (u′, v′) +
+ta
+K
+(v′ − v)J a
+v (u′′, v′′)Φ(u, v)
+�
+X⟩ + 0
+= I1 + I2
+(E.1)
+(Where, second line is (6.1), third line is the Non-Lorentzian Sugawara construction, fourth
+line uses definition of normal ordering and the fact that OPE = contractions + normal
+ordered product + . . . )
+Now,
+I1 = − ta
+k
+2k2
+�
+v
+dv′
+2πi
+�
+v′
+dv′′
+2πi
+1
+(v′′ − v′)(v′′ − v)⟨Φ(u, v)J a
+v (u′, v′)X⟩
+= − ta
+k
+2k2
+�
+v
+dv′
+2πi
+1
+(v′ − v)⟨Φ(u, v)J a
+v (u′, v′)X⟩
+= ta
+k
+2k2
+�
+j
+�
+vj
+dv′
+2πi
+1
+(v′ − v)⟨Φ(u, v)Φ1(u1, v1)...
+�
+J a
+v (u′, v′)Φj(uj, vj)
+�
+. . . Φn(un, vn)⟩
+=
+1
+2k2
+�
+j
+�
+vj
+dv′
+2πi
+ta
+K ⊗ ta
+K,j
+(v′ − v)(v′ − vj)⟨Φ(u, v)Φ1(u1, v1)...Φj(uj, vj) . . . Φn(un, vn)⟩
+=
+1
+2k2
+�
+j
+ta
+K ⊗ ta
+K,j
+(vj − v) ⟨Φ(u, v)Φ1(u1, v1)...Φj(uj, vj) . . . Φn(un, vn)⟩
+(E.2)
+– 49 –
+
+(third line involves a change in contour)
+and similarly,
+I2 = − ta
+k
+2k2
+�
+v
+dv′
+2πi
+�
+v′
+dv′′
+2πi
+1
+(v′′ − v′)(v′ − v)⟨J a
+v (v′′, x′′)Φ(u, v)X⟩
+=
+1
+2k2
+�
+j
+ta
+K ⊗ ta
+K,j
+(vj − v) ⟨Φ(u, v)Φ1(u1, v1)...Φj(uj, vj) . . . Φn(un, vn)⟩
+= I1
+(E.3)
+Using the above two results,we get one of the Carrollian K-Z equations
+�
+∂ui + 1
+k2
+�
+j̸=i
+ta
+Ri,K ⊗ ta
+Rj,K
+vij
+�
+⟨Φ1(u1, v1) . . . Φn(un, vn)⟩ = 0
+(E.4)
+The other equation can be obtained similarly.
+F
+NL KZ equation as a limit
+Taking the following linear combination of the equations (7.30a) and (7.30b),
+(7.30a) × ⟨¯φ¯r1¯R1( ¯w1)...¯φ¯rN
+¯RN ( ¯wN)⟩ + ((7.30b)) × ⟨φr1
+R1(w1)...φrN
+RN (wN)⟩ = 0
+(F.1)
+gives us,
+�
+�∂wi + ∂ ¯wi − 2γ
+�
+j̸=i
+�
+a(ta
+Ri ⊗′ ¯I)ri,¯ri
+si,¯si(ta
+Rj ⊗′ ¯I)rj,¯rj
+sj,¯sj
+wi − wj
+− 2¯γ
+�
+j̸=i
+�
+a(I ⊗′ ¯ta¯Ri)ri,¯ri
+si,¯si(I ⊗′ ¯ta¯Rj)rj,¯rj
+sj,¯sj
+¯wi − ¯wj
+�
+�
+⟨...Φsi,¯si
+Ri, ¯Ri(wi, ¯wi)...Φsj,¯sj
+Rj, ¯Rj(wj, ¯wj)...⟩ = 0
+(F.2)
+We have (wi, ¯wi) = t ± ϵx ⇒ ∂wi = 1
+2(∂ti + ∂xi
+ϵ ); ∂ ¯wi = 1
+2(∂ti − ∂xi
+ϵ ) such that the Carrollian
+limit is achieved by taking ϵ → 0. Using these we get,
+⇒
+�
+�∂ti − 2
+�
+j̸=i
+X
+�
+� ⟨...Φsi,¯si
+Ri, ¯Ri(xi, ti)...Φsj,¯sj
+Rj, ¯Rj(xj, tj)...⟩ = 0
+(F.3)
+where(with tij = ti − tj and xij = xi − xj), we have,
+X =
+�
+γ
+tij + ϵxij
+�
+a
+(ta
+Ri ⊗′ ¯I)ri,¯ri
+si,¯si(ta
+Rj ⊗′ ¯I)rj,¯rj
+sj,¯sj +
+¯γ
+tij − ϵxij
+�
+a
+(I ⊗′ ¯ta¯Ri)ri,¯ri
+si,¯si(I ⊗′ ¯ta¯Rj)rj,¯rj
+sj,¯sj
+�
+(F.4)
+Inverting relations (1.11), we have,
+ta
+Ri ⊗′ ¯I = 1
+2(ta
+Ri,J −
+ta
+Ri,K
+ϵ
+) ; I ⊗′ ¯ta¯Ri = 1
+2(ta
+Ri,J +
+ta
+Ri,K
+ϵ
+)
+(F.5)
+– 50 –
+
+Hence,
+X = 1
+4
+�
+a
+�
+γ
+tij + ϵxij
+�
+ta
+Ri,J −
+ta
+Ri,K
+ϵ
+�ri,¯ri
+si,¯si
+�
+ta
+Rj,J −
+ta
+Rj,K
+ϵ
+�rj,¯rj
+sj,¯sj
++
+¯γ
+tij − ϵxij
+�
+ta
+Ri,J +
+ta
+Ri,K
+ϵ
+�ri,¯ri
+si,¯si
+�
+ta
+Rj,J +
+ta
+Rj,K
+ϵ
+�rj,¯rj
+sj,¯sj
+�
+= 1
+4
+�
+a
+�
+γ
+tij + ϵxij
+�
+ta
+Ri,J ⊗ ta
+Rj,J − 1
+ϵ (ta
+Ri,J ⊗ ta
+Rj,K + ta
+Ri,K ⊗ ta
+Rj,J)
++ 1
+ϵ2 ta
+Ri,K ⊗ ta
+Rj,K
+�ri,¯ri;rj,¯rj
+si,¯si;sj,¯sj
++
+¯γ
+tij − ϵxij
+�
+ta
+Ri,J ⊗ ta
+Rj,J + 1
+ϵ (ta
+Ri,J ⊗ ta
+Rj,K + ta
+Ri,K ⊗ ta
+Rj,J)
++ 1
+ϵ2 ta
+Ri,K ⊗ ta
+Rj,K
+�ri,¯ri;rj,¯rj
+si,¯si;sj,¯sj
+�
+(F.6)
+We introduce notation for convenience,
+�
+a
+(ta
+Ri,A ⊗ ta
+Rj,B)ri,¯ri;rj,¯rj
+si,¯si;sj,¯sj = tij
+AB (where A, B = J, K)
+(F.7)
+Therefore,
+X =1
+4
+� γ
+tij
+(1 − ϵxij
+tij
++ ϵ2 x2
+ij
+t2
+ij
++ ...)(tij
+JJ − 1
+ϵ (tij
+JK + tij
+KJ) + 1
+ϵ2 tij
+KK)
++ ¯γ
+tij
+(1 + ϵxij
+tij
++ ϵ2 x2
+ij
+t2
+ij
++ ...)(tij
+JJ + 1
+ϵ (tij
+JK + tij
+KJ) + 1
+ϵ2 tij
+KK)
+�
+⇒ X =
+1
+4tij
+�
+− (γ − ¯γ)
+ϵ
+(tij
+JK + tij
+KJ) + γ + ¯γ
+ϵ2
+tij
+KK) − xij
+4t2
+ij
+(γ − ¯γ)
+ϵ
+tij
+KK
+�
+(F.8)
+In limit ϵ → 0 and using (7.25), we get,
+⇒ X =
+1
+4tij
+� 2
+k2
+(tij
+JK + tij
+KJ) − 2(k1 + 2Cg)
+k2
+2
+tij
+KK) + xij
+4t2
+ij
+2
+k2
+tij
+KK
+�
+⇒ X =
+1
+2k2
+�(tij
+JK + tij
+KJ)
+tij
++ (xij
+t2
+ij
+− (k1 + 2Cg)
+k2tij
+)tij
+KK
+�
+(F.9)
+Therefore, (F.1) can be finally written as( in limit ϵ → 0),
+⇒
+�
+∂ti −
+�
+j̸=i
+1
+k2
+�(tij
+JK + tij
+KJ)
+tij
++
+�xij
+t2
+ij
+− (k1 + 2Cg)
+k2tij
+�
+tij
+KK
+��
+⟨...Φsi,¯si
+xi,ti(xi, ti)...Φsj,¯sj
+Rj, ¯Rj(xj, tj)...⟩ = 0
+⇒
+�
+∂ti − 1
+k2
+�
+j̸=i
+��
+a(ta
+Ri,J ⊗ ta
+Rj,K + ta
+Ri,K ⊗ ta
+Rj,J)ri,¯ri;rj,¯rj
+si,¯si;sj,¯sj
+tij
+– 51 –
+
++
+�xij
+t2
+ij
+− (k1 + 2Cg)
+k2tij
+� �
+a
+(ta
+Ri,K ⊗ ta
+Rj,K)ri,¯ri;rj,¯rj
+si,¯si;sj,¯sj
+��
+⟨...Φsi,¯si
+Ri, ¯Ri(xi, ti)...Φsj,¯sj
+Rj, ¯Rj(xj, tj)...⟩ = 0
+⇒
+�
+∂ti − 1
+k2
+�
+j̸=i
+��
+a(ta
+Ri,J ⊗ ta
+Rj,K + ta
+Ri,K ⊗ ta
+Rj,J)
+tij
++
+�xij
+t2
+ij
+− (k1 + 2Cg)
+k2tij
+� �
+a
+(ta
+Ri,K ⊗ ta
+Rj,K)
+��
+⟨...ΦRi, ¯Ri(xi, ti)...ΦRj, ¯
+Rj(xj, tj)...⟩ = 0
+(F.10)
+which is in agreement with (6.3).
+Now,we consider another linear combination,
+ϵ{(7.30a) × ⟨¯φ¯r1¯R1( ¯w1)...¯φ¯rN
+¯RN ( ¯wN)⟩ − (7.30b) × ⟨φr1
+R1(w1)...φrN
+RN (wN)⟩} = 0
+(F.11)
+Again following similar steps as before,
+⇒
+�
+∂xi − 2
+�
+j̸=i
+Y
+�
+⟨...Φsi,¯si
+Ri, ¯Ri(xi, ti)...Φsj,¯sj
+Rj, ¯Rj(xj, tj)...⟩ = 0
+(F.12)
+where,
+Y = ϵ
+�
+γ
+tij + ϵxij
+�
+a
+(ta
+Ri ⊗′ ¯I)ri,¯ri
+si,¯si(ta
+Rj ⊗′ ¯I)rj,¯rj
+sj,¯sj −
+¯γ
+tij − ϵxij
+�
+a
+(I ⊗′ ¯ta¯Ri)ri,¯ri
+si,¯si(I ⊗′ ¯ta¯Rj)rj,¯rj
+sj,¯sj
+�
+(F.13)
+Similar to what we did for X, we get an equation analogous to (F.8) using (F.5) and (F.7),
+Y = ϵ
+4{ γ
+tij
+(1 − ϵxij
+tij
++ ϵ2 x2
+ij
+t2
+ij
++ ...)(tij
+JJ − 1
+ϵ (tij
+JK + tij
+KJ) + 1
+ϵ2 tij
+KK)
+− ¯γ
+tij
+(1 + ϵxij
+tij
++ ϵ2 x2
+ij
+t2
+ij
++ ...)(tij
+JJ + 1
+ϵ (tij
+JK + tij
+KJ) + 1
+ϵ2 tij
+KK)}
+(F.14)
+Again using (7.25) and collecting the finite terms in the limit ϵ → 0, we get,
+Y = − 1
+2k2
+tij
+KK
+tij
+(F.15)
+Putting back in (F.12),
+�
+�∂xi + 1
+k2
+�
+j̸=i
+�
+a(ta
+Ri,K ⊗ ta
+Rj,K)
+tij
+�
+� ⟨...ΦRi, ¯Ri(xi, ti)...ΦRj, ¯Rj(xj, tj)...⟩ = 0
+(F.16)
+which is in agreement with (6.5).
+– 52 –
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diff --git a/ItE3T4oBgHgl3EQfugv_/content/tmp_files/load_file.txt b/ItE3T4oBgHgl3EQfugv_/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..22f8831b1b33c3f5fe340e21168c5945328cc08a
--- /dev/null
+++ b/ItE3T4oBgHgl3EQfugv_/content/tmp_files/load_file.txt
@@ -0,0 +1,1761 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf,len=1760
+page_content='Non-Lorentzian Kaˇc-Moody Algebras  Arjun Bagchi,a,b Ritankar Chatterjee,a Rishabh Kaushik,a,c Amartya Saha,a and  Debmalya Sarkar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='a,c  aIndian Institute of Technology Kanpur, Kanpur 208016, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  bCentre de Physique Theorique, Ecole Polytechnique de Paris, 91128 Palaiseau Cedex, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  cInternational Centre for Theoretical Sciences (ICTS-TIFR), Bengaluru 560089, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  E-mail: (abagchi, ritankar, amartyas)@iit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='in,  rishabh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='kaushik@icts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='in, sarkardebmalya01@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='com  Abstract: We investigate two dimensional (2d) quantum field theories which exhibit Non-  Lorentzian Kaˇc-Moody (NLKM) algebras as their underlying symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Our investigations  encompass both 2d Galilean (speed of light c → ∞) and Carrollian (c → 0) CFTs with ad-  ditional number of infinite non-Abelian currents, stemming from an isomorphism between  the two algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We alternate between an intrinsic and a limiting analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Our NLKM  algebra is constructed first through a contraction and then derived from an intrinsically  Carrollian perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We then go on to use the symmetries to derive a Non-Lorentzian  (NL) Sugawara construction and ultimately write down the NL equivalent of the Knizhnik  Zamolodchikov equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' All of these are also derived from contractions, thus providing  a robust cross-check of our analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='04686v1  [hep-th]  11 Jan 2023  Contents  1  Introduction  2  2  Non-Lorentzian Kaˇc-Moody algebra in 2d  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1  Carrollian and Galilean CFTs  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2  NL Affine Lie algebras  7  3  An intrinsic Carrollian derivation  10  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1  An Infinity of Conserved Quantities  10  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2  Current Ward Identities  11  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3  Current-primary fields  13  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4  Current-Current OPEs  15  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5  Global internal symmetry  18  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='6  EM tensor-current OPEs  19  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='7  The Algebra of Modes  22  4  Sugawara Construction  24  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1  Intrinsic construction from algebra  24  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2  Consistency with OPEs  27  5  Tensionless String as NLKM  29  6  Non-Lorentzian Knizhnik Zamolodchikov Equations  31  7  NLKM from Contractions  32  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1  A brief detour to representations of BMS  32  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2  Contraction of the affine parameters  33  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3  Contracting the Sugawara construction  35  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4  NLKZ equations from Contraction  36  8  Conclusions  38  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1  Summary  38  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2  Discussions and future directions  39  A Carroll Multiplets  40  B Calculation of Sugawara Construction Commutators  42  C Modified Sugawara Construction  46  D Details of OPE calculations  47  E K-Z equation in field theory approach  49  – 1 –  F NL KZ equation as a limit  50  1  Introduction  Relativistic conformal field theory (CFT) is one of the most potent tools of modern theo-  retical physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' with applications ranging from statistical mechanics of phase transitions to  quantum gravity through holography and string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Especially powerful are methods  of two dimensional (2d) CFTs [1] where symmetries enhance to two copies of the infinite  dimensional Virasoro algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The ideas and methods of 2d CFTs are of particular impor-  tance to the success of string theory, where this arises as residual symmetry on the string  worldsheet after the fixing of conformal gauge [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Non-abelian current algebras arise on the string worldsheet when one considers strings  moving on arbitrary group manifolds [3], generalising the abelian versions which arise  for strings propagating on flat backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' These Kaˇc-Moody algebras give rise to the  worldsheet 2d CFT by the Sugawara construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The construction of strings on arbitrary  backgrounds is thus intimately linked to these Kaˇc-Moody algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Kaˇc-Moody (KM) algebras also arise when we think of 2d CFTs augmented by additional  symmetries [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' For example, a CFT with additional U(1) global symmetry arises in a  number of places, including the study of black holes in AdS3 with charged U(1) hair that  are solutions to Einstein-Chern Simons theory [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Virasoro with U(1) KM symmetry  forms the chiral algebra of a large number of theories including N = 2 superconformal field  theories and theories with W1+∞ symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  In this paper, we will be interested in the construction of Non-Lorentzian (NL) versions  of Kaˇc-Moody algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Specifically, we are concerned with Galilean and Carrollian CFTs  in 2d with additional symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Galilean and Carrollian CFTs are obtained from their  relativistic counterparts by a process of contraction where the speed of light is taken to  infinity (Galilean theory) or zero (Carrollian theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' In two dimensions, the symmetry  algebras turn out to be isomorphic and it is in this 2d case we will focus our attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We  will obtain the algebras of interest by a contraction and then construct various properties  of these algebras by methods that have no connections with the limiting procedure and can  be thought of as completely intrinsic analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We also show that suitable singular limits  also reproduce our intrinsic answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Possible applications  We have a variety of applications in mind for our algebraic explorations in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Holography of flat spacetimes  Following closely related observation in [6], it was shown in [7] that d-dimensional Con-  formal Carrollian algebras are isomorphic to asymptotic symmetry algebras of (d + 1)  – 2 –  dimensional flat spacetimes discovered first by Bondi, van der Burg, Metzner [8] and Sachs  [9] and called BMS algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The Carroll CFTs can thus act as putative duals to asypm-  totically flat spacetimes [6, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Some important evidence for this duality has been  provided in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' [12–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Of particular importance is the recent result [19] that links 3d  Carroll CFTs and scattering amplitudes in 4d asymptotically flat spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Our explorations in this paper are of direct importance in the context of 3d flatspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Here, the U(1) version of NLKM symmetries are of interest for the study of Flat Space  Cosmological solutions [20] with U(1) hair, which are solutions of Einstein-Chern-Simons  theory, like the AdS case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' This was addressed recently in [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  The above approach to holography in asymptotically flat spacetimes goes under the name  of Carrollian holography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' There is an alternate formulation called Celestial holography  which posits that there is a 2d (relativistic) CFT that computes S-matrix elements in 4d  asymptotically flat spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' This has been instrumental in the uncovering of many new  results in asymptotic symmetries and scattering amplitudes in 4d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The interested reader is  pointed to the excellent reviews [23–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' For connections between Celestial and Carrollian  holography, we point to [19, 26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Interestingly, of late there have been studies of tree-level massless scattering amplitudes  which suggest that the asymptotic symmetries are far richer than the extended BMS group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  In [28, 29], it was shown that there is an SL(2) current algebra at level zero underlying the  symmetries of tree-level graviton amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' More recently, massless scattering amplitudes  have revealed an infinite dimensional w1+∞ algebra [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  If we assume that the field theory duals with these additional symmetries would be related  to a co-dimension one holographic description of 4d flatspace,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' and that these theories should  live on the whole of the null boundary and not only on the celestial sphere,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' the structure  emerging from the above discussions should only be a part of the whole symmetries,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' very  much like the relation between the two copies of the Virasoro algebra that make up the  Celestial CFT and the whole (extended) BMS4 algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' It is very likely that the algebras  of interest would then be the 3d versions of the Carrollian Kac-Moody algebras that we  discuss at length in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Tensionless strings  The tensionless limit of strings, which is analogous to the massless limit of point particles,  is an important sector of string theory that remains relatively less explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' In this limit,  the string worldsheet becomes null [31, 32] and the 2d relativistic conformal symmetry that  arises on the tensile worldsheet is replaced by 2d Carrollian Conformal symmetry in the  tensionless theory [33–35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The study of tensionless strings on arbitrary group manifolds  would naturally incorporate the NLKM algebras we study in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We would, in  future, attempt a construction of a Wess-Zumino-Witten model with the NLKM algebras  we discuss in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  It is of interest to mention that in [36], it was argued that tensionless strings appeared when  the level of the affine algebra corresponding to the WZW model of the strings propagating  in a group manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We believe that the intrinsic formulation of such strings should involve  – 3 –  the NLKM algebras we are studying here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' A direction of future work is the connection of  these two ideas and the aim would be to show that the NLKM algebras appear when the  level of the relativistic affine algebras are dialled to their critical value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Other applications  There are, of course, other natural applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The Galilean version of our story is of  relevance for understanding 2d Galilean CFTs with additional symmetry which may appear  in real life non-relativistic systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' These would be of interest also in understanding non-  relativistic strings in curved Newton-Cartan backgrounds [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Interesting enough, Carrollian structures also arise condensed matter systems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' in the  physics of flat bands that is of relevance in the context of “magic” superconductivity in  bi-layer graphene and also in fractional quantum Hall systems [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' It is easy to envision  condensed matter systems with additional symmetry and hence our methods in this paper  which lay the foundation for systems with Carrollian (and Galilean) affine Lie algebras,  should have applicability in a wide number of condensed matter systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Finally, a U(1)  affine algebra was also found to emerge in studies of the BMS scalar field theory in 2d [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  It would be of interest to figure out if this is a feature of all free BMS field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Our  methods outlined in this paper should then be useful even for free BMS field theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Outline of the paper  In section 2, we will start with a brief review of Carrollian and Galilean CFTs in gen-  eral dimensions, subsequently specializing the discussion to two dimensions followed by  a brief introduction to Non Lorentzian Kac Moody (NLKM) algebras via contraction of  relativistic Affine Lie Algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' In section 3, we present an intrinsic carrollian derivation  to the NLKM Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Then, we formulate the non-Lorentzian version of the Sugawara  construction for these algebras in section 4 and verify its validity by showing its consis-  tency with the OPEs we obtained in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' In section 5, we work out the example of  tensionless strings on a flat background geometry which exhibits the U(1) NLKM algebra  as the current algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Section 6 contains the derivation of the Non-Lorentzian analog  of Knizhnik Zamolodchikov(KZ) equations using the OPE definition of the primaries in  section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' And finally in section 7, we derive NLKM Algebra, Sugawara Construction and  the Non Lorentzian KZ equations through contractions in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' There are six appendices  which collect the details of various calculations that have been skipped in the main text  for the ease of readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  – 4 –  2  Non-Lorentzian Kaˇc-Moody algebra in 2d  In this section, we will begin our analysis by building the algebra we wish to study in the  remainder of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The NLKM algebra will be defined as a limit from the relativistic  Virasoro KM algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We will see later how the current part of the algebra can be used  to generate the entire algebra by a NL version of the Suwagara construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We begin by  reminding the reader of the Galilean (c → ∞) and the Carrollian (c → 0) contractions of  relativistic CFTs in generic dimensions and the fact that this leads to isomorphic algebras  in d = 2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1  Carrollian and Galilean CFTs  The Galilean (c → ∞) and Carrollian (c → 0) limits of the Poincare algebra leads to two  different contractions of the parent relativistic algebra and two different algebras in the  limit in general dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The Poincare algebra in D dimension ISO(D − 1, 1) is given  by  [Pµ, Pν] = 0,  [Mµν, Pρ] = −2ηρ[µPν],  [Mµν, Mρσ] = 4η[µρMσ]ν],  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1)  where µ = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=', D − 1, Pµ = −∂µ are translation generators and Mµν = xµ∂ν − xν∂µ  are Lorentz generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Galilean limit is achieved by taking c → ∞ limit, alternatively by  taking the contraction t → t, xi → ϵxi, ϵ → 0 limit where i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=', D − 1} [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Under this limit we see that  M0i = t∂i + xi∂t → 1  ϵ t∂i + ϵxi∂t  =⇒ Bi = lim  ϵ→0 ϵM0i = t∂i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2)  The spatial rotation generators Mij remains same under this contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Doing this  contraction, we end up with the Galilei algebra, where all the non-zero commutators are  given by  [Mij, Mkl] = 4δi[kMl]j], [Mij, Pk] = −2δk[iPj], [Mij, Bk] = 2δk[iBj], [Bi, H] = −Pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3)  Carrollian limit is achieved by taking c → 0 limit, alternatively by taking the contraction  t → ϵt, xi → xi, ϵ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Under this limit, we have  M0i = ϵt∂i + xi∂t → ϵt∂i + 1  ϵ xi∂t  =⇒ Bi = lim  ϵ→0 ϵM0i = xi∂t,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4)  with Mij intact again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' This gives us the Carroll algebra where the non-zero commutators  are given by  [Mij, Mkl] = 4δi[kMl]j], [Mij, Pk] = −2δk[iPj], [Mij, Bk] = −2δk[iBj] [Pi, Bj] = δijH, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5)  where H = −∂t is the Hamiltonian which has now become a central element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  The tale of the two contractions is also true for the relativistic conformal algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  In  relativistic Conformal symmetry group there are two additional generators  D = −xµ∂µ  Kµ = −(2xµxν∂ν − x2∂µ),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='6)  – 5 –  giving us the following additional commutators along with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1)  [D, Pµ] = Pµ, [D, Kµ] = −Kµ, [Kµ, Pν] = 2(ηµνD − Mµν), [Kρ, Lµν] = 2ηρ[µKν].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='7)  Taking the non-relativistic limit this time, we will end up with the following additional  generators along with the generators of Galilei algebra  D = −(xi∂i + t∂t)  K = K0 = −(2txi∂i + t2∂t)  Ki = t2∂i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='8)  These generators, along with the generators of the Galilei algebra gives us the Galilean  Conformal Algebra (GCA) in D dimensions [40] where non-zero commutators apart from  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3) are given by  [K, Bi] = Ki,  [K, Pi] = 2Bi  [Mij, Kr] = −2K[iδj]r,  [H, Ki] = −2Bi  [D, Ki] = −2Ki,  [D, Pi] = Pi  [D, H] = H,  [H, K] = −2D,  [D, K] = −K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='9)  When we take the Carrollian limit of the relativistic Conformal algebra, we get the following  new generators apart form the Carroll generators previously encountered:  D = −(xi∂i + t∂t)  K = K0 = −xixi∂t  Kj = −2xj(xi∂i + t∂t) + xixi∂j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='10)  These additional generators give us the following non-vanishing commutators along with  those given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5)  [Mij, Kk] = δk[jKi]  [Bi, Kj] = δijK,  [D, K] = −K  [K, Pi] = −2Bi  [Ki, Pj] = −2Dδij − 2Mij,  [H, Ki] = 2Bi  [D, H] = −H,  [D, Pi] = −Pi  [D, Ki] = Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='11)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='11) together form Carrollian Conformal Algebra (CCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We see that in general  dimensions the two different contractions give us different algebras that are not isomorphic  to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Let us now move to the interesting case of d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' In 2d, the relativistic conformal algebra  becomes infinite dimensional and is given by two copies of the Virasoro algebra:  [Lm, Ln] = (m − n)Lm+n + c  12(m3 − m)δm+n,0,  [ ¯Lm, ¯Ln] = (m − n) ¯Lm+n + ¯c  12(m3 − m)δm+n,0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12)  [Lm, ¯Ln] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Here c, ¯c are central charges of the Virasoro algebra (not to be confused with the speed  of light which we also had called c earlier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The Galilean contraction [42] of the above is  given by  Ln = Ln + ¯Ln,  Mn = ϵ( ¯Ln − Ln),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='13)  – 6 –  This contraction of the Virasoro algebra leads to  [Lm, Ln] = (m − n)Lm+n + cL  12(m3 − m)δn+m,0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='14a)  [Lm, Mn] = (m − n)Mm+n + cM  12 (m3 − m)δn+m,0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='14b)  [Mm, Mn] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='14c)  The way to see that this combination yields the non-relativistic limit, it is instructive to  write the generators of the Virasoro algebra in cylindrical cooridinates  Ln = einω∂ω,  ¯Ln = ein¯ω∂¯ω,  ω, ¯ω = τ ± σ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='15)  The limit (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='13) then translates to  σ → ϵσ, τ → τ, ϵ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='16)  which essentially means scaling velocities to be very small compared to 1 and since we are  doing all of this in units of speed of light c = 1, this is indeed the non-relativistic limit  c → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  On the other hand, the Carrollian contraction of the Virasoro algebra is given by  Ln = Ln − ¯L−n,  Mn = ϵ(Ln − ¯L−n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='17)  Again if we go back to the cylindrical coordinates, this limit translates to  σ → σ, τ → ϵτ, ϵ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='18)  The velocities are very large compared to 1 now and this means v/c → ∞, which equiv-  alently translates to c → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' This is thus the Carrollian limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The surprising thing is  that even this contraction yields the same algebra (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' In order to avoid confusion with  Galilean or Carrollian notions, we will exclusively call this the BMS3 algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Galilei and  Carroll contractions in d = 2 yield isomorphic algebras and this is down to the fact that  there is only one contracted and one uncontracted direction in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The algebra does  not differentiate between a contracted spatial and a contracted temporal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2  NL Affine Lie algebras  We start with the two copies of Virasoro Kac-Moody algebra whose holomorphic part is,  [Lm, Ln] = (m − n)Lm+n + c  12(m3 − m)δm+n,0,  [Lm, ja  n] = −nja  m+n  [ja  m, jb  n] = i  dim(g)  �  c=1  fabcjc  m+n + mkδm+n,0δab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='19)  There is an equivalent anti-holomorphic part with ¯fabc, ¯c and ¯k in place of fabc, c and k  respectively which are not necessarily equal to their holomorphic counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We are  take fabc ̸= ¯fabc for generality, but the dimensions of the two Lie groups are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  – 7 –  We will take a contraction of the algebra as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We will work with the following linear  combinations of the relativistic KM generators:  Ln = Ln + ¯Ln,  Mn = ϵ( ¯Ln − Ln),  Ja  m = ja  m + ¯ja  m,  Ka  m = ϵ(¯ja  m − ja  m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='20)  We will the consider the limit ϵ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The contracted algebra is given by:  [Lm, Ln] = (m − n)Lm+n + cL  12(m3 − m)δn+m,0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21a)  [Lm, Mn] = (m − n)Mm+n + cM  12 (m3 − m)δn+m,0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21b)  [Lm, Ja  n] = −nJa  m+n , [Lm, Ka  n] = −nKa  m+n , [Mm, Ja  n] = −nKa  m+n  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21c)  [Ja  m, Jb  n] = i  dim(g)  �  c=1  F abcJc  m+n + i  dim(g)  �  c=1  GabcKc  m+n + mk1δabδm+n,0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21d)  [Ja  m, Kb  n] = i  dim(g)  �  c=1  F abcKc  m+n + mk2δabδm+n,0,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21e)  with rest of the commutators vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We recognise the first two lines as the familiar  BMS3 algebra, equivalently the 2d Galilean or 2d Carrollian Conformal Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Through-  out the paper, we will call this sub-algebra the BMS algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' In the above, the structure  constants are related to their relativistic counterparts by  F abc = 1  2  �  fabc + ¯fabc�  , Gabc = 1  2ϵ  �  ¯fabc − fabc�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='22)  while the central terms are given by  cL = c + ¯c,  cM = ϵ(¯c − c), k1 = ¯k + k, k2 = ϵ(¯k − k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='23)  We will call the algebra (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21) the 2d Non-Lorentzian Kaˇc-Moody algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We can also carry  out the contraction ultrarelativistically for which we take the following linear combinations  of generators,  Ln = Ln − ¯L−n,  Mn = ϵ(Ln + ¯L−n),  Ja  n = (ja  n + ¯ja  −n),  Ka  n = ϵ(ja  n − ¯ja  −n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='24)  Contracted algebra will be same as (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21) with relations analogous to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='22) and(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='23)  taking the following form:  F abc = 1  2  �  fabc + ¯fabc  �  , Gabc = 1  2ϵ  �  fabc − ¯fabc  �  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='25)  cL = c − ¯c,  cM = ϵ(c + ¯c),  k1 = k − ¯k,  k2 = ϵ(k + ¯k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='26)  This (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21) will be the algebra of interest for the rest of our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Note that if we start with 2 identical Kac-Moody algebras for the holomorphic and anti-  holomorphic sections, then after contraction we get  fabc = ¯fabc ⇒ F abc = fabc, Gabc = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='27)  – 8 –  For most of our analysis, we will use this simplified algebra, but the results can be easily  generalised for the general algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  We should also clarify something about the new Lie algebra structure and our notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  We started with (the affine version of) two copies of a Lie algebra g of dimension  n = dim(g),  constructed from the generators {ja, a ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' , n}} and {¯ja, a ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' , n}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Via  contraction we obtain a (non-semisimple) Lie algebra ˜g of dimension  ˜n = dim(˜g) = 2 dim(g)  consisting of generators {Ja, Ka, a ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' , n}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  So in all our notation, the indices  a, b, c run from 1 to n = dim(g) (not ˜n), and whenever we write dim(g) (like in expressions  of central charge cL later in the paper), we mean the dimension of the parent algebra g,  which is actually half of the dimension of the new algebra ˜g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' This is done for the tidiness  of expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Before we conclude this section, we would like to comment on the choice of contraction of  the currents to get to the NLKM algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Note that the chosen linear combinations of the  Kac Moody generators are motivated by the Galilean/Carrollian contractions analogous to  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='16) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The generators in cylindrical coordinates look like,  ja  n = ja ⊗ einω,  ¯ja  n = ¯ja ⊗ ein¯ω,  ω, ¯ω = τ ± σ  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='28)  Galilean limit will correspond to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='16) and Carrollian limit corresponds to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  In  Galilean case, we have (upto linear order in ϵ),  ja  n + ¯ja  n = (ja + ¯ja) ⊗ einτ + inσϵ(ja − ¯ja) ⊗ einτ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='29a)  ja  n − ¯ja  n = (ja − ¯ja) ⊗ einτ + inσϵ(ja + ¯ja) ⊗ einτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='29b)  We now have two choices, keeping the BMS contraction the same:  Ja = ja + ¯ja, Ka = ϵ(¯ja − ja) ⇒ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Ja = ¯ja − ja, Ka = ϵ(¯ja + ja) ⇒ Ja  m = ¯ja  m − ja  m, Ka  m = ϵ(¯ja  m + ja  m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Both of these choices lead to relations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='23) but (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='22) need to be altered for the second  choice to:  F abc = 1  2( ¯fabc − fabc), Gabc = 1  2ϵ(fabc + ¯fabc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='30)  Similarly, in Carrollian case, we have two choices: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='24) and  Ja  n = ja  n − ja  −n,  Ka  n = ϵ(ja  n + ja  −n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  in which case (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='26) remains same but we have  F abc = 1  2(fabc − ¯fabc),  Gabc = 1  2ϵ(fabc + ¯fabc)  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='31)  – 9 –  instead of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Since we will be considering the special case when fabc = ¯fabc, we will  stick to the first choice in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' For the intrinsic analysis, which only uses the algebra  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21), of course these choices do not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' But when we derive results from the limit,  in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' 7, it is important to state everything works for the other contraction as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Sugawara construction through contraction (Sec 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3) can be carried out for the second  choice if we specialize to the case ¯fabc = −fabc in case of Galilean contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  3  An intrinsic Carrollian derivation  Having derived the non-Lorentzian current algebra through a contraction, we now go on to  present an intrinsically Carrollian derivation of the same, where we would not be alluding  to a limiting procedure at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' This section heavily borrows from the machinery detailed in  [43], some of the important features of which are described in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Although we  will try and be self consistent so that the section (with the help of the related appendix A)  stands on its own, for any details that we may have inadvertently skipped in what follows,  the reader is referred back to [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1  An Infinity of Conserved Quantities  A 2D Carrollian CFT on the flat Carrollian background (t, x) is invariant under an infinite  number of 2D Carrollian conformal (CC) transformations whose infinitesimal versions are  given as:  x′ = x + ϵxf(x) ,  t′ = t + ϵxtf′(x) + ϵtg(x)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1)  As a consequence, the EM tensor components classically satisfy the following conditions  [43]:  ∂µT µ  ν = 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  T x  t = 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  T µ  µ = 0  =⇒ ∂tT t  t = 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  ∂tT t  x = ∂xT t  t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2)  This allows for an infinite number of Noether currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The two conserved currents corre-  sponding to the symmetry transformation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1) are noted below:  jµ  t =  �  jt  t , jx  t  �  =  �  g(x)T t  t , 0  �  ,  jµ  x =  �  jt  x , jx  x  �  =  �  f(x)T t  x + tf′(x)T t  t , −f(x)T t  t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3)  Now, we suppose that there are some other pairs of fields {J a  x , J a  t } in the theory that obey  the following conditions analogous to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2):  ∂tJ a  t (t, x) = 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  ∂tJ a  x (t, x) = ∂xJ a  t (t, x)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4)  where a is to be thought of as a ‘flavor’ index (but t and x in subscript are not tensor  indices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Using these fields, we can construct an infinite number of conserved quantities:  kaµ =  �  kat , kax�  = (g(x)J a  t , 0) ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5a)  jaµ =  �  jat , jax�  =  �  f(x)J a  x + tf′(x)J a  t , −f(x)J a  t  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5b)  We shall regard these conserved quantities as the Noether currents associated to some  ‘internal’ symmetries of the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  – 10 –  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2  Current Ward Identities  We now consider an infinitesimal internal transformation of a (possibly multi-component)  field Φ(t, x):  Φ(t, x) → ˜Φ(t, x) = Φ(t, x) + ϵa (Fa · Φ) (t, x)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='6)  where Fa · Φ denotes the functional changes of the (multi-component) field Φ under in-  finitesimal transformations labeled by ϵa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' So, the generator GaΦ of this transformation is  given by:  −iϵaGaΦ(t, x) := ˜Φ(t, x) − Φ(t, x) = ϵa (Fa · Φ) (t, x)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='7)  We shall now find the Ward identity corresponding to this internal transformation which  is assumed to be a symmetry of the 2D Carrollian CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' For this purpose, we analytically  continue the real space variable x ∈ R∪{∞} to the complex plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' thus, the Ward identity  reads [43]:  ∂µ⟨jµ  a(t, x)X⟩ ∼ −i  n  �  i=1  δ(t − ti)  �  �−(Fa)i · ⟨X⟩  x − xi  +  �  k≥2  ⟨Y (k)  a  ⟩i(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='xn)  (x − xi)k  �  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='8)  where the yet unknown correlators ⟨Y (k)  a  ⟩i depend on the transformation properties of the  fields in the string-of-fields X and the transformation itself and ∼ denotes ‘modulo terms  holomorphic in x inside [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' ]’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We also use the shorthand x1 = (t1, x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' All the correlators  are time-ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Let us now assume that the symmetry transformation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='6) is associated to the following  conserved current operator:  kaµ = (J a  t , 0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='9)  The corresponding Ward identity then is:  ∂t⟨J a  t (t, x)X⟩ ∼ −i  n  �  i=1  δ(t − ti)  �  �−(Fa)i · ⟨X⟩  x − xi  +  �  k≥2  ⟨Y (k)  a  ⟩i(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='xn)  (x − xi)k  �  �  ⇒ ⟨J a  t (t, x)X⟩ = −i  n  �  i=1  θ(t − ti)  �  �−(Fa)i · ⟨X⟩  x − xi  +  �  k≥2  ⟨Y (k)  a  ⟩i(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='xn)  (x − xi)k  �  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='10)  where, following [43], the initial condition has been taken to be:  lim  t→−∞⟨J a  t (t, x)X⟩ = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='11)  and as in 2D relativistic CFT [4], ⟨J a  t (t, x)X⟩ is assumed to be finite whenever x ̸= {xi};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  this condition makes the holomorphic terms inside [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='] vanish in this Ward identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  – 11 –  Similarly, if the conserved current:  jaµ = (J a  x , −J a  t )  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12)  is associated to another internal symmetry transformation:  Φ(t, x) → ˜Φ(t, x) = Φ(t, x) + ϵa (Ga · Φ) (t, x)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='13)  the corresponding Ward identity is:  ⟨(∂tJ a  x (t, x) − ∂xJ a  t (t, x)) X⟩ ∼ −i  n  �  i=1  δ(t − ti)  �  �−(Ga)i · ⟨X⟩  x − xi  +  �  k≥2  ⟨Y (k)  a  ⟩i(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='xn)  (x − xi)k  �  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='14)  Assuming the initial condition:  lim  t→−∞⟨J a  x (t, x)X⟩ = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='15)  and the finite-ness property of ⟨J a  x (t, x)X⟩ for x ̸= {xi}, this Ward identity together with  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='10) lead us to:  ⟨J a  x (t, x)X⟩ = −i  n  �  i=1  θ(t − ti)  �  �−(Ga)i · ⟨X⟩  x − xi  +  �  k≥2  ⟨Z(k)  a ⟩i(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='xn)  (x − xi)k  −(t − ti)  �  �−(Fa)i · ⟨X⟩  (x − xi)2  +  �  k≥2  k⟨Y (k)  a  ⟩i(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='xn)  (x − xi)k+1  �  �  �  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='16)  Again, the correlators ⟨Z(k)  a ⟩i can not be determined without knowing the explicit internal  transformation properties of the fields in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  For future references, we note the iϵ-form [43] of the Ward identities (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='16)  below with ∆˜xp := x − xp − iϵ(t − tp) 1 :  i⟨J a  t (t, x)X⟩ = lim  ϵ→0+  n  �  i=1  �  �−(Fa)i · ⟨X⟩  ∆˜xi  +  �  k≥2  ⟨Y (k)  a  ⟩i  (∆˜xi)k  �  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='17)  i⟨J a  x (t, x)X⟩ = lim  ϵ→0+  n  �  i=1  �  �−(Ga)i · ⟨X⟩  ∆˜xi  +  �  k≥2  ⟨Z(k)  a ⟩i  (∆˜xi)k  −(t − ti)  �  �−(Fa)i · ⟨X⟩  (∆˜xi)2  +  �  k≥2  k ⟨Y (k)  a  ⟩i  (∆˜xi)k+1  �  �  �  � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='18)  1We hope the reader does not confuse the delta appearing in ∆˜xp with the conformal weight ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The ∆  appearing in the difference of coordinates would always appear with a coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  – 12 –  Thus, a general (possibly multi-component) 2D Carrollian conformal field Φ(t, x) has the  following OPEs (in the iϵ-form) with the current-vector components (∆˜x′ := x′−x−iϵ(t′−  t)):  iJ a  t (t′, x′)Φ(t, x) ∼ lim  ϵ→0+  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' + −(Fa · Φ)(t, x)  ∆˜x′  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='19)  iJ a  x (t′, x′)Φ(t, x) ∼ lim  ϵ→0+  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' + −(Ga · Φ)(t, x)  ∆˜x′  − (t′ − t)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' + −(Fa · Φ)(t, x)  (∆˜x′)2  ��  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='20)  where .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' represents higher order poles at x′ = x and ∼ denotes ‘modulo terms holomorphic  in x′ that have vanishing VEVs’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  In this work, we shall only consider currents with scaling dimension ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Comparing  the behavior of both sides of the OPEs (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='19) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='20) under dilatation, we then infer  that the scaling dimensions of the local fields Fa · Φ and Ga · Φ must be same as that of Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Since this is true for any arbitrary local field Φ, we conclude that Fa · Φ and Ga · Φ must  be linear combinations of the components of the multi-component field Φ that ‘internally’  transforms under a bi-matrix representation of the global current symmetry algebra;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' all of  these components must have an equal scaling dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Our goal is to find this global  algebra and its infinite extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  We now express Fa · Φ and Ga · Φ as explicit linear combinations:  (Fa · Φ)ii′ = (ta  K)i  j Φji′ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (Ga · Φ)ii′ = Φij′ (ta  J) i′  j′  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21)  where ta  J and ta  K are just two matrices as of now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Later, we shall relate them to the  generators of the internal symmetry algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3  Current-primary fields  In the operator formalism of a QFT, the conserved charge QA is the generator of an  infinitesimal symmetry transformation on the space of the quantum fields:  ˜Φ(x) − Φ(x) = −iϵA[QA , Φ(x)]  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='22)  In 2D Carroll CFT, the above generator equation for any conserved charge operator QA is  related to the following contour integral prescription involving an OPE [43]2:  QA =  1  2πi  �  Cu  dx jt  A(t, x)  generates  [QA , Φ(t, x)] =  1  2πi  �  x  dx′ jt  A(t+, x′)Φ(t, x)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='23)  where t+ > t and the counter-clockwise contour Cu encloses the upper half-plane along  with the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The contour around x must not enclose any possible singularities of the  vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  2Section 5 of this reference contains a derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  – 13 –  The conserved quantum charges Qa  t [g] and Qa  x[f] of the respective currents (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5a) and  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5b) are thus given by:  Qa  t [g] =  1  2πi  �  Cu  dx g(x)J a  t (t, x)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='24)  Qa  x[f] =  1  2πi  �  Cu  dx  �  f(x)J a  x (t, x) + tf′(x)J a  t (t, x)  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='25)  The conserved charges of all flavors collectively induce the following infinitesimal changes  to a generic quantum field, as deduced from the OPEs (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='19) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='20):  −i  �  a  ϵa [Qa  t [ga] , Φ(x)] = − 1  2π  �  a  ϵa  �  x  dx′ ga(x′)J a  t (t+, x′)Φ(t, x)  =  �  a  ϵa [ga(x) (ta  K · Φ) (t, x) + (h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=')]  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='26)  −i  �  a  ϵa [Qa  x[fa] , Φ(x)] = − 1  2π  �  a  ϵa  �  x  dx′ �  fa(x′)J a  x (t+, x′) + t+fa′(x′)J a  t (t+, x′)  �  Φ(t, x)  =  �  a  ϵa �  fa(x) (Φ · ta  J) (t, x) + tfa′(x) (ta  K · Φ) (t, x) + (h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=')  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='27)  where h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' denotes terms necessarily containing derivatives (of order at least 1) of ga(x)  and fa(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  For the currents (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12), we simply have f(x) = 1 = g(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' For any arbitrary field  Φ(t, x) this immediately leads to:  − i  �  a  ϵa [Qa  t [1] , Φ(x)] =  �  a  ϵa (ta  K · Φ) (t, x)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='28a)  − i  �  a  ϵa [Qa  x[1] , Φ(x)] =  �  a  ϵa (Φ · ta  J) (t, x)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='28b)  Thus, the finite internal transformation that is generated by the charges {Qa  k[1]} is:  Φ(t, x) → ˜Φ(t, x) = e−i �  a ϵaQa  t [1]Φ(t, x)ei �  a ϵaQa  t [1] = e  �  a ϵata  K · Φ(t, x)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='29)  which is obtained by using the BCH lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Similarly, the charges {Qa  x[1]} generate the  following finite transformation:  Φ(t, x) → ˜Φ(t, x) = Φ(t, x) · e  �  a ϵata  J  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='30)  On the other hand, as derived from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='26), an arbitrary field finitely transforms under the  action of the charges {Qa  t [ga]} as:  Φ(t, x) → ˜Φ(t, x) = e  �  a ϵaga(x)ta  K · Φ(t, x) + extra terms  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='31)  – 14 –  while (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='27) leads to the following finite action of the charges {Qa  x[fa]}:  Φ(t, x) → ˜Φ(t, x) = e  �  a ϵatfa′(x)ta  K · Φ(t, x) · e  �  a ϵafa(x)ta  J + extra terms  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='32)  In view of the above discussion, we emphasize that while a generic field transforms co-  variantly under the action of the charges associated to the conserved currents (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='9) and  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12), that is not the case for the generic conserved currents (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5a) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' A field  that transforms covariantly (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' for which the extra terms in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='31) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='32) vanish)  even under the action of the charges of any currents of the form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5a) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5b) is called  a current-primary field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Consequently, there is no h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='26) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='27) appropriate  for a current-primary field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' This enables us to completely specify the pole structures of the  current-primary OPEs for a primary field Φ(t, x):  J a  t (t′, x′)Φ(t, x) ∼ lim  ϵ→0+ ita  K · Φ(t, x)  ∆˜x′  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='33)  J a  x (t′, x′)Φ(t, x) ∼ lim  ϵ→0+ i  �Φ(t, x) · ta  J  ∆˜x′  − (t′ − t)ta  K · Φ(t, x)  (∆˜x′)2  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='34)  that immediately imply the following Ward identities for a string X of primary fields:  ⟨J a  t (t, x)X⟩ = lim  ϵ→0+ i  n  �  i=1  (ta  K)i · ⟨X⟩  ∆˜xi  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='35)  ⟨J a  x (t, x)X⟩ = lim  ϵ→0+ i  n  �  i=1  �⟨X⟩ · (ta  J)i  ∆˜xi  − (t − ti)(ta  K)i · ⟨X⟩  (∆˜xi)2  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='36)  where (ta  J)i and (ta  K)i denotes transformation-matrices appropriate for the i-th primary  field in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4  Current-Current OPEs  To derive the current-current OPEs using the machinery just developed, we shall assume  that no field in the theory has negative scaling dimension with the identity being the only  field with ∆ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Under these assumptions, the OPEs between the EM tensor components were derived using  only symmetry arguments in [43];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' the results are:  T t  t(t′, x′)T t  t(t, x) ∼ 0  T t  x(t′, x′)T t  t(t, x) ∼ lim  ϵ→0+ −i  � −i cM  2  (∆˜x′)4 + 2T t  t(t, x)  (∆˜x′)2  + ∂xT t  t(t, x)  ∆˜x′  �  T t  t(t′, x′)T t  x(t, x) ∼ lim  ϵ→0+ −i  � −i cM  2  (∆˜x′)4 + 2T t  t(t, x)  (∆˜x′)2  + ∂tT t  x(t, x)  ∆˜x′  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='37)  T t  x(t′, x′)T t  x(t, x) ∼ lim  ϵ→0+ −i  � −i cL  2  (∆˜x′)4 + 2T t  x(t, x)  (∆˜x′)2  + ∂xT t  x(t, x)  ∆˜x′  −(t′ − t)  �−2icM  (∆˜x′)5 + 4T t  t(t, x)  (∆˜x′)3  + ∂tT t  x(t, x)  (∆˜x′)2  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  – 15 –  The constants cL and cM are the central charges of the 2D Carrollian conformal QFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  We shall use the same technique to find the current-current OPEs below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Keeping in mind  that here we are dealing with currents with scaling dimension ∆ = 1, below we note the  most general allowed form of the Jt − Jt OPE compatible with the general OPE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='19):  J a  t (t′, x′)J b  t (t, x) ∼ lim  ϵ→0+ i  �  Aab  (∆˜x′)2 + (ta  K · Jt)b (t, x)  ∆˜x′  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='38)  where Aab is a field proportional to identity so that it has vanishing scaling dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Clearly, (ta  K · Jt)b (t, x) must have scaling dimension ∆ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' So, in a generic 2D CCFT,  (ta  K · Jt)b must be a linear combination of {J a  x , J a  t }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Correspondingly to the classical conservation equation ∂tJ a  t (t, x) = 0 , in the QFT we  should have:  J a  t (t′, x′)∂tJ b  t (t, x) ∼ 0  ⇒  ∂t (ta  K · Jt)b (t, x) = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='39)  which means that {J a  x } can not contribute to the linear combination (ta  K · Jt)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Since the currents have scaling dimension ∆ = 1, they must satisfy the bosonic3 exchange  property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' For the J a  t (t, x) field, it is:  J a  t (t′, x′)J b  t (t, x) = J b  t (t, x)J a  t (t′, x′)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='40)  This condition implies the following restrictions:  Aab = Aba  and  (ta  K · Jt)b = −  �  tb  K · Jt  �a  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='41)  Looking at (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='20), we write the allowed form of a Jx − Jt OPE:  J a  x (t′, x′)J b  t (t, x) ∼ lim  ϵ→0+ i  �  Bab  (∆˜x′)2 + (Jt · ta  J)b (t, x)  ∆˜x′  − (t′ − t)  �  2Aab  (∆˜x′)3 + (ta  K · Jt)b (t, x)  (∆˜x′)2  ��  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='42)  where Bab are some constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Now, we have the following restrictions:  J a  x (t′, x′)∂tJ b  t (t, x) ∼ 0  =⇒  Aab = 0  and  (ta  K · Jt)b = 0  and  ∂t (Jt · ta  J)b (t, x) = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='43)  Thus, again {J a  x } do not contribute to the linear combination (Jt · ta  J)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  On the other hand, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='19), we get the following Jt − Jx OPE:  J b  t (t′, x′)J a  x (t, x) ∼ lim  ϵ→0+ i  �  Cba  (∆˜x′)2 +  �  tb  K · Jx  �a (t, x)  ∆˜x′  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='44)  3Since, as will be shown later, the currents are 2D CC primary fields with integer scaling dimension,  this statement is justified [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  – 16 –  with Cab being constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Using the following bosonic exchange property:  J b  t (t′, x′)J a  x (t, x) = J a  x (t, x)J b  t (t′, x′)  to compare the OPE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='42) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='44), we get the following conditions:  Bab = Cba  and  (Jt · ta  J)b = −  �  tb  K · Jx  �a  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='45)  which implies that {J a  x } do not appear also in the linear combination  �  tb  K · Jx  �a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Finally, we write the Jx − Jx OPE in accordance with the general form (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='20):  J a  x (t′, x′)J b  x(t, x) ∼ lim  ϵ→0+ i  �  Dab  (∆˜x′)2 + (Jx · ta  J)b (t, x)  ∆˜x′  − (t′ − t)  �  2Cab  (∆˜x′)3 + (ta  K · Jx)b (t, x)  (∆˜x′)2  ��  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='46)  from which, the bosonic exchange property:  J b  x(t′, x′)J a  x (t, x) = J a  x (t, x)J b  x(t′, x′)  leads to the following conditions:  Dab = Dba ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Cab = Cba  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='47)  (Jx · ta  J)b = −  �  Jx · tb  J  �a  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (ta  K · Jx)b = −  �  tb  K · Jx  �a  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  ∂t (Jx · ta  J)b = ∂x (ta  K · Jx)b  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='48)  It can be readily checked that these conditions are compatible with the quantum versions  (in the OPE language) of the classical conservation laws (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  We now explicitly write the allowed forms of the linear combinations appearing in the  above OPEs:  (Jt · ta  J)b = (ta  K · Jx)b = F abcJ c  t  with  F abc = −F bac,  (Jx · ta  J)b = F abcJ c  x + GabcJ c  t  with  Gabc = −Gbac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='49)  Thus, the final forms of the current-current OPEs are:  J a  t (t′, x′)J b  t (t, x) ∼ 0  J a  x (t′, x′)J b  t (t, x) ∼ lim  ϵ→0+ i  � Cab  (∆˜x′)2 + F abcJ c  t (t, x)  ∆˜x′  �  J a  t (t′, x′)J b  x(t, x) ∼ lim  ϵ→0+ i  � Cab  (∆˜x′)2 + F abcJ c  t (t, x)  ∆˜x′  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='50)  J a  x (t′, x′)J b  x(t, x) ∼ lim  ϵ→0+ i  �  Dab  (∆˜x′)2 +  �  F abcJ c  x + GabcJ c  t  �  (t, x)  ∆˜x′  − (t′ − t)  � 2Cab  (∆˜x′)3 + F abcJ c  t (t, x)  (∆˜x′)2  ��  These OPEs imply that the currents themselves are not current-primary fields in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  – 17 –  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5  Global internal symmetry  All the correlation functions in the theory must be invariant under the global internal  transformations associated to which are the conserved currents (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' This fact  will put constraints on Cab and Dab, as we will now see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  We begin by noting the following 2-point correlators between the currents, from the above  OPEs:  �  J a  t (t′, x′)J b  t (t, x)  �  = 0  �  J a  t (t′, x′)J b  x(t, x)  �  =  �  J a  x (t′, x′)J b  t (t, x)  �  = lim  ϵ→0+ i Cab  (∆˜x′)2  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='51)  �  J a  x (t′, x′)J b  x(t, x)  �  = lim  ϵ→0+ i  � Dab  (∆˜x′)2 − (t′ − t) 2Cab  (∆˜x′)3  �  since the currents, with ∆ = 1, must have vanishing VEVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Next, due to the global internal symmetry, an arbitrary n-point correlator in the theory  must satisfy:  n  �  i=1  ⟨Φ1(t1, x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' (Φi · ta  J) (ti, xi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Φn(tn, xn)⟩ = 0  n  �  i=1  ⟨Φ1(t1, x1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' (ta  K · Φi) (ti, xi) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Φn(tn, xn)⟩ = 0  Thus the 2-point current correlators explicitly satisfy:  �  (Jx · ta  J)b (t1, x1)J c  x(t2, x2)  �  +  �  J b  x(t1, x1) (Jx · ta  J)c (t2, x2)  �  = 0  =⇒ F abdDdc + F acdDdb + GabdCdc + GacdCdb = 0  and  F abdCdc + F acdCdb = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='52)  The analogues relations obtained from the invariance of the 3-point current correlators are:  F abeF ecfCfd + F caeF ebfCfd + F adeF bcfCfe = 0  F abeF ecfDfd + F caeF ebfDfd + F adeF bcfDfe +  �  F abeGecf + GabeF ecf�  Cfd  +  �  F caeGebf + GcaeF ebf�  Cfd +  �  F adeGbcf + GadeF bcf�  Cfe = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='53)  In what follows, we shall see that {F abc} and {Gabc} must also obey the following constraints  arising as the Jacobi identity of the infinite-dimensional Lie algebra of the current-modes,  which we had previously described in our initial algebraic description from the contraction  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21) and also will derive independently later in this section (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='80):  F abeF ecd + F caeF ebd + F bceF ead = 0  – 18 –  F abeGecd + GabeF ecd + F caeGebd + GcaeF ebd + F bceGead + GbceF ead = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='54)  No new constraint for {F abc} and {Gabc} arises from the global internal invariance of  n-point current correlators for n ≥ 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Upon comparison, we notice that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='53) reduces to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='54) if we choose:  Dab = −ik1δab  and  Cab = −ik2δab with k2 ̸= 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='55)  In that case, F abc and Gabc are anti-symmetric in all indices, as seen from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='6  EM tensor-current OPEs  We now show under the assumption that no field in the theory has negative scaling dimen-  sion with the identity field being the only one with ∆ = 0 , that the currents J a  x (t, x) and  J a  t (t, x) must transform as a rank-1  2 primary multiplet under 2D CC transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  From [43], we recall the OPEs of a general 2D CC (multi-component) field Φ(l)(t, x) having  scaling dimension ∆, Carrollian boost-charge ξ and boost rank l with the EM tensor  components:  T t  x(t′, x′)Φ(l)(t, x) ∼ lim  ϵ→0+ −i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' + ∆Φ(l)(t, x)  (∆˜x′)2  + ∂xΦ(l)(t, x)  ∆˜x′  −(t′ − t)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' + 2  �  ξ · Φ(l)  �  (t, x)  (∆˜x′)3  + ∂tΦ(l)(t, x)  (∆˜x′)2  ��  T t  t(t′, x′)Φ(l)(t, x) ∼ lim  ϵ→0+ −i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' +  �  ξ · Φ(l)  �  (t, x)  (∆˜x′)2  + ∂tΦ(l)(t, x)  ∆˜x′  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='56)  The defining feature of 2D CC primary fields is the vanishing of the higher order poles in  the above OPEs that leads to:  T t  x(t′, x′)Φ(l)(t, x) ∼ lim  ϵ→0+ −i  �∆Φ(l)(t, x)  (∆˜x′)2  + ∂xΦ(l)(t, x)  ∆˜x′  −(t′ − t)  �  2  �  ξ · Φ(l)  �  (t, x)  (∆˜x′)3  + ∂tΦ(l)(t, x)  (∆˜x′)2  ��  T t  t(t′, x′)Φ(l)(t, x) ∼ lim  ϵ→0+ −i  ��  ξ · Φ(l)  �  (t, x)  (∆˜x′)2  + ∂tΦ(l)(t, x)  ∆˜x′  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='57)  Now, along with the above assumption, the classical time-independence of the field J a  t  restricts the general OPE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='56) to the following form:  T t  t(t′, x′)J a  t (t, x) ∼ lim  ϵ→0+ −i  �  Aa  1  (∆˜x′)3 + (ξ · J a  t ) (t, x)  (∆˜x′)2  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='58)  where the field Aa  1 is proportional to the identity field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Similarly, we write the most general  allowed form of the following OPE from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='56):  T t  x(t′, x′)J a  t (t, x) ∼ lim  ϵ→0+ −i  �  Aa  2  (∆˜x′)3 + J a  t (t, x)  (∆˜x′)2 + ∂xJ a  t (t, x)  ∆˜x′  − (t′ − t)  � 3Aa  1  (∆˜x′)4 + 2 (ξ · J a  t ) (t, x)  (∆˜x′)3  ��  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='59)  – 19 –  with Aa  2 being another constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' But, the quantum counterpart of the conservation law  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4) leads to:  T t  t(t′, x′)∂tJ a  t (t, x) ∼ 0 =⇒ ∂t (ξ · J a  t ) (t, x) = 0,  T t  x(t′, x′)∂tJ a  t (t, x) ∼ 0 =⇒ Aa  1 = 0  and  ξ · J a  t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='60)  The OPEs for J a  x (t, x) may have the most general form given below:  T t  t(t′, x′)J a  x (t, x) ∼ lim  ϵ→0+ −i  �  Aa  3  (∆˜x′)3 + (ξ · J a  x ) (t, x)  (∆˜x′)2  + ∂tJ a  x (t, x)  ∆˜x′  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='61)  with the conservation law (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4) forcing:  ∂t (ξ · J a  x ) (t, x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='62)  The remaining one OPE is given below:  T t  x(t′, x′)J a  x (t, x) ∼ lim  ϵ→0+ −i  �  Aa  4  (∆˜x′)3 + J a  x (t, x)  (∆˜x′)2 + ∂xJ a  x (t, x)  ∆˜x′  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='63)  −(t′ − t)  � 3Aa  3  (∆˜x′)4 + 2 (ξ · J a  x ) (t, x)  (∆˜x′)3  + ∂tJ a  x (t, x)  (∆˜x′)2  ��  where Aa  3 and Aa  4 are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' From the quantum version of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4), one then obtains:  T t  x(t′, x′) [∂tJ a  x (t, x) − ∂xJ a  t (t, x)] ∼ 0  =⇒  ξ · J a  x = J a  t  and  Aa  2 = Aa  3  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='64)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' the currents J a  x (t, x) and J a  t (t, x) transform under Carrollian boost as a rank-1  2 mul-  tiplet with boost charge ξ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  The OPEs for the currents then are:  T t  t(t′, x′)J a  t (t, x) ∼ 0  T t  x(t′, x′)J a  t (t, x) ∼ lim  ϵ→0+ −i  �  Aa  3  (∆˜x′)3 + J a  t (t, x)  (∆˜x′)2 + ∂xJ a  t (t, x)  ∆˜x′  �  T t  t(t′, x′)J a  x (t, x) ∼ lim  ϵ→0+ −i  �  Aa  3  (∆˜x′)3 + J a  t (t, x)  (∆˜x′)2 + ∂tJ a  x (t, x)  ∆˜x′  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='65)  T t  x(t′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x′)J a  x (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x) ∼ lim  ϵ→0+ −i  �  Aa  4  (∆˜x′)3 + J a  x (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x)  (∆˜x′)2 + ∂xJ a  x (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x)  ∆˜x′  −(t′ − t)  � 3Aa  3  (∆˜x′)4 + 2J a  t (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x)  (∆˜x′)3  + ∂tJ a  x (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x)  (∆˜x′)2  ��  On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' applying the bosonic exchange property between the currents and the  EM tensor components,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' we obtain:  J a  t (t′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x′)T t  t(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x) ∼ 0  – 20 –  J a  t (t′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x′)T t  x(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x) ∼ lim  ϵ→0+ −i  �  −  Aa  3  (∆˜x′)3 + J a  t (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x)  (∆˜x′)2  �  J a  x (t′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x′)T t  t(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x) ∼ lim  ϵ→0+ −i  �  −  Aa  3  (∆˜x′)3 + J a  t (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x)  (∆˜x′)2  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='66)  J a  x (t′, x′)T t  x(t, x) ∼ lim  ϵ→0+ −i  �  −  Aa  4  (∆˜x′)3 + J a  x (t, x)  (∆˜x′)2 − (t′ − t)  �  − 3Aa  3  (∆˜x′)4 + 2J a  t (t, x)  (∆˜x′)3  ��  Comparing these with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='19) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='20), we immediately note the following:  �  ta  K · T t  t  �  (t, x) =  �  ta  K · T t  x  �  (t, x) =  �  T t  t · ta  J  �  (t, x) =  �  T t  x · ta  J  �  (t, x) = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='67)  which implies that the EM tensor components actually transform under the singlet repre-  sentation of the global internal symmetry algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Next we look at the consequences of the global internal symmetry on the 2-point correlators  between the currents and the EM tensor components to find possible constraints on Aa  3  and Aa  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' It suffices to consider the following:  �  (Jx · ta  J)b (t1, x1)T t  x(t2, x2)  �  +  �  J b  x(t1, x1)  �  T t  x · ta  J  �  (t2, x2)  �  = 0  =⇒  Aa  3 = Aa  4 = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='68)  Causing the vanishing of the poles of appropriate orders in the OPEs (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='65), we finally  conclude, comparing with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='57), that:  the currents tranform as a 2D CC primary multiplet of rank-1  2 with ∆ = ξ = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Thus, the currents have the following infinitesimal 2D CC transformation property under  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1), as is obtained using the prescription (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='23) for the corresponding conserved currents  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3):  − iGxJ a  t (t, x) = −[f(x)∂x + f′(x)]J a  t (t, x)  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  −iGtJ a  t (t, x) = 0  − iGtJ a  x (t, x) = −g(x)∂tJ a  x (t, x) − g′(x)J a  t (t, x)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='69)  − iGxJ a  x (t, x) = −[f(x)∂x + tf′(x)∂t + f′(x)]J a  x (t, x) − tf′′(x)J a  t (t, x)  As an aside, from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='66) we note down below the infinitesimal internal transformation  properties of the EM tensor components, generated by the conserved charges of the currents  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5a) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5b):  − iGJT t  t(t, x) = −fa′(x)J a  t (t, x)  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  −iGKT t  t(t, x) = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='70)  − iGKT t  x(t, x) = −ga′(x)J a  t (t, x)  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  −iGJT t  x(t, x) = −fa′(x)J a  x (t, x) − tfa′′(x)J a  t (t, x)  – 21 –  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='7  The Algebra of Modes  The EM tensor components have the following mode-expansions:  T t  t(t, x) = −i  �  n∈Z  x−n−2Mn  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  T t  x(t, x) = −i  �  n∈Z  x−n−2 [Ln − (n + 2) t  xMn]  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='71)  =⇒ Mn = i  �  0  dx  2πi xn+1 T t  t(t, x)  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Ln = i  �  0  dx  2πi  �  xn+1T t  x(t, x) + (n + 1)xnt T t  t(t, x)  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='72)  In [43], it was shown from the EM tensor OPEs (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='37) that the EM tensor modes indeed  generate the centrally extended BMS3 algebra:  [Mn , Mm] = 0  [Ln , Mm] = (n − m)Mn+m + cM  12 (n3 − n)δn+m,0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='73)  [Ln , Lm] = (n − m)Ln+m + cL  12(n3 − n)δn+m,0  The infinitesimal 2D CC transformation properties of the currents are expressed in the  operator language [43] using the prescription (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='23) for the charges in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='72):  [Ln , J a  x (t, x)] = [xn+1∂x + t(n + 1)xn∂t + (n + 1)xn]J a  x (t, x) + t(n + 1)nxn−1J a  t (t, x)  [Mn , J a  x (t, x)] = xn+1∂tJ a  x (t, x) + (n + 1)xnJ a  t (t, x)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='74)  [Ln , J a  t (t, x)] = xn+1∂xJ a  t (t, x) + (n + 1)xnJ a  t (t, x)  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [Mn , J a  t (t, x)] = 0  Next, the classical conservation laws (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4) imply the following space-time dependence of  the fields {J a  t } and {J a  x }:  ∂tJ a  t (t, x) = 0  =⇒  J a  t (t, x) = J a  t (x)  ∂tJ a  x (t, x) = ∂xJ a  t (x)  =⇒  J a  x (t, x) = t∂xJ a  t (x) + Ra(x)  with Ra(x) being arbitrary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Guided by the above functional dependence and using the fact that the pair of the fields  J a  x and J a  t forms a 2D CC primary rank-1  2 multiplet with scaling dimension ∆ = 1, we  write down the mode-expansions following [43]:  J a  t (x) =  �  n∈Z  x−n−1Ka  n  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  J a  x (t, x) =  �  n∈Z  x−n−1  �  Ja  n − (n + 1) t  xKa  n  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='75)  Ka  n =  �  Cu  dx  2πi xnJ a  t (x)  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Ja  n =  �  Cu  dx  2πi  �  xnJ a  x (t, x) + nxn−1tJ a  t (x)  �  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='76)  where the counter-clockwise contour Cu encloses the upper half-plane and the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  – 22 –  Comparing (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='76) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='24) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='25), we immediately see that:  Ja  n is the conserved charge of the current jaµ =  �  xnJ a  x + tnxn−1J a  t , −xnJ a  t  �  Ka  n is the conserved charge of the current kaµ = (xnJ a  t , 0)  Thus, using the prescription (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='23) on the OPEs (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='33) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='34) we reach the definition  of a current primary field Φ(t, x) in the operator formalism (for any n ∈ Z):  [Ka  n , Φ(t, x)] = ixn (ta  K · Φ) (t, x),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='77a)  [Ja  n , Φ(t, x)] = i  �  xn (Φ · ta  J) + tnxn−1 (ta  K · Φ)  �  (t, x)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='77b)  For the EM tensor components, the analogues commutation relations are found directly  from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='70):  [Ka  n , T t  t(t, x)] = 0, [Ja  n , T t  t(t, x)] = −inxn−1J a  t (t, x), [Ka  n , T t  x(t, x)] = −inxn−1J a  t (t, x),  [Ja  n , T t  x(t, x)] = −i  �  nxn−1J a  x + tn(n − 1)xn−2J a  t  �  (t, x)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='78)  Substituting the current mode-expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='75) and EM tensor mode expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='71) in  these operator relations, we obtain the cross-commutation relations between the modes of  the EM tensor and the currents:  [Ln , Ja  m] = −mJa  m+n  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [Mn , Ja  m] = −mKa  m+n = [Ln , Ka  m] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [Mn , Ka  m] = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='79)  Similarly, from the current-current OPEs (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='50), using the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='55) and the current  mode-expansion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='75), we reach the Lie algebra of the current modes:  [Ja  n , Jb  m] = iF abcJc  n+m + iGabcKc  n+m + nk1δabδn+m,0  [Ja  n , Kb  m] = iF abcKc  n+m + nk2δabδn+m,0  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [Ka  n , Kb  m] = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='80)  The global internal symmetry is governed by the subalgebra of the zero-modes:  [Ja  0 , Jb  0] = iF abcJc  0 + iGabcKc  0  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [Ja  0 , Kb  0] = iF abcKc  0  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [Ka  0 , Kb  0] = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='81)  This is precisely the algebra we have obtained for the NL currents in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  – 23 –  4  Sugawara Construction  As is well known, in 2d CFT, the Sugawara construction is employed to construct (the  modes of) the energy momentum tensor in terms of (the modes of) the currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' In this  section, here we will attempt to construct a NL version of the Sugawara construction and  express Lm and Mm in terms of Jm and Km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  We shall see that the Lm’s and Mm’s  constructed in such manner indeed satisfy the BMS algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Similar constructions have  been studied earlier in [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1  Intrinsic construction from algebra  We begin here by assuming we have only the algebra of the NL currents, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [Ja  m, Jb  n] = ifabcJc  m+n + mk1δabδm+n,0,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1a)  [Ja  m, Kb  n] = ifabcKc  m+n + mk2δabδm+n,0,  [Ka  m, Kb  n] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1b)  In the above, the sum over the group indices e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' fabcJc  m+n = �dim(¯g)  c=1  fabcJc  m+n is implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Comparing with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21), as stressed before, we work with the case where F abc = fabc, Gabc =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Let us first consider the zero modes of J and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Putting n = m = 0 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1), the algebra  for the zero modes become  [Ja  0 , Jb  0] = ifabcJc  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [Ja  0 , Kb  0] = ifabcKc  0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [Ka  0, Kb  0] = 0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2)  Now let us look for quadratic Casimir operators for the above algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  The possible  combinations are  �  a  Ja  0 Ja  0 ,  �  a  Ja  0 Ka  0 ,  �  a  Ka  0Ja  0 ,  �  a  Ka  0Ka  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3)  Keeping in mind that Casimir operators must commute with all the generators, we can  exclude �  a Ja  0 Ja  0 since it does not commute with Ka  0  � �  a  Ja  0 Ja  0 , Kb  0  �  = i  �  a,c  fabc(JaKc + KcJa) ̸= 0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4)  All other combinations in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3) commute with all Ja  0 ’s and Ka  0’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Hence a generic Casimir  operator constructed from the zero modes of J and K will be a linear combination of the  above three combinations of Ja  0 ’s and Ka  0’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We therefore want to construct the zero level  generator L0, M0 from these combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Since L0 and M0 are expected to be quadratic in terms of Ja  n’s and Ka  n’s, we can write  down the following generic expression for them (The term � JJ is excluded since we have  already seen that the term �  a Ja  0 Ja  0 does not contribute to the zero mode part of L0 and  M0)  L0 = α  �  a,l,n  Ja  l Ka  n + β  �  a,l,n  Ka  l Ja  n + ρ  �  a,l,n  Ka  l Ka  n  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5a)  M0 = µ  �  a,l,n  Ja  l Ka  n + ν  �  a,l,n  Ka  l Ja  n + η  �  a,l,n  Ka  l Ka  n  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5b)  – 24 –  Now putting the conditions that  [L0, Ja  n] = −nJa  n, [M0, Ja  n] = −nKa  n  and looking at just the level of current generators on the RHS, we can see that only (l = −n)  terms should contribute, so we obtain  L0 = α  �  a,n  Ja  nKa  −n + β  �  a,n  Ka  nJa  −n + ρ  �  a,n  Ka  nKa  −n = αX0 + βY0 + ρZ0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='6a)  M0 = µ  �  a,l  Ja  l Ka  −l + ν  �  a,l  Ka  l Ja  −l + η  �  a,l  Ka  l Ka  −l = µX0 + νY0 + ηZ0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='6b)  Now take  Y0 =  �  a,l  Ka  l Ja  −l =  �  a,l  Ja  −lKa  l + k2  dimg  2  ∞  �  l=−∞  l = X0 + c,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='7)  where c is a (possibly infinite) constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Since we always have the independence of redefin-  ing BMS generators by constant shifts, we can define level 0 BMS generators as  L0 ≡ L′  0 = α + β  2  (X0 + Y0) + ρZ0,  M0 ≡ M′  0 = µ + ν  2  (X0 + Y0) + ηZ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='8)  The normal ordering of the operators L0, M0 is achieved by individual normal ordering  of X0, Y0, Z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We generalise the definition to the other BMS generators as (with normal  ordering)  Lm =  dimg  �  a=1  �  �α + β  2  �  �  �  �  l≤−1  (Ja  l Ka  m−l + Ka  l Ja  m−l) +  �  l>−1  (Ja  m−lKa  l + Ka  m−lJa  l )  �  �  � + ρ  �  l  Ka  l Ka  m−l  �  �  Mm =  dimg  �  a=1  �  �µ + ν  2  �  �  �  �  l≤−1  (Ja  l Ka  m−l + Ka  l Ja  m−l) +  �  l>−1  (Ja  m−lKa  l + Ka  m−lJa  l )  �  �  � + η  �  l  Ka  l Ka  m−l  �  �  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='9)  Using the form of the BMS generators in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='9), and substituting in the BMS-current cross  commutators, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [Lm, Ja  n] = −nJa  m+n ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' [Lm, Ka  n] = −nKa  m+n ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' [Mm, Ja  n] = −nKa  m+n,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='10)  we obtain the following values for the coefficients  α + β = 1  k2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' ρ = −k1 + 2Cg  2k2  2  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  µ + ν = 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' η =  1  2k2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='11)  So we obtain the final form for our NL Sugawara construction  Lm =  1  2k2  dim(g)  �  a=1  �  �  �  �  l≤−1  (Ja  l Ka  m−l + Ka  l Ja  m−l) +  �  l>−1  (Ja  m−lKa  l + Ka  m−lJa  l ) − (k1 + 2Cg)  k2  �  l  Ka  l Ka  m−l  �  �  �  – 25 –  Mm =  1  2k2  dim(g)  �  a=1  �  l  Ka  l Ka  m−l  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12)  As a check of the validity of the analysis, we compute the algebra of the L and M generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  These satisfy the following algebra (see Appendix B for detailed calculations)  [Lm, Ln] = (m − n)Lm+n + dim(g)  6  (m3 − m)δm+n,0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' [Lm, Mn] = (m − n)Mm+n  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='13)  Hence, we see that by doing the NL Sugawara construction of NL currents, we can find  BMS algebra with  cL = 2dim(g),  cM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='14)  Non-zero cM:  We can obtain a non-zero cM with a slight modification to the Sugawara  construction presented above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' In case of Virasoro algebra with additional symmetries, we  can define new Virasoro generators [46] as  ˜Ln = LS  n + inθaja  n + 1  2kθ2δn,0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='15)  where LS  n is the Virasoro generators obtained from Sugawara construction and θ = θata is a  vector belonging to the Lie algebra with generators ta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' It can be showed that (see Appendix  C) if Ln satisfy Virasoro algebra with central charge c then ˜Ln will satisfy Virasoro algebra  with shifted central charge ˜c where  ˜c = c + 12kθ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='16)  In case of NL Kac-Moody algebra, inspired by this, we introduce the following redefinitions  ˜Ln = LS  n + inθaJa  n + 1  2k1θ2δn,0  ˜  Mn = MS  n + inθaKa  n + 1  2k2θ2δn,0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='17)  where LS  n and MS  n are given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' This redefinition will give us the following algebra  (see Appendix C)  [˜Lm, ˜Ln] = (m − n)˜Lm+n + cL  12(m3 − m)δn+m,0  [˜Lm, ˜  Mn] = (m − n) ˜  Mm+n + cM  12 (m3 − m)δn+m,0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='18)  where the central charges are given by  cL = 2dim(g) + 12k1θ2  cM = 12k2θ2  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='19)  Hence we can obtain the fully centrally extended BMS algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  – 26 –  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2  Consistency with OPEs  We will recast the NL Sugawara construction in terms of the NL EM tensor that we  introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' 3 instead of the generators of the BMS: {Ln, Mn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' {Ln, Mn} are of  course modes of the NL EM tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Hence the calculations in this subsection provide a  sanity check for our analysis making sure the various formulations are consistent with each  other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  The NL Sugawara construction gave us (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  We now normal order the products to  rewrite this as  Ln =  1  2k2  �  l,a  �  : Ja  l Ka  n−l : + : Ka  l Ja  n−l : −k1 + 2Cg  k2  : Ka  l Ka  n−l :  �  Mn =  1  2k2  �  l,a  : Ka  l Ka  n−l :  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='20)  Here, : Al : is a shorthand for normal ordered products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Now substituting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='20) in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='71)  and rearranging the terms, we get uv to xt  Tu(u, v) =  1  2k2  �  a  ��  (J a  u J a  v )(u, v) + (J a  v J a  u )(u, v)  �  − k1 + 2Cg  k2  (J a  v J a  v )(v)  �  Tv(v) =  1  2k2  �  a  (J a  v J a  v )(v)  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21)  Here (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' ) means normal ordered products of fields, which can be expressed in terms of  contour integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Here we are using u, v coordinates instead of x, t of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' 3 because  this construction applies to both Galilean and Carrollian CFTs, because of the similarity  between the 2 theories mentioned in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' So by substituting u → x, v → t or  u → t, v → x, we can obtain the Galilean or Carrollian theory expressions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  So (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21) gives the field expression for the NL Sugawara construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' For the modified  Sugawara construction as defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='17), the EM tensor fields turn out to be (by doing  the mode expansion using the new generators)  ˜Tu(u, v) = Tu(u, v) − iθa∂vJ a  u (u, v) − iθa J a  u (u, v)  v  − iθa uJ a  v (v)  v2  + 1  2  k1θ2  v2  + uk2θ2  v3  ˜Tv(v) = Tv(v) − iθa∂vJ a  v (v) − iθa J a  v (v)  v  + 1  2  k2θ2  v2  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='22)  where, Tu(u, v), Tv(u, v) refers to the quantities in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' the normal NL Sugawara  construction expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  T − J OPE:  Now, for a consistency check of our analysis so far, we will rederive the  OPEs from the above definitions of the NL Sugawara construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We begin with the T-J  OPEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Taking the definitions as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21) and contracting with the currents, we get the  – 27 –  following expressions (See Appendix D for detailed calculation):  Tv(u1, v1)J b  v (u2, v2) ∼ regular  Tu(u1, v1)J b  v (u2, v2) ∼ J b  v (u2, v2)  v2  12  + ∂vJ b  v (u2, v2)  v12  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Tv(u1, v1)J a  u (u2, v2) ∼ J a  v (u2, v2)  v2  12  + ∂vJ a  v (u2, v2)  v12  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Tu(u1, v1)J a  u (u2, v2) ∼ J a  u (u2, v2)  v2  12  + ∂vJ a  u (u2, v2)  v12  + u12  v2  12  ∂uJ a  u (u2, v2)  + 2u12  v12  �J a  v (u2, v2)  v2  12  + ∂vJ a  v (u2, v2)  v12  �  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='23)  Which are equivalent to the [Lm, Ja  n] type commutation relations given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Clearly  these relation satisfy the OPE relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='66) that the currents and the EM tensors of a  theory are supposed to satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  T − T OPE:  Next we use the definition of the Sugawara construction to compute the  T-T type OPEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Doing the calculations (see Appendix D for detailed Calculations), we  get,  Tu(u1, v1)Tu(u2, v2) ∼2Tu(u2, v2)  v2  12  + ∂vTu(u2, v2)  v12  + u12  v2  12  ∂uTu(u2, v2)  + 2u12  v12  �2Tv(u2, v2)  v2  12  + ∂vTv(u2, v2)  v12  �  + dim(g)  v4  12  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Tu(u1, v1)Tv(u2, v2) ∼2Tv(u2, v2)  v2  12  + ∂vTv(u2, v2)  v12  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='24)  Tv(u1, v1)Tv(u2, v2) ∼ regular  These results (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='24) match with the OPEs of the energy momentum tensors of a 2d Carrol-  lian or Galilean CFT, as given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' So, the NL Sugawara construction really gives the  EM tensors of a 2d NL CFT (proved both at algebra level and now at field level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Also from  the OPE expressions, we can verify the earlier results of cL = 2dim(g), cM = 0 obtained ear-  lier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Again if we start with the modified Sugawara construction (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='22), we can obtain same  OPE relations as above, but with modified central charges cL = 2dim(g) + 12k1θ2, cM =  12k2θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' To see this, let’s look at, for example, the ˜Tu(u1, v1) ˜Tv(v2) OPE and focus our  attention to a specific term  ˜Tu(u1, v1) ˜Tv(v2) ∼ −θaθb∂vJ a  u (u1, v1)∂vJ a  v (v2) + · · · ∼ −θ2∂v1∂v2  k2  v2  12  + · · · ∼ 6k2θ2  v4  12  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Matching the above expression with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='37), we can see cM = 12k2θ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Similarly we can  obtain the other shifted central charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  – 28 –  5  Tensionless String as NLKM  The tensionless limit of string theory is the limit that is diametrically opposite to the  usual point particle limit where known supergravity appears from strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' This limit that  explores the very strong gravity regime and, when quantized, the very highly quantum and  highly stringy regime of string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' In this section, we explore how this is connected  to the NLKM structures we have discussed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We will see that the currents  satisfying the U(1) NLKM algebra come up intrinsically when we are looking at tensionless  strings propagating in flat spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  As is very well known, the action for a tensile relativistic string propagating in a flat  d-dimensional spacetime is given by the Polyakov action:  SP = T  �  d2ξ √γγαβ∂αXµ∂βXνηµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1)  In order to take the tensionless limit, it is helpful to work with the phase space action of  string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' One can then systematically take the limit [] and the resulting action can be cast  in a Polyakov like form, which we will call the ILST action after the authors:  SILST =  �  d2ξ V αV β∂αXµ∂βXνηµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2)  Here Xµ are the coordinates in the background flat space ηµν which are scalar fields on  the worldsheet parametrized by ξa = σ, τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The worldsheet metric γαβ degenerates in the  limit and the following replacement is made:  T√γγαβ → V αV β,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3)  where V α is a vector density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The action is invariant under worldsheet diffeomorphisms  and hence one needs to fix a gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' It is helpful to go to the equivalent of the conformal  gauge  V α = (v, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4)  There is some symmetry left over even after this gauge fixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' In the usual tensile string,  the residual symmetry gives two copies of the Virasoro algebra and the appearance of  a 2d CFT on the worldsheet is the central reason why we understand string theory as  well as we do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Now in the tensionless case, the residual gauge symmetry that appears on  the worldsheet is the BMS3 algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' This now dictates the theory of tensionless strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  The reason behind the appearance of the BMS algebra is that the tensionless string in  flat spacetimes is actually a null string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' This is the string equivalent of a massless point  particle which is constrained to travel on a null geodesic of the ambient spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The  string sweeps out a worldsheet which is null and since there are no mass terms, the resulting  action has to have conformal Carrollian symmetry in d = 2 or equivalently BMS3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Now, let us connect to our discussion in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' In the favourable conformal gauge  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4), the equations of motion and constraints arising from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2) take the form  EOM:  ¨Xµ = 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Constraints:  ˙X2 = 0 ,  ˙X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='X′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5)  – 29 –  A convenient form of the solution is given by  Xµ(σ, τ) = xµ +  √  2c′�  Jµ  0 σ + Kµ  0 τ + i  �  n̸=0  1  n(Jµ  n − inτKµ  n)e−inσ�  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='6)  Here σ, τ are the coordinates on the cylinder, which are related to the planar Non-Lorentzian  coordinates as  u = e−iσ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v = −τe−iσ  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='7)  The periodic condition Xµ(σ + 2π, τ) = Xµ(σ, τ) forces us to have Jµ  0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='6), we  obtain  Jµ  τ = ∂σXµ =  √  2c′ �  n  (Jµ  n − inτKµ  n)e−inσ,  Jµ  σ = −∂τXµ = −  √  2c′ �  n  Kµ  ne−inσ  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='8)  Clearly the currents in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='8) satisfy similar relations as (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  ∂τJµ  σ = 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' ∂τJµ  τ + ∂σJµ  σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='9)  If we canonically quantise the system by demanding [Xµ(τ, σ), Πν(τ, σ′)] = iδ(σ − σ′)ηµν,  we obtain the following relations for the current modes  [Jµ  m, Jν  n] = [Kµ  m, Kν  n] = 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' [Jµ  m, Kν  n] = 2mδm+n,0ηµν  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='10)  Clearly, the current modes for the same spacetime index µ follow a special case of U(1) NL  Kac-Moody algebra (look at (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21) for comparison), with k(µν)  1  = 0, k(µν)  2  = 2ηµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Next if we look into the (classical) energy momentum tensor of this theory, their mode  expansion coefficients are given by  Ln = 1  2  �  m  Jµ  mKν  n−mηµν ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Mn = 1  4  �  m  Kµ  mKν  n−mηµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='11)  This matches classically with the relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12) for the NL Sugawara construction with  k1 = 0,  k2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12)  Note that Cg = 0 for U(1) algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The generators in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='11) satisfy the (centreless) BMS3  algebra, which is expected as the action (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2) has BMS symmetry as a residual gauge  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Thus we can see that the Non-Lorentzian U(1) Kac-Moody algebra and the associated NL  Sugawara construction come up in the theory of tensionless or null strings propagating on  a flat background geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' An outstanding question is what happens when we look at the  propagation of null strings on arbitrary curved manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' These can be viewed as group  manifolds and there will be the NL equivalent of a Wess-Zumino-Witten model appearing  for tensionless strings in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We expect that the non-abelian NLKM algebras  explored in this work would naturally appear on these NL WZW models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' This is work in  progress and we hope to report on this in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  – 30 –  6  Non-Lorentzian Knizhnik Zamolodchikov Equations  In usual Lorentzian Kac-Moody algebras, the Knzhnik-Zamolodchikov equations are the  linear differential equations that are satisfied by the correlation functions of these CFTs  endowed with additional affine Lie symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' In this section, we write down the non-  Lorentzian analogues of these KZ equations based on our construction of the NLKM algebra  and its representations in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  In what follows, we will outline the steps to get to the NL Knizhnik Zamolodchikov equa-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The details of the calculation are presented in a separate appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  We begin with the OPE definition of BMS primary field (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='57) and obtain  ∂uΦ(u′, v′) = −  �  v′  dv  2πi  �  u′  du  2πi(u − u′)−1Tv(u, v)Φ(u′, v′) = −  �  v′  dv  2πiTv(v)Φ(u′, v′)  ∂vΦ(u′, v′) =  �  v′  dv  2πi  �  u′  du  2πi(u − u′)−1Tu(u, v)Φ(u′, v′) =  �  v′  dv  2πiTu(u′, v)Φ(u′, v′)  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1)  Next, we take the correlation function of a string of primary fields as shown below  ⟨∂uΦ(u, v)Φ1(u1, v1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Φn(un, vn)⟩ = ⟨∂uΦ(u, v)X({ui, vi})⟩ =  �  v  dv′  2πi⟨Tv(u′, v′)Φ(u, v)X({ui, vi})⟩  =  �  v  dv′  2πi⟨ 1  2k2  �  (J a  v J a  v )(u′, v′)  �  Φ(u, v)X({ui, vi})⟩  and use relations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='57, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1) to finally get the following expression (details in Ap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' E)  ⟨∂uΦ(u, v)X({ui, vi})⟩ = − 1  k2  ��  j  ta  R,K ⊗ ta  Rj,K  (v − vj)  �  ⟨Φ(u, v)X({ui, vi})⟩  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2)  which can be written as  �  ∂ui + 1  k2  �  j̸=i  ta  Ri,K ⊗ ta  Rj,K  (vi − vj)  �  ⟨Φ1(u1, v1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Φn(un, vn)⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3)  This is one of the Non-Lorentzian Knizhnik Zamolodchikov equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Similarly we can start with  ⟨∂vΦ(u, v)Φ1(u1, v1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Φn(un, vn)⟩ = ⟨∂vΦ(u, v)X({ui, vi})⟩ =  �  v  dv′  2πi⟨Tu(u′, v′)Φ(u, v)X({ui, vi})⟩  =  �  v  dv′  2πi⟨ 1  2k2  �  (J a  v J a  u )(u′, v′) + (J a  u J a  v )(u′, v′) − k1 + 2Cg  k2  (J a  v J a  v )(u′, v′)  �  Φ(u, v)X({ui, vi})⟩  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4)  From here, we can proceed in the same way as Appendix E to obtain the other equation  �  ∂vi − 1  k2  �  j̸=i  1  (vi − vj)  dimg  �  a=1  ��  ta  Ri,J ⊗ ta  Rj,K + ta  Ri,K ⊗ ta  Rj,J  �  +  �ui − uj  vi − vj  − k1 + 2Cg  k2  �  ta  Ri,K ⊗ ta  Rj,K  ��  ⟨ΦR1(u1, v1)ΦR2(u2, v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='ΦRn(un, vn)⟩ = 0  (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5)  – 31 –  This above equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5) is the second of the Non-Lorentzian Knizhnik Zamolodchikov  equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The solutions to these two equations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5) would give the correlation  functions of the underlying NLKM theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' For usual relativistic theories, the KZ equations  are difficult to solve in general, but one can do it for the four-point functions yielding a  closed solution in terms of hypergeometric functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  It would be instructive to check  whether something similar happens here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' If one can find the solutions to the NLKM four  point functions, it would also be a nice exercise to check whether these can be arrived  at as a limit of the relativistic answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' This process of attempting to generate the non-  Lorentzian answers from the relativistic ones is something we outline in detail in the section  that follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  7  NLKM from Contractions  In this section, we rederive various results we have obtained earlier in the paper through  a systematic limit on the algebraic structures obtained in the relativistic set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Before  moving to recovering the answers, we spend a bit of time understanding which limit is  appropriate for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1  A brief detour to representations of BMS  We said in the introduction that there were two distinct contractions that land us up on  the BMS3 algebra starting from two copies of the Virasoro algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  One of them was  a Carrollian or ultra-relativistic limit (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='17), where there was a mixing of positive and  negative Virasoro modes creating the BMS generators, while the other was a Galilean or  non-relativistic (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='13), where no mixing took place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' This mixing of modes is critical for the  understanding of the representations in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  We begin with the Galilean contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  In the parent relativistic CFT, the theory is  best described in terms of the highest weight representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The states of the theory are  labelled by the zero modes:  L0|h, ¯h⟩ = h|h, ¯h⟩,  ¯L0|h, ¯h⟩ = ¯h|h, ¯h⟩  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1)  There is a class of states called primary states which are annihilated by all positive modes:  Ln|h, ¯h⟩p = 0,  ¯Ln|h, ¯h⟩p = 0,  ∀n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2)  The Virasoro modules are built on these primary states by acting with raising operators  L−n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Now, looking back at the Galilean contraction (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='13), we see that  Ln = 1  2  �  Ln + 1  ϵ Mn  �  ,  ¯Ln = 1  2  �  Ln − 1  ϵ Mn  �  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3)  The 2d CFT primary conditions then boil down to:  L0|∆, ξ⟩ = ∆|∆, ξ⟩, M0|∆, ξ⟩ = ξ|∆, ξ⟩;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Ln|∆, ξ⟩p = 0, Mn|∆, ξ⟩p = 0, ∀n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4)  – 32 –  In the above, the assumption is that the state |h, ¯h⟩ goes to the state |∆, ξ⟩ in the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' So  we see that highest weight states map to highest weight states in the Galilean contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  This analysis holds in a similar way when we consider the full NLKM algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We will be  using this for the analysis in the rest of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  But before we get there, let us point out why we are not using the Carroll limit (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='17) for  this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' In the Carroll contraction, we can read off  Ln = 1  2  �  Ln + 1  ϵ Mn  �  ,  ¯Ln = 1  2  �  −L−n + 1  ϵ M−n  �  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5)  The 2d CFT primary conditions in the Carroll limit become:  M0|M, s⟩ = M|M, s⟩,  L0|M, s⟩ = s|M, s⟩,  Mn|M, s⟩ = 0,  ∀n ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='6)  In the above, the state |h, ¯h⟩ in the Carroll limit becomes the state |M, s⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' This is clearly  not a highest weight state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The set of these states form what is called the induced repre-  sentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The story is again similar for the full NLKM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' In the analysis that follows, we  will not focus on the induced representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' It would be of interest to consider them in  future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  We should stress however, that the answers we have obtained in the intrinsic Carrollian  form earlier are for highest weight representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The Galilean limit would be an effective  way of reproducing these answers with a flip of space and time directions at the end of the  analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2  Contraction of the affine parameters  In this section we will establish the mode expansion of the NLKM currents (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='76) from  another approach, by doing a contraction of the original loop extended algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Starting  from the finite Lie algebra  [ja, jb] = ifabcjc ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' [¯ja, ¯jb] = ifabc¯jc  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='7)  we can define loop extended generators  ja  n = ja ⊗ zn ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' ¯ja  n = ¯ja ⊗ ¯zn,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='8)  which satisfy  [ja  n, jb  m] = ifabcjc ⊗ zn+m = ifabcjc  n+m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='9)  Loop extended algebra obtained above admits a central extension to give us the current  algebra in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Now, we can do a contraction of the affine parameter as z = t + ϵx and  ¯z = t − ϵx in the Galilean limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We then obtain (keeping upto first order in ϵ),  ja  n = ja ⊗ (tn + nϵxtn−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=') ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' ¯ja  n = ¯ja ⊗ (tn − nϵxtn−1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=')  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='10)  Now we can define new contracted generators of the finite algebra as follows:  Ja = ja + ¯ja ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Ka = ϵ(¯ja − ja)  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='11)  – 33 –  Above definition can be extended to general modes of J and K using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='10),  Ja  n = lim  ϵ→0(ja  n + ¯ja  n) = Ja ⊗ tn − nKa ⊗ xtn−1  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12a)  Ka  n = lim  ϵ→0 ϵ(¯ja  n − ja  n) = Ka ⊗ tn  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12b)  In this way, we get the structure of the power series expansion of J and K in terms of x  and t, which is the Galilean analog of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='76), or identical upto the flip of temporal and  spatial directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  If we define primary field φ(x, t) as a representation of the finite algebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' in terms of  the action of zero modes of the generators J and K,  [Ja  0 , φ(x, t)] ≡ [Ja, φ(x, t)] = ta  Jφ(x, t)  [Ka  0, φ(x, t)] ≡ [Ka, φ(x, t)] = ta  Kφ(x, t)  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='13)  We can get the action of general modes using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12),  [Ja  n, φ(x, t)] = ta  Jφ(x, t) ⊗ tn − nta  Kφ(x, t) ⊗ xtn−1 ≡ tnta  Jφ(x, t) − nxtn−1ta  Kφ(x, t)  [Ka  n, φ(x, t)] = ta  Kφ(x, t) ⊗ tn ≡ tnta  Kφ(x, t)  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='14)  This definition of a primary field agrees with our earlier result (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='77).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  This action of Ja  n and Ka  n on primary fields also appears when we draw motivation from  [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We can define the primary field as,  [Ja  0 , φ(0, 0)] = ta  Jφ(0, 0) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' [Ka  0, φ(0, 0)] = ta  kφ(0, 0),  [Ja  n, φ(0, 0)] = 0 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' [Ka  n, φ(0, 0)] = 0  ∀ n > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='15)  For n ≥ 0 and U = etL−1−xM−1,  [Ja  n, φ(x, t)] = [Ja  n, Uφ(0)U −1] = U[U −1Ja  nU, φ(0)]U −1  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='16)  Consider,  U −1Ja  nU = e−tL−1+xM−1Ja  netL−1−xM−1  =  n  �  k=0  tk  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='Ja  n−k − nx  n−1  �  k=0  tk  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (n − k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='Ka  n−k−1  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='17)  where we have used the Baker-Campbell-Hausdorff(BCH) formula twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Putting this back  in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='16), we finally get,  [Ja  n, φ(x, t)] = U  � n  �  k=0  tk  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' [Ja  n−k, φ(0)] − nx  n−1  �  k=0  tk  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (n − k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' [Ka  n−k−1, φ(0)]  �  U −1  = (tnta  J − nxtn−1ta  K)φ(x, t)  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='18)  where only the terms corresponding to k = n and k = n − 1 survived respectively in the  first and the second sum because of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Similarly for Ka  n, we get,  [Ka  n, φ(x, t)] = U  � n  �  k=0  tk  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' [Ka  n−k, φ(0)]  �  U −1 = tnta  Kφ(x, t)  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='19)  These results verify (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='14) again from a different perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  – 34 –  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3  Contracting the Sugawara construction  In section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1, we constructed the BMS generators by taking quadratic products of the  NL currents and ended up with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' In this subsection, we shall reproduce the same  from the Galilean limit of Sugawara construction for CFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We start from the following  expression for virasoro modes Lm and ¯Lm in relativistic sugawara construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Lm = γ  dim(g)  �  a=1  (  �  l≤−1  ja  l ja  m−l +  �  l>−1  ja  m−lja  l )  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='20a)  ¯Lm = ¯γ  dim(g)  �  a=1  (  �  l≤−1  ¯ja  l ¯ja  m−l +  �  l>−1  ¯ja  m−l¯ja  l )  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='20b)  where γ =  1  2(k + Cg),¯γ =  1  2(¯k + Cg) and Cg = −  1  2dim(g)  �  b,c  fbacfbcd  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='20c)  We can now take the Galilean limit by using the inverted version of relations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='20) and  get the following form of Virasoro Generators by collecting the terms with same order in ϵ,  Lm = γ  4(Am − Bm  ϵ  + Cm  ϵ2 ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' ¯Lm = ¯γ  4(Am + Bm  ϵ  + Cm  ϵ2 )  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21)  where,  Am =  dim(g)  �  a=1  �  l≤−1  Ja  l Ja  m−l +  �  l>−1  Ja  m−lJa  l  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='22a)  Bm =  dim(g)  �  a=1  {  �  l≤−1  (Ja  l Ka  m−l + Ka  l Ja  m−l) +  �  l>−1  (Ja  m−lKa  l + Ka  m−lJa  l )}  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='22b)  Cm =  dim(g)  �  a=1  �  l  Ka  l Ka  m−l  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='22c)  where we have employed the commutativity of K’s in order to write 1  ϵ2 term(Cm) as a sum  from l = −∞ to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We can use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='13) and the above relations to obtain:  Lm = 1  4  �  (γ + ¯γ)Am − (γ − ¯γ)  ϵ  Bm + (γ + ¯γ)  ϵ2  Cm  �  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='23a)  Mm = −1  4  �  ϵ(γ − ¯γ)Am − (γ + ¯γ)Bm + (γ − ¯γ)  ϵ  Cm  �  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='23b)  Inverting the relations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='20), we can write k and ¯k in terms of k1 and k2,  k = 1  2(k1 − k2  ϵ ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' ¯k = 1  2(k1 + k2  ϵ )  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='24)  Using the definitions of γ and ¯γ in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='20) and using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='24), we can write,  lim  ϵ→0  γ + ¯γ  ϵ2  = −2(k1 + 2Cg)  k2  2  and lim  ϵ→0  γ − ¯γ  ϵ  = − 2  k2  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='25)  – 35 –  Hence, in limit ϵ → 0 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='23) can be written as,  Lm =  1  2k2  �  Bm − (k1 + 2Cg)  k2  Cm  �  and Mm =  1  2k2  Cm  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='26)  which agrees with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  As we have already seen in the previous section that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12) satisfies BMS algebra with  central charges cL = 2dim(g) and cM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' This fact can also be verified using the following  definitions of central charges in the relativistic Sugawara construction,  c = kdim(g)  k + Cg  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' ¯c =  ¯kdim(g)  ¯k + Cg  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='27)  Using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='27) and following similar steps as before using the (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='24), we can get the following,  cL = lim  ϵ→0(c + ¯c) = lim  ϵ→0  �  dim(g)  �  1  2(k1 − k2  ϵ )  1  2(k1 − k2  ϵ ) + Cg  +  1  2(k1 + k2  ϵ )  1  2(k1 + k2  ϵ ) + Cg  ��  ⇒ cL = 2dim(g)  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='28)  Similarly, we can get,  cM = lim  ϵ→0 ϵ(¯c − c) = 0  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='29)  These are the same values of central charges appeared in the L, M commutation relations  as we got in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  The modification we have done in order to get non-zero cM too can be retrieved from  contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' For this, we can start from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='15) and its conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Now if we take the limit  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='20), we shall again retrieve the redefined Lns and Mns as we have seen in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='17), and  the commutators will be same as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='18) with central charges (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4  NLKZ equations from Contraction  Finally, we show how to obtain the non-Lorentzian Knizhnik Zamolodchikov equations  from a limit of the ones for a 2d CFT with additional symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Some of the details of the  analysis are contained in Appendix F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We start with the original Knizhnik Zamolodchikov  equations:  �  �∂wi − 2γ  �  j̸=i  �  a(ta  Ri)ri  si(ta  Rj)rj  sj  wi − wj  �  � ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='φsi  Ri(wj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='φsj  Rj(wj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='⟩ = 0  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='30a)  �  �∂ ¯wi − 2¯γ  �  j̸=i  �  a(¯ta¯Ri)¯ri¯si(¯ta¯Rj)¯rj  ¯sj  ¯wi − ¯wj  �  � ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='¯φ¯si¯Ri( ¯wj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='¯φ¯sj  ¯Rj( ¯wj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='⟩ = 0  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='30b)  where  (ta  Ri)ri  si(ta  Rj)rj  sj⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='φsi  Ri(wj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='φsj  Rj(wj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='⟩ = ((ta  Ri ⊗ ta  Rj)⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='φRi(wj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='φRj(wj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='⟩)ri,rj  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='31)  – 36 –  Using the separability of the primary fields, Φr,¯r  R, ¯R(w, ¯w) = φr  R(w) ⊗′ ¯φ¯r¯R( ¯w), we can write,  ⟨Φr1,¯r1  R1, ¯R1(w1, ¯w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='ΦrN,¯rN  RN, ¯RN (wN, ¯wN)⟩ = ⟨φr1  R1(w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='φrN  RN (wN)⟩⟨¯φ¯r1¯R1( ¯w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='¯φ¯rN  ¯RN ( ¯wN)⟩ (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='32)  where the primed tensor product(⊗′) ensures independent action of operators on chiral and  anti-chiral sectors whereas the unprimed tensor product(⊗) ensures independent action on  ith and jth insertion in the n-point function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Using the following linear combinations in limit ϵ → 0,  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='30a) × ⟨¯φ¯r1¯R1( ¯w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='¯φ¯rN  ¯RN ( ¯wN)⟩ + (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='30b) × ⟨φr1  R1(w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='φrN  RN (wN)⟩ = 0  ϵ{(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='30a) × ⟨¯φ¯r1¯R1( ¯w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='¯φ¯rN  ¯RN ( ¯wN)⟩ − (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='30b) × ⟨φr1  R1(w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='φrN  RN (wN)⟩} = 0  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='33)  We get (shown in details in Appendix F),  �  ∂ti − 1  k2  �  j̸=i  ��  a(ta  Ri,J ⊗ ta  Rj,K + ta  Ri,K ⊗ ta  Rj,J)  tij  +  �xij  t2  ij  − (k1 + 2Cg)  k2tij  � �  a  (ta  Ri,K ⊗ ta  Rj,K)  ��  ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='ΦRi, ¯Ri(xi, ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='ΦRj, ¯  Rj(xj, tj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='⟩ = 0  �  ∂xi + 1  k2  �  j̸=i  �  a(ta  Ri,K ⊗ ta  Rj,K)  tij  �  ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='ΦRi, ¯Ri(xi, ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='ΦRj, ¯Rj(xj, tj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='⟩ = 0  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='34)  where, ta  Ri,J = ta  Ri ⊗′ ¯I +I ⊗′ ¯ta¯Ri and ta  Ri,J = ϵ(I ⊗′ ¯ta¯Ri −ta  Ri ⊗′ ¯I) in limit ϵ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The above  equations are same as what we got form intrinsic analysis i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5), with t → v  and x → u as its supposed to for a Galilean contracted result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  – 37 –  8  Conclusions  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1  Summary  In this paper we have explored aspects of Non-Lorentzian CFTs with additional Lie Al-  gebraic symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' First we have reproduced the Non-Lorentzian Kaˇc-Moody algebra by  taking singular limit from the Virasoro Kaˇc-moody algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  After this we attempt to construct the same from intrinsic viewpoint without any knowledge  of the parent algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We see that 2d Carrollian Conformal symmetry allows an infinite  number of Noether currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We then take a few pairs of conserved currents (introducing  flavour indices to distinguish them) satisfying the conditions for EM tensor components in  2d Carrollian Conformal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' After this we derive the Ward identities associated with  those conserved currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We also derive OPEs of a general 2d Carrollian Conformal field  with the current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' After this we find the conserved charge operators associated with these  currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The transformation generated by these charges on a generic field is also derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Here we first encounter the transformation matrices which we encounter in relativistic  CFT with Kac-Moody algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We introduce current primary fields which turn out to  be analogous to the Virasoro Kaˇc-Moody primary fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' After this the current current  OPEs are derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' While doing so, the structure constants emerge from the action of the  transformation matrices on the currents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' After this, global internal symmetry is applied on  two point and three point correlation and it turns out that the structure constants we have  encountered while calculating the current current OPEs satisfy Jacobi identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' After this  we derive the OPE between the Energy Momentum Tensor components and the current  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Using all these OPEs we derive the algebra of the current modes and the  Energy Momentum tensor modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The algebra transpire to be identical to the algebra  obtained from limits of Virasoro Kaˇc-Moody algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Later in the paper, we attempt to construct the EM tensor modes (which forms the BMS  algebra) from the current modes through Sugawara construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Here we see that the  Sugawara construction only takes us to the BMS algebra with one of the central charges  to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We needed another modification to the Sugawara construction in order to get  a fully centrally extended BMS algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Using the expression of the EM tensor modes in  terms of the current modes, we calculate the OPEs of EM tensor fields with themselves and  with currents and see that the OPEs thus derived matches with the OPEs in the earlier  section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  After this we have a brief look at the tensionless string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  When we look at the mode  expansion of the coordinates (treating them as scalar fields), we see that the modes satisfy  a special case of Non Lorentzian U(1) Kac-Moody algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We also look at the expression  of modes of the classical energy momentum tensor in terms of these U(1) modes and see  that classically this matches with the expression we have earlier derived for Non-Lorentzian  Sugawara construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Hence we see that U(1) NLKM currents are intrinsically present  in the tensionless string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  In section 6, we take a correlation function of a string of BMS current primary fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Using  the OPE definition of the BMS primary field, we arrive at the Non-Lorentzian version of the  – 38 –  Knizhnik Zamolodchikov equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Finally, in section 7, we derive all the earlier results  by taking limits from the parent algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2  Discussions and future directions  We mentioned in the introduction that our results in this paper lay the groundwork for a  large number of applications, most importantly to the construction of a holographic dictio-  nary for asymptotic flat spacetimes and also the understanding of tensionless strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We  have built the underlying algebraic structures in this paper which would be of importance  to the quantum field theories that are at the heart of these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  One of the most important and immediate next steps is to construct a Non-Lorentzian  Wess-Zumino-Witten model that realises these symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' This would be central to the  understanding of tensionless null strings moving on arbitrarily curved manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The work  on this is currently underway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  The tensionless limit of string theory on arbitrary backgrounds is intimately related to this  and as mentioned in the introduction, we wish to revisit the construction of [36] in the  light of our findings in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We believe that this limit would lead to null tensionless  strings in AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' This is to be contrasted with tensionless strings in AdS considered in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [47] and subsequent work in this direction, which are tensionless but not null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' A better  understanding of the differences and perhaps similarities between the two approaches would  be important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  A generalisation of the methods outlined in this paper would be carried out for 3d Carrollian  and 3d Galilean theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The structures are expected to remain similar for the 3d Galilean  theories, as the infinite dimensional structure of the algebra without the extra currents  remains intact when generalised to higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' But for Carrollian theories, owing  to the fundamental difference between BMS3 and BMS4, where one copy of the Virasoro  in the 3d case gets enhanced to two Virasoros in the 4d case and supertranslations develop  two legs instead of one, the construction of the quantum field theories with additional  non-abelian currents would be more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Acknowledgements  We thank Aritra Banerjee, Rudranil Basu and Niels Obers for interesting discussions and  comments on an initial version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  The work of AB is partially supported by a Swarnajayanti fellowship (SB/SJF/2019-  20/08) from the Science and Engineering Research Board (SERB) India, the SERB grant  (CRG/2020/002035), and a visiting professorship at ´Ecole Polytechnique Paris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' AB also  acknowledges the warm hospitality of the Niels Bohr Institute, Copenhagen during later  stages of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' RC is supported by the CSIR grant File No: 09/092(0991)/2018-EMR-  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' AS is financially supported by a PMRF fellowship, MHRD, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' RK acknowledges the  support of the Department of Atomic Energy, Government of India, under project number  RTI4001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' DS would like to thank ICTS, Bengaluru for hospitality during the course of this  project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  – 39 –  APPENDICES  A  Carroll Multiplets  In this appendix, we review the construction of Carrollian boost multiplets in two dimen-  sions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  In two space-time dimensions, the Carrollian boost (CB) transformation is defined as:  x → x′ = x , t → t′ = t + vx ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' or equivalently, as:  �  x  t  �  −→  �  x′  t′  �  =  �  exp  ��  0 0  v 0  ��� �  x  t  �  ⇐⇒ xµ → x′µ =  �  evB(2)�µ  ν xν  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1)  with  B(2) :=  �  0 0  1 0  �  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2)  being the 2D representation of the CB generator B that is clearly not diagonalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Taking a cue from the Lorentz covariance of Lorentz tensors, it was postulated in [] that  a rank-n Carrollian Cartesian tensor field Φ with ‘boost-charge’ ξ transforms under the  Carrollian boost as:  Φµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='µn(t, x) −→ ˜Φµ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='µn(t′, x′) =  �  e−ξvB(2)  �µ1  ν1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  �  e−ξvB(2)  �µn  νnΦν1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='νn(t, x)  ⇐⇒  Φ(t, x) −→ ˜Φ(t′, x′) =  � n  �  i=1  e−ξvB(2)  �  Φ(t, x) = e  −ξv  n  �  i=1  B(2)Φ(t, x)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3)  where µi, νi are Carrollian space-time indices and for matrices, the left index denotes row  while the right one denotes column;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' repeated indices are summed over and, in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3), indices  are suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' It is to be noted that the up/down appearance of a tensor-index does not  matter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' only the left/right ordering is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Clearly, the Carrollian Cartesian tensors defined above are decomposible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  So, we now  construct indecomposible Carrollian multiplets from these tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' We begin by recognizing  that:  B(2) = J−  (l= 1  2 )  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4)  which is the lowering ladder operator in the SU(2) spin-1  2 representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Thus,  n�  i=1  B(2) in  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3) can be decomposed into indecomposable representations of J− using the technique  of ‘addition of n spin-1  2 angular momenta’ in quantum mechanics, such that:  B(d) ≡ J−  (l= d−1  2 )  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5)  It is evident that the representations B(d) of the classical CB generator are indecomposable  since their only generalized eigenvalue is 0 and it has geometric multiplicity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' But, these  representations are reducible for d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  – 40 –  A multi-component field transforming under the d-dimensional representation of CB, B(d),  will be called a Carrollian multiplet of rank d−1  2  with d components, denoted by  Φm  (l= d−1  2 )  with  m = 1 − d  2  , 3 − d  2  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=', d − 1  2  By treating the µ = t index as spin-1  2 up-state and the µ = x index as spin-1  2 down-state,  components Φm  (l) of a Carrollian multiplet arise precisely as such linear combinations (with  proper Clebsch-Gordon coefficients) of the components of a Cartesian tensor of an allowed  rank n that would appear while expanding the |l, m⟩ states in an allowed |s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=', sn⟩  basis (where |si⟩ are Jz  ( 1  2 ) eigenstates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' So, as a linear combination of the components of  a rank-n Cartesian tensor, one can obtain multipltes of ranks: 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=', n  2 for even n and  1  2, 3  2, 5  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=', n  2 for odd n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' As an example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' we see how Carrollian multiplets of ranks 1  2 and 3  2  are constructed from a rank-3 Cartesian tensor:  Φ  3  2  ( 3  2 )(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x) := Φttt(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x)  Φ  1  2  ( 3  2 )(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x) :=  1  √  3  �  Φttx + Φtxt + Φxtt�  (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x)  Φ  − 1  2  ( 3  2 )(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x) :=  1  √  3  �  Φtxx + Φxtx + Φxxt�  (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x)  Φ  − 3  2  ( 3  2 )(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x) := Φxxx(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x)  Φ  1  2  ( 1  2 )(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x) :=  1  √  a2 + b2 + c2  �  aΦttx + bΦtxt + cΦxtt�  (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x)  with a + b + c = 0  Φ  − 1  2  ( 1  2 )(t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x) :=  1  √  a2 + b2 + c2  �  aΦxxt + bΦxtx + cΦtxx�  (t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' x)  (As the tuple (a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' c) in R3 lies on the plane a + b + c = 0 which is spanned by two basis  vectors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' two linearly independent rank-1  2 multiplets arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=')  A rank-l Carrollian multiplet with boost-charge ξ thus transforms under the 2l + 1 dimen-  sional representation of the CB as:  Φm  (l)(t, x) −→ ˜Φm  (l)(t′, x′) =  �  e−ξvJ−  (l)  �m  m′Φm′  (l)(t, x)  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='6)  After constructing the Carrollian multiplets from the Cartesian tensors as demonstrated  above, the components of the multiplets can always be redefined such that:  in (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='6), J−  (l) is replaced by M(l) := sub-diag (1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=', 1)2l+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Hence, instead of the actual J−  (l) matrix, only the indecomposable Jordan-block structure  is important for defining the CB transformation property of the Carrollian multiplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  We conclude this appendix with the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Since the finite dimensional  indecomposable representations of B are not symmetric (or Hermitian), one can start  with:  �  t  x  �  −→  �  t′  x′  �  =  �  exp  ��  0 v  0 0  ��� �  t  x  �  ⇐⇒ xµ → x′µ =  �  evB′  (2)  �µ  ν xν  – 41 –  where  B′  (2) :=  �  0 1  0 0  �  and follow the preceding argument to construct  B′  (d) ≡ J+  (l= d−1  2 )  which is the SU(2) raising ladder operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' But, as J+  (l) = (J−  (l))T, the raising and lowering  operators’ representation matrices are related to each other by the similarity transforma-  tion:  S = anti-diag (1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=', 1)2l+1  and consequently, B and B′ furnish two equivalent representations of the CB generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  B  Calculation of Sugawara Construction Commutators  In this appendix, we provide the details of the calculation of commutators of the NLKM  algebra from the Sugawara construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Calculating [Mm, Jb  n]  [Mm, Jb  n] =  1  2k2  �  a  �  l  [Ka  l Ka  m−l, Jb  n]  (Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12))  =  1  2k2  �  a  �  l  �  [Ka  l , Jb  n]Ka  m−l + Ka  l [Ka  m−l, Jb  n]  �  =  1  2k2  �  a  �  l  �  (−i  �  c  fbacKc  n+l − nk2δn+l,0δab)Ka  m−l  + Ka  l (−i  �  c  fbacKc  m+n−l − nk2δm+n−l,0δab)  �  (Using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21))  =  i  2k2  �  a,c  �  l  fabc�  Kc  n+lKa  m−l + Ka  l Kc  m+n−l  �  − nKb  m+n  (∵ −fbac = fabc)  where we have omitted the limits in summation over a and c and it is understood that  it runs over 1 to dim(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Now, �  l Kc  n+lKa  m−l = �  l Kc  m+n−lKa  l simply by translating l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Hence,  [Mm, Jb  n] = i  k2  �  a,c  �  l  fabcKa  l Kc  m+n−l − nKb  m+n  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1)  Now, again using antisymmetry property of fabc  �  a,c  �  l  fabcKa  l Kc  m+n−l =  �  a,c  �  l  fabcKa  m+n−lKc  l  – 42 –  =  �  a,c  �  l  fcbaKc  m+n−lKa  l  = −  �  a,c  �  l  fabcKa  l Kc  m+n−l  ⇒  �  a,c  �  l  fabcKa  l Kc  m+n−l = 0  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2)  Hence, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1) can be written as,  [Mm, Jb  n] = −nKb  m+n  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3)  Calculating [Lm, Kb  n]  Again, using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='21),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' we can get the following,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [Lm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Kb  n] =  1  2k2  �  a  � �  l≤−1  ([Ja  l ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Kb  n]Ka  m−l + Ka  l [Ja  m−l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Kb  n])  +  �  l>−1  ([Ja  m−l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Kb  n]Ka  l + Ka  m−l[Ja  l ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Kb  n])  �  =  1  2k2  �  a  � �  l≤−1  �  i  �  c  fabc(Kc  l+nKa  m−l + Ka  l Kc  m+n−l) + k2lKa  m−lδl+n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='0δab  + k2(m − l)Ka  l δm+n−l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='0δab  �  +  �  l>−1  �  i  �  c  fabc(Kc  m+n−lKa  l + Ka  m−lKc  l+n)  + k2(m − l)Ka  l δm+n−l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='0δab + k2lKa  m−lδl+n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='0δab  ��  = i  k2  �  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='c  �  l  fabc(Kc  m+n−lKa  l + Ka  m−lKc  l+n)  +  1  2k2  �  a  �  l  �  k2(m − l)Ka  l δm+n−l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='0δab + k2lKa  m−lδl+n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='0δab  �  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4)  First term vanishes again because of the antisymmetry of f and commutativity of K’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Therefore,  [Lm, Kb  n] = −nKb  m+n  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5)  Calculating [Lm, Jb  n]  Following similar steps as before, we get,  [Lm, Jb  n] = 1  2k2  �  a  � �  l≤−1  ([Ja  l Ka  m−l, Jb  n] + [Ka  l Ja  m−l, Jb  n]) +  �  l>−1  ([Ja  m−lKa  l , Jb  n]  + [Ka  m−lJa  l , Jb  n])  �  − (k1 + 2Cg)  k2  [Mm, Jb  n]  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='6)  – 43 –  First term can be simplified as,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  �  a  �  l≤−1  ([Ja  l Ka  m−l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Jb  n] + [Ka  l Ja  m−l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Jb  n])  =  �  a  �  l≤−1  �  Ja  l [Ka  m−l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Jb  n] + [Ja  l ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Jb  n]Ka  m−l + Ka  l [Ja  m−l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Jb  n] + [Ka  l ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Jb  n]Ja  m−l  �  =  �  a  �  l≤−1  �  i  �  c  fabcJa  l Kc  m+n−l + i  �  c  fabcJc  l+nKa  m−l + i  �  c  fabcKa  l Jc  m+n−l  + i  �  c  fabcKc  l+nJa  m−l − k2nJa  l δm+n−l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='0δab + k1lKa  m−lδl+n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='0δab  + k1(m − l)Ka  l δm+n−l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='0δab − k2nJa  m−lδl+n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='0δab  �  =  �  l≤−1  �  i  �  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='c  fabcJa  l Kc  m+n−l + i  �  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='c  fabcJc  l+nKa  m−l + i  �  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='c  fabcKa  l Jc  m+n−l  + i  �  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='c  fabcKc  l+nJa  m−l  �  − n  �  l≤−1  (k2Jb  m+n + k1Kb  m+n)(δm+n−l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='0 + δl+n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='0) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='7)  Similarly, the second term looks like,  �  a  �  l>−1  ([Ka  m−lJa  l , Jb  n] + [Ja  m−lKa  l , Jb  n])  =  �  l>−1  �  i  �  a,c  fabcKa  m−lJc  l+n + i  �  a,c  fabcKc  m+n−lJa  l + i  �  a,c  fabcJa  m−lKc  l+n  + i  �  a,c  fabcJc  m+n−lKa  l  �  − n  �  l>−1  (k2Jb  m+n + k1Kb  m+n)(δm+n−l,0 + δl+n,0) (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='8)  Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [Lm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Jb  n] =  1  2k2  �  i  �  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='c  fabc  �  �  0≤l≤n−1  Jc  l Ka  m+n−l −  �  0≤l≤n−1  Ka  l Jc  m+n−l −  �  0≤l≤n−1  Ka  m+n−lJc  l  +  �  0≤l≤n−1  Jc  m+n−lKa  l  ��  − nJb  m+n + 2Cg  k2  nKb  m+n  =  1  2k2  �  i  �  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='c  fabc  �  �  0≤l≤n−1  [Jc  l ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Ka  m+n−l] +  �  0≤l≤n−1  [Jc  m+n−l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Ka  l ]  ��  − nJb  m+n + 2Cg  k2  nKb  m+n  =  1  2k2  �  − 2n  �  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='d  fabcfcadKd  m+n + 2ik2  �  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='c  �  0≤l≤n−1  fabcδm+n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='0δab  �  − nJb  m+n + 2Cg  k2  nKb  m+n  = −nJb  m+n + n  k2  (2CgKb  m+n −  �  a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='d  fabcfcadKd  m+n)  – 44 –  = −nJb  m+n  Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [Lm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Jb  n] = −nJb  m+n  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='9)  Calculating [Lm, Mn]  [Lm, Mn] =  1  2k2  �  a  �  l  �  [Lm, Ka  l ]Ka  n−l + Ka  l [Lm, Ka  n−l]  �  =  1  2k2  �  a  �  l  �  − lKa  m+lKa  n−l − (n − l)Ka  l Ka  m+n−l  �  (Using (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5))  =  1  2k2  �  a  �  l  {−(l − m)Ka  l Ka  n+m−l − (n − l)Ka  l Ka  m+n−l}  = (m − n) 1  2k2  �  a  �  l  Ka  l Ka  n+m−l  where , we have done the re-labelling l → l − m in the second step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Hence, we get,  [Lm, Mn] = (m − n)Mm+n  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='10)  Calculating [Lm, Ln]  It can be carried out in similar fashion by writing one of the L’s in terms of J’s and K’s  using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12) and then using (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5), (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='9) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' we can do the following,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [Lm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Ln] = 1  2k2  �  a  � �  l≤−1  ([Lm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Ja  l ]Ka  n−l + Ja  l [Lm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Ka  n−l] + [Lm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Ka  l ]Ja  n−l + Ka  l [Lm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Ja  n−l])  +  �  l>−1  ([Lm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Ja  n−l]Ka  l + Ja  n−l[Lm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Ka  l ] + [Lm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Ka  n−l]Ja  l + Ka  n−l[Lm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Ja  l ])  �  − (k1 + 2Cg)  k2  [Lm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' 1  2k2  �  l  Ka  l Ka  n−l]  =  1  2k2  �  a  � �  l≤−1  (−lJa  m+lKa  n−l + lJa  l Ka  m+n−l − lKa  m+lJa  n−l + lKa  l Ja  m+n−l)  +  �  l>−1  (lJa  m+n−lKa  l − lJa  n−lKa  m+l + lKa  m+n−lJa  l − lKa  n−lJa  m+l)  �  − m(k1 + 2Cg)  k2  1  2k2  �  a  �  l  Ka  l Ka  m+n−l − nLm+n  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='11)  Changing the index l → (l − m) in the negative terms in the curly brackets and then  simplifying, we can get,  [Lm, Ln] =  1  2k2  �  a  � �  l≤−1  m(Ja  l Ka  n+m−l + Ka  l Ja  m+n−l) +  �  l>−1  m(Ja  n+m−lKa  l + Ka  n+m−lJa  l )  – 45 –  +  m−1  �  l=0  (m − l)([Ja  l , Ka  n+m−l] − [Ja  n+m−l, Ka  l ])  �  − m(k1 + 2Cg)  k2  1  2k2  �  a  �  l  Ka  l Ka  m+n−l − nLm+n  =  1  2k2  �  a  � m−1  �  l=0  (m − l)(k2lδm+n,0 − k2(n + m − l)δm+n,0)  �  + (m − n)Lm+n  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12)  which upon further simplification gives,  [Lm, Ln] = (m − n)Lm+n + dim(g)  6  m(m2 − 1)δm+n,0  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='13)  Other commutation relations of type [Mm, Kb  n] and [Mm, Mn] vanish trivially because of  the vanishing [Ka  m, Kb  n] commutator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  C  Modified Sugawara Construction  In this appendix, we give details of the modified Sugawara construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' First we start  with the modified construction in the relativistic case and then explain the construction in  the non-Lorentzian case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  We begin by calculating [ ˜Lm, ˜Ln] with ˜Lm given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='15)  [ ˜Lm, ˜Ln] = [LS  m + imθaja  m + 1  2kθ2δm,0, LS  n + inθbjb  n + 1  2kθ2δn,0]  = [LS  m, LS  n] + inθb[LS  m, jb  n] + imθa[ja  m, LS  n] − mnθaθb[ja  m, jb  n]  = (m − n)LS  m+n − in2θbjb  m+n + im2θaja  m+n − mnθaθb�  ifabcjc  m+n + mkδm+nδab  �  = (m − n) ˜Lm+n − i(m2 − n2)θaja  m+n − 1  2kθ2(m − n)δm+n,0 + i(m2 − n2)θaja  m+n  − imnθaθbfabcjc  m+n + m3kθ2δm+n,0 + c  12(m3 − m)δm+n,0  = (m − n) ˜Lm+n +  � c  12 + kθ2�  (m3 − m)δm+n,0  = (m − n) ˜Lm+n + ˜c  12(m3 − m)δm+n,0  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1)  Here ˜c = c + 12kθ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' In the fifth line we have used the fact that θaθbfabc vanishes due to  antisymmetry of fabc over indices a and b, also the fact that nδm+n,0 = −mδm+n,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Now defining ˜L and ˜  M as in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='17) we would like to calculate [˜Lm, ˜Ln] and [˜Lm, ˜  Mn].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' The  calculation of [˜Lm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' ˜Ln] will be exactly similar to that of [ ˜Lm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' ˜Ln],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' while that of [˜Lm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' ˜  Mn]  will be as following  [˜Lm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' ˜  Mn] = [LS  m + imθaJa  m + 1  2k2θ2δn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' MS  n + inθbKb  n + 1  2k2θ2δn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='0]  = [LS  m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' MS  n ] + imθa[LS  m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Ka  n] + inθb[Ja  m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' MS  n ] − mnθaθb[Jb  m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Ka  n]  – 46 –  = (m − n)MS  m+n − in2θbKb  m+n + im2θaKa  m+n − mnθaθb�  ifabcKc  m+n + mk2δm+nδab  �  = (m − n)MS  m+n − i(m2 − n2)θaKa  m+n − 1  2k2θ2(m − n)δm+n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='0 + i(m2 − n2)θaKa  m+n  − imnθaθbfabcKc  m+n + m3k2θ2δm+n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='0  = (m − n)Mm+n + k2θ2(m3 − m)δm+n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='0  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2)  Hence, just by doing a slight modification to the Sugawara construction we end up with  fully centrally extended BMS algebra with central charges given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  D  Details of OPE calculations  Calculation of T-J OPE  First consider  J a  v (u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v1)  �  b  �  (J b  uJ b  v )(u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v2)  �  =  �  v2  dv′  v′ − v2  �  u2  du′  u′ − u2  �  b  �  J a  v (u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v1)J b  u(u′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v′)J b  v (u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v2) + J b  u(u′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v′)J a  v (u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v1)J b  v (u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v2)  �  =  �  v2  dv′  v′ − v2  �  u2  du′  u′ − u2  �  b  �  ifabc J c  v (u′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v′)J b  v (u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v2)  (v1 − v′)  + k2δab J b  v (u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v2)  (v1 − v′)2 + J b  u(u′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v′) × 0  �  =  �  v2  dv′  v′ − v2  �  u2  du′  u′ − u2  ��  b  ifabc (J c  v J b  v )(u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v2)  (v1 − v′)  + k2  J a  v (u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v2)  (v1 − v′)2  �  =  �  b  ifabc (J c  v J b  v )(u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v2)  v12  + k2  J a  v (u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v2)  v2  12  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1)  Similarly we get  J a  v (u1, v1)  �  b  �  (J b  v J b  u)(u2, v2)  �  =  �  v2  dv′  v′ − v2  �  u2  du′  u′ − u2  ��  b  ifabc (J b  v J c  v )(u2, v2)  v12  + k2  J a  v (u′, v′)  v2  12  �  =  �  b  iF abc (J b  v J c  v )(u2, v2)  v12  + k2  J a  v (u2, v2)  v2  12  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2)  Summing up (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='66) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2), we see that the first terms in the expressions cancel each  other due to the antisymmetry of the structure constant, so we get  J a  v (u1, v1)  �  b  �  (J b  uJ b  v )(u2, v2) + (J b  v J b  u)(u2, v2)  �  = 2k2  J a  v (u2, v2)  v2  12  = 2k2  �J a  v (u1, v1)  v2  21  + ∂vJ a  v (u1, v1)  v21  �  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3)  – 47 –  Also it can be trivially shown that  J a  v (u1, v1)  �  b  �  (J a  v J a  v )(u2, v2)  �  = 0  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4)  From (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4), we can determine  Tv(u1, v1)J b  v (u2, v2) ∼ regular  Tu(u1, v1)J b  v (u2, v2) ∼ J b  v (u2, v2)  v2  12  + ∂vJ b  v (u2, v2)  v12  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5)  Similarly we get  Tv(u1, v1)J a  u (u2, v2) ∼ J a  v (u2, v2)  v2  12  + ∂vJ a  v (u2, v2)  v12  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Tu(u1, v1)J a  u (u2, v2) ∼ J a  u (u2, v2)  v2  12  + ∂vJ a  u (u2, v2)  v12  + u12  v2  12  ∂uJ a  u (u2, v2)  + 2u12  v12  �J a  v (u1, v1)  v2  12  + ∂vJ a  v (u2, v2)  v2  12  �  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='6)  Calculation of T-T OPE  First consider  Tu(u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v1)(J a  v J a  v )(u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v2)  =  �  v2  dv′  v′ − v2  �  u2  du′  u′ − u2  �  Tu(u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v1)J a  v (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v)J a  v (u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v2) + J a  v (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v)Tu(u1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v1)v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' u2  �  =  �  v2  dv′  v′ − v2  �  u2  du′  u′ − u2  �J a  v (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v)J a  v (u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v2)  (v1 − v′)2  + ∂v′J a  v (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v)J a  v (u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v2)  (v1 − v′)  + J a  v (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v)J a  v (u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v2)  v2  12  + J a  v (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v)∂v2J a  v (u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v2)  v12  �  = 2(J a  v J a  v )(u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v2)  v2  12  +  �  v2  dv′  v′ − v2  �  u2  du′  u′ − u2  �∂v′[(J a  v J a  v )(u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v2) + (∂vJ a  v J a  v )(u2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' v2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' ]  (v1 − v′)  + ∂v2[(J a  v J a  v )(u2, v2) + (∂vJ a  v J a  v )(u2, v2) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' ]  v12  �  = 2(J a  v J a  v )(u2, v2)  v2  12  + ∂v(J a  v J a  v )(u2, v2)  v12  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='7)  So we obtain  Tu(u1, v1)Tv(u2, v2) ∼ 2(J a  v J a  v )(u2, v2)  v2  12  + ∂v(J a  v J a  v )(u2, v2)  v12  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='8)  Similarly, we can check  Tv(u1, v1)Tv(u2, v2) ∼ regular,  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='9)  – 48 –  and  Tu(u1, v1)Tu(u2, v2) ∼ 2Tu(u2, v2)  v2  12  +∂vTu(u2, v2)  v12  + u12  v2  12  ∂uTu(u2, v2)  + 2u12  v12  �  2Tv(u2, v2)  v2  12  + ∂vTv(u2, v2)  v12  �  + dim(g)  v4  12  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='10)  E  K-Z equation in field theory approach  We start with  ⟨∂uΦ(u, v)Φ1(u1, v1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Φn(un, vn)⟩ = ⟨∂uΦ(u, v)X⟩  = −  �  v  dv′  2πi⟨Tv(u′, v′)Φ(u, v)X⟩  = −  �  v  dv′  2πi  1  2k2  ⟨(J a  v J a  v )(u′, v′)Φ(u, v)X⟩  = −  �  v  dv′  2πi  1  2k2  �  v′  dv′′  2πi  1  (v′′ − v′)⟨  �  J a  v (u′′, v′′)Φ(u, v)J a  v (u′, v′) + J a  v (u′′, v′′)J a  v (u′, v′)Φ(u, v)  �  X⟩  −  �  v  dv′  2πi  1  2k2  ⟨(J a  v J a  v Φ)(u, v)X⟩ + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  = −  �  v  dv′  2πi  1  2k2  �  v′  dv′′  2πi  1  (v′′ − v′)⟨  �  ta  k  (v′′ − v)Φ(u, v)J a  v (u′, v′) +  ta  K  (v′ − v)J a  v (u′′, v′′)Φ(u, v)  �  X⟩ + 0  = I1 + I2  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1)  (Where, second line is (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1), third line is the Non-Lorentzian Sugawara construction, fourth  line uses definition of normal ordering and the fact that OPE = contractions + normal  ordered product + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' )  Now,  I1 = − ta  k  2k2  �  v  dv′  2πi  �  v′  dv′′  2πi  1  (v′′ − v′)(v′′ − v)⟨Φ(u, v)J a  v (u′, v′)X⟩  = − ta  k  2k2  �  v  dv′  2πi  1  (v′ − v)⟨Φ(u, v)J a  v (u′, v′)X⟩  = ta  k  2k2  �  j  �  vj  dv′  2πi  1  (v′ − v)⟨Φ(u, v)Φ1(u1, v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  �  J a  v (u′, v′)Φj(uj, vj)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Φn(un, vn)⟩  =  1  2k2  �  j  �  vj  dv′  2πi  ta  K ⊗ ta  K,j  (v′ − v)(v′ − vj)⟨Φ(u, v)Φ1(u1, v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='Φj(uj, vj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Φn(un, vn)⟩  =  1  2k2  �  j  ta  K ⊗ ta  K,j  (vj − v) ⟨Φ(u, v)Φ1(u1, v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='Φj(uj, vj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Φn(un, vn)⟩  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2)  – 49 –  (third line involves a change in contour)  and similarly,  I2 = − ta  k  2k2  �  v  dv′  2πi  �  v′  dv′′  2πi  1  (v′′ − v′)(v′ − v)⟨J a  v (v′′, x′′)Φ(u, v)X⟩  =  1  2k2  �  j  ta  K ⊗ ta  K,j  (vj − v) ⟨Φ(u, v)Φ1(u1, v1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='Φj(uj, vj) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Φn(un, vn)⟩  = I1  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3)  Using the above two results,we get one of the Carrollian K-Z equations  �  ∂ui + 1  k2  �  j̸=i  ta  Ri,K ⊗ ta  Rj,K  vij  �  ⟨Φ1(u1, v1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Φn(un, vn)⟩ = 0  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4)  The other equation can be obtained similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  F  NL KZ equation as a limit  Taking the following linear combination of the equations (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='30a) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='30b),  (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='30a) × ⟨¯φ¯r1¯R1( ¯w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='¯φ¯rN  ¯RN ( ¯wN)⟩ + ((7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='30b)) × ⟨φr1  R1(w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='φrN  RN (wN)⟩ = 0  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1)  gives us,  �  �∂wi + ∂ ¯wi − 2γ  �  j̸=i  �  a(ta  Ri ⊗′ ¯I)ri,¯ri  si,¯si(ta  Rj ⊗′ ¯I)rj,¯rj  sj,¯sj  wi − wj  − 2¯γ  �  j̸=i  �  a(I ⊗′ ¯ta¯Ri)ri,¯ri  si,¯si(I ⊗′ ¯ta¯Rj)rj,¯rj  sj,¯sj  ¯wi − ¯wj  �  �  ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='Φsi,¯si  Ri, ¯Ri(wi, ¯wi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='Φsj,¯sj  Rj, ¯Rj(wj, ¯wj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='⟩ = 0  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='2)  We have (wi, ¯wi) = t ± ϵx ⇒ ∂wi = 1  2(∂ti + ∂xi  ϵ );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' ∂ ¯wi = 1  2(∂ti − ∂xi  ϵ ) such that the Carrollian  limit is achieved by taking ϵ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Using these we get,  ⇒  �  �∂ti − 2  �  j̸=i  X  �  � ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='Φsi,¯si  Ri, ¯Ri(xi, ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='Φsj,¯sj  Rj, ¯Rj(xj, tj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='⟩ = 0  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3)  where(with tij = ti − tj and xij = xi − xj), we have,  X =  �  γ  tij + ϵxij  �  a  (ta  Ri ⊗′ ¯I)ri,¯ri  si,¯si(ta  Rj ⊗′ ¯I)rj,¯rj  sj,¯sj +  ¯γ  tij − ϵxij  �  a  (I ⊗′ ¯ta¯Ri)ri,¯ri  si,¯si(I ⊗′ ¯ta¯Rj)rj,¯rj  sj,¯sj  �  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='4)  Inverting relations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='11), we have,  ta  Ri ⊗′ ¯I = 1  2(ta  Ri,J −  ta  Ri,K  ϵ  ) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' I ⊗′ ¯ta¯Ri = 1  2(ta  Ri,J +  ta  Ri,K  ϵ  )  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5)  – 50 –  Hence,  X = 1  4  �  a  �  γ  tij + ϵxij  �  ta  Ri,J −  ta  Ri,K  ϵ  �ri,¯ri  si,¯si  �  ta  Rj,J −  ta  Rj,K  ϵ  �rj,¯rj  sj,¯sj  +  ¯γ  tij − ϵxij  �  ta  Ri,J +  ta  Ri,K  ϵ  �ri,¯ri  si,¯si  �  ta  Rj,J +  ta  Rj,K  ϵ  �rj,¯rj  sj,¯sj  �  = 1  4  �  a  �  γ  tij + ϵxij  �  ta  Ri,J ⊗ ta  Rj,J − 1  ϵ (ta  Ri,J ⊗ ta  Rj,K + ta  Ri,K ⊗ ta  Rj,J)  + 1  ϵ2 ta  Ri,K ⊗ ta  Rj,K  �ri,¯ri;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='rj,¯rj  si,¯si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='sj,¯sj  +  ¯γ  tij − ϵxij  �  ta  Ri,J ⊗ ta  Rj,J + 1  ϵ (ta  Ri,J ⊗ ta  Rj,K + ta  Ri,K ⊗ ta  Rj,J)  + 1  ϵ2 ta  Ri,K ⊗ ta  Rj,K  �ri,¯ri;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='rj,¯rj  si,¯si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='sj,¯sj  �  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='6)  We introduce notation for convenience,  �  a  (ta  Ri,A ⊗ ta  Rj,B)ri,¯ri;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='rj,¯rj  si,¯si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='sj,¯sj = tij  AB (where A, B = J, K)  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='7)  Therefore,  X =1  4  � γ  tij  (1 − ϵxij  tij  + ϵ2 x2  ij  t2  ij  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=')(tij  JJ − 1  ϵ (tij  JK + tij  KJ) + 1  ϵ2 tij  KK)  + ¯γ  tij  (1 + ϵxij  tij  + ϵ2 x2  ij  t2  ij  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=')(tij  JJ + 1  ϵ (tij  JK + tij  KJ) + 1  ϵ2 tij  KK)  �  ⇒ X =  1  4tij  �  − (γ − ¯γ)  ϵ  (tij  JK + tij  KJ) + γ + ¯γ  ϵ2  tij  KK) − xij  4t2  ij  (γ − ¯γ)  ϵ  tij  KK  �  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='8)  In limit ϵ → 0 and using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='25), we get,  ⇒ X =  1  4tij  � 2  k2  (tij  JK + tij  KJ) − 2(k1 + 2Cg)  k2  2  tij  KK) + xij  4t2  ij  2  k2  tij  KK  �  ⇒ X =  1  2k2  �(tij  JK + tij  KJ)  tij  + (xij  t2  ij  − (k1 + 2Cg)  k2tij  )tij  KK  �  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='9)  Therefore, (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='1) can be finally written as( in limit ϵ → 0),  ⇒  �  ∂ti −  �  j̸=i  1  k2  �(tij  JK + tij  KJ)  tij  +  �xij  t2  ij  − (k1 + 2Cg)  k2tij  �  tij  KK  ��  ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='Φsi,¯si  xi,ti(xi, ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='Φsj,¯sj  Rj, ¯Rj(xj, tj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='⟩ = 0  ⇒  �  ∂ti − 1  k2  �  j̸=i  ��  a(ta  Ri,J ⊗ ta  Rj,K + ta  Ri,K ⊗ ta  Rj,J)ri,¯ri;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='rj,¯rj  si,¯si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='sj,¯sj  tij  – 51 –  +  �xij  t2  ij  − (k1 + 2Cg)  k2tij  � �  a  (ta  Ri,K ⊗ ta  Rj,K)ri,¯ri;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='rj,¯rj  si,¯si;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='sj,¯sj  ��  ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='Φsi,¯si  Ri, ¯Ri(xi, ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='Φsj,¯sj  Rj, ¯Rj(xj, tj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='⟩ = 0  ⇒  �  ∂ti − 1  k2  �  j̸=i  ��  a(ta  Ri,J ⊗ ta  Rj,K + ta  Ri,K ⊗ ta  Rj,J)  tij  +  �xij  t2  ij  − (k1 + 2Cg)  k2tij  � �  a  (ta  Ri,K ⊗ ta  Rj,K)  ��  ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='ΦRi, ¯Ri(xi, ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='ΦRj, ¯  Rj(xj, tj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='⟩ = 0  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='10)  which is in agreement with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  Now,we consider another linear combination,  ϵ{(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='30a) × ⟨¯φ¯r1¯R1( ¯w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='¯φ¯rN  ¯RN ( ¯wN)⟩ − (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='30b) × ⟨φr1  R1(w1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='φrN  RN (wN)⟩} = 0  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='11)  Again following similar steps as before,  ⇒  �  ∂xi − 2  �  j̸=i  Y  �  ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='Φsi,¯si  Ri, ¯Ri(xi, ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='Φsj,¯sj  Rj, ¯Rj(xj, tj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='⟩ = 0  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12)  where,  Y = ϵ  �  γ  tij + ϵxij  �  a  (ta  Ri ⊗′ ¯I)ri,¯ri  si,¯si(ta  Rj ⊗′ ¯I)rj,¯rj  sj,¯sj −  ¯γ  tij − ϵxij  �  a  (I ⊗′ ¯ta¯Ri)ri,¯ri  si,¯si(I ⊗′ ¯ta¯Rj)rj,¯rj  sj,¯sj  �  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='13)  Similar to what we did for X, we get an equation analogous to (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='8) using (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5) and (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='7),  Y = ϵ  4{ γ  tij  (1 − ϵxij  tij  + ϵ2 x2  ij  t2  ij  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=')(tij  JJ − 1  ϵ (tij  JK + tij  KJ) + 1  ϵ2 tij  KK)  − ¯γ  tij  (1 + ϵxij  tij  + ϵ2 x2  ij  t2  ij  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=')(tij  JJ + 1  ϵ (tij  JK + tij  KJ) + 1  ϵ2 tij  KK)}  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='14)  Again using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='25) and collecting the finite terms in the limit ϵ → 0, we get,  Y = − 1  2k2  tij  KK  tij  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='15)  Putting back in (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='12),  �  �∂xi + 1  k2  �  j̸=i  �  a(ta  Ri,K ⊗ ta  Rj,K)  tij  �  � ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='ΦRi, ¯Ri(xi, ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='ΦRj, ¯Rj(xj, tj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='⟩ = 0  (F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='16)  which is in agreement with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  – 52 –  References  [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Belavin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Polyakov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Zamolodchikov, Infinite Conformal Symmetry in  Two-Dimensional Quantum Field Theory, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' B241 (1984) 333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [2] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Friedan, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Martinec and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Shenker, Conformal Invariance, Supersymmetry and  String Theory, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' B 271 (1986) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Gepner and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Witten, String Theory on Group Manifolds, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' B 278 (1986) 493.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Zamolodchikov, Infinite Additional Symmetries in Two-Dimensional Conformal  Quantum Field Theory, Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' 65 (1985) 1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [5] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Das, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Datta and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Pal, Charged structure constants from modularity, JHEP 11 (2017)  183 [1706.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='04612].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Bagchi, Correspondence between Asymptotically Flat Spacetimes and Nonrelativistic  Conformal Field Theories, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' 105 (2010) 171601 [1006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='3354].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [7] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Duval, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
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+page_content=' Salgado-Rebolledo, Generalizing the bms3 and  2D-conformal algebras by expanding the Virasoro algebra, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' C 78 (2018) 262  [1707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='07209].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  [47] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Gaberdiel and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content=' Gopakumar, Tensionless string spectra on AdS3, JHEP 05 (2018)  085 [1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='04423].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
+page_content='  – 55 –' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItE3T4oBgHgl3EQfugv_/content/2301.04686v1.pdf'}
diff --git a/J9E1T4oBgHgl3EQfsQWj/content/tmp_files/load_file.txt b/J9E1T4oBgHgl3EQfsQWj/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf,len=382
+page_content='Towards an AI-enabled Connected Industry: AGV  Communication and Sensor Measurement Datasets  Rodrigo Hernang´omez∗, Alexandros Palaios§, Cara Watermann§, Daniel Sch¨aufele∗,  Philipp Geuer§, Rafail Ismayilov∗, Mohammad Parvini†, Anton Krause†,  Martin Kasparick∗, Thomas Neugebauer¶, Oscar D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Ramos-Cantor‡, Hugues Tchouankem‡,  Jose Leon Calvo§, Bo Chen∗∗, Sławomir Sta´nczak∗∥, Gerhard Fettweis†  ∗Fraunhofer Heinrich Hertz Institute, Germany, {firstname.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='lastname}@hhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='fraunhofer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='de  §Ericsson Research, Germany, {alex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='palaios, cara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='watermann, philipp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='geuer}@ericsson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='com  †Vodafone Chair, Technische Universit¨at Dresden, Germany, {mohammad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='parvini, anton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='krause, gerhard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='fettweis}@tu-dresden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='de  ‡Corporate Research, Robert Bosch GmbH, Germany, {oscardario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='ramoscantor, huguesnarcisse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='tchouankem}@de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='bosch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='com  ∥Network Information Theory Group, Technische Universit¨at Berlin, Germany  ¶G¨otting KG, Germany, neugebauer@goetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='de  ∗∗Enway GmbH, Germany, bo@enway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='ai  Abstract—This paper presents two wireless measurement cam-  paigns in industrial testbeds: industrial Vehicle-to-vehicle (iV2V)  and industrial Vehicle-to-infrastructure plus Sensor (iV2I+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' De-  tailed information about the two captured datasets is provided  as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' iV2V covers sidelink communication scenarios between  Automated Guided Vehicles (AGVs), while iV2I+ is conducted  at an industrial setting where an autonomous cleaning robot is  connected to a private cellular network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The combination of dif-  ferent communication technologies, together with a common mea-  surement methodology, provides insights that can be exploited  by Machine Learning (ML) for tasks such as fingerprinting,  line-of-sight detection, prediction of quality of service or link  selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Moreover, the datasets are labelled and pre-filtered for  fast on-boarding and applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The corresponding testbeds  and measurements are also presented in detail for both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Index Terms—Measurement data, QoS prediction, AGV, drive  tests, V2X, campus networks, wireless communications  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' INTRODUCTION  It is anticipated that the next generation of wireless commu-  nication systems (5G and beyond) will bring about an upsurge  in the number of new services and applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' each of which  demanding for a specific Quality of Service (QoS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' In parallel,  there is a resurgence of interest in promoting the concept of  predictive Quality of Service (pQoS), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=', QoS estimation for a  given time instance in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' This can be done in different  prediction horizons, ranging from milliseconds to hours or  even days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' pQoS can pave the way to satisfy a very demanding  set of QoS requirements, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=', very low latency, minimum  Signal-to-Noise Ratio (SNR), delay, packet error rate, or huge  Uplink (UL) or Downlink (DL) throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  pQoS can be particularly important for wireless networks  in the industrial domain, where communication needs to be  highly reliable due to, among other reasons, its integration into  This work was supported by the Federal Ministry of Education and Re-  search (BMBF) of the Federal Republic of Germany as part of the AI4Mobile  project (16KIS1170K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The authors alone are responsible for the content of  the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  control loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Wireless links are especially relevant in mobile  setups, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=', with one or more Automated Guided Vehicles  (AGVs) connected in a Vehicle-to-vehicle (V2V), Vehicle-to-  infrastructure (V2I) or Vehicle-to-everything (V2X) manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  In this regard, some datasets are available for automotive sce-  narios to train and test Machine Learning (ML) algorithms and  thus enhance such schemes [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' However, the availability of  datasets from industrial and indoor measurement campaigns,  such as [2], is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' With proper knowledge of the upcoming  QoS conditions, pQoS can facilitate the proper operation  of industrial applications to guarantee human-machine safe  interaction or robot cooperation to fulfill a common task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Other use cases may include tele-operated driving, high-  density platooning, and High Definition (HD) map collecting  and sharing for optimal route selection [3], [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  In this manner, we can see a growing tendency toward  applying deep learning algorithms for pQoS applications, such  as [5]–[8], to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' A consolidated overview of the ML-  enabled throughput prediction scenarios is presented in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  In the same vein, [6] investigates a ML-model to predict the  throughput in a non-standalone 5G network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  ML has been a very active research area in the past few  years and there is ample literature around it;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' however, its real-  world implementation or validation has remained elusive for  industrial communication due to its high dependency on avail-  able datasets to test, validate and generalize the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Therefore, creating a reference dataset from experimental  testbeds or practical simulations is paramount to evaluate the  underlying theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  In this paper, we will describe the industrial Vehicle-  to-vehicle (iV2V) dataset and the industrial Vehicle-to-  infrastructure plus Sensor (iV2I+) dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' These two datasets  aim to pave the way for future advancement in the experi-  mentation of mobile networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The measurement campaigns  that were conducted here are part of a bigger measurement  framework and procedure that is described in detail in [9]  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='03364v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='NI]  20 Dec 2022  with some first results being reported in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  The remainder of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' In  Section II and Section III, we describe the iV2V and iV2I+  testbed and datasets and we elaborate on their details and  components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' We conclude with an overview of possible future  research directions in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' THE IV2V TESTBED AND DATASET  In this section, we present the first of the two collected  datasets and the considerations that have been taken into  account for its measurement campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' We give a brief in-  troduction to the sidelink technology, continue with a detailed  description of the testbed, and finally describe the processing  and resulting data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Figure 1a depicts one of the AGVs, carrying the measure-  ment and communication hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The AGVs communicate  directly in a V2V manner, using the sidelink technology as  introduced by 3rd Generation Partnership Project (3GPP) in  Release 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' In the sidelink setup, every AGV acts both as  transmitter and sender (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Testbed Components  1) Sidelink: Sidelink has been standardized in 3GPP during  4G and 5G mobile networks to define a framework where  communication is possible with and without network coverage  and with varying degrees of interaction between the devices  and the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Two modes of resource allocation are defined  [11]:  Network-based resource allocation (Mode 1 in 5G  sidelink and Mode 3 in 4G sidelink): This mode is only  available when all the devices are in network coverage,  and the network selects the resources and other transmit  parameters used by the devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Autonomous resource allocation (Mode 2 in 5G sidelink  and Mode 4 in 4G sidelink): This mode offers a com-  pletely decentralized solution in which the User Equip-  ments (UEs) autonomously select the resources and other  transmit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Overall, network-based resource allocation can outperform  the autonomous case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' This is due to the network controlling  the resources to be used by each of the UEs involving UL  signaling from the UE to the network to obtain a grant for  transmissions [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  On the other hand, autonomous resource allocation is mainly  useful when there is no possibility of having network coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  The basic operation of autonomous resource selection involves  the device performing sensing within a pre-configured resource  pool, detecting which resources are not in use by other devices  with higher-priority traffic, and choosing some of these free  resources for its transmissions [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The autonomous resource  allocation is more prone to collisions while also suffering from  hidden node and half duplex problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Solutions have been  considered in 3GPP to mitigate these issues [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  For the measurement campaign, we use a full stack,  software-based, standard-compliant and open implementation  of the 3GPP Release 14 PC5 Mode 4 standard [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The plat-  form allows research concepts and standard features to be val-  idated in hardware testbeds and it provides interfaces and tools  for recording measurement data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Changes and adjustments are  possible at every layer, which allows a realistic verification of  new features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The sidelink software (all layers incl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' baseband  processing) can be run on standard general purpose computing  hardware in connection with suitable Software Defined Radio  (SDR) hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' We opted for a full stack implementation,  thus providing a standard based IP to IP (one to all) interface  for any application, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=', all protocols on OSI layer 3 and higher  can be transferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The hardware setup is shown in Figure 1c  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' [14] for further details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  2) Localization: Precise position of the communicating  devices is required to link the environmental conditions with  the measured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' For that purpose, the position information  provided by the AGVs, carrying the communication entities,  was recorded during the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Two types of local-  ization methods are used by different AGVs in the testbed,  namely, marker/track-based, and Simultaneous Localization  and Mapping (SLAM)-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' In the former method, the AGVs  follow a track on the floor with help of an onboard camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Additionally, Radio-frequency identification (RFID) tags are  placed on the track to provide the exact position information  to the AGVs, when they pass over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Between the RFID tags,  the AGVs estimate their position by using odometry, which  describes a method of estimating the position and orientation  of a mobile system using data from its propulsion system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Wheel-driven systems use the measurement of the wheel rota-  tions for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' In combination with dead reckoning, odometry  is a basic navigation method for ground-based vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Since  the AGVs do not leave the track, the transversal error is in the  order of few mm, while the longitudinal error depends on the  separation distance between the RFID tags and the positional  accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  For our testbed, the longitudinal error was in the order of  few cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' In the latter localization method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=', SLAM-based,  the AGVs are equipped with a laser scanner to detect and  estimate the distance to landmarks (reference points) in the  testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' These landmarks are also defined in the map of the  AGV system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Hence, the AGVs can estimate their position in  the map through a combination of information from several  landmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' In the testbed, we achieve a position accuracy in  the order of few cm with the SLAM-based method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  The reported AGV position was timestamped during the  measurements, so that a combination with other measured data  is possible during post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Unless otherwise stated,  the measurements scenarios presented below consider that the  SLAM-based AGVs were static and the marker/track-based  AGVs were moving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Since the moving AGV is guided by an optical line, the real  lateral position is better than ±2 mm (3σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The along track  (longitudinal) error while passing an RFID tag has a timing  uncertainty of up to 30 ms, which gives an error depending  on the actual speed (up to 30 mm at 1m/s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Dead reckoning  results in additional errors being displayed due to the route  (a) AGV testbed in industrial environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' (b) Schematic illustration of sidelink mea-  surements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  (c) Sidelink SDR platform with all components inte-  grated in 19 inch case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' 1: iV2V testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  and steering angle sensors not being perfectly adjusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The  longitudinal error increases with the length of the unsupported  route driven without an RFID tag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The repeatability of the  position information is less than ±2 mm transversally and less  than +2 cm longitudinally at the driven speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  3) Time Synchronization: To enable accurate evaluation  of network latency and other QoS properties, a proper time  synchronization is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The time was synchronized across  sidelink devices by running NTP over Ethernet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The error is  typically in the order of several µs with a worst case error of  1 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' However, precisely quantifying this is difficult without  specialized measurement equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  4) Controlled Packet Generation: To ensure a highly pre-  cise packet generation, we used a network packet generator  tool based on Real-time User Datagram Protocol (UDP) Data  Emitter (RUDE) & Collector for RUDE (CRUDE) which  is able to produce heterogeneous UDP network traffic for  realistic network workloads [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' It consists of two main  modules: RUDE generates traffic to the network, which is then  received and logged by the other module of the network with  CRUDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' We extended the packet generator tool by enabling  the capability to log all channel information for successfully  received packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  5) Automatic Gain Control: In order to be able to assess  the received signal quality, which is an elementary quantity for  assessing the QoS, the function of the Automatic Gain Control  (AGC) in a receiver must be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' AGC is a feature in an  RF receiver signal path that is used to keep the received signal  magnitude at a suitable level for subsequent signal processing  so that signals are not clipped and the receive path with good  sensitivity is operated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The function of the AGC is technically  realized by controllable amplifiers in the received signal path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  For this purpose, the signal level is determined in the signal  processing during special preambles at the beginning of a  defined data frame and regulated within a specified range  by setting the AGC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' A criterion for the regulation can be  the evaluation of the preamble of the Orthogonal Frequency-  Division Multiplexing (OFDM)-based signal obtained from  the I/Q samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' However, the corresponding values at the  antenna input, namely Received Signal Strength Indicator  (RSSI) and Reference Signal Received Power (RSRP), are of  interest for signal evaluation and a corresponding indication of  comparable level values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' With knowledge of the amplification  and attenuation of individual components in the front-end and  the AGC setting, these can be determined from the measured  values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' from the total gain between the antenna input and  the evaluation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The AGC calculations have already been  performed before the output and the values supplied for pre-  processing are the correct values and can be used as is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Measurement Scenarios  We collected data for roughly 10 hours over the course  of two days to acquire almost 50 GB communication data  between up to three industrial AGVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' A schematic of the test  area and its surroundings is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The dotted  gray line depicts the track used by AGV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Several obstacles,  depicted in different blue tones in the figure, were located  within the test area to achieve different radio propagation  conditions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=', Line-of-Sight (LOS) and Non-Line-of-Sight  (NLOS) conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The obstacles were rearranged during the  measurement campaign to create two scenarios, A and B, with  different N/LOS characteristics, as marked in light blue in the  figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The Dataset  1) Captured Sidelink Data: For each scenario illustrated  in Figure 2, we capture the sidelink channel parameters for  every transmitter/receiver pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The selected sidelink channel  parameters and their description are presented in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The  parameters in the table are obtained/estimated from the De-  modulation Reference Signal (DMRS) of the Physical Sidelink  Shared Channel (PSSCH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  The AGV localization data is provided as x and y coordi-  nates in a local coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  2) Dataset Pre-processing: In this section, we describe the  pre-processing of captured sidelink data, and we present a  dataset constructed with the pre-processed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The dataset  is constructed in a tabular format where each row represents  a sample and the columns contain the value of the mea-  sured sidelink channel parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Note that the side channel  RF Frontend  3)  ise  Shecker  SAt  BasebandPC  SDR46cm  X  Wall Height ~3,10  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='5m  22m  65cm  1m  123cm  (0,0)  AGV  2  170cm  AGV  3  263cm  251cm  2  3  AGV  1  1  AGV Test Track  Glass door/window  Foam wall  Metallic wall  Foam wall in scenario B only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Replaced by metallic wall  in scenario A  Foam wall in scenario A only  2  3  1  AGV 1 moves along  the test track  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' 2: Illustration of measurement scenarios A & B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  TABLE I: Selected iV2V Data Features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Parameter  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Description  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='SNR [dB]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Derived from noise and power estimations of DMRS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='RSRP [dBm]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Average energy per carrier/RE for DMRS  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='RSSI [dBm]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Signal power over the whole band  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Noise Power  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Estimated on DMRS in decoded subframe  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Time [sec]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Receive time of first IQ-Sample of decoded subframe  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Frame Number  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='System frame number  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Subframe Number  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='System subframe number  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='UHD Rx Gain [dB]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Receive antenna gain  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='SCI FRL N  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Starting subchannel of decoded PSSCH  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='SCI FRL L  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Number of used subchannels for PSSCH  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='RLC SN  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Sequence number of radio link control header  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Location  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='Local x and y coordinates of AGV 1 on the track  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='parameters and AGV location are measured independently  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='and simultaneously in different devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' With this setting,  each measuring device embeds its own timestamp into the  measured parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Since the processing of the received  signal in different devices requires different lengths of time,  the embedded timestamps in these devices also have some  differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' We align the timestamps between the location  data and the sidelink data as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Given the timestamp  tloc  n corresponding to the measured location Ln = [xn, yn] of  the AGV at nth time step, we find the timestamp tsl  n from the  measured sidelink such that  ��tloc  n − tsl  n  �� ≤ γ, where γ denotes  the alignment tolerance, and we use γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='005s to construct  the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  In addition, we present indicators Mn ∈ Z and Sn ∈ Z,  where Mn indicates the measurement scenario (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=', the place-  ment of obstacles), and Sn denotes the source of the received  sidelink signal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=', AGV1 receives a signal from AGV2 or  from AGV3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' With the parameters described in Table I, the nth  row of the tabular dataset is denoted by Rn, and it contains the  parameters as Rn =  �  tloc  n , tsl  n,  ��tloc  n − tsl  n  �� , Ln, Pn, Sn, Mn  �  ,  where Pn ∈ RK denotes the K measured parameters of  sidelink channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' THE IV2I+ TESTBED AND DATASET  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Testbed Components  The testbed for the measurements is located in an industrial  co-working space in Berlin, with a layout as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  The hall had a gateway, which allowed the AGV to drive  outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  1) The AGV: The AGV used in the testbed is an au-  tonomous cleaning robot from the company Enway, as shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' They are specially designed for use under the  operating conditions of the manufacturing industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  The sweeper has all the necessary navigation data saved on  a digital map, and drives over the cleaning area autonomously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Thanks to high-performance sensors and control software  from Enway, the AGV navigates the environment completely  independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Using a combination of laser distance mea-  surement and cameras, the robot captures the environment  in three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' This 360-degree view enables very safe  navigation between people, complex production lines, and  overhanging systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The AGV immediately detects obstacles  that suddenly appear along the route, and drives around them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Additional equipment such as floor markings, QR codes, or  magnetic tracks are not required for navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' In the event  that the AGV encounters an unsolvable situation, the robot  stops and reports automatically to Enway headquarters via a  data connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The remote team monitors every movement  of the device around the clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The specialists can end  autonomous journeys at any time, and can take control from a  distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Collisions with people, production systems, vehicles,  and stored goods are thus avoided at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The machine  can also be navigated manually by the operating personnel  on site, if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Because the AGV is a retrofitted ride-  on sweeper, it can be controlled from the driver’s seat in  the traditional sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Ongoing use of the autonomous sweeper  can be monitored, controlled, and then evaluated using mobile  devices such as smartphones, laptops, or tablets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The software  completely logs the cleaning trips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  2) Cellular Network: The mobile network used for the  measurements in this test bed corresponded to a standardized  4G campus network with TDD medium access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The bandwidth  was 20 MHz in the frequency band between 3700 and 3800  MHz approved by the Federal Network Agency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' A correspond-  ing frequency assignment was applied for the period of the  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The hardware consisted of a server running  the LTE core and a radio base station with integrated antennas  connected to the server via Gbit LAN and powered via Power  over Ethernet (PoE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The location of the base station is marked  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' 3c with a yellow circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' A Mini-PC with the Linux  operating system was used as the UE to carry out QoS-relevant  measurements on the AGV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' A Quectel RM500Q-GL card was  used as the radio device, which was connected to the Mini-PC  via USB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' External antennas were connected to the radio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  The server provides a service interface to which the appli-  cations required for the measurements can be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The  stationary applications can communicate with mobile applica-  tions running in the Mini-PC via the service interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' This  (a) The Autonomous Cleaning Robot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  (b) Architecture of the iV2I+ Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' (c) Map of the Environment as captured by  the LIDAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Walls and AGV tracking route  are shown with red and black, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  The yellow circle is the location of the base  station.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' 3: iV2I+ testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  way, the location-dependent data rate and latency parameters  relevant for evaluating the QoS can be determined at the Mini-  PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  3) Time Synchronization: The AGV and the server were  time synchronized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' 3b visualizes the communication set-  up, where the dotted arrow shows the wireless connection  and the solid line the cable connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Both the server and  the Mini-PC on the AGV were connected to a GPS receiver,  allowing accurate time synchronization by using Pulse-per-  Second (PPS) signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The maximum error is typically in µs-  range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' However, the AGV only had consistent GPS reception  at the start point and the outdoor area, which could lead to  inaccuracies the ms-range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  4) LTE Modem Access: We used Mobile Insight, an open-  source cross-platform application for mobile network moni-  toring and analytics to capture mobile network data at the  Mini-PC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' It collects mobile network information across several  cellular protocols, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Radio Resource Control (RRC) or  EPS Mobility Management (EMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' During the measurement  campaign, the available information was logged every 40 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Additionally, a Python script that accesses the modem  via a virtual serial interface, a few radio parameters (RSSI,  RSRP, Reference Signal Received Quality (RSRQ), Signal-  to-Interference-plus-Noise Ratio (SINR)) were logged every  200 ms by the modem and written to a file with a time stamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  The data can then be linked to other data, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' the location,  via the common time stamps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  5) Controlled Packet Generation: We used the application  iperf3 on the Mini-PC and the server to generate UDP traffic  in either UL or DL direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Apart from the generated iperf3  log files, tcpdump was used both on the Mini-PC and server  to capture all incoming and outgoing packets at the respective  network interfaces, allowing a more detailed evaluation on a  packet level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Measurement Scenarios  For the measurement campaign, one AGV, equipped with  the sensors and measurement devices described in the prior  section, was driving through the testbed area over the course  of three days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Overall, 16 hours of data were collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  UL and DL communication was measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Two types of  packet flows were established to generate high and medium  throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' In DL, high throughput measurements were con-  ducted with a throughput target of 80 Mbps, in UL with  25 Mbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Approximately twice as many high throughput  measurements as medium throughput measurements were col-  lected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The route contained a diverse set of radio conditions,  namely: LOS and NLOS situations, coverage loss as well  as indoor and outdoor measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The measured path is  shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The Dataset  For each scenario, data from the described network compo-  nents and the sensors from the AGV was collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' A subset  of the captured data is presented in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  TABLE II: Selected iV2I+ Data Features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Parameter  Description  SINR [dB]  Derived from noise and power estimations of DMRS  RSRP [dBm]  Average energy per carrier/RE for DMRS  RSSI [dBm]  Signal power over the whole band  Throughput  Acquired throughput in respective link direction  Ping [ms]  Time in ms until a ping reply was received  Jitter  Delay variation measured over 1s  Odometry  Fused position,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' orientation and speed of the AGV  Map static elevation  Single pre-computed map of the whole area  Near/far map obstacles  36 m2/400 m2 obstacle map around the AGV  LIDAR  3D point cloud with obstacles  1) AGV-Sensor Data: The AGV delivers a series of sensor  data via its Robot Operating System (ROS),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' including the last  fourth rows in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Except for Light Detection and Rang-  ing (LIDAR), all ROS topics shown here are obtained through  MENWAY  IENWAY  21178  ITENWAY  100% AUTONOMOUS | 100% ELECTRIC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='sensor fusion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=', through techniques such as Extended  Kalman Filter (EKF)) and provide the relevant information  from a wireless perspective: position, orientation and speed  of the AGV and location of walls and obstacles in different  formats (2D, 3D, offline and online, near and far).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' The raw  input for the sensor fusion comes from sources including pure  wheel odometry, drive commands, an Inertial Measurement  Unit (IMU) and the already mentioned LIDAR, all of them  also available in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Pre-Processing  Similar to what was described in Section II-C2 the data was  pre-processed to further simplify the work with the collected  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Moreover, and since the position of the base station is  known and fixed, the distance and clearance of the wireless  link can be easily inferred from the sensor data and is provided  as part of the pre-processed dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  We have merged the GPS logs, the LTE stack measurements  and the throughput measurement together with the sensor-  based link distance and link clearance into a single dataframe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  As these data streams have different sampling frequencies, we  re-sampled as needed to 1 second before the final merge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' CONCLUSION  In this paper, we have describe industrial Vehicle-to-vehicle  (iV2V) and industrial Vehicle-to-infrastructure plus Sensor  (iV2I+), two testbeds for wireless communications in indus-  trial settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' We have provided detailed information about the  components of the testbeds, together with the initial concept  and captured scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' As mentioned before, the described  datasets are publicly available for ML research, what we  consider a valuable contribution to the available industrial  datasets both in terms of size and quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  iV2V and iV2I+ contain extensive and complete data that  we believe to be highly useful to answer questions regarding  the use and generalisation of ML for mobile use cases in  industrial environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Indeed, both datasets can be used  to train and evaluate methods for pQoS, which is a crucial  enabler for high reliability in wireless industrial applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='  Moreover, pQoS in general, and our data in particular can be  used as an ingredient for ML algorithms that optimize the  network itself, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=', by performing proactive radio resource  management (RRM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' This type of new network capabilities  will be an important part of the evolution of wireless com-  munications, such as the 6G cellular evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Finally, the  addition of AGV sensor data and localization opens the gate  to advanced techniques like fingerprinting or channel charting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+page_content='  [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
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+page_content=' Prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' (2022) RUDE-CRUDE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Accessed on:  2022-10-27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content=' Available: https://rude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='sourceforge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
+page_content='net/' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/J9E1T4oBgHgl3EQfsQWj/content/2301.03364v1.pdf'}
diff --git a/JNE4T4oBgHgl3EQfhg0d/content/tmp_files/2301.05125v1.pdf.txt b/JNE4T4oBgHgl3EQfhg0d/content/tmp_files/2301.05125v1.pdf.txt
new file mode 100644
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+Adaptive Dynamic Global Illumination
+SAYANTAN DATTA, McGill Unviversity, Canada
+NEGAR GOLI, Huawei/AMD, Canada
+JERRY ZHANG, Huawei, Canada
+𝑡 = start
+𝑡 = start + 4𝑠
+𝑡 = start + 8𝑠
+Direct only
+Direct + Indirect
+Tunnel interior (D + I)
+SSIM/MSE:
+0.989/0.007
+SSIM/MSE:
+0.974/0.008
+SSIM/MSE:
+0.970/0.008
+SSIM/MSE:
+0.957/0.008
+SSIM/MSE:
+0.942/0.009
+SSIM/MSE:
+0.939/0.010
+SSIM/MSE:
+0.993/0.000
+SSIM/MSE:
+0.953/0.003
+SSIM/MSE:
+0.932/0.004
+SSIM/MSE:
+0.837/0.003
+SSIM/MSE:
+0.826/0.009
+SSIM/MSE:
+0.875/0.010
+Left: Ours@15.6ms
+Right: Q-DDGI@26.8ms
+Fig. 1. Our technique demonstrated on a modified Bistro Exterior scene containing 192 × 64 × 192 probes.
+The third row shows the changes inside the tunnel as the gate opens over time. Our techniques responds faster
+to a dynamic stimuli and offers 1.7-times higher performance compared to the Q-DDGI implementation even
+with large probe grid containing excess of 2.3 million probes. Q-DDGI, detailed in section 5, is an extension of
+vanilla DDGI making it more competitive and comparable against our approach.
+We present an adaptive extension of probe based global illumination solution that enhances the response
+to dynamic changes in the scene while while also enabling an order of magnitude increase in probe count.
+Our adaptive sampling strategy carefully places samples in regions where we detect time varying changes in
+radiosity either due to a change in lighting, geometry or both. Even with large number of probes, our technique
+robustly updates the irradiance and visibility cache to reflect the most up to date changes without stalling
+the overall algorithm. Our bandwidth aware approach is largely an improvement over the original Dynamic
+Diffuse Global Illumination while also remaining orthogonal to the recent advancements in the technique. 1 2
+CCS Concepts: • Computing methodologies → Ray tracing; Rasterization.
+Additional Key Words and Phrases: Adaptive sampling, irradiance probes, global illumination, real-time
+1
+INTRODUCTION
+Global illumination (GI) strikingly improves the realism of a virtual scene, but its high computational
+cost has been a long-standing challenge in its application to real-time rendering [22].
+1Project Page
+2Poster
+1
+arXiv:2301.05125v1  [cs.GR]  12 Jan 2023
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+Several real-time GI solutions have been proposed, such as screen space [43] techniques, which
+support fully dynamic scenes but suffer from quality issues due to the limited availability of
+information in screen space. On the other hand, baked texture light-maps only support static
+geometry but remain popular due to their simplicity, low run-time cost, and quality. Precomputed
+Radiance Transfer [51] combined with light probes [31] and light-maps [15] solved some of the issues
+plaguing static light maps; in particular, these approaches support semi-dynamic geometry and self-
+occlusion while adhering to a strict compute budget. The advent of real-time ray-tracing hardware
+set the stage for modern fully dynamic GI. Dynamic real-time GI methods build upon the decades
+of research in sampling, and amortization of shading and visibility across space (pixel/world), angle,
+and time to improve convergence [40]. Adaptations of several offline techniques such as photon
+mapping [17], many-light rendering [20, 62], and radiosity maps [54] have also been explored
+in the context of modern [26, 27] ray-tracing capable hardware. However, presence of noise in
+sampled algorithms require the use of strong denoisers. Machine learning denoisers [6, 66] have
+demonstrable advantages in terms of quality compared to more traditional frequency [32] or
+variance [46] based denoisers. However, the prospect of training a neural network, the added
+complexity of integrating machine learning inference with traditional graphics pipeline, and the
+proprietary nature of machine learning frameworks have stalled the industry-wide adoption of
+these techniques. The recent probe-based algorithm, Dynamic Diffuse Global Illumination (DDGI)
+[28], extending the classic irradiance probes, still remains an excellent choice due to its relative
+simplicity, quality, and cloud streaming capabilities [14, 53]. However, scaling of DDGI in its original
+formulation is limited, and approaches such as multi-grid hierarchy and probe rolling [29] are
+necessary to scale it across large environments. Our adaptive approach focuses on dynamic contents
+in environments containing millions of probes in a single hierarchy.
+We propose Adaptive Dynamic GI (ADGI) algorithm where we trace a few pilot rays per frame to
+scan the environment and build a coarse representative model of the dynamic events. Using Markov-
+Chain sampling, we dynamically allocate resources to the critical areas, improving convergence in
+those regions. While DDGI allocates a fixed number of samples per probe and uniformly distributes
+samples across directions, ADGI non-uniformly samples the joint spatio-angular domain of the
+discretized 5D light-field represented by the probes. Our approach essentially decouples resource
+allocation from the number of probes resulting in a user-controlled performance target (FPS)
+and improved scaling even with millions of probes. Additionally, our approach results in faster
+convergence in static and dynamic environments given equal render time. Our approach is drop-in
+compatible with the original implementation and its several other extensions such as probe rolling
+and probe volume hierarchies [29].
+We achieve these objectives by formulating a guided function approximation technique, which is
+purposefully accurate in specific regions highlighted by our guiding function and thus eliminates
+the need for uniform resource allocation. Furthermore, we develop a sampling methodology based
+on temporal Markov-chain, which adapts naturally to a dynamic environment while also enabling
+scaling across large number of probes. Finally, we discuss memory and bandwidth preserving color
+compression schemes tailored specifically for our purpose.
+2
+RELATED WORK
+Probe-base approaches: Modern games rely extensively on light probes for static and dynamic
+global illumination due to their ease of integration into the game engine pipeline at low run-time
+cost. Some advocate a uniform grid probe placement due to their simplicity while others have
+proposed non-unform probes due to their efficiency. Probe based techniques are usually prone to
+light leakage. As such, uniform grid approaches [28, 31] use additional information, stored in the
+probes to determine whether a probe is visible from a shade point. Non-uniform approaches may
+2
+
+Adaptive Dynamic Global Illumination
+High Performance Graphics, Poster, July 11–14, 2022,
+Guiding function
+ℎ(𝑥) = 𝑝 (𝑥) × 𝑙 (𝑥)
+𝑝 (𝑥)
+𝑙 (𝑥)
+𝑥
+𝑥
+State of
+Environment
+Sample
+Feedback
+(a) Construct the guide ℎ(𝑥).
+𝑥
+Metropolis sampling
+𝑥𝑖
+ℎ(𝑥)
+(b) Sample 𝑥𝑖 ∼ ℎ(𝑥).
+𝑥
+𝑥𝑖
+𝑔(𝑥)
+(c) Evaluate objective 𝑔(𝑥𝑖).
+𝑥
+𝑥𝑖
+𝐸𝑟
+𝑔
+^𝑔
+^𝑔(𝑥)
+(d) ^𝑔(𝑥) - Reconstruct 𝑔(𝑥)
+from 𝑔(𝑥𝑖). 𝐸𝑟 indicates the
+reconstruction error.
+Fig. 2. Figure showing the steps in our adaptive-sampling strategy. We define a guiding function ℎ(𝑥) that
+highlights (in yellow) the interesting regions of the domain. The samples 𝑥𝑖 obtained from ℎ(𝑥) are used to
+evaluate the objective 𝑔(𝑥𝑖). Our goal is to obtain an approximate representation of 𝑔(𝑥), denoted as ^𝑔(𝑥),
+from the (𝑥𝑖,𝑔(𝑥𝑖)) pairs. As more samples are obtained from the highlighted region, the reconstruction error
+is lower in the yellow area, as shown in sub-figure (d).
+use carefully curated probe placement [63] combined with spatial data-structures like octrees to
+determine the visibility of a probe from a surfel. McGuire et al. [28, 31] stores the depth values
+of the surrounding geometry from a probe and use a similar idea as Variance-Shadow-Mapping
+[9] to approximate visibility. However, non-uniform approaches has been mostly limited to static
+geometry due to their high initial construction cost. Some approaches use rasterization [31, 63]
+while other may use ray-tracing [28] to compute the probe content. Probe based techniques also
+differ on how they store the information in the probes. Some use discrete textures [28, 31] while
+other may use a compressed basis representation such as Spherical Harmonics [14, 55]. Spherical
+harmonics implicitly pre-filters the content before storage but may cause light and dark ringing
+issues. Memory bandwidth required for reading and writing from the probes is also a major concern.
+Texture compression [31, 53] is usually the preferred choice to minimize memory bandwidth.
+Bandwidth is also crucial for cloud streaming of probe data. In such scenarios, Spherical Harmonics
+[14] representation may be preferable as they provide excellent compression for low frequency
+data. At run-time, dynamic probe based [28] GI solutions uniformly distributes rays across probes
+to update their content; this quickly becomes a bottleneck as the number of probes increases. Our
+approach on the other hand, focuses on the optimal distribution of resources to maximize visual
+fidelity. Various extensions have also been proposed to increase scalablity [29] of uniform grid
+approaches such as multiple-volume hierarchies and probe rolling. Our approach remains largely
+orthogonal and fully compatible with these extensions.
+Adaptive sampling: Adaptive sampling has been used in the context of screen-space ray-traced
+global illumination where more samples are accumulated in regions with high noise and high
+frequency [12]. Adaptive sampling is also useful for filtering soft shadows [32], where pilot-rays
+model the spatial frequency of shadow-penumbra and provide the number of additional samples
+required at each pixel to improved convergence. Neural versions [13] of adaptive sampling has
+also been proposed where a neural network generates a sampling-map that is tightly coupled to a
+post-process neural-denoiser. Conceptually our approach is similar, but our execution is tailored
+for the problem of temporally coherent sampling of probes. We refer readers to section 8 for an
+extend related work in irradiance-caching, screen-space GI and MCMC techniques
+3
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+3
+OVERVIEW
+We focus on two primary issues with DDGI in its original formulation. First, the technique does
+not allow for the non-uniform allocation of resources, resulting in unnecessary probe updates in
+regions that are not crucial for visual fidelity. Seconds, it does not update the probes quick enough
+to reflect transient changes in the scene environment. Our adaptive strategy involves detecting
+the changes in the environment and allocating resources driven by the detected changes. While
+the detection phase requires allocating additional resources, our empirical evaluations suggest our
+non-uniform adaptive sampling compensates for the lost efficiency in the detection phase. Our
+detection phase also enables fast probe updates for capturing transient changes in the scene. We
+model our technique as guided function approximation where we approximate a continuous function
+(e.g. 5D light-field) using a discrete (e.g. probes) representation driven by a guiding function.
+A naive approach to approximate a continuous function is to discretize the domain and reserve a
+representative sample for each discretization. The strategy is useful when the domain is relatively
+small; however, as the domain gets larger or the number of discretizations increases, it is prohibi-
+tively expensive to update all discretizations in real-time. This is one of the issues plaguing the
+original DDGI technique. In many applications, it is not necessary to update the entire domain
+uniformly; instead, we can tolerate more approximation errors in some regions than others. A
+simple example is foveated rendering, where errors in the periphery are less intrusive than those
+near the gaze center. In our case, we need the most accuracy in probes contributing to final shading.
+We introduce the notion of guiding function, which highlights the regions where a higher
+reconstruction accuracy is desired. We define the guide using a product of terms - the first term
+represents the current state of the environment while the second term is a feedback from the
+sampled cache. We sample the guide using a temporally coherent Markov-chain and use the
+samples to update our approximate representation using a parallel thread-safe approach. Thus our
+approach is summarized in three steps - defining a guiding function, sampling the guide, and using
+the samples to update the approximate representation. We describe these steps in sections 3.2, 3.3
+and 3.4 while we discuss various implementation specific details in section 4. See figure 2.
+Our approach provides two distinct advantages compared to the original DDGI - approximation
+quality and scalability. At any time, we concentrate our resources on a potentially challenging
+area as opposed to the entire domain. Provided our guide correctly identifies the challenging
+regions, the quality is improved due to a higher concentration of resources in the appropriate
+region. Since we sample the guide independent of the number of discretizations, the decoupling
+allows for a high number of statically allocated probes without affecting run-time performance.
+Increased discretizations improve approximation quality while the independence of sampling from
+the number of discretizations improves scalability. More specifically, we transparently increase
+the number of discrete probes without affecting performance. The run-time performance depends
+on the number of samples we generate; the samples are channeled to the appropriate areas by the
+guiding function. Our Markov-chain sampling is highly parallel, temporally coherent, and scalable,
+making it suitable for real-time temporally distributed reconstruction of large probe grids.
+3.1
+Background
+Here we briefly describe the original DDGI algorithm. DDGI consists of a 3D grid of directionally
+resolved irradiance probes that are updated in real-time through hardware ray-tracing. The probes
+also contains visibility information to prevent light leakage. The probe representation has many
+benefits, it performs optimally for diffuse indirect transport and is relatively inexpensive to encode
+and decode information to and from the probes. The algorithm evenly distributes ray-samples
+outwards from the probe center at each active probe in a stochastic rotated spiral pattern. DDGI is
+4
+
+Adaptive Dynamic Global Illumination
+High Performance Graphics, Poster, July 11–14, 2022,
+Uniform
+probe grid
+placement
+Generating &
+tracing rays
+Evenly distribute
+Probe state update
+Update irr &
+vis 2𝐷 atlas
+Shade each
+point
+8 cage probe
+Fig. 3. The figure illustrates the main steps of DDGI algorithm. Algorithm defines a uniform grid of probes
+and trace uniform-random rays in all direction from each probe. Based on the hit information, we compute
+the visibility (vis) and irradiance (irr) and update the 2𝐷 atlas. We also update the probe states based on
+visibility information (back-face hit ratio). Finally, for each shade-point, we query the eight bounding probes
+surrounding it and interpolate them to compute incoming indirect illumination.
+a two step algorithm. First, it updates the shading on the probe texels. Next a screen-space pass
+where the up-to-date probe content is used for shading the camera-pixels. The probe texel values
+are encoded into a spherical-mapped diffuse irradiance-texture with 8 × 8 resolution. Probes also
+captures the average ray-hit distance, and squared distances to the nearest geometry at 16 × 16
+resolution. DDGI temporally filters the probe texels by blending in the new values using a fixed
+hysteresis. The visibility data is used to decide whether a probe is visible at a shade-point and also
+used to infer whether a probe is inside a geometry and deactivated. The probe’s state is not limited
+to on or off and can vary with scenarios [29]. The world-position of the screen-space pixel is used
+as a key to the probe-texture lookup. The lookup interpolates the corresponding eight probes of
+the grid voxel containing the shade-point. The algorithm is illustrated in figure 3. DDGI algorithm
+is suitable for diffuse and slow changing phenomena in time. Therefore DDGI, combined with our
+adaptive-sampling strategy is a reasonable real-time GI approximation for dynamic scenes.
+3.2
+Guiding function
+As summarised in section 3 and figure 2, a guiding function highlights the important areas in
+the domain, i.e., challenging regions where more resources are required. These highlighted areas
+receive more adaptive samples, reducing the approximation error in those regions. Mathematically,
+the domain of the guiding function ℎ : 𝑅𝑑 → 𝑅 is the continuous 5D light field. Upon query, the
+guide function returns a scalar value indicating the importance of a sampled point. In our case,
+𝑑 = 5 as the domain is a 5-dimensional space of world-space positions and directions, and the guide
+encodes the importance of sampling a direction on a probe (texel’s importance).
+We model the guiding function (ℎ) as a product of two terms. The first term, we call 𝑓 : 𝑅𝑑 → 𝑅,
+represents the value in sampling a texel based on our understanding (limited) of whether such a
+texel would contribute towards the final screen-space shading. The second term is the observed
+sampled evidence (a.k.a irradiance cache) as they become available. Initially, the irradiance cache is
+empty but filled progressively through sampling. We define the first term based on some heuristics
+that describes our understanding of the probe-environment:
+• Probes closer to the camera,
+• Probes closer to geometric surfaces,
+• Directions on the probes facing away from geometric surfaces,
+• Directions on the probe with higher incoming irradiance,
+• Directions with temporal change in irradiance and visibility
+We trace pilot rays from the probes to generate the information necessary to quantify the above
+heuristics. We also call it the detection phase where we pre-scan the scene environment for changes.
+We denote the individual heuristics as 𝑓𝑖 : 𝑅𝑑 → 𝑅, and compose them into its final form 𝑓 as
+shown in equation 1, where 𝜙 represents a composition function. The composition function is
+5
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+Algorithm 1: Metropolis algorithm
+Input: ℎ: Guide distribution, 𝑀 : No. of iterations
+Input: 𝐾: No. of initial samples to reject
+Output: 𝑥 : Sample
+Ensure: 𝑀 ≥ 2, and 𝐾 < 𝑀
+1 𝑗 ← ShaderInvocationIndex()
+2 𝑥0 ← 𝑆[𝑗]
+// Initialize Markov-chain from memory
+3 while 𝑖 ← 0 to 𝑀 − 1 do
+4
+𝑥𝑖+1 ← RandomWalk (𝑥𝑖,ℎ(𝑥𝑖))
+// Random walk step, algorithm 5
+5
+if 𝑖 > 𝐾 then
+/* Use sample 𝑥𝑖+1 for probe updates, see algorithm 2
+*/
+6 𝑆[𝑗] ← 𝑥𝑖 + 1
+// Save Markov-chain state
+simply a recipe to appropriately combine the individual heuristics. We quantify the individual
+heuristics (𝑓𝑖) in section 4.1 and the composition (𝜙) in section 4.2.
+𝑓 = 𝜙(𝑓0, 𝑓1, ..., 𝑓𝑖).
+(1)
+The second term uses the stored irradiance in the probes, denoted by ^𝑔, to modulate the first
+term. We model the second term as - 𝑒𝑥𝑝 (𝛼 · ^𝑔(𝑥)/𝑓 (𝑥)), where the scalar 𝛼 ∈ [0, ∞) indicates our
+confidence in the irradiance probe content; a higher value indicating greater confidence. Note that a
+stored texel with high irradiance value may or may not have a high contribution to the final shading.
+Example - in a dynamic environment the probe content from the last frame is quickly outdated
+and thus less useful. The parameter 𝛼 models this uncertainty. The term 𝑓 (𝑥) in the denominator
+ensures that we only trust ^𝑔(𝑥) when 𝑓 (𝑥) is low. Finally, we define the guiding function as:
+ℎ(𝑥) = 𝑒𝑥𝑝
+�
+𝛼 · ^𝑔(𝑥)
+𝑓 (𝑥)
+�
+· 𝑓 (𝑥).
+(2)
+3.3
+Sampling the guide
+Next we sample the guiding function (equation 2). Mathematically, given an unnormalized distribu-
+tion ℎ : 𝑅𝑑 → 𝑅, our goal is to obtain samples 𝑥𝑖 from ℎ(𝑥), where 𝑥𝑖 ∈ 𝑅𝑑.
+Our sampling algorithm is straightforward. We use the Metropolis sampling, as shown in al-
+gorithm 1 to sample ℎ. The algorithm randomly initializes a state (𝑥0 ∈ 𝑅𝑑) and moves the state
+forward based on the acceptance of a newly proposed state. We generate the proposed states by
+perturbing the current state with a zero-mean Gaussian noise, also known as Random-walk [5].
+Parallelism: Note that algorithm 1 runs as a shader invocation, meaning several instances of
+the chain run in parallel. Each instance is independent with its own memory to load and store the
+chain state (denoted by S[] in algorithm 1). The instances generate thousands of samples per frame.
+As an input to our algorithm, we explicitly specify the number of chains that run in parallel, thus
+controlling the number of adaptive samples and performance. Contrasting with the original DDGI,
+the number of samples in the original implementation is proportional to the number of probes
+which increases cubically with scene dimensions. As such, it is difficult to scale up when the scene
+gets larger or when using a denser probe grid. Our approach is independent of the discretization
+resolution and scales better to higher probe counts without compromising approximation quality.
+Mixing-time: Initially, a Markov chain requires many iterations for the chain to generate
+samples from the target distribution (here ℎ(𝑥)), a phenomenon known as mixing time. We avoid
+6
+
+Adaptive Dynamic Global Illumination
+High Performance Graphics, Poster, July 11–14, 2022,
+Table 1. List of symbols
+Symbol
+Description
+Remarks
+𝑓
+Heuristics model
+Section 3.2
+ℎ
+Guiding function/Target distribution
+Section 3.2, 3.3
+𝑔
+Objective function
+Symbolic proxy for 𝑔𝑟, 𝑔𝑐.
+Section 3.4
+^𝑔
+Approximation of objective function
+Symbolic proxy for ^𝑔𝑟, ^𝑔𝑐.
+Section 3.4
+𝑔𝑟
+5D Light field
+Section 3.4
+𝑔𝑐
+Chebychev visibility
+Section3.4
+^𝑔𝑟
+Approximation of 5D light field
+(Irradiance cache)
+Section 3.4, 4.5
+^𝑔𝑐
+Approximation of Chebychev visibility
+(Visibility cache)
+Section 3.4, 4.6
+𝑥 or 𝑥𝑖
+Markov-chain samples
+Symbolic proxy for 𝑝𝑖, 𝜔𝑖.
+Section 3.3
+𝑝𝑖
+Positional (∈ 𝑅3) component of 𝑥𝑖
+–
+𝜔𝑖
+Directional (∈ 𝑅2) component of 𝑥𝑖
+–
+this problem by bootstrapping the initial chain state from the last frame. As such, we keep the
+number of iterations per frame small, but over frames, the chain effectively accrues many iterations.
+Distribution stationarity: Markov chain sampling requires the target distribution ℎ(𝑥) remain
+stationary. Due to a dynamic scene environment, the stationarity condition is seemingly violated.
+This may affect the approximation quality of our technique if the distribution changes rapidly
+between frames. However, we have several contingencies to deal with the issue. First, we target high
+frame-rates, which minimizes the change in the target distribution between consecutive frames.
+As an additional margin of safety, we reject initial 𝐾 samples per frame as shown in algorithm 1,
+line 5. This ensures our usable samples are obtained closer to the target distribution. Note that
+the evaluation time for ℎ(𝑥) negligible and thus rejecting few initial samples per frame does not
+significantly impact performance. We also smooth out the target distribution (see section 4.1.4)
+using spatio-temporal convolution to minimize abrupt changes in the target across frames.
+Temporal tracking: Since our target distribution may vary with time, we require the samples
+generated from the Markov-chain to closely follow the distribution to capture the transient changes
+in the environment. We make some crucial modifications to our sampling algorithm to allow for
+fast tracking of the target distribution, which we discuss in detail in section 4.9.
+3.4
+Approximation
+With samples obtained from the highlighted (figure 2(b)) parts of the domain, we focus on using
+the samples to evaluate (figure 2(c)) and reconstruct (figure 2(d)) our objective function. The term
+objective function refers to the quantity we aim to approximate. Mathematically, we denote our
+objective function as 𝑔 : 𝑅𝑑 → 𝑅𝑐, and its approximate reconstruction as ^𝑔. For ADGI, we have
+two objective functions - the light field 𝑔𝑟 : 𝑅5 → 𝑅3, and Chebychev-visibility 𝑔𝑐 : 𝑅5 → 𝑅2
+surrounding the probes. We denote their approximate reconstructions as the irradiance cache ^𝑔𝑟,
+and the visibility cache - ^𝑔𝑐 respectively. See section 4.5 and 4.6 for more details.
+Updating ^g: We evaluate the continuous objective function 𝑔 at collected sample points 𝑥𝑖 and
+store the evaluations - 𝑔(𝑥𝑖) into ^𝑔, as shown in algorithm 2. For ADGI, the evaluation step involves
+7
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+Algorithm 2: Approximation algorithm
+Input: 𝑥: Markov-chain samples
+1 function UpdateRepresentation(𝑥):
+2
+𝑣 ← 𝑔(𝑥)
+// Evaluate sample, ray-trace
+3
+AtomicMovingAvg(𝑥, 𝑣)
+// Populate ^𝑔, see algorithm 4
+tracing a ray to query the local light field and visibility. At each Metropolis iteration, the evaluated
+samples update the closest entry in the probes (^𝑔) within a critical section construct.
+Representing ^g: Prior work represent ^𝑔 as either as discrete LUTs [28], continuous Spherical
+Harmonics [14], Neural Networks [36], or any combination. In our case, the choice to use a discrete
+representation is based on several factors. First, multiple parallel streams of Markov-chain samples
+may update the same memory location in ^𝑔. As such, provisions are necessary to prevent race
+conditions. We also need a representation that handles temporal accumulation and quickly update
+itself to reflect any transient changes in the scene. Finally, the representation must be bandwidth
+efficient to improve the read and write performance. We refer to section 4.5 and 4.9 for details.
+3.5
+MCMC analysis
+In this section, we analyze our adaptive sampling algorithm in the context of MCMC (Markov
+Chain Monte Carlo). Note that our goal is not variance reduction through importance sampling;
+rather the focus is guided approximation of the objective function via sampling the target function.
+As such, unlike importance sampling, the sampling function is not necessarily correlated to the
+integrand. With this distinction in mind, we first look at the equation driving importance sampling
+using MCMC and then repurpose it for guided function approximation.
+The following equation shows a typical case of importance sampling where the objective is to
+compute the integral
+∫
+ℎ(𝑥)𝑔(𝑥)𝑑𝑥 and there exists a strategy to sample from h(x). In many typical
+scenarios (e.g. full Bayesian inference), the distribution ℎ(𝑥) is a proper distribution (
+∫
+ℎ(𝑥)𝑑𝑥 = 1)
+but does not have an efficient sampling mechanism. This where Markov Chain MC is useful.
+∫
+ℎ(𝑥)𝑔(𝑥)𝑑𝑥 ≈
+�
+1
+𝑀
+𝑀−1
+∑︁
+𝑖=0
+𝑔(𝑥𝑖)
+� ∫
+ℎ(𝑥)𝑑𝑥, 𝑥𝑖 ∼ ℎ(𝑥).
+(3)
+In contrast, our choice of Markov Chain (Metropolis) is primarily technical - simplicity, GPU
+parallelism and temporal sample tracking. Nevertheless, the same equations provide meaningful
+insight - albeit in a different context of adaptive sampling. In our algorithm, we simply sum the
+samples obtained from the target distribution without taking into account the sample density. This
+is equivalent to computing the following:
+𝐼 = 1
+𝑀
+𝑀−1
+∑︁
+𝑖=0
+𝑔(𝑥𝑖), 𝑥𝑖 ∼ ℎ(𝑥).
+(4)
+While our goal is to estimate
+∫
+Ω 𝑔(𝑥)𝑑𝑥, the expectation of 𝐼 (rearranging equation 3) is:
+E [𝐼] =
+∫
+Ω ℎ(𝑥)𝑔(𝑥)𝑑𝑥
+∫
+Ω ℎ(𝑥)𝑑𝑥
+,
+(5)
+where Ω is the domain of integration. Clearly, the expected value of 𝐼 does not converge to the
+correct estimate -
+∫
+Ω 𝑔(𝑥)𝑑𝑥. However, there are two factors to consider - size of the domain Ω and
+8
+
+Adaptive Dynamic Global Illumination
+High Performance Graphics, Poster, July 11–14, 2022,
+shape of ℎ(𝑥) in the domain. First consider the limit case where Ω → 0. In this case, the integrals
+collapses to a point evaluation and indeed the expected value of 𝐼 equals the unbiased estimate as
+shown below.
+𝐿.𝐻.𝑆. = lim
+Ω→0
+∫
+Ω ℎ(𝑥)𝑔(𝑥)𝑑𝑥
+∫
+Ω ℎ(𝑥)𝑑𝑥
+=
+∫
+Ω ℎ(𝑥)𝑔(𝑥)𝛿(𝑥 − 𝑥0)𝑑𝑥
+∫
+Ω ℎ(𝑥)𝛿(𝑥 − 𝑥0)𝑑𝑥
+= 𝑔(𝑥0).
+(6)
+𝑅.𝐻.𝑆. = lim
+Ω→0
+∫
+Ω
+𝑔(𝑥)𝑑𝑥 =
+∫
+Ω
+𝑔(𝑥)𝛿(𝑥 − 𝑥0)𝑑𝑥 = 𝑔(𝑥0).
+(7)
+In the above equation, 𝛿 is the Kronecker delta. The result is important as it shows with increasing
+probe resolution, bias is reduced. However, reducing texel size is not always practical as more rays
+and memory are required to populate and store a high resolution probe. Notice how the term ℎ(𝑥)
+is cancelled in equation 6. When the domain of integration is sufficiently small, ℎ(𝑥) is practically
+constant and the term cancels out in the denominator and numerator.
+We now consider the shape of ℎ(𝑥). While the target h(x) varies globally, it is piece-wise constant
+at a local scale due to its tabular nature. More crucially, the target ℎ(𝑥) is stored at a much lower
+resolution compared to the irradiance probe ^𝑔(𝑥). This implies ℎ(𝑥) is practically constant across a
+texel of the irradiance probe. The expected value of 𝐼 for the 𝑘𝑡ℎ texel is thus given by:
+E [𝐼𝑘] =
+∫
+𝑇𝑘 ℎ(𝑥)𝑔(𝑥)𝑑𝑥
+∫
+𝑇𝑘 ℎ(𝑥)𝑑𝑥
+=
+∫
+𝑇𝑘 𝑐𝑘𝑔(𝑥)𝑑𝑥
+∫
+𝑇𝑘 𝑐𝑘𝑑𝑥
+=
+∫
+𝑇𝑘 𝑔(𝑥)𝑑𝑥
+∫
+𝑇𝑘 𝑑𝑥
+,
+(8)
+where 𝑇𝑘 represents the domain of 𝑘𝑡ℎ texel and 𝑐𝑘 represents the piece-wise constant value of
+ℎ(𝑥) when 𝑥 ∈ 𝑇𝑘. The area estimate
+∫
+𝑇𝑘 𝑑𝑥 is fixed for all texels and equivalent to 4𝜋/#𝑟𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛.
+Thus, due to the tabular nature of our target function, the estimates of irradiance texels remain
+un-biased. While performing texture filtering over irradiance texels, it is possible to compute an
+unbiased estimate by weighing the texel values with 𝑐𝑘 as follows:
+𝐼 𝑓 𝑖𝑙𝑡𝑒𝑟
+𝑘
+=
+∑︁
+𝑗 ∈N𝑘
+𝑤𝑘−𝑗𝐼𝑘−𝑗, 𝑤𝑖 = 𝑐𝑖/
+∑︁
+𝑗 ∈N𝑘
+𝑐 𝑗,
+(9)
+where N𝑘 represents the texels in the neighbourhood of texel 𝑘. The values 𝑐𝑖 are obtained by
+querying the probes storing ℎ(𝑥). Note that bias is unavoidable as we blend samples temporally
+in a dynamic environment. In a dynamic environment, the objective is evolving and the bias
+manifests itself as temporal lag. Practically however, within a small time window, both ℎ(𝑡) and
+𝑔(𝑡) are assumed constant and the samples can be blended using a windowed moving average.
+Note that windowed moving average requires storing historical information. A cheaper but biased
+approximation to windowed moving average is exponential moving.
+4
+IMPLEMENTATION DETAILS
+This section provides the several implementation details with a brief summary in figure 4.
+4.1
+Heuristics construction
+The section describes the construction of 𝑓 using the heuristics discussed in section 3.2. Our goal is
+to measure and quantify the heuristics that highlight the probes which actively contribute to the
+final shading and require additional resources for faster convergence. We represent the heuristics
+either parametrically (equation 10) or using an explicit LUT representation as shown in figure 5(a).
+The LUT is constructed such that each probe has eight texels corresponding to an octant. We trace
+a ray for each octant; the rays return the hit distance and incoming irradiance at the hit-point.
+From this information, we compute several quantities (equation 11 - 18) and store them in the
+9
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+World space
+Trace 8
+pilot rays
+Octahedral
+Quantify heuristics
+𝜙 (𝑓0, 𝑓1, ..., 𝑓𝑖)
+Model feedback
+𝑒𝑥𝑝 (𝛼 · ^𝑔/𝑓 )
+Guide function
+𝑝 (𝑥) × 𝑙 (𝑥)
+Metropolis sampling
+ℎ(𝑝,𝜔)
+(𝑝𝑖,𝜔𝑖)
+Irradiance
+Visibility
+𝜔𝑖
+−𝜔𝑖
+𝑝𝑖
+𝑝𝑖
+Trace ray
+Update irradiance, visibility
+cache
+Irradiance ( ^𝑔𝑟 )
+8 × 8
+Visibility ( ^𝑔𝑐)
+16 × 16
+Deferred shading
+Fig. 4. This figure illustrates our overall algorithm. We trace 8 pilot rays, one from each octant on the
+probe and approximate the heuristic model 𝑓 (𝑝,𝜔). Using the heuristic and feedback, we define the guide
+ℎ(𝑝,𝜔) and sample it using Metropolis sampling. The sampled (𝑝𝑖,𝜔𝑖) are used to trace more adaptive ray
+samples, gathering hit-distance and irradiance at the sample points. We update the probe-cache (^𝑔) with
+adaptive-samples. The cache is used in the next shader and also looped back as feedback to model the target.
+LUT/texture mapped to the probe octants. We define and evaluate the following heuristics for a
+probe at position 𝑝 and a direction 𝜔.
+4.1.1
+Distance from camera. A probe far away from the camera is less likely to contribute to the
+final shading. We represent this parametrically as described in equation 10, where 𝑝 represents
+probe position, 𝑐 camera position and 𝑘 is a threshold set by the user.
+𝑓𝑐 (𝑝,𝜔) =
+�
+1
+if ||𝑝 − 𝑐|| < 𝑘 ,
+𝑒−( ||𝑝−𝑐 ||−𝑘)
+otherwise.
+(10)
+4.1.2
+Probe visibility. Only the probes encompassing a geometry participates in the deferred
+shading. Thus, probes closer to a geometric surface are more important. Similarly, texels facing
+away from the surface are queried more often for shading. We express both quantities together in
+equation 11, where 𝑝 represents probe location and 𝑡 = 𝑡𝑟𝑎𝑐𝑒(𝑝, −𝜔). The function trace returns
+the distance of the nearest surface hit, and the scalar 𝑠 is the diagonal distance of a grid voxel.
+𝑓𝑣(𝑝,𝜔) = 𝑒−2𝑡/𝑠
+(11)
+4.1.3
+Incoming radiance. We consider directions with high incoming radiance as more impor-
+tant. To identify those directions, we query the radiance along each probe octant and use it as a
+representative for incoming radiance.
+𝑓𝑟 (𝑝,𝜔) = 𝑚𝑖𝑛(𝑟, 𝛽)
+𝛽
+,
+(12)
+where 𝑟 = 𝑙𝑢𝑚(𝑝,𝜔). The function lum returns the incoming luminance using direct illumination
+at the surface hit point. The parameter 𝛽 controls the dynamic range and we set 𝛽 = 5.
+4.1.4
+Probe visibility change . Detection of dynamic geometry is crucial for increased resource
+allocation in regions affected by these changes. We detect dynamic geometry by computing a
+temporal gradient of probe visibility followed by a spatio-temporal smoothing operation.
+𝑓0(𝑝,𝜔) = 𝑓 𝑡
+𝑣 (𝑝,𝜔) − 𝑓 𝑡−1
+𝑣
+(𝑝,𝜔),
+(13)
+where 𝑓 𝑡
+𝑣 , 𝑓 𝑡−1
+𝑣
+represent visibility in the current and last time step respectively. Equation 13
+implicitly states we keep the position and the direction fixed when measuring the time difference
+across frames to avoid noisy gradients. The gradient is passed through a temporal trigger (𝑇𝑟) as:
+10
+
+Adaptive Dynamic Global Illumination
+High Performance Graphics, Poster, July 11–14, 2022,
+a. Probes storing prior-information (𝑓 ).
+octant-mapped
+pilot-rays
+b. Irradiance cache ( ^𝑔𝑟 ).
+c. Visibility cache ( ^𝑔𝑐).
+Fig. 5. Figure showing various probe-mapped textures and LUT in our technique.
+𝑓1(𝑝,𝜔) = 𝑇𝑟 (𝑓0(𝑝,𝜔),𝜃) ,
+(14)
+where 𝑇𝑟 converts a pulse in time to a decaying signal controlled by the parameter 𝜃 as shown
+in figure 6(a). For simplicity, we drop the time axis from the function 𝑇𝑟. The function minimizes
+temporal discontinuities, thus helping the Markov-chain to closely follow the target distribution
+(ℎ) across frames. Finally, we perform a spatial convolution as follows:
+𝑓Δ𝑣(𝑝,𝜔) =
+∑︁
+𝑖,𝑗
+𝑓1(𝑝 − 𝑝𝑖,𝜔 − 𝜔𝑗).
+(15)
+The convolution step smooths out uncertainties in a single texel and also serves as a weak
+predictor of possible locations of the dynamic geometry in the next frame. We use a 5 × 5 × 5 and
+3 × 3 convolution in space and direction, respectively.
+4.1.5
+Probe radiance change. Similar to the previous section, we detect a change in radiosity
+using a temporal gradient of the probe radiance. We apply the same temporal trigger and spatial
+convolution operator as in the previous section. The corresponding equations are as follows:
+𝑓2(𝑝,𝜔) = 𝑓 𝑡
+𝑟 (𝑝,𝜔) − 𝑓 𝑡−1
+𝑟
+(𝑝,𝜔),
+(16)
+𝑓3(𝑝,𝜔) = 𝑇𝑟 (𝑓2(𝑝,𝜔),𝜃) ,
+(17)
+𝑓Δ𝑟 (𝑝,𝜔) =
+∑︁
+𝑖,𝑗
+𝑓3(𝑝 − 𝑝𝑖,𝜔 − 𝜔𝑗).
+(18)
+4.2
+Heuristics composition
+Now that the individual heuristics are defined, as described in equation 1, we compose them for
+the static and dynamic cases as follows:
+𝑓𝑠 (𝑝,𝜔) =
+𝑠𝑡𝑎𝑡𝑖𝑐
+����
+𝑓𝑐 𝑓𝑣 ,
+(19)
+𝑓𝑑 (𝑝,𝜔) =
+𝑑𝑦𝑛𝑎𝑚𝑖𝑐
+��������������������������������
+𝑓𝑐 𝑓𝑣(𝑓Δ𝑣 + 𝜇𝑓Δ𝑟) .
+(20)
+When the environment is static, we sample according to the camera and probe-to-surface distance
+heuristics denoted by 𝑓𝑐 and 𝑓𝑣 in equation 19. In the dynamic case represented by equation 20, we
+modulate the changes in the environment by the static term 𝑓𝑐 𝑓𝑣. The modulation indicates we are
+more interested in changes close to the camera and geometric surfaces. The factor 𝜇 weighs the
+strength of change in geometry versus change in lighting. We use 𝜇 = 2 in all our experiments.
+11
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+𝑇𝑟 (𝑡; 𝑣,𝜃)
+𝑣
+Temporal-pulse
+𝜃
+Linear decay
+start
+𝑡
+a. Transform temporal-pulse to a decaying signal.
+𝑓𝑐 ≥ 0.75
+𝑓𝑐 ≥ 0.5
+|𝑐𝑙𝑖𝑝𝑥𝑦 | ≤ 1.2
+|𝑐𝑙𝑖𝑝𝑥𝑦 | ≤ 1.4
+b. Defining clip volumes for probes.
+Fig. 6. Figure (a) shows the construction of temporal-trigger𝑇𝑟 (𝑣,𝜃). In figure (b), we call the volume bounded
+by the blue frustum and black boundary as inner volume 𝑉𝑖𝑛. Similarly, outer volume 𝑉𝑜𝑢𝑡 is the volume
+bounded by green frustum and outer grey boundary. All probes in 𝑉𝑜𝑢𝑡 participate in the heuristic modelling,
+as described in section 4.4. Probes inside the blue frustum participate in adaptive sampling as described in
+section 4.8, 4.9. We set the probe state 𝑁 = 16 for all probes outside 𝑉𝑖𝑛 but inside 𝑉𝑜𝑢𝑡, refer section 4.8.
+4.3
+Heuristics storage
+We store the quantities 𝑓𝑣, 𝑓𝑠, 𝑓𝑑 as a 6-10-10 bit encoded 32 bit integer at each octant of the probes.
+The remaining 6 bits are used for other flags. When querying the LUT/texture, we use a mapping
+function that maps the continuous position 𝑝 and direction 𝜔 to the corresponding texel in the
+LUT. We note that 𝑓𝑐 is implicitly defined, hence do not require additional storage.
+4.4
+Improving construction efficiency
+The heuristics construction step is a potential bottleneck if we trace 8 rays per probe for all probes
+in the scene. As such, we restrict the pilot-rays to the probes that are contained within an extended
+camera frustum as shown in figure 6(b). To maximize the efficiency of our algorithm, we further
+reuse the samples collected from the 8 pilot-rays to populate the irradiance ( ^𝑔𝑟) and visibility ( ^𝑔𝑐)
+caches. We change the ray-directions at alternate frames in an AABBCCDD... pattern, improving
+the detection of temporally varying light-field surrounding the probes. We measure the time-delta
+(equation 13 and 16) between two frames with identical set of ray-queries, avoiding noisy gradients.
+However, this effectively halves the detection frequency (frame-rate / 2) but improves the spatial
+awareness. We use a stratified-random ray-direction such that there is always one ray per octant.
+We update the irradiance and visibility cache at each alternate frame.
+4.5
+Probe irradiance cache
+As shown in figure 5(b), the irradiance cache ( ^𝑔𝑟) is represented as a uniform probe grid in space
+where each probe stores the surrounding diffuse irradiance at a 8 × 8 texel resolution using a
+spherical mapping. At each texel, we store the irradiance in a custom RGB encoding with 9-9-8 bits
+for the three channels. The remaining 6 bits (out of 32bit) store the sample accumulation count
+(N), used for computing the moving average (see algorithm 3) of a sample stream in time. We
+take several considerations into account for the choice of our encoding. Our encoding should be
+bandwidth efficient and must support atomic updates on a commodity GPU. We found both DX12
+and GLSL supports atomic operations on 32 bit integers. Finally, our encoding must faithfully
+12
+
+Adaptive Dynamic Global Illumination
+High Performance Graphics, Poster, July 11–14, 2022,
+Algorithm 3: Moving Average algorithm
+Input: 𝑥: Update location, 𝑣 : New sample, 𝑁𝑚𝑎𝑥 : Max sample count
+Output: 𝑉 : Updated value, 𝑁: Sample count
+1 function MovingAvgUpdate(𝑥, 𝑣, 𝑁𝑚𝑎𝑥):
+2
+𝑛 ← ^𝑔[𝑥].𝑁
+// Cumulative sample count
+3
+𝑜 ← ^𝑔[𝑥].𝑉
+// Cumulative value
+4
+𝑉 ←
+𝑣
+𝑛+1 + 𝑛·𝑜
+𝑛+1
+// Update cumulative value
+5
+𝑁 ← 𝑚𝑖𝑛(𝑛 + 1, 𝑁𝑚𝑎𝑥)
+// Increment sample count
+6
+return 𝑉, 𝑁
+encode intensities beyond the standard definition. We apply a non-linear color compression across
+the three color channels, 𝑖 ∈ [0..2] as shown in the equation below.
+𝑢𝑖 = 𝑚𝑖𝑛 (𝑙𝑛(𝛾 · 𝑣𝑖 + 1), 𝛽)
+𝛽
+.
+(21)
+We apply an inverse transform (𝑒𝑥𝑝(𝛽 · 𝑢𝑖) − 1) /𝛾 while decoding where 𝛽 = 5 and 𝛾 = 15.
+More details regarding our choice of compression scheme is provided in appendix B and figure 11.
+4.6
+Probe visibility cache
+As shown in figure 5(c), texels in the visibility probes store the mean distances and mean squared
+distances to the nearest geometry at 16x16 texel resolution. We call this ^𝑔𝑐 - our visibility cache.
+Each texel stores the two channels with 13 bits of precision each while the rest 6 bits are used for
+sample accumulation count. We normalize the distances with probe cage diagonal length. Similar to
+irradiance cache, we apply a logarithmic encoding as per equation 21 for efficient use of available
+precision. We use (𝛽,𝛾) values of (5, 15) and (8, 20) for the linear and squared channels respectively.
+4.7
+Temporal sample accumulation mecahnism
+We use a moving-average accumulation to store the samples in the irradiance and visibility caches. In
+the algorithm 3, we have two parameters 𝑁 and 𝑁𝑚𝑎𝑥 to control the moving-average accumulation.
+As we start accumulating samples, 𝑁 is incremented and the algorithm performs like a true moving
+average. However, as 𝑁 approaches 𝑁𝑚𝑎𝑥 − 1, the algorithm switches to an exponential moving
+average form with hysteresis (𝑁𝑚𝑎𝑥 − 1)/𝑁𝑚𝑎𝑥. Also, note that when the value of 𝑁 is low, the
+cache updates itself quickly, but the stored values may be noisy. As 𝑁 increases, the new samples
+are weighed less in their contribution to the cache. We exploit these parameters to control the
+learning rate and noise in the static and dynamic cases as discussed in the following sections.
+4.8
+Adaptive sampling - static
+We split our adaptive sampling strategy into two stages - static and dynamic. We have two separate
+Markov-chain sets, each focusing on different aspects of capturing the surrounding light-field.
+While the static chain focuses more on the accuracy, the dynamic chain is tuned for capturing the
+transient responses. We discuss the dynamic chain in detail in the next section.
+We set up equation 2 as - ℎ = 𝑒𝑥𝑝(𝑚𝑖𝑛(^𝑔𝑟/𝑓𝑠, 1)) · 𝑓𝑠. The feedback from irradiance cache ^𝑔𝑟 is
+obtained from the previous frame and from a higher mip-level (also used in deferred shader). The
+lowest mip-level ^𝑔𝑟 is continuously updated and thus avoided as feedback due to possible violation
+of stationarity condition within a frame. We use the Metropolis sampling, algorithm 1, to generate
+the samples 𝑥𝑖 ≡ (𝑝𝑖,𝜔𝑖). As summarized in the algorithm, 2, we use the samples to evaluate the
+13
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+Algorithm 4: Atomic moving average algorithm
+Input: 𝑥: Update location, 𝑣 : New update value
+Output: Update ^𝑔[𝑥]
+1 function AtomicMovingAvg(𝑥, 𝑣):
+2
+current ← ^𝑔[𝑥]
+/* Repeat until destination value stops changing
+*/
+3
+do
+4
+expected ← current
+5
+next ← MovingAvgUpdate(𝑥, 𝑣, 64)
+6
+InterlockedCompareExchange(^𝑔[𝑥], expected, next, current)
+// Refer HLSL
+7
+while current ≠ expected
+continuous light field 𝑔𝑟, which involves tracing a ray originating at 𝑝𝑖 along the direction 𝜔𝑖.
+We trace an additional shadow-ray per sample to compute the visibility in the opposite direction
+(−𝜔𝑖) as the probe queries in the deferred shader for visibility is exactly 180◦ out of phase w.r.t
+irradiance. Next we store the irradiance and visibility values in the irradiance ( ^𝑔𝑟) and visibility ( ^𝑔𝑐)
+caches using an atomic update rule as presented in the algorithm 4. Atomic updates are required
+as multiple invocations of the chain may update the same location in the irradiance and visibility
+caches. Figure 4 summarizes the overall idea.
+We set the random walk step size, denoted by 𝜎 ∈ 𝑅5 in algorithm 5, proportional to the size of
+discretization in the irradiance and visibility cache. Thus positional step size is proportional to the
+size of a voxel in the probe grid, while angular step size is roughly
+√︁
+𝜋/256. Due to the small step
+size, texels in the cache may accumulate more than one sample per texel, thereby accumulating
+a large sample count over time. We also note that our cache behaves like a true moving average
+between sample count 𝑁 = 0 to 64, which also contributes to better accuracy.
+The static adaptive samples are useful for improving convergence in a static scene and for slow
+changes that are undetected during prior construction. For example, slow changes in lighting such
+as day-night cycles in games. We lower the hysteresis by setting 𝑁 = 16 for all probes in the region
+{𝑉𝑜𝑢𝑡} − {𝑉𝑖𝑛} in figure 6(b). This enables the probe to quickly catch-up to the most recent values.
+4.9
+Adaptive sampling - dynamic
+We run a second set of Markov-chain when dynamic content is detected in the scene. When there
+are dynamic elements, especially moving geometry, we run into two main issues. The generated
+samples are not well distributed in the region of interest i.e. the areas where time varying changes
+are present. When the step size is small, the chain cannot track the target distribution fast enough
+to generate samples from the target, causing the samples to lag the moving target distribution.
+The second problem is noise due to multi-sampling of the irradiance texel. Potentially, this can be
+solved by increasing the hysteresis to improve temporal sample reuse. However, the reduced noise
+comes at the cost of introducing objectionable temporal blur.
+We solve the first issue by increasing the chain step size and by coarsening the target function
+(𝑓𝑑). Practically, this amounts to grouping the heuristics-probes into virtual proxies. In our case, a
+virtual proxy represents a group the 3 × 3 × 3 probes. This virtual probe has 8 directions and each
+direction represents an axis-aligned octant. The value of a texel of the virtual probe is the max of
+all 27 probes it represents along the corresponding direction. We also drop the sampled evidence by
+setting 𝛼 = 0 in equation 2, as the stale irradiance cache ( ^𝑔𝑟) provide little useful information for
+14
+
+Adaptive Dynamic Global Illumination
+High Performance Graphics, Poster, July 11–14, 2022,
+Table 2. Table showing probe grid details for various scenes used in our technique.
+Scene
+Probe Grid
+Probe spacing
+(in meters)
+Irradiance ( ^𝑔𝑟)
+Cache Resolution
+Visibility ( ^𝑔𝑐)
+Cache Resolution
+Bistro - Exterior
+192 × 64 × 192
+0.5 × 0.5 × 0.5
+8 × 8
+16 × 16
+Sponza - Diffuse
+192 × 64 × 192
+0.5 × 0.5 × 0.5
+8 × 8
+16 × 16
+Sponza - Glossy
+192 × 64 × 192
+0.1 × 0.1 × 0.1
+16 × 16
+16 × 16
+Table 3. Table showing probe encoding details for the various techniques we use in our comparison.
+Technique
+Irradiance
+Cache Encoding
+Visibility
+Cache Encoding
+Temporal
+Hysteresis
+Ours
+⌊R9⌋⌊G9⌋⌊B8⌋ − N
+[R13][G13] − N
+Static: 0.98 (𝑁𝑚𝑎𝑥 = 63)
+Dyna: 0.91 (𝑁𝑚𝑎𝑥 = 10)
+Q-DDGI
+⌊R11⌋⌊G11⌋⌊B10⌋ − N
+[R16][G16] − N
+0.94
+Reference
+RGB32f
+RG32f
+N/A
+sampling a time varying region. The chain step size is 3x, and 6x larger for position and directions,
+respectively w.r.t the static case.
+Since each sample from the coarse chain represents an entire octant, we trace 64 rays for the
+octant for all underlying 3x3x3 probes in the group. We make the tracing step more efficient by
+culling probes that are not used in deferred shading. The scheduling of ray-direction is deterministic,
+passing through the center of a texel in the irradiance cache ( ^𝑔𝑟). This solves the problem of sampling
+noise and also affords the opportunity to simplify the atomic updates. Since the rays are not random,
+we do not benefit from multiple shader invocations updating the same octant. As such, the first
+invocation to update the octant marks (atomically) it updated such that other invocations do not
+repeat the same work move to the next.
+We run the dynamic sampling after the static sampling step. During static sampling, if a probe
+has non-zero dynamic component(𝑓𝑑 > 0), we quantize the ray directions to go through the
+irradiance/visibility cache texel center to avoid injecting sampling noise in the texels.
+5
+RESULTS AND COMPARISONS
+We compare our results with Q-DDGI and a reference probe-based implementation in different
+scenarios - static scene (fig. 7), dynamic geometry (fig. 1, 8, 10), and dynamic lighting (fig. 9).
+Q-DDGI: Quantized-DDGI or Q-DDGI is a performance enhanced extension of original DDGI
+[28], achieved without major modifications to the base algorithm. Q-DDGI is equipped with a more
+compact irradiance and visibility cache representation that closely resembles ours. See table 3. We
+also enable camera-frustum culling of probes in Q-DDGI as described in section 4.4 and figure 6.
+These modifications allow Q-DDGI to have similar performance (table 4) at same probe count (table
+2) as ours across different scenes. We believe these modifications make our comparisons more fair.
+We use 32 rays per probe for a total ray budget of 800-1600k (depending on scene) rays per frame.
+Reference: Reference implementation uses a standard FP32 representation for irradiance and
+visibility caches as shown in table 3. We also use a higher resolution 32×32 irradiance and visibility
+cache. Due to memory constraints, we are limited to a smaller probe-grid of size 32 × 32 × 32 using
+same probe spacing (table 2) as other techniques. For each frame, we discard any previous values
+in the probes and accumulate samples using a true-average with 64 rays per texel.
+Ours: We use 4096 instances of static chain invocations and 1024 instances of dynamic chain
+invocations. Overall, we use use between 500-900k (depending on scene) rays per frame.
+15
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+Table 4. Performance breakdown of our technique and Q-DDGI. Our probe sampling stage is divided into
+three sub-stages - heuristic construction (P), static adaptive sampling (S), and dynamic adaptive sampling (D).
+Scene
+Ours (in milliseconds)
+Q-DDGI (in milliseconds)
+Probe Sampling
+(P + S + D)
+Deferred
+Total
+Probe sampling
+Deferred
+Total
+Bistro - Exterior
+4.01 + 2.23 + 4.73
+= 11.0
+4.63
+15.6
+22.3
+4.47
+26.8
+Sponza - Diffuse
+1.21 + 1.85 + 3.18
+=6.24
+3.62
+9.86
+9.69
+3.51
+13.2
+Sponza - Glossy
+4.83 + 2.11 + 4.33
+=11.27
+6.44
+19.7
+24.9
+6.71
+31.6
+Figure 1 and 8 shows a large scene (Bistro Exterior), with the tunnel’s entry and exit modified
+with dynamic gates. The tunnel interior walls are illuminated by indirect illumination alone,
+controlled by the direct light bouncing off the floor. The direct illumination on the floor is controlled
+by the dynamic entry gate. The scene tests the tracking capabilities of our algorithm; the dynamic
+Markov-chain should sample the probes close to the moving door. The scene also tests our color
+compression scheme under low-light and moving-average accumulation.
+Figure 9 shows the Sponza scene under dynamic lighting, testing the detection capabilities of
+ADGI in the absence of dynamic geometry. Figure 7 shows a static scene without dynamic geometry
+or lighting, testing the convergence of our static adaptive sampling when no dynamism is detected
+or the dynamic changes are too slow to detect, such as day-night cycles in games.
+Figure 10 shows a dynamic geometry (Stanford Buddha) under glossy indirect illumination
+with ambient lighting as direct component. The scene is stressful as the camera frustum contains
+many times more probes compared to other scenes due to the increased probe density required
+for glossy illumination. This scene tests the transient response of a dynamic geometry on a glossy
+floor. Thus the scene is less forgiving of spatio-temporal blurring.
+We measured the results on a desktop with Nvidia 2080Ti GPU and AMD 5600X CPU at
+1920 × 1080 resolution. The performance numbers cited in table 4 are only for ADGI and Q-
+DDGI algorithms. The GBuffer and direct-illumination passes require an additional 2ms and 3ms,
+respectively.
+6
+LIMITATIONS
+We inherit similar limitations as the vanilla DDGI algorithm. The probe visibility from a shade-point
+is only approximate and requires modifications such as probe movement to minimize light leakage.
+The probe representation is not efficient in capturing glossy light-transport and requires a dense
+spatio-angular discretization of irradiance cache to capture glossy reflections.
+Accurate detection of transient spatio-temporal changes in a scene are difficult. The accuracy of
+detecting dynamic geometry reduces with the distance of the dynamic object from a probe. The
+same is true for dynamic lighting; especially high frequency localized lighting that is far from a
+probe is difficult to detect. Also, for the Markov chain to track the target distribution, the speed
+of motion should be capped comparable to the product of Markov-chain step size and average
+frame-rate. While many game engines keep track of the dynamic objects, facilitating the detection
+of changing in visibility, we still need ray-tracing to detect dynamic radiosity.
+16
+
+Adaptive Dynamic Global Illumination
+High Performance Graphics, Poster, July 11–14, 2022,
+7
+CONCLUSION
+Our adaptive sampling approach improves upon the efficiency of the original DDGI algorithm. Our
+approach non-uniformly allocates resources in regions with time varying phenomena and captures
+transient localized changes in an environment containing millions of probes. By contrast, DDGI’s
+uniform allocation policy dilutes resource concentration in critical regions, especially when a large
+number of probes are present. These improvements reduce temporal lag and minimizes reliance on
+temporal blur to reduce noise. Our probe encoding scheme minimizes memory requirements by 4x
+(and by extension memory bandwidth) with minimal impact on quality while also enabling millions
+of probes in a scene. Our adaptive sampling stages have a fixed upper bound on the compute
+requirement and also decouples sampling from the number of probes, further reducing memory
+bandwidth requirement. These changes enable improved probe-based rendering while also enabling
+1.5-2x performance improvements.
+8
+RELATED WORK EXTENSION
+Irradiance caching Irradiance caching is another line of techniques attempting to overcome
+the high computation cost of GI. The irradiance caching method assumes that irradiance vary
+smoothly across the scene, and texture detail can be recovered using albedo modulation [64]. The
+interpolation and location of the various cache records is a critical, especially when the assumptions
+on smoothness do not hold. While robust, principled offline solutions exist [16, 24], real-time
+applications often resort to complex heuristics and impose harsh constraints to achieve online
+GI. Compression [56], sparse interpolation [49], pre-convolved environment maps [42, 45], spatial
+hashing [3] and using neural network [37] are instances of advancements in real-time irradiance
+caching. Although these approaches aim for real-time performance, their complexity and constraints
+make them challenging to implement and deploy.
+𝑡 = 32 ms
+64 ms
+96 ms
+128 ms
+32 ms
+64 ms
+96 ms
+128 ms
+Ours
+Q-DDGI
+MSE: 0.072
+0.031
+0.022
+0.014
+SSIM: 0.750
+0.894
+0.910
+0.940
+MSE:0.085
+0.064
+0.049
+0.031
+SSIM:
+0.561
+0.758
+0.828
+0.888
+SSIM: 0.732
+0.895
+0.908
+0.914
+SSIM: 0.524
+0.745
+0.838
+0.874
+MSE : 0.076
+0.038
+0.025
+0.020
+MSE : 0.087
+0.067
+0.052
+0.041
+Green: Diminished luminance
+Red: Excess luminance
+Fig. 7. Comparing the convergence of our technique over time on a static Bistro Exterior scene. The figure
+demonstrates the effectiveness of our static adaptive sampling step. The two rows measure the difference in
+luminance w.r.t reference and highlight the error in red and green color.
+17
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+𝑡 = start
+𝑡 = start + 4𝑠
+𝑡 = start + 8𝑠
+Direct + Indirect
+Tunnel interior (D + I)
+SSIM/MSE:
+0.971/0.008
+SSIM/MSE:
+0.974/0.008
+SSIM/MSE:
+0.982/0.007
+SSIM/MSE:
+0.951/0.010
+SSIM/MSE:
+0.951/0.009
+SSIM/MSE:
+0.967/0.009
+SSIM/MSE:
+0.931/0.004
+SSIM/MSE:
+0.948/0.004
+SSIM/MSE:
+0.994/0.000
+SSIM/MSE:
+0.913/0.009
+SSIM/MSE:
+0.871/0.005
+SSIM/MSE:
+0.804/0.004
+Left: Ours@15.6ms
+Right: Q-DDGI@26.8ms
+Fig. 8. Our technique compared with Q-DDGI on a modified Bistro Exterior scene augmented with a
+moving door. The scene has 192 × 64 × 192 probes and shows the convergence of the two techniques near
+a dynamic area in the scene. The second row shows the changes inside the tunnel as the door closes over
+time. Our technique is better able to allocate the resources closer to the dynamic areas resulting in faster
+convergence and higher performance.
+Path tracing The flexibility and generality offered by path tracing [18] is highly desirable for
+real-time rendering. However, path tracing has been out of reach for real-time applications due to its
+substantial computational requirements. Even with the advent of hardware-accelerated ray tracing
+[23], it is only possible to trace a few tens of rays at each pixel in real-time. Therefore, effective
+sampling strategies and high-quality denoising algorithms [38, 46, 47] are essential. Many sampling
+methods try to learn the representation of incident illumination during rendering [1, 8, 34, 44, 60].
+While these approaches can provide substantial error reduction, constructing these structures in
+parallel on a GPU incurs a significant overhead that seem unsuitable for real-time applications.
+Recently proposed ReSTIR GI [41] provides an efficient real-time sampling strategy by reusing the
+paths spatially and temporally but the algorithm becomes complicated after second bounce and still
+requires denoising for the final stage. Deep learning has also been applied to path guiding, including
+work by [35, 36]. These approaches demonstrated a substantial reduction in error due to more
+effective path sampling, though their performance remain insufficient for real-time applications.
+Screen space approaches: Approximating physically plausible illumination at real-time frame
+rates with screen space methods is popular in games. Screen space methods are fast, GPU-friendly,
+and simple to implement. Screen space ambient occlusion (SSAO) [2, 33] is part of many real-time
+rendering engines. Following SSAO, screen Space Directional Occlusion (SSDO) [43] is used for
+near-field direct and indirect diffuse lighting. Sousa et al. [52] proposed Screen Space Reflections
+(SSR) using a 2D ray-tracing approach directly in screen space to obtain the indirect specular
+component. Recently Screen-Space Global Illumination (SSGI) [43, 50, 52] methods offer a viable
+solution to real-time GI. However, these methods are limited by the information visible from the
+observer’s position, thus making it difficult to engineer a robust solution.
+Importance sampling and Bayesian modeling: Importance sampling provides a tool to
+reduce the cost of brute force integration by selectively evaluating elements of the integrand based
+on prior knowledge, i.e. an educated guess. Previous works in importance sampling proposed
+different methods to apply importance sampling to various Monte-Carlo integration existing in
+rendering equations [21, 48, 57]. Although Markov Chain Monte Carlo(MCMC) methods have been
+18
+
+Adaptive Dynamic Global Illumination
+High Performance Graphics, Poster, July 11–14, 2022,
+𝑡 = start
+𝑡 = start + 2𝑠
+𝑡 = start + 3𝑠
+Direct only
+Indirect - Ours
+Indirect - Q-DDGI
+SSIM/MSE: 0.920/0.005
+SSIM/MSE: 0.966/0.006
+SSIM/MSE:0.957/0.007
+SSIM/MSE: 0.868/0.012
+SSIM/MSE: 0.763/0.023
+SSIM/MSE: 0.794/0.021
+Difference in luminance w.r.t reference
+Error - Ours
+Error - Q-DDGI
+Green: Diminished luminance
+Red: Excess luminance
+Ours@9.86ms
+Q-DDGI@13.2ms
+Fig. 9. Figure comparing the convergence of our technique under dynamic lighting controlled by the direct
+component shown in the first row. The last two rows measure the difference in luminance w.r.t reference and
+highlight the error in red and green color.
+used in Bayesian learning from the early days of neural networks [39], and Stochastic-Gradient
+MCMC has been proposed [65] with various applications [25], our approach is neither Monte
+Carlo-based nor Neural-network learning. We exploit Bayesian inference and Markov Chains as our
+mathematical means to sample the important texels on the probe, by defining our guide function
+(prior), likelihood, and posterior.
+Markov Chain: Markov Chains are used broadly in Monte Carlo path-tracing. For example,
+Veach and Guibas [58] used Metropolis Sampling to explore the space of all possible paths. Kelemen
+et al. [19] later applied the exact sampling in the space of random numbers, i.e., in Primary Sample
+19
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+𝑡 = start
+𝑡 = start + 10𝑠
+𝑡 = start + 15𝑠
+Glossy indirect - Ours
+Glossy indirect without texture
+Ours
+Q-DDGI
+SSIM/MSE:0.996/0.017
+SSIM/MSE:0.995/0.019
+SSIM/MSE:0.992/0.018
+SSIM/MSE:0.990/0.019
+SSIM/MSE:0.989/0.020
+SSIM/MSE:0.987/0.028
+Ours
+Reference
+Q-DDGI
+Ours
+Reference
+Q-DDGI
+Ours
+Reference
+Q-DDGI
+SSIM: 0.979
+0.981
+SSIM: 0.995
+0.993
+SSIM: 0.998
+0.992
+MSE : 0.013
+0.028
+MSE : 0.016
+0.034
+MSE : 0.014
+0.058
+Ours@19.7ms
+Q-DDGI@31.6ms
+Fig. 10. Figure comparing glossy indirect reflection on a scene lit by ambient lighting. The scene tests transient
+response due to the moving Buddha geometry over a glossy floor.
+Space. The most recent work by Bitterli et al. [4] combines a simple path tracing integrator with
+MCMC by using the random seeds of high variance paths as starting points for the Markov Chains.
+Although Markov Chains are encountered extensively beneficial in solving Monte Carlo sampling,
+our point of view on sampling and employing the Markov Chain to draw samples from the guide
+function is distinct.
+Bayesian inference: Bayesian modeling is a widespread methodology in computer vision and
+graphics. Brouillat et al. [5] and Marques et al. [30] pioneered the use of Bayesian Monte Carlo
+(BMC) [11] in light transport simulation. In contrast, [59] keep the efficient classic, frequentist
+MC approach and apply Bayesian modeling to optimize their sampling distributions for direct
+illumination estimates across the scene. Similar approach is used by Vorba et al. [61], who employ
+a maximum a posteriori (MAP) formulation to regularize training of parametric mixture models for
+optimized indirect illumination sampling. Our approach uses Bayesian modeling in the context of
+light-probes to detect important probes and directions based on sampled evidence.
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+23
+
+High Performance Graphics, Poster, July 11–14, 2022,
+Datta et al.
+A
+METROPOLIS-HASTINGS
+Markov Chain Monte Carlo (MCMC) allows sampling from the posterior without computing
+the marginal. [10]. Metropolis-Hastings (Metropolis), which we exploit in this work, is a specific
+implementation of MCMC [7]. The Metropolis–Hastings algorithm can draw samples from any
+probability distribution with probability density 𝑃(𝑥), provided a function ℎ(𝑥) proportional to
+the density 𝑃(𝑥). The Metropolis algorithm works by generating a sequence of sample values so
+that, as more samples are produced, the distribution of samples more closely approximates the
+desired distribution. These sample values are produced iteratively, meaning the next sample being
+dependent on the current sample (thus making the sequence of samples into a chain). Let ℎ(𝑥)
+be a function that is proportional to the desired probability density function 𝑃(𝑥) (a.k.a. a target
+distribution). The Metropolis Markov Chain algorithm with random walk is defined as follows:
+Algorithm 5: Random-walk algorithm
+Input: 𝑥𝑖: Current state, 𝑦𝑖 : Probability of current state
+Input: 𝜎 : Step size or std-dev of Gaussian noise
+Output: 𝑥𝑖+1: Next state, 𝑦𝑖+1 : Probability of next state
+1 function RandomWalk(𝑥𝑖, 𝑦𝑖):
+2
+𝑥𝑖+1 ← 𝑥𝑖 + N (𝜎)
+// Propose a new state
+3
+𝑦𝑖+1 ← ℎ(𝑥𝑖+1)
+4
+𝜇 ← min
+�
+𝑦𝑖+1
+𝑦𝑖 , 1
+�
+// Compute acceptance ratio
+5
+𝜖 ∼ 𝑈 (0, 1)
+// Sample uniform distribution
+6
+if 𝜖 > 𝜇 then
+/* Reject proposed state
+*/
+7
+𝑥𝑖+1 ← 𝑥𝑖
+8
+𝑦𝑖+1 ← 𝑦𝑖
+9
+return 𝑥𝑖+1,𝑦𝑖+1
+Initialization: Choose an arbitrary point 𝑥𝑖−1 as the initial observation in the sample-space and
+choose an arbitrary probability density N (𝑥𝑖 | 𝑥𝑖−1) that suggests the next sample candidate 𝑥𝑖,
+given the previous sample value 𝑥𝑖−1. In our work, N is assumed to be symmetric. A usual choice
+is to let N (𝑥𝑖 | 𝑥𝑖−1) be a Gaussian distribution centered at 𝑥𝑖−1, so that points closer to 𝑥𝑖−1 are
+more likely to be visited next, making the sequence of samples resemble a random walk [7]. The
+random walk algorithm is described in algorithm 5.
+B
+PROBE COMPRESSION
+We tested several 26-bit encoding and settled on a non-linear RGB encoding represented by
+⌊R9⌋⌊G9⌋⌊B8⌋ −N in figure 11. In this encoding, the RGB color is first passed through a logarithmic
+non-linearity as per equation 21 such that the quantization errors are distributed evenly across
+intensities. We perform a round-to-lowest-integer (⌊⌋) quantization for all channels, although round-
+to-nearest-integer ([ ] ) is more accurate. Our quantization scheme ensures the moving-average
+updates produce dark colors when the intensity of new samples are low. In a round-to-nearest set-
+ting, due to a round-up error, the colors may never go to zero. Interestingly, YCbCr encoding allows
+round-to-lowest for the Y channel and round round-to-nearest for Cb and Cr channels, however,
+they perform poorly in both luminance and color preservation metrics as shown in figure 11.
+24
+
+Adaptive Dynamic Global Illumination
+High Performance Graphics, Poster, July 11–14, 2022,
+Intensity:
+[0 - 0.25]*
+[0.25 - 1.0]
+[1.0 - 5.0]*
+Decoded RGB
+Error
+Decoded RGB
+Error
+Decoded RGB
+Error
+⌊Y8⌋[ Cb9] [ Cr9] − N
+⌊Y8⌋[ Cb9] [ Cr9]
+⌊R9⌋ ⌊G9⌋ ⌊B8⌋ − N
+⌊R9⌋ ⌊G9⌋ ⌊B8⌋
+LUM: 0.0024
+LUM: 0.0025
+LUM: 0.0025
+LUM: 0.0004
+LUM: 0.0011
+LUM: 0.0043
+LUM: 0.0045
+LUM: 0.0049
+LUM: 0.0050
+LUM: 0.0007
+LUM: 0.0021
+LUM: 0.0086
+COR: 0.9972
+COR: 0.9999
+COR: 1.0000
+COR: 1.0000
+COR: 1.0000
+COR: 1.0000
+COR: 0.9944
+COR: 0.9997
+COR: 1.0000
+COR: 1.0000
+COR: 1.0000
+COR: 1.0000
+Fig. 11. Figure comparing 26-bit color encodings on slices of the 3D color-space with dynamic range. We
+compare the reconstruction error measured in Luminance and Color Correlation with RGB32f reference. The
+log-non-linear encodings marked with - N suffix shifts the bit error from lower to higher intensities - which
+are less frequent in indirect illumination. ⌊⌋ and [ ] denotes round-low and round-nearest quantizations
+respectively. * Color map visualizations are normalized.
+The parameters in equation 21 are obtained by performing a grid search minimizing the recon-
+struction error w.r.t RGB32f reference across various color and intensity combinations as shown in
+figure 11. Luminance error is the r.m.s. value of the difference between the two color-maps. Color
+accuracy is measured using a normalized dot product between the two flattened color-maps.
+25
+
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+page_content='010  Left: Ours@15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='6ms  Right: Q-DDGI@26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='8ms  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our technique demonstrated on a modified Bistro Exterior scene containing 192 × 64 × 192 probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  The third row shows the changes inside the tunnel as the gate opens over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our techniques responds faster  to a dynamic stimuli and offers 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='7-times higher performance compared to the Q-DDGI implementation even  with large probe grid containing excess of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='3 million probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Q-DDGI, detailed in section 5, is an extension of  vanilla DDGI making it more competitive and comparable against our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  We present an adaptive extension of probe based global illumination solution that enhances the response  to dynamic changes in the scene while while also enabling an order of magnitude increase in probe count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Our adaptive sampling strategy carefully places samples in regions where we detect time varying changes in  radiosity either due to a change in lighting, geometry or both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Even with large number of probes, our technique  robustly updates the irradiance and visibility cache to reflect the most up to date changes without stalling  the overall algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our bandwidth aware approach is largely an improvement over the original Dynamic  Diffuse Global Illumination while also remaining orthogonal to the recent advancements in the technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 1 2  CCS Concepts: • Computing methodologies → Ray tracing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Rasterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Additional Key Words and Phrases: Adaptive sampling, irradiance probes, global illumination, real-time  1  INTRODUCTION  Global illumination (GI) strikingly improves the realism of a virtual scene, but its high computational  cost has been a long-standing challenge in its application to real-time rendering [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  1Project Page  2Poster  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='05125v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='GR]  12 Jan 2023  High Performance Graphics, Poster, July 11–14, 2022,  Datta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Several real-time GI solutions have been proposed, such as screen space [43] techniques, which  support fully dynamic scenes but suffer from quality issues due to the limited availability of  information in screen space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' On the other hand, baked texture light-maps only support static  geometry but remain popular due to their simplicity, low run-time cost, and quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Precomputed  Radiance Transfer [51] combined with light probes [31] and light-maps [15] solved some of the issues  plaguing static light maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' in particular, these approaches support semi-dynamic geometry and self-  occlusion while adhering to a strict compute budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The advent of real-time ray-tracing hardware  set the stage for modern fully dynamic GI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Dynamic real-time GI methods build upon the decades  of research in sampling, and amortization of shading and visibility across space (pixel/world), angle,  and time to improve convergence [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Adaptations of several offline techniques such as photon  mapping [17], many-light rendering [20, 62], and radiosity maps [54] have also been explored  in the context of modern [26, 27] ray-tracing capable hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' However, presence of noise in  sampled algorithms require the use of strong denoisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Machine learning denoisers [6, 66] have  demonstrable advantages in terms of quality compared to more traditional frequency [32] or  variance [46] based denoisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' However, the prospect of training a neural network, the added  complexity of integrating machine learning inference with traditional graphics pipeline, and the  proprietary nature of machine learning frameworks have stalled the industry-wide adoption of  these techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The recent probe-based algorithm, Dynamic Diffuse Global Illumination (DDGI)  [28], extending the classic irradiance probes, still remains an excellent choice due to its relative  simplicity, quality, and cloud streaming capabilities [14, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' However, scaling of DDGI in its original  formulation is limited, and approaches such as multi-grid hierarchy and probe rolling [29] are  necessary to scale it across large environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our adaptive approach focuses on dynamic contents  in environments containing millions of probes in a single hierarchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  We propose Adaptive Dynamic GI (ADGI) algorithm where we trace a few pilot rays per frame to  scan the environment and build a coarse representative model of the dynamic events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Using Markov-  Chain sampling, we dynamically allocate resources to the critical areas, improving convergence in  those regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' While DDGI allocates a fixed number of samples per probe and uniformly distributes  samples across directions, ADGI non-uniformly samples the joint spatio-angular domain of the  discretized 5D light-field represented by the probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our approach essentially decouples resource  allocation from the number of probes resulting in a user-controlled performance target (FPS)  and improved scaling even with millions of probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Additionally, our approach results in faster  convergence in static and dynamic environments given equal render time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our approach is drop-in  compatible with the original implementation and its several other extensions such as probe rolling  and probe volume hierarchies [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  We achieve these objectives by formulating a guided function approximation technique, which is  purposefully accurate in specific regions highlighted by our guiding function and thus eliminates  the need for uniform resource allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Furthermore, we develop a sampling methodology based  on temporal Markov-chain, which adapts naturally to a dynamic environment while also enabling  scaling across large number of probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Finally, we discuss memory and bandwidth preserving color  compression schemes tailored specifically for our purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  2  RELATED WORK  Probe-base approaches: Modern games rely extensively on light probes for static and dynamic  global illumination due to their ease of integration into the game engine pipeline at low run-time  cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Some advocate a uniform grid probe placement due to their simplicity while others have  proposed non-unform probes due to their efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Probe based techniques are usually prone to  light leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' As such, uniform grid approaches [28, 31] use additional information, stored in the  probes to determine whether a probe is visible from a shade point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Non-uniform approaches may  2  Adaptive Dynamic Global Illumination  High Performance Graphics, Poster, July 11–14, 2022,  Guiding function  ℎ(𝑥) = 𝑝 (𝑥) × 𝑙 (𝑥)  𝑝 (𝑥)  𝑙 (𝑥)  𝑥  𝑥  State of  Environment  Sample  Feedback  (a) Construct the guide ℎ(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  𝑥  Metropolis sampling  𝑥𝑖  ℎ(𝑥)  (b) Sample 𝑥𝑖 ∼ ℎ(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  𝑥  𝑥𝑖  𝑔(𝑥)  (c) Evaluate objective 𝑔(𝑥𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  𝑥  𝑥𝑖  𝐸𝑟  𝑔  ^𝑔  ^𝑔(𝑥)  (d) ^𝑔(𝑥) - Reconstruct 𝑔(𝑥)  from 𝑔(𝑥𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 𝐸𝑟 indicates the  reconstruction error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Figure showing the steps in our adaptive-sampling strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We define a guiding function ℎ(𝑥) that  highlights (in yellow) the interesting regions of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The samples 𝑥𝑖 obtained from ℎ(𝑥) are used to  evaluate the objective 𝑔(𝑥𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our goal is to obtain an approximate representation of 𝑔(𝑥), denoted as ^𝑔(𝑥),  from the (𝑥𝑖,𝑔(𝑥𝑖)) pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' As more samples are obtained from the highlighted region, the reconstruction error  is lower in the yellow area, as shown in sub-figure (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  use carefully curated probe placement [63] combined with spatial data-structures like octrees to  determine the visibility of a probe from a surfel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' McGuire et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' [28, 31] stores the depth values  of the surrounding geometry from a probe and use a similar idea as Variance-Shadow-Mapping  [9] to approximate visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' However, non-uniform approaches has been mostly limited to static  geometry due to their high initial construction cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Some approaches use rasterization [31, 63]  while other may use ray-tracing [28] to compute the probe content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Probe based techniques also  differ on how they store the information in the probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Some use discrete textures [28, 31] while  other may use a compressed basis representation such as Spherical Harmonics [14, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Spherical  harmonics implicitly pre-filters the content before storage but may cause light and dark ringing  issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Memory bandwidth required for reading and writing from the probes is also a major concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Texture compression [31, 53] is usually the preferred choice to minimize memory bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Bandwidth is also crucial for cloud streaming of probe data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' In such scenarios, Spherical Harmonics  [14] representation may be preferable as they provide excellent compression for low frequency  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' At run-time, dynamic probe based [28] GI solutions uniformly distributes rays across probes  to update their content;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' this quickly becomes a bottleneck as the number of probes increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our  approach on the other hand, focuses on the optimal distribution of resources to maximize visual  fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Various extensions have also been proposed to increase scalablity [29] of uniform grid  approaches such as multiple-volume hierarchies and probe rolling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our approach remains largely  orthogonal and fully compatible with these extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Adaptive sampling: Adaptive sampling has been used in the context of screen-space ray-traced  global illumination where more samples are accumulated in regions with high noise and high  frequency [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Adaptive sampling is also useful for filtering soft shadows [32], where pilot-rays  model the spatial frequency of shadow-penumbra and provide the number of additional samples  required at each pixel to improved convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Neural versions [13] of adaptive sampling has  also been proposed where a neural network generates a sampling-map that is tightly coupled to a  post-process neural-denoiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Conceptually our approach is similar, but our execution is tailored  for the problem of temporally coherent sampling of probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We refer readers to section 8 for an  extend related work in irradiance-caching, screen-space GI and MCMC techniques  3  High Performance Graphics, Poster, July 11–14, 2022,  Datta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  3  OVERVIEW  We focus on two primary issues with DDGI in its original formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' First, the technique does  not allow for the non-uniform allocation of resources, resulting in unnecessary probe updates in  regions that are not crucial for visual fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Seconds, it does not update the probes quick enough  to reflect transient changes in the scene environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our adaptive strategy involves detecting  the changes in the environment and allocating resources driven by the detected changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' While  the detection phase requires allocating additional resources, our empirical evaluations suggest our  non-uniform adaptive sampling compensates for the lost efficiency in the detection phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our  detection phase also enables fast probe updates for capturing transient changes in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We  model our technique as guided function approximation where we approximate a continuous function  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 5D light-field) using a discrete (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' probes) representation driven by a guiding function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  A naive approach to approximate a continuous function is to discretize the domain and reserve a  representative sample for each discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The strategy is useful when the domain is relatively  small;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' however, as the domain gets larger or the number of discretizations increases, it is prohibi-  tively expensive to update all discretizations in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' This is one of the issues plaguing the  original DDGI technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' In many applications, it is not necessary to update the entire domain  uniformly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' instead, we can tolerate more approximation errors in some regions than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' A  simple example is foveated rendering, where errors in the periphery are less intrusive than those  near the gaze center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' In our case, we need the most accuracy in probes contributing to final shading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  We introduce the notion of guiding function, which highlights the regions where a higher  reconstruction accuracy is desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We define the guide using a product of terms - the first term  represents the current state of the environment while the second term is a feedback from the  sampled cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We sample the guide using a temporally coherent Markov-chain and use the  samples to update our approximate representation using a parallel thread-safe approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Thus our  approach is summarized in three steps - defining a guiding function, sampling the guide, and using  the samples to update the approximate representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We describe these steps in sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='3  and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='4 while we discuss various implementation specific details in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' See figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Our approach provides two distinct advantages compared to the original DDGI - approximation  quality and scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' At any time, we concentrate our resources on a potentially challenging  area as opposed to the entire domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Provided our guide correctly identifies the challenging  regions, the quality is improved due to a higher concentration of resources in the appropriate  region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Since we sample the guide independent of the number of discretizations, the decoupling  allows for a high number of statically allocated probes without affecting run-time performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Increased discretizations improve approximation quality while the independence of sampling from  the number of discretizations improves scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' More specifically, we transparently increase  the number of discrete probes without affecting performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The run-time performance depends  on the number of samples we generate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' the samples are channeled to the appropriate areas by the  guiding function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our Markov-chain sampling is highly parallel, temporally coherent, and scalable,  making it suitable for real-time temporally distributed reconstruction of large probe grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='1  Background  Here we briefly describe the original DDGI algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' DDGI consists of a 3D grid of directionally  resolved irradiance probes that are updated in real-time through hardware ray-tracing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The probes  also contains visibility information to prevent light leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The probe representation has many  benefits, it performs optimally for diffuse indirect transport and is relatively inexpensive to encode  and decode information to and from the probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The algorithm evenly distributes ray-samples  outwards from the probe center at each active probe in a stochastic rotated spiral pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' DDGI is  4  Adaptive Dynamic Global Illumination  High Performance Graphics, Poster, July 11–14, 2022,  Uniform  probe grid  placement  Generating &  tracing rays  Evenly distribute  Probe state update  Update irr &  vis 2𝐷 atlas  Shade each  point  8 cage probe  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The figure illustrates the main steps of DDGI algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Algorithm defines a uniform grid of probes  and trace uniform-random rays in all direction from each probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Based on the hit information, we compute  the visibility (vis) and irradiance (irr) and update the 2𝐷 atlas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We also update the probe states based on  visibility information (back-face hit ratio).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Finally, for each shade-point, we query the eight bounding probes  surrounding it and interpolate them to compute incoming indirect illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  a two step algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' First, it updates the shading on the probe texels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Next a screen-space pass  where the up-to-date probe content is used for shading the camera-pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The probe texel values  are encoded into a spherical-mapped diffuse irradiance-texture with 8 × 8 resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Probes also  captures the average ray-hit distance, and squared distances to the nearest geometry at 16 × 16  resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' DDGI temporally filters the probe texels by blending in the new values using a fixed  hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The visibility data is used to decide whether a probe is visible at a shade-point and also  used to infer whether a probe is inside a geometry and deactivated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The probe’s state is not limited  to on or off and can vary with scenarios [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The world-position of the screen-space pixel is used  as a key to the probe-texture lookup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The lookup interpolates the corresponding eight probes of  the grid voxel containing the shade-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The algorithm is illustrated in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' DDGI algorithm  is suitable for diffuse and slow changing phenomena in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Therefore DDGI, combined with our  adaptive-sampling strategy is a reasonable real-time GI approximation for dynamic scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='2  Guiding function  As summarised in section 3 and figure 2, a guiding function highlights the important areas in  the domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=', challenging regions where more resources are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' These highlighted areas  receive more adaptive samples, reducing the approximation error in those regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Mathematically,  the domain of the guiding function ℎ : 𝑅𝑑 → 𝑅 is the continuous 5D light field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Upon query, the  guide function returns a scalar value indicating the importance of a sampled point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' In our case,  𝑑 = 5 as the domain is a 5-dimensional space of world-space positions and directions, and the guide  encodes the importance of sampling a direction on a probe (texel’s importance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  We model the guiding function (ℎ) as a product of two terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The first term, we call 𝑓 : 𝑅𝑑 → 𝑅,  represents the value in sampling a texel based on our understanding (limited) of whether such a  texel would contribute towards the final screen-space shading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The second term is the observed  sampled evidence (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='a irradiance cache) as they become available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Initially, the irradiance cache is  empty but filled progressively through sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We define the first term based on some heuristics  that describes our understanding of the probe-environment:  Probes closer to the camera,  Probes closer to geometric surfaces,  Directions on the probes facing away from geometric surfaces,  Directions on the probe with higher incoming irradiance,  Directions with temporal change in irradiance and visibility  We trace pilot rays from the probes to generate the information necessary to quantify the above  heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We also call it the detection phase where we pre-scan the scene environment for changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  We denote the individual heuristics as 𝑓𝑖 : 𝑅𝑑 → 𝑅, and compose them into its final form 𝑓 as  shown in equation 1, where 𝜙 represents a composition function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The composition function is  5  High Performance Graphics, Poster, July 11–14, 2022,  Datta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Algorithm 1: Metropolis algorithm  Input: ℎ: Guide distribution, 𝑀 : No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' of iterations  Input: 𝐾: No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' of initial samples to reject  Output: 𝑥 : Sample  Ensure: 𝑀 ≥ 2, and 𝐾 < 𝑀  1 𝑗 ← ShaderInvocationIndex()  2 𝑥0 ← 𝑆[𝑗]  // Initialize Markov-chain from memory  3 while 𝑖 ← 0 to 𝑀 − 1 do  4  𝑥𝑖+1 ← RandomWalk (𝑥𝑖,ℎ(𝑥𝑖))  // Random walk step, algorithm 5  5  if 𝑖 > 𝐾 then  /* Use sample 𝑥𝑖+1 for probe updates, see algorithm 2  /  6 𝑆[𝑗] ← 𝑥𝑖 + 1  // Save Markov-chain state  simply a recipe to appropriately combine the individual heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We quantify the individual  heuristics (𝑓𝑖) in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='1 and the composition (𝜙) in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  𝑓 = 𝜙(𝑓0, 𝑓1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=', 𝑓𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  (1)  The second term uses the stored irradiance in the probes, denoted by ^𝑔, to modulate the first  term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We model the second term as - 𝑒𝑥𝑝 (𝛼 · ^𝑔(𝑥)/𝑓 (𝑥)), where the scalar 𝛼 ∈ [0, ∞) indicates our  confidence in the irradiance probe content;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' a higher value indicating greater confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Note that a  stored texel with high irradiance value may or may not have a high contribution to the final shading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Example - in a dynamic environment the probe content from the last frame is quickly outdated  and thus less useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The parameter 𝛼 models this uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The term 𝑓 (𝑥) in the denominator  ensures that we only trust ^𝑔(𝑥) when 𝑓 (𝑥) is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Finally, we define the guiding function as:  ℎ(𝑥) = 𝑒𝑥𝑝  �  𝛼 · ^𝑔(𝑥)  𝑓 (𝑥)  �  𝑓 (𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  (2)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='3  Sampling the guide  Next we sample the guiding function (equation 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Mathematically, given an unnormalized distribu-  tion ℎ : 𝑅𝑑 → 𝑅, our goal is to obtain samples 𝑥𝑖 from ℎ(𝑥), where 𝑥𝑖 ∈ 𝑅𝑑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Our sampling algorithm is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We use the Metropolis sampling, as shown in al-  gorithm 1 to sample ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The algorithm randomly initializes a state (𝑥0 ∈ 𝑅𝑑) and moves the state  forward based on the acceptance of a newly proposed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We generate the proposed states by  perturbing the current state with a zero-mean Gaussian noise, also known as Random-walk [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Parallelism: Note that algorithm 1 runs as a shader invocation, meaning several instances of  the chain run in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Each instance is independent with its own memory to load and store the  chain state (denoted by S[] in algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The instances generate thousands of samples per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  As an input to our algorithm, we explicitly specify the number of chains that run in parallel, thus  controlling the number of adaptive samples and performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Contrasting with the original DDGI,  the number of samples in the original implementation is proportional to the number of probes  which increases cubically with scene dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' As such, it is difficult to scale up when the scene  gets larger or when using a denser probe grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our approach is independent of the discretization  resolution and scales better to higher probe counts without compromising approximation quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Mixing-time: Initially, a Markov chain requires many iterations for the chain to generate  samples from the target distribution (here ℎ(𝑥)), a phenomenon known as mixing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We avoid  6  Adaptive Dynamic Global Illumination  High Performance Graphics, Poster, July 11–14, 2022,  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' List of symbols  Symbol  Description  Remarks  𝑓  Heuristics model  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='2  ℎ  Guiding function/Target distribution  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='3  𝑔  Objective function  Symbolic proxy for 𝑔𝑟, 𝑔𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='4  ^𝑔  Approximation of objective function  Symbolic proxy for ^𝑔𝑟, ^𝑔𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='4  𝑔𝑟  5D Light field  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='4  𝑔𝑐  Chebychev visibility  Section3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='4  ^𝑔𝑟  Approximation of 5D light field  (Irradiance cache)  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='5  ^𝑔𝑐  Approximation of Chebychev visibility  (Visibility cache)  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='4, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='6  𝑥 or 𝑥𝑖  Markov-chain samples  Symbolic proxy for 𝑝𝑖, 𝜔𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='3  𝑝𝑖  Positional (∈ 𝑅3) component of 𝑥𝑖  –  𝜔𝑖  Directional (∈ 𝑅2) component of 𝑥𝑖  –  this problem by bootstrapping the initial chain state from the last frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' As such, we keep the  number of iterations per frame small, but over frames, the chain effectively accrues many iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Distribution stationarity: Markov chain sampling requires the target distribution ℎ(𝑥) remain  stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Due to a dynamic scene environment, the stationarity condition is seemingly violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  This may affect the approximation quality of our technique if the distribution changes rapidly  between frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' However, we have several contingencies to deal with the issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' First, we target high  frame-rates, which minimizes the change in the target distribution between consecutive frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  As an additional margin of safety, we reject initial 𝐾 samples per frame as shown in algorithm 1,  line 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' This ensures our usable samples are obtained closer to the target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Note that  the evaluation time for ℎ(𝑥) negligible and thus rejecting few initial samples per frame does not  significantly impact performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We also smooth out the target distribution (see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='4)  using spatio-temporal convolution to minimize abrupt changes in the target across frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Temporal tracking: Since our target distribution may vary with time, we require the samples  generated from the Markov-chain to closely follow the distribution to capture the transient changes  in the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We make some crucial modifications to our sampling algorithm to allow for  fast tracking of the target distribution, which we discuss in detail in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='4  Approximation  With samples obtained from the highlighted (figure 2(b)) parts of the domain, we focus on using  the samples to evaluate (figure 2(c)) and reconstruct (figure 2(d)) our objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The term  objective function refers to the quantity we aim to approximate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Mathematically, we denote our  objective function as 𝑔 : 𝑅𝑑 → 𝑅𝑐, and its approximate reconstruction as ^𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' For ADGI, we have  two objective functions - the light field 𝑔𝑟 : 𝑅5 → 𝑅3, and Chebychev-visibility 𝑔𝑐 : 𝑅5 → 𝑅2  surrounding the probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We denote their approximate reconstructions as the irradiance cache ^𝑔𝑟,  and the visibility cache - ^𝑔𝑐 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' See section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='6 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Updating ^g: We evaluate the continuous objective function 𝑔 at collected sample points 𝑥𝑖 and  store the evaluations - 𝑔(𝑥𝑖) into ^𝑔, as shown in algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' For ADGI, the evaluation step involves  7  High Performance Graphics, Poster, July 11–14, 2022,  Datta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Algorithm 2: Approximation algorithm  Input: 𝑥: Markov-chain samples  1 function UpdateRepresentation(𝑥):  2  𝑣 ← 𝑔(𝑥)  // Evaluate sample, ray-trace  3  AtomicMovingAvg(𝑥, 𝑣)  // Populate ^𝑔, see algorithm 4  tracing a ray to query the local light field and visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' At each Metropolis iteration, the evaluated  samples update the closest entry in the probes (^𝑔) within a critical section construct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Representing ^g: Prior work represent ^𝑔 as either as discrete LUTs [28], continuous Spherical  Harmonics [14], Neural Networks [36], or any combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' In our case, the choice to use a discrete  representation is based on several factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' First, multiple parallel streams of Markov-chain samples  may update the same memory location in ^𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' As such, provisions are necessary to prevent race  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We also need a representation that handles temporal accumulation and quickly update  itself to reflect any transient changes in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Finally, the representation must be bandwidth  efficient to improve the read and write performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We refer to section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='5 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='9 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='5  MCMC analysis  In this section, we analyze our adaptive sampling algorithm in the context of MCMC (Markov  Chain Monte Carlo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Note that our goal is not variance reduction through importance sampling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  rather the focus is guided approximation of the objective function via sampling the target function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  As such, unlike importance sampling, the sampling function is not necessarily correlated to the  integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' With this distinction in mind, we first look at the equation driving importance sampling  using MCMC and then repurpose it for guided function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  The following equation shows a typical case of importance sampling where the objective is to  compute the integral  ∫  ℎ(𝑥)𝑔(𝑥)𝑑𝑥 and there exists a strategy to sample from h(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' In many typical  scenarios (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' full Bayesian inference), the distribution ℎ(𝑥) is a proper distribution (  ∫  ℎ(𝑥)𝑑𝑥 = 1)  but does not have an efficient sampling mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' This where Markov Chain MC is useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  ∫  ℎ(𝑥)𝑔(𝑥)𝑑𝑥 ≈  �  1  𝑀  𝑀−1  ∑︁  𝑖=0  𝑔(𝑥𝑖)  � ∫  ℎ(𝑥)𝑑𝑥, 𝑥𝑖 ∼ ℎ(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  (3)  In contrast, our choice of Markov Chain (Metropolis) is primarily technical - simplicity, GPU  parallelism and temporal sample tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Nevertheless, the same equations provide meaningful  insight - albeit in a different context of adaptive sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' In our algorithm, we simply sum the  samples obtained from the target distribution without taking into account the sample density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' This  is equivalent to computing the following:  𝐼 = 1  𝑀  𝑀−1  ∑︁  𝑖=0  𝑔(𝑥𝑖), 𝑥𝑖 ∼ ℎ(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  (4)  While our goal is to estimate  ∫  Ω 𝑔(𝑥)𝑑𝑥, the expectation of 𝐼 (rearranging equation 3) is:  E [𝐼] =  ∫  Ω ℎ(𝑥)𝑔(𝑥)𝑑𝑥  ∫  Ω ℎ(𝑥)𝑑𝑥  ,  (5)  where Ω is the domain of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Clearly, the expected value of 𝐼 does not converge to the  correct estimate -  ∫  Ω 𝑔(𝑥)𝑑𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' However, there are two factors to consider - size of the domain Ω and  8  Adaptive Dynamic Global Illumination  High Performance Graphics, Poster, July 11–14, 2022,  shape of ℎ(𝑥) in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' First consider the limit case where Ω → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' In this case, the integrals  collapses to a point evaluation and indeed the expected value of 𝐼 equals the unbiased estimate as  shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' = lim  Ω→0  ∫  Ω ℎ(𝑥)𝑔(𝑥)𝑑𝑥  ∫  Ω ℎ(𝑥)𝑑𝑥  =  ∫  Ω ℎ(𝑥)𝑔(𝑥)𝛿(𝑥 − 𝑥0)𝑑𝑥  ∫  Ω ℎ(𝑥)𝛿(𝑥 − 𝑥0)𝑑𝑥  = 𝑔(𝑥0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  (6)  𝑅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='𝐻.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='𝑆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' = lim  Ω→0  ∫  Ω  𝑔(𝑥)𝑑𝑥 =  ∫  Ω  𝑔(𝑥)𝛿(𝑥 − 𝑥0)𝑑𝑥 = 𝑔(𝑥0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  (7)  In the above equation, 𝛿 is the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The result is important as it shows with increasing  probe resolution, bias is reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' However, reducing texel size is not always practical as more rays  and memory are required to populate and store a high resolution probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Notice how the term ℎ(𝑥)  is cancelled in equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' When the domain of integration is sufficiently small, ℎ(𝑥) is practically  constant and the term cancels out in the denominator and numerator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  We now consider the shape of ℎ(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' While the target h(x) varies globally, it is piece-wise constant  at a local scale due to its tabular nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' More crucially, the target ℎ(𝑥) is stored at a much lower  resolution compared to the irradiance probe ^𝑔(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' This implies ℎ(𝑥) is practically constant across a  texel of the irradiance probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The expected value of 𝐼 for the 𝑘𝑡ℎ texel is thus given by:  E [𝐼𝑘] =  ∫  𝑇𝑘 ℎ(𝑥)𝑔(𝑥)𝑑𝑥  ∫  𝑇𝑘 ℎ(𝑥)𝑑𝑥  =  ∫  𝑇𝑘 𝑐𝑘𝑔(𝑥)𝑑𝑥  ∫  𝑇𝑘 𝑐𝑘𝑑𝑥  =  ∫  𝑇𝑘 𝑔(𝑥)𝑑𝑥  ∫  𝑇𝑘 𝑑𝑥  ,  (8)  where 𝑇𝑘 represents the domain of 𝑘𝑡ℎ texel and 𝑐𝑘 represents the piece-wise constant value of  ℎ(𝑥) when 𝑥 ∈ 𝑇𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The area estimate  ∫  𝑇𝑘 𝑑𝑥 is fixed for all texels and equivalent to 4𝜋/#𝑟𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Thus, due to the tabular nature of our target function, the estimates of irradiance texels remain  un-biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' While performing texture filtering over irradiance texels, it is possible to compute an  unbiased estimate by weighing the texel values with 𝑐𝑘 as follows:  𝐼 𝑓 𝑖𝑙𝑡𝑒𝑟  𝑘  =  ∑︁  𝑗 ∈N𝑘  𝑤𝑘−𝑗𝐼𝑘−𝑗, 𝑤𝑖 = 𝑐𝑖/  ∑︁  𝑗 ∈N𝑘  𝑐 𝑗,  (9)  where N𝑘 represents the texels in the neighbourhood of texel 𝑘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The values 𝑐𝑖 are obtained by  querying the probes storing ℎ(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Note that bias is unavoidable as we blend samples temporally  in a dynamic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' In a dynamic environment, the objective is evolving and the bias  manifests itself as temporal lag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Practically however, within a small time window, both ℎ(𝑡) and  𝑔(𝑡) are assumed constant and the samples can be blended using a windowed moving average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Note that windowed moving average requires storing historical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' A cheaper but biased  approximation to windowed moving average is exponential moving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  4  IMPLEMENTATION DETAILS  This section provides the several implementation details with a brief summary in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='1  Heuristics construction  The section describes the construction of 𝑓 using the heuristics discussed in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our goal is  to measure and quantify the heuristics that highlight the probes which actively contribute to the  final shading and require additional resources for faster convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We represent the heuristics  either parametrically (equation 10) or using an explicit LUT representation as shown in figure 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  The LUT is constructed such that each probe has eight texels corresponding to an octant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We trace  a ray for each octant;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' the rays return the hit distance and incoming irradiance at the hit-point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  From this information, we compute several quantities (equation 11 - 18) and store them in the  9  High Performance Graphics, Poster, July 11–14, 2022,  Datta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  World space  Trace 8  pilot rays  Octahedral  Quantify heuristics  𝜙 (𝑓0, 𝑓1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=', 𝑓𝑖)  Model feedback  𝑒𝑥𝑝 (𝛼 · ^𝑔/𝑓 )  Guide function  𝑝 (𝑥) × 𝑙 (𝑥)  Metropolis sampling  ℎ(𝑝,𝜔)  (𝑝𝑖,𝜔𝑖)  Irradiance  Visibility  𝜔𝑖  −𝜔𝑖  𝑝𝑖  𝑝𝑖  Trace ray  Update irradiance, visibility  cache  Irradiance ( ^𝑔𝑟 )  8 × 8  Visibility ( ^𝑔𝑐)  16 × 16  Deferred shading  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' This figure illustrates our overall algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We trace 8 pilot rays, one from each octant on the  probe and approximate the heuristic model 𝑓 (𝑝,𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Using the heuristic and feedback, we define the guide  ℎ(𝑝,𝜔) and sample it using Metropolis sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The sampled (𝑝𝑖,𝜔𝑖) are used to trace more adaptive ray  samples, gathering hit-distance and irradiance at the sample points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We update the probe-cache (^𝑔) with  adaptive-samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The cache is used in the next shader and also looped back as feedback to model the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  LUT/texture mapped to the probe octants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We define and evaluate the following heuristics for a  probe at position 𝑝 and a direction 𝜔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='1  Distance from camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' A probe far away from the camera is less likely to contribute to the  final shading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We represent this parametrically as described in equation 10, where 𝑝 represents  probe position, 𝑐 camera position and 𝑘 is a threshold set by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  𝑓𝑐 (𝑝,𝜔) =  �  1  if ||𝑝 − 𝑐|| < 𝑘 ,  𝑒−( ||𝑝−𝑐 ||−𝑘)  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  (10)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='2  Probe visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Only the probes encompassing a geometry participates in the deferred  shading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Thus, probes closer to a geometric surface are more important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Similarly, texels facing  away from the surface are queried more often for shading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We express both quantities together in  equation 11, where 𝑝 represents probe location and 𝑡 = 𝑡𝑟𝑎𝑐𝑒(𝑝, −𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The function trace returns  the distance of the nearest surface hit, and the scalar 𝑠 is the diagonal distance of a grid voxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  𝑓𝑣(𝑝,𝜔) = 𝑒−2𝑡/𝑠  (11)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='3  Incoming radiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We consider directions with high incoming radiance as more impor-  tant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' To identify those directions, we query the radiance along each probe octant and use it as a  representative for incoming radiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  𝑓𝑟 (𝑝,𝜔) = 𝑚𝑖𝑛(𝑟, 𝛽)  𝛽  ,  (12)  where 𝑟 = 𝑙𝑢𝑚(𝑝,𝜔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The function lum returns the incoming luminance using direct illumination  at the surface hit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The parameter 𝛽 controls the dynamic range and we set 𝛽 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='4  Probe visibility change .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Detection of dynamic geometry is crucial for increased resource  allocation in regions affected by these changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We detect dynamic geometry by computing a  temporal gradient of probe visibility followed by a spatio-temporal smoothing operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  𝑓0(𝑝,𝜔) = 𝑓 𝑡  𝑣 (𝑝,𝜔) − 𝑓 𝑡−1  𝑣  (𝑝,𝜔),  (13)  where 𝑓 𝑡  𝑣 , 𝑓 𝑡−1  𝑣  represent visibility in the current and last time step respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Equation 13  implicitly states we keep the position and the direction fixed when measuring the time difference  across frames to avoid noisy gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The gradient is passed through a temporal trigger (𝑇𝑟) as:  10  Adaptive Dynamic Global Illumination  High Performance Graphics, Poster, July 11–14, 2022,  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Probes storing prior-information (𝑓 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  octant-mapped  pilot-rays  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Irradiance cache ( ^𝑔𝑟 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Visibility cache ( ^𝑔𝑐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Figure showing various probe-mapped textures and LUT in our technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  𝑓1(𝑝,𝜔) = 𝑇𝑟 (𝑓0(𝑝,𝜔),𝜃) ,  (14)  where 𝑇𝑟 converts a pulse in time to a decaying signal controlled by the parameter 𝜃 as shown  in figure 6(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' For simplicity, we drop the time axis from the function 𝑇𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The function minimizes  temporal discontinuities, thus helping the Markov-chain to closely follow the target distribution  (ℎ) across frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Finally, we perform a spatial convolution as follows:  𝑓Δ𝑣(𝑝,𝜔) =  ∑︁  𝑖,𝑗  𝑓1(𝑝 − 𝑝𝑖,𝜔 − 𝜔𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  (15)  The convolution step smooths out uncertainties in a single texel and also serves as a weak  predictor of possible locations of the dynamic geometry in the next frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We use a 5 × 5 × 5 and  3 × 3 convolution in space and direction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='5  Probe radiance change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Similar to the previous section, we detect a change in radiosity  using a temporal gradient of the probe radiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We apply the same temporal trigger and spatial  convolution operator as in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The corresponding equations are as follows:  𝑓2(𝑝,𝜔) = 𝑓 𝑡  𝑟 (𝑝,𝜔) − 𝑓 𝑡−1  𝑟  (𝑝,𝜔),  (16)  𝑓3(𝑝,𝜔) = 𝑇𝑟 (𝑓2(𝑝,𝜔),𝜃) ,  (17)  𝑓Δ𝑟 (𝑝,𝜔) =  ∑︁  𝑖,𝑗  𝑓3(𝑝 − 𝑝𝑖,𝜔 − 𝜔𝑗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  (18)  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='2  Heuristics composition  Now that the individual heuristics are defined, as described in equation 1, we compose them for  the static and dynamic cases as follows:  𝑓𝑠 (𝑝,𝜔) =  𝑠𝑡𝑎𝑡𝑖𝑐  ����  𝑓𝑐 𝑓𝑣 ,  (19)  𝑓𝑑 (𝑝,𝜔) =  𝑑𝑦𝑛𝑎𝑚𝑖𝑐  ��������������������������������  𝑓𝑐 𝑓𝑣(𝑓Δ𝑣 + 𝜇𝑓Δ𝑟) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  (20)  When the environment is static, we sample according to the camera and probe-to-surface distance  heuristics denoted by 𝑓𝑐 and 𝑓𝑣 in equation 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' In the dynamic case represented by equation 20, we  modulate the changes in the environment by the static term 𝑓𝑐 𝑓𝑣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The modulation indicates we are  more interested in changes close to the camera and geometric surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The factor 𝜇 weighs the  strength of change in geometry versus change in lighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We use 𝜇 = 2 in all our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  11  High Performance Graphics, Poster, July 11–14, 2022,  Datta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  𝑇𝑟 (𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 𝑣,𝜃)  𝑣  Temporal-pulse  𝜃  Linear decay  start  𝑡  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Transform temporal-pulse to a decaying signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  𝑓𝑐 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='75  𝑓𝑐 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='5  |𝑐𝑙𝑖𝑝𝑥𝑦 | ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='2  |𝑐𝑙𝑖𝑝𝑥𝑦 | ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='4  b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Defining clip volumes for probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Figure (a) shows the construction of temporal-trigger𝑇𝑟 (𝑣,𝜃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' In figure (b), we call the volume bounded  by the blue frustum and black boundary as inner volume 𝑉𝑖𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Similarly, outer volume 𝑉𝑜𝑢𝑡 is the volume  bounded by green frustum and outer grey boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' All probes in 𝑉𝑜𝑢𝑡 participate in the heuristic modelling,  as described in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Probes inside the blue frustum participate in adaptive sampling as described in  section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='8, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We set the probe state 𝑁 = 16 for all probes outside 𝑉𝑖𝑛 but inside 𝑉𝑜𝑢𝑡, refer section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='3  Heuristics storage  We store the quantities 𝑓𝑣, 𝑓𝑠, 𝑓𝑑 as a 6-10-10 bit encoded 32 bit integer at each octant of the probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  The remaining 6 bits are used for other flags.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' When querying the LUT/texture, we use a mapping  function that maps the continuous position 𝑝 and direction 𝜔 to the corresponding texel in the  LUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We note that 𝑓𝑐 is implicitly defined, hence do not require additional storage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='4  Improving construction efficiency  The heuristics construction step is a potential bottleneck if we trace 8 rays per probe for all probes  in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' As such, we restrict the pilot-rays to the probes that are contained within an extended  camera frustum as shown in figure 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' To maximize the efficiency of our algorithm, we further  reuse the samples collected from the 8 pilot-rays to populate the irradiance ( ^𝑔𝑟) and visibility ( ^𝑔𝑐)  caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We change the ray-directions at alternate frames in an AABBCCDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' pattern, improving  the detection of temporally varying light-field surrounding the probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We measure the time-delta  (equation 13 and 16) between two frames with identical set of ray-queries, avoiding noisy gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  However, this effectively halves the detection frequency (frame-rate / 2) but improves the spatial  awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We use a stratified-random ray-direction such that there is always one ray per octant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  We update the irradiance and visibility cache at each alternate frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='5  Probe irradiance cache  As shown in figure 5(b), the irradiance cache ( ^𝑔𝑟) is represented as a uniform probe grid in space  where each probe stores the surrounding diffuse irradiance at a 8 × 8 texel resolution using a  spherical mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' At each texel, we store the irradiance in a custom RGB encoding with 9-9-8 bits  for the three channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The remaining 6 bits (out of 32bit) store the sample accumulation count  (N), used for computing the moving average (see algorithm 3) of a sample stream in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We  take several considerations into account for the choice of our encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our encoding should be  bandwidth efficient and must support atomic updates on a commodity GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We found both DX12  and GLSL supports atomic operations on 32 bit integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Finally, our encoding must faithfully  12  Adaptive Dynamic Global Illumination  High Performance Graphics, Poster, July 11–14, 2022,  Algorithm 3: Moving Average algorithm  Input: 𝑥: Update location, 𝑣 : New sample, 𝑁𝑚𝑎𝑥 : Max sample count  Output: 𝑉 : Updated value, 𝑁: Sample count  1 function MovingAvgUpdate(𝑥, 𝑣, 𝑁𝑚𝑎𝑥):  2  𝑛 ← ^𝑔[𝑥].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='𝑁  // Cumulative sample count  3  𝑜 ← ^𝑔[𝑥].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='𝑉  // Cumulative value  4  𝑉 ←  𝑣  𝑛+1 + 𝑛·𝑜  𝑛+1  // Update cumulative value  5  𝑁 ← 𝑚𝑖𝑛(𝑛 + 1, 𝑁𝑚𝑎𝑥)  // Increment sample count  6  return 𝑉, 𝑁  encode intensities beyond the standard definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We apply a non-linear color compression across  the three color channels, 𝑖 ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='.2] as shown in the equation below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  𝑢𝑖 = 𝑚𝑖𝑛 (𝑙𝑛(𝛾 · 𝑣𝑖 + 1), 𝛽)  𝛽  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  (21)  We apply an inverse transform (𝑒𝑥𝑝(𝛽 · 𝑢𝑖) − 1) /𝛾 while decoding where 𝛽 = 5 and 𝛾 = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  More details regarding our choice of compression scheme is provided in appendix B and figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='6  Probe visibility cache  As shown in figure 5(c), texels in the visibility probes store the mean distances and mean squared  distances to the nearest geometry at 16x16 texel resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We call this ^𝑔𝑐 - our visibility cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Each texel stores the two channels with 13 bits of precision each while the rest 6 bits are used for  sample accumulation count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We normalize the distances with probe cage diagonal length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Similar to  irradiance cache, we apply a logarithmic encoding as per equation 21 for efficient use of available  precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We use (𝛽,𝛾) values of (5, 15) and (8, 20) for the linear and squared channels respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='7  Temporal sample accumulation mecahnism  We use a moving-average accumulation to store the samples in the irradiance and visibility caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' In  the algorithm 3, we have two parameters 𝑁 and 𝑁𝑚𝑎𝑥 to control the moving-average accumulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  As we start accumulating samples, 𝑁 is incremented and the algorithm performs like a true moving  average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' However, as 𝑁 approaches 𝑁𝑚𝑎𝑥 − 1, the algorithm switches to an exponential moving  average form with hysteresis (𝑁𝑚𝑎𝑥 − 1)/𝑁𝑚𝑎𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Also, note that when the value of 𝑁 is low, the  cache updates itself quickly, but the stored values may be noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' As 𝑁 increases, the new samples  are weighed less in their contribution to the cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We exploit these parameters to control the  learning rate and noise in the static and dynamic cases as discussed in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='8  Adaptive sampling - static  We split our adaptive sampling strategy into two stages - static and dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We have two separate  Markov-chain sets, each focusing on different aspects of capturing the surrounding light-field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  While the static chain focuses more on the accuracy, the dynamic chain is tuned for capturing the  transient responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We discuss the dynamic chain in detail in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  We set up equation 2 as - ℎ = 𝑒𝑥𝑝(𝑚𝑖𝑛(^𝑔𝑟/𝑓𝑠, 1)) · 𝑓𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The feedback from irradiance cache ^𝑔𝑟 is  obtained from the previous frame and from a higher mip-level (also used in deferred shader).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The  lowest mip-level ^𝑔𝑟 is continuously updated and thus avoided as feedback due to possible violation  of stationarity condition within a frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We use the Metropolis sampling, algorithm 1, to generate  the samples 𝑥𝑖 ≡ (𝑝𝑖,𝜔𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' As summarized in the algorithm, 2, we use the samples to evaluate the  13  High Performance Graphics, Poster, July 11–14, 2022,  Datta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Algorithm 4: Atomic moving average algorithm  Input: 𝑥: Update location, 𝑣 : New update value  Output: Update ^𝑔[𝑥]  1 function AtomicMovingAvg(𝑥, 𝑣):  2  current ← ^𝑔[𝑥]  /* Repeat until destination value stops changing  /  3  do  4  expected ← current  5  next ← MovingAvgUpdate(𝑥, 𝑣, 64)  6  InterlockedCompareExchange(^𝑔[𝑥], expected, next, current)  // Refer HLSL  7  while current ≠ expected  continuous light field 𝑔𝑟, which involves tracing a ray originating at 𝑝𝑖 along the direction 𝜔𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  We trace an additional shadow-ray per sample to compute the visibility in the opposite direction  (−𝜔𝑖) as the probe queries in the deferred shader for visibility is exactly 180◦ out of phase w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='t  irradiance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Next we store the irradiance and visibility values in the irradiance ( ^𝑔𝑟) and visibility ( ^𝑔𝑐)  caches using an atomic update rule as presented in the algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Atomic updates are required  as multiple invocations of the chain may update the same location in the irradiance and visibility  caches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Figure 4 summarizes the overall idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  We set the random walk step size, denoted by 𝜎 ∈ 𝑅5 in algorithm 5, proportional to the size of  discretization in the irradiance and visibility cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Thus positional step size is proportional to the  size of a voxel in the probe grid, while angular step size is roughly  √︁  𝜋/256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Due to the small step  size, texels in the cache may accumulate more than one sample per texel, thereby accumulating  a large sample count over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We also note that our cache behaves like a true moving average  between sample count 𝑁 = 0 to 64, which also contributes to better accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  The static adaptive samples are useful for improving convergence in a static scene and for slow  changes that are undetected during prior construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' For example, slow changes in lighting such  as day-night cycles in games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We lower the hysteresis by setting 𝑁 = 16 for all probes in the region  {𝑉𝑜𝑢𝑡} − {𝑉𝑖𝑛} in figure 6(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' This enables the probe to quickly catch-up to the most recent values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='9  Adaptive sampling - dynamic  We run a second set of Markov-chain when dynamic content is detected in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' When there  are dynamic elements, especially moving geometry, we run into two main issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The generated  samples are not well distributed in the region of interest i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' the areas where time varying changes  are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' When the step size is small, the chain cannot track the target distribution fast enough  to generate samples from the target, causing the samples to lag the moving target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  The second problem is noise due to multi-sampling of the irradiance texel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Potentially, this can be  solved by increasing the hysteresis to improve temporal sample reuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' However, the reduced noise  comes at the cost of introducing objectionable temporal blur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  We solve the first issue by increasing the chain step size and by coarsening the target function  (𝑓𝑑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Practically, this amounts to grouping the heuristics-probes into virtual proxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' In our case, a  virtual proxy represents a group the 3 × 3 × 3 probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' This virtual probe has 8 directions and each  direction represents an axis-aligned octant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The value of a texel of the virtual probe is the max of  all 27 probes it represents along the corresponding direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We also drop the sampled evidence by  setting 𝛼 = 0 in equation 2, as the stale irradiance cache ( ^𝑔𝑟) provide little useful information for  14  Adaptive Dynamic Global Illumination  High Performance Graphics, Poster, July 11–14, 2022,  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Table showing probe grid details for various scenes used in our technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Scene  Probe Grid  Probe spacing  (in meters)  Irradiance ( ^𝑔𝑟)  Cache Resolution  Visibility ( ^𝑔𝑐)  Cache Resolution  Bistro - Exterior  192 × 64 × 192  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='5 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='5 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='5  8 × 8  16 × 16  Sponza - Diffuse  192 × 64 × 192  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='5 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='5 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='5  8 × 8  16 × 16  Sponza - Glossy  192 × 64 × 192  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='1 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='1 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='1  16 × 16  16 × 16  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Table showing probe encoding details for the various techniques we use in our comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Technique  Irradiance  Cache Encoding  Visibility  Cache Encoding  Temporal  Hysteresis  Ours  ⌊R9⌋⌊G9⌋⌊B8⌋ − N  [R13][G13] − N  Static: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='98 (𝑁𝑚𝑎𝑥 = 63)  Dyna: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='91 (𝑁𝑚𝑎𝑥 = 10)  Q-DDGI  ⌊R11⌋⌊G11⌋⌊B10⌋ − N  [R16][G16] − N  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='94  Reference  RGB32f  RG32f  N/A  sampling a time varying region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The chain step size is 3x, and 6x larger for position and directions,  respectively w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='t the static case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Since each sample from the coarse chain represents an entire octant, we trace 64 rays for the  octant for all underlying 3x3x3 probes in the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We make the tracing step more efficient by  culling probes that are not used in deferred shading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The scheduling of ray-direction is deterministic,  passing through the center of a texel in the irradiance cache ( ^𝑔𝑟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' This solves the problem of sampling  noise and also affords the opportunity to simplify the atomic updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Since the rays are not random,  we do not benefit from multiple shader invocations updating the same octant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' As such, the first  invocation to update the octant marks (atomically) it updated such that other invocations do not  repeat the same work move to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  We run the dynamic sampling after the static sampling step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' During static sampling, if a probe  has non-zero dynamic component(𝑓𝑑 > 0), we quantize the ray directions to go through the  irradiance/visibility cache texel center to avoid injecting sampling noise in the texels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  5  RESULTS AND COMPARISONS  We compare our results with Q-DDGI and a reference probe-based implementation in different  scenarios - static scene (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 7), dynamic geometry (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 1, 8, 10), and dynamic lighting (fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Q-DDGI: Quantized-DDGI or Q-DDGI is a performance enhanced extension of original DDGI  [28], achieved without major modifications to the base algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Q-DDGI is equipped with a more  compact irradiance and visibility cache representation that closely resembles ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' See table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We  also enable camera-frustum culling of probes in Q-DDGI as described in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='4 and figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  These modifications allow Q-DDGI to have similar performance (table 4) at same probe count (table  2) as ours across different scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We believe these modifications make our comparisons more fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  We use 32 rays per probe for a total ray budget of 800-1600k (depending on scene) rays per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Reference: Reference implementation uses a standard FP32 representation for irradiance and  visibility caches as shown in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We also use a higher resolution 32×32 irradiance and visibility  cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Due to memory constraints, we are limited to a smaller probe-grid of size 32 × 32 × 32 using  same probe spacing (table 2) as other techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' For each frame, we discard any previous values  in the probes and accumulate samples using a true-average with 64 rays per texel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Ours: We use 4096 instances of static chain invocations and 1024 instances of dynamic chain  invocations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Overall, we use use between 500-900k (depending on scene) rays per frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  15  High Performance Graphics, Poster, July 11–14, 2022,  Datta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Performance breakdown of our technique and Q-DDGI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our probe sampling stage is divided into  three sub-stages - heuristic construction (P), static adaptive sampling (S), and dynamic adaptive sampling (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Scene  Ours (in milliseconds)  Q-DDGI (in milliseconds)  Probe Sampling  (P + S + D)  Deferred  Total  Probe sampling  Deferred  Total  Bistro - Exterior  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='01 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='23 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='73  = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='63  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='6  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='3  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='47  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='8  Sponza - Diffuse  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='21 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='85 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='18  =6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='24  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='62  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='86  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='69  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='51  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='2  Sponza - Glossy  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='83 + 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='11 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='33  =11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='27  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='44  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='7  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='9  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='71  31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='6  Figure 1 and 8 shows a large scene (Bistro Exterior), with the tunnel’s entry and exit modified  with dynamic gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The tunnel interior walls are illuminated by indirect illumination alone,  controlled by the direct light bouncing off the floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The direct illumination on the floor is controlled  by the dynamic entry gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The scene tests the tracking capabilities of our algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' the dynamic  Markov-chain should sample the probes close to the moving door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The scene also tests our color  compression scheme under low-light and moving-average accumulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Figure 9 shows the Sponza scene under dynamic lighting, testing the detection capabilities of  ADGI in the absence of dynamic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Figure 7 shows a static scene without dynamic geometry  or lighting, testing the convergence of our static adaptive sampling when no dynamism is detected  or the dynamic changes are too slow to detect, such as day-night cycles in games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Figure 10 shows a dynamic geometry (Stanford Buddha) under glossy indirect illumination  with ambient lighting as direct component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The scene is stressful as the camera frustum contains  many times more probes compared to other scenes due to the increased probe density required  for glossy illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' This scene tests the transient response of a dynamic geometry on a glossy  floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Thus the scene is less forgiving of spatio-temporal blurring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  We measured the results on a desktop with Nvidia 2080Ti GPU and AMD 5600X CPU at  1920 × 1080 resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The performance numbers cited in table 4 are only for ADGI and Q-  DDGI algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The GBuffer and direct-illumination passes require an additional 2ms and 3ms,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  6  LIMITATIONS  We inherit similar limitations as the vanilla DDGI algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The probe visibility from a shade-point  is only approximate and requires modifications such as probe movement to minimize light leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  The probe representation is not efficient in capturing glossy light-transport and requires a dense  spatio-angular discretization of irradiance cache to capture glossy reflections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Accurate detection of transient spatio-temporal changes in a scene are difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The accuracy of  detecting dynamic geometry reduces with the distance of the dynamic object from a probe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The  same is true for dynamic lighting;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' especially high frequency localized lighting that is far from a  probe is difficult to detect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Also, for the Markov chain to track the target distribution, the speed  of motion should be capped comparable to the product of Markov-chain step size and average  frame-rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' While many game engines keep track of the dynamic objects, facilitating the detection  of changing in visibility, we still need ray-tracing to detect dynamic radiosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  16  Adaptive Dynamic Global Illumination  High Performance Graphics, Poster, July 11–14, 2022,  7  CONCLUSION  Our adaptive sampling approach improves upon the efficiency of the original DDGI algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our  approach non-uniformly allocates resources in regions with time varying phenomena and captures  transient localized changes in an environment containing millions of probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' By contrast, DDGI’s  uniform allocation policy dilutes resource concentration in critical regions, especially when a large  number of probes are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' These improvements reduce temporal lag and minimizes reliance on  temporal blur to reduce noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our probe encoding scheme minimizes memory requirements by 4x  (and by extension memory bandwidth) with minimal impact on quality while also enabling millions  of probes in a scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our adaptive sampling stages have a fixed upper bound on the compute  requirement and also decouples sampling from the number of probes, further reducing memory  bandwidth requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' These changes enable improved probe-based rendering while also enabling  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='5-2x performance improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  8  RELATED WORK EXTENSION  Irradiance caching Irradiance caching is another line of techniques attempting to overcome  the high computation cost of GI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The irradiance caching method assumes that irradiance vary  smoothly across the scene, and texture detail can be recovered using albedo modulation [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The  interpolation and location of the various cache records is a critical, especially when the assumptions  on smoothness do not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' While robust, principled offline solutions exist [16, 24], real-time  applications often resort to complex heuristics and impose harsh constraints to achieve online  GI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Compression [56], sparse interpolation [49], pre-convolved environment maps [42, 45], spatial  hashing [3] and using neural network [37] are instances of advancements in real-time irradiance  caching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Although these approaches aim for real-time performance, their complexity and constraints  make them challenging to implement and deploy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  𝑡 = 32 ms  64 ms  96 ms  128 ms  32 ms  64 ms  96 ms  128 ms  Ours  Q-DDGI  MSE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
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+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Comparing the convergence of our technique over time on a static Bistro Exterior scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The figure  demonstrates the effectiveness of our static adaptive sampling step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
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+page_content='t reference and highlight the error in red and green color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
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+page_content='004  Left: Ours@15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='6ms  Right: Q-DDGI@26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='8ms  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our technique compared with Q-DDGI on a modified Bistro Exterior scene augmented with a  moving door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The scene has 192 × 64 × 192 probes and shows the convergence of the two techniques near  a dynamic area in the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The second row shows the changes inside the tunnel as the door closes over  time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our technique is better able to allocate the resources closer to the dynamic areas resulting in faster  convergence and higher performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Path tracing The flexibility and generality offered by path tracing [18] is highly desirable for  real-time rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' However, path tracing has been out of reach for real-time applications due to its  substantial computational requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Even with the advent of hardware-accelerated ray tracing  [23], it is only possible to trace a few tens of rays at each pixel in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Therefore, effective  sampling strategies and high-quality denoising algorithms [38, 46, 47] are essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Many sampling  methods try to learn the representation of incident illumination during rendering [1, 8, 34, 44, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  While these approaches can provide substantial error reduction, constructing these structures in  parallel on a GPU incurs a significant overhead that seem unsuitable for real-time applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Recently proposed ReSTIR GI [41] provides an efficient real-time sampling strategy by reusing the  paths spatially and temporally but the algorithm becomes complicated after second bounce and still  requires denoising for the final stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Deep learning has also been applied to path guiding, including  work by [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' These approaches demonstrated a substantial reduction in error due to more  effective path sampling, though their performance remain insufficient for real-time applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Screen space approaches: Approximating physically plausible illumination at real-time frame  rates with screen space methods is popular in games.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Screen space methods are fast, GPU-friendly,  and simple to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Screen space ambient occlusion (SSAO) [2, 33] is part of many real-time  rendering engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Following SSAO, screen Space Directional Occlusion (SSDO) [43] is used for  near-field direct and indirect diffuse lighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Sousa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' [52] proposed Screen Space Reflections  (SSR) using a 2D ray-tracing approach directly in screen space to obtain the indirect specular  component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Recently Screen-Space Global Illumination (SSGI) [43, 50, 52] methods offer a viable  solution to real-time GI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' However, these methods are limited by the information visible from the  observer’s position, thus making it difficult to engineer a robust solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Importance sampling and Bayesian modeling: Importance sampling provides a tool to  reduce the cost of brute force integration by selectively evaluating elements of the integrand based  on prior knowledge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' an educated guess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Previous works in importance sampling proposed  different methods to apply importance sampling to various Monte-Carlo integration existing in  rendering equations [21, 48, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Although Markov Chain Monte Carlo(MCMC) methods have been  18  Adaptive Dynamic Global Illumination  High Performance Graphics, Poster, July 11–14, 2022,  𝑡 = start  𝑡 = start + 2𝑠  𝑡 = start + 3𝑠  Direct only  Indirect - Ours  Indirect - Q-DDGI  SSIM/MSE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
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+page_content='021  Difference in luminance w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='t reference  Error - Ours  Error - Q-DDGI  Green: Diminished luminance  Red: Excess luminance  Ours@9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='86ms  Q-DDGI@13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='2ms  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Figure comparing the convergence of our technique under dynamic lighting controlled by the direct  component shown in the first row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The last two rows measure the difference in luminance w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='t reference and  highlight the error in red and green color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  used in Bayesian learning from the early days of neural networks [39], and Stochastic-Gradient  MCMC has been proposed [65] with various applications [25], our approach is neither Monte  Carlo-based nor Neural-network learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We exploit Bayesian inference and Markov Chains as our  mathematical means to sample the important texels on the probe, by defining our guide function  (prior), likelihood, and posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Markov Chain: Markov Chains are used broadly in Monte Carlo path-tracing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' For example,  Veach and Guibas [58] used Metropolis Sampling to explore the space of all possible paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Kelemen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' [19] later applied the exact sampling in the space of random numbers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=', in Primary Sample  19  High Performance Graphics, Poster, July 11–14, 2022,  Datta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  𝑡 = start  𝑡 = start + 10𝑠  𝑡 = start + 15𝑠  Glossy indirect - Ours  Glossy indirect without texture  Ours  Q-DDGI  SSIM/MSE:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
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+page_content='028  Ours  Reference  Q-DDGI  Ours  Reference  Q-DDGI  Ours  Reference  Q-DDGI  SSIM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
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+page_content='058  Ours@19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='7ms  Q-DDGI@31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='6ms  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Figure comparing glossy indirect reflection on a scene lit by ambient lighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The scene tests transient  response due to the moving Buddha geometry over a glossy floor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The most recent work by Bitterli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' [4] combines a simple path tracing integrator with  MCMC by using the random seeds of high variance paths as starting points for the Markov Chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Although Markov Chains are encountered extensively beneficial in solving Monte Carlo sampling,  our point of view on sampling and employing the Markov Chain to draw samples from the guide  function is distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Bayesian inference: Bayesian modeling is a widespread methodology in computer vision and  graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Brouillat et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' [5] and Marques et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' [30] pioneered the use of Bayesian Monte Carlo  (BMC) [11] in light transport simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' In contrast, [59] keep the efficient classic, frequentist  MC approach and apply Bayesian modeling to optimize their sampling distributions for direct  illumination estimates across the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Similar approach is used by Vorba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' [61], who employ  a maximum a posteriori (MAP) formulation to regularize training of parametric mixture models for  optimized indirect illumination sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our approach uses Bayesian modeling in the context of  light-probes to detect important probes and directions based on sampled evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  REFERENCES  [1] 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' SIGGRAPH ’19: ACM SIGGRAPH 2019 Production Sessions (Los Angeles, California).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Association for Computing  Machinery, New York, NY, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  20  Adaptive Dynamic Global Illumination  High Performance Graphics, Poster, July 11–14, 2022,  [2] Louis Bavoil, Miguel Sainz, and Rouslan Dimitrov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Image-Space Horizon-Based Ambient Occlusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' In ACM  SIGGRAPH 2008 Talks (Los Angeles, California) (SIGGRAPH ’08).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Association for Computing Machinery, New York,  NY, USA, Article 22, 1 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='1145/1401032.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='1401061  [3] Nikolaus Binder, Sascha Fricke, and Alexander Keller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Fast Path Space Filtering by Jittered Spatial Hashing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  In ACM SIGGRAPH 2018 Talks (Vancouver, British Columbia, Canada) (SIGGRAPH ’18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
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+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
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+page_content=' ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
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+page_content=' In Proceedings of High Performance Graphics (Los  Angeles, California) (HPG ’17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
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+page_content=' In Proceedings of the ACM SIGGRAPH Symposium on Interactive 3D Graphics and Games (Montreal,  Quebec, Canada) (I3D ’19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
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+page_content='3317024  [64] Gregory J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Ward, Francis M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Rubinstein, and Robert D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' A Ray Tracing Solution for Diffuse Interreflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  SIGGRAPH Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 22, 4 (jun 1988), 85–92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='1145/378456.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='378490  [65] Max Welling and Yee Whye Teh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Bayesian Learning via Stochastic Gradient Langevin Dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' In Proceedings  of the 28th International Conference on International Conference on Machine Learning (Bellevue, Washington, USA)  (ICML’11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Omnipress, Madison, WI, USA, 681–688.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  [66] Lei Xiao, Salah Nouri, Matt Chapman, Alexander Fix, Douglas Lanman, and Anton Kaplanyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  Neural  Supersampling for Real-Time Rendering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  ACM Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 39, 4, Article 142 (July 2020), 12 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  https:  //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='1145/3386569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='3392376  23  High Performance Graphics, Poster, July 11–14, 2022,  Datta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  A  METROPOLIS-HASTINGS  Markov Chain Monte Carlo (MCMC) allows sampling from the posterior without computing  the marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Metropolis-Hastings (Metropolis), which we exploit in this work, is a specific  implementation of MCMC [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The Metropolis–Hastings algorithm can draw samples from any  probability distribution with probability density 𝑃(𝑥), provided a function ℎ(𝑥) proportional to  the density 𝑃(𝑥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The Metropolis algorithm works by generating a sequence of sample values so  that, as more samples are produced, the distribution of samples more closely approximates the  desired distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' These sample values are produced iteratively, meaning the next sample being  dependent on the current sample (thus making the sequence of samples into a chain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Let ℎ(𝑥)  be a function that is proportional to the desired probability density function 𝑃(𝑥) (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' a target  distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The Metropolis Markov Chain algorithm with random walk is defined as follows:  Algorithm 5: Random-walk algorithm  Input: 𝑥𝑖: Current state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 𝑦𝑖 : Probability of current state  Input: 𝜎 : Step size or std-dev of Gaussian noise  Output: 𝑥𝑖+1: Next state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 𝑦𝑖+1 : Probability of next state  1 function RandomWalk(𝑥𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 𝑦𝑖):  2  𝑥𝑖+1 ← 𝑥𝑖 + N (𝜎)  // Propose a new state  3  𝑦𝑖+1 ← ℎ(𝑥𝑖+1)  4  𝜇 ← min  �  𝑦𝑖+1  𝑦𝑖 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 1  �  // Compute acceptance ratio  5  𝜖 ∼ 𝑈 (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 1)  // Sample uniform distribution  6  if 𝜖 > 𝜇 then  /* Reject proposed state  /  7  𝑥𝑖+1 ← 𝑥𝑖  8  𝑦𝑖+1 ← 𝑦𝑖  9  return 𝑥𝑖+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='𝑦𝑖+1  Initialization: Choose an arbitrary point 𝑥𝑖−1 as the initial observation in the sample-space and  choose an arbitrary probability density N (𝑥𝑖 | 𝑥𝑖−1) that suggests the next sample candidate 𝑥𝑖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  given the previous sample value 𝑥𝑖−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' In our work, N is assumed to be symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' A usual choice  is to let N (𝑥𝑖 | 𝑥𝑖−1) be a Gaussian distribution centered at 𝑥𝑖−1, so that points closer to 𝑥𝑖−1 are  more likely to be visited next, making the sequence of samples resemble a random walk [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The  random walk algorithm is described in algorithm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  B  PROBE COMPRESSION  We tested several 26-bit encoding and settled on a non-linear RGB encoding represented by  ⌊R9⌋⌊G9⌋⌊B8⌋ −N in figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' In this encoding, the RGB color is first passed through a logarithmic  non-linearity as per equation 21 such that the quantization errors are distributed evenly across  intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We perform a round-to-lowest-integer (⌊⌋) quantization for all channels, although round-  to-nearest-integer ([ ] ) is more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Our quantization scheme ensures the moving-average  updates produce dark colors when the intensity of new samples are low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' In a round-to-nearest set-  ting, due to a round-up error, the colors may never go to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Interestingly, YCbCr encoding allows  round-to-lowest for the Y channel and round round-to-nearest for Cb and Cr channels, however,  they perform poorly in both luminance and color preservation metrics as shown in figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  24  Adaptive Dynamic Global Illumination  High Performance Graphics, Poster, July 11–14, 2022,  Intensity:  [0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='25]*  [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='25 - 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0]  [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0 - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0]*  Decoded RGB  Error  Decoded RGB  Error  Decoded RGB  Error  ⌊Y8⌋[ Cb9] [ Cr9] − N  ⌊Y8⌋[ Cb9] [ Cr9]  ⌊R9⌋ ⌊G9⌋ ⌊B8⌋ − N  ⌊R9⌋ ⌊G9⌋ ⌊B8⌋  LUM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0024  LUM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0025  LUM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0025  LUM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0004  LUM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0011  LUM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0043  LUM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0045  LUM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0049  LUM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0050  LUM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0007  LUM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0021  LUM: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0086  COR: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='9972  COR: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='9999  COR: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0000  COR: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0000  COR: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0000  COR: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0000  COR: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='9944  COR: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='9997  COR: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0000  COR: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0000  COR: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0000  COR: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='0000  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Figure comparing 26-bit color encodings on slices of the 3D color-space with dynamic range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' We  compare the reconstruction error measured in Luminance and Color Correlation with RGB32f reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' The  log-non-linear encodings marked with - N suffix shifts the bit error from lower to higher intensities - which  are less frequent in indirect illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' ⌊⌋ and [ ] denotes round-low and round-nearest quantizations  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' * Color map visualizations are normalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  The parameters in equation 21 are obtained by performing a grid search minimizing the recon-  struction error w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='t RGB32f reference across various color and intensity combinations as shown in  figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Luminance error is the r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' value of the difference between the two color-maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content=' Color  accuracy is measured using a normalized dot product between the two flattened color-maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
+page_content='  25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JNE4T4oBgHgl3EQfhg0d/content/2301.05125v1.pdf'}
diff --git a/JtFQT4oBgHgl3EQfSTYo/content/tmp_files/2301.13289v1.pdf.txt b/JtFQT4oBgHgl3EQfSTYo/content/tmp_files/2301.13289v1.pdf.txt
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@@ -0,0 +1,2090 @@
+On the Statistical Benefits of Temporal Difference Learning
+David Cheikhi
+Daniel Russo
+Columbia University
+Abstract
+Given a dataset on actions and resulting long-term rewards, a direct estimation approach fits
+value functions that minimize prediction error on the training data. Temporal difference learning
+(TD) methods instead fit value functions by minimizing the degree of temporal inconsistency
+between estimates made at successive time-steps. Focusing on finite state Markov chains, we
+provide a crisp asymptotic theory of the statistical advantages of this approach. First, we show that
+an intuitive inverse trajectory pooling coefficient completely characterizes the percent reduction in
+mean-squared error of value estimates. Depending on problem structure, the reduction could be
+enormous or nonexistent. Next, we prove that there can be dramatic improvements in estimates of
+the difference in value-to-go for two states: TD’s errors are bounded in terms of a novel measure
+— the problem’s trajectory crossing time — which can be much smaller than the problem’s time
+horizon.
+1
+Introduction
+Temporal difference learning is a distinctive approach to estimation in long-term optimization problems.
+Its importance to reinforcement learning is hard to overstate. In their seminal book, Sutton & Barto
+(2018) write: If one had to identify one idea as central and novel to reinforcement learning, it would
+undoubtedly be temporal difference learning.
+Competing with temporal difference (TD) learning is a straightforward direct-estimation approach.
+There, one proceeds by collecting data on past decisions and the cumulative long-term ‘reward’ that
+followed them. If actions were chosen with some experimental randomness, then regression of long-term
+rewards on the draw of actions would – with enough data – correctly identify actions’ causal impacts.
+The direct approach has two significant drawbacks. The first is delay: actions can only be evaluated
+after their full long-term effects realize. The second is variance: long-term outcomes can be extremely
+noisy and individual actions often have a small impact on them.
+TD aims to alleviate these challenges by leveraging data on intermediate outcomes – those observed
+after the decision but before final outcomes are realized. The availability of such data is increasingly
+common. Robots collect regular sensor measurements, recommendation systems log sequential user
+interactions, and digital devices can track patient’s health metrics across time. Sutton (1988) observed
+that successive predictions, updated as information is gathered, should be temporally consistent. He
+proposed to fit maximally consistent value estimates by iteratively minimizing ‘temporal difference
+errors’. TD has since become an intellectual pillar of the reinforcement learning literature. It is used
+in most successful applications.
+Our Contributions.
+We aim to provide crisp insight into the statistical benefits of TD. This paper
+focuses on the simplest possible setting, where training data consists of a batch of trajectories sampled
+independently from a finite Markov reward process. We compare the asymptotic scaling of mean
+squared error under TD and direct value estimation. Two main findings emerge:
+1. The relative benefits of TD are determined by a natural ‘inverse trajectory pooling coefficient.’
+TD uses value-to-go at intermediate states as a surrogate Athey et al. (2019); Prentice (1989).
+1
+arXiv:2301.13289v1  [cs.LG]  30 Jan 2023
+
+This is beneficial exactly when trajectories that originate with distinct states/actions tend to
+reach common intermediate states. We present simple examples illustrating when the benefits of
+TD are enormous and when they vanish entirely.
+2. TD is especially beneficial for advantage estimation; That is, when estimating the difference
+in value-to-go from one state/action versus another. While the mean squared error of direct
+advantage estimation generally scales with the length of the problem horizon, we show that TD’s
+errors are bounded by a smaller trajectory crossing time. This novel notion of effective horizon
+can be small even in some problems with unbounded mixing time.
+On the Markov assumption and state representation.
+Our focus on Markov models is standard
+in the academic literature on reinforcement learning. This is, in a certain technical sense, an innocuous
+assumption. One could always use the entire sequence of observations so far as a Markov state
+Puterman (2014). But practical algorithms need to use appropriate compression of the history. The
+choice of representation has a subtle interplay with the benefit of TD. Indeed, we comment in section
+9 that the benefits of TD can vanish when the state representation is too rich and trajectory pooling
+vanishes.
+2
+Related works
+TD has been a central idea in RL since it was first proposed. It is deceptively simple, has intriguing
+connections to neuroscience Schultz et al. (1997), and seems to be routed in dynamic programming
+theory. In the 1990s, researchers gathered both limited convergence guarantees Dayan & Sejnowski
+(1994); Jaakkola et al. (1993) and examples of divergence Baird (1995). Tsitsiklis & Van Roy (1997)
+offered a clarifying theory of when TD converges, and characterized the TD fixed point it reaches.
+Maei et al. (2009); Sutton et al. (2009) proposed methods to reach the TD fixed point in off-policy
+settings or when nonlinear function approximation is used.
+While illuminating, this theory does not clarify why TD should be preferred over direct value
+function estimation (dubbed ‘Monte-Carlo’ or ‘MC’). In fact, the main guarantee is convergence to an
+approximate value function whose mean-squared error is larger than the one MC reaches. Folklore,
+intuition, and experiments suggest TD often converges to its limit at a faster rate.
+The literature has emphasized the distinction between online and batch TD algorithms. The
+convergence speed of batch TD methods, like LSTD Bradtke & Barto (1996), is a statistical question.
+With purely online algorithms, each observation is used to make a single stochastic gradient type
+update and then immediately discarded, so issues of memory, compute, and data efficiency are conflated.
+The deep RL literature has adopted experience replay Andrychowicz et al. (2017); Mnih et al. (2015);
+Schaul et al. (2015); Wang et al. (2016), which blurs the line between batch and online implementations
+by recording observations in a dataset and resampling them many times.
+We give a complete and intuitive characterization of the efficiency benefits of TD in the simplest
+possible setting: a batch variant applied without function approximation in a finite state Markov Reward
+Process. Here it is straightforward to show TD is more efficient that MC, but more subtle to understand
+when the efficiency gains are large. Grunewalder et al. (2007) and Gr¨unew¨alder & Obermayer (2011)
+make progress in this direction. They prove that LSTD is at least as statistically efficient as MC,
+without quantifying the improvement. They also display cases where the two procedures have the same
+performance. Textbooks by Sutton & Barto (2018) and Szepesv´ari (2010) give illuminating examples,
+but no theory.
+A number of papers bound the data requirements of TD. See for example Mannor et al. (2004), Lu
+(2005)), Lazaric et al. (2010), Pires & Szepesv´ari (2012), Tagorti & Scherrer (2015)), Bhandari et al.
+(2018), Pananjady & Wainwright (2020), Khamaru et al. (2020), Chen et al. (2020), or Farias et al.
+(2022). These show certain problem instances have low data requirements, but do not clarify when
+2
+
+enforcing temporal consistency in value estimates produces large efficiency gains. To the best of our
+knowledge, our insights about advantage estimation are new (See Sec 8).
+3
+Problem Formulation
+We first describe the problem of value function estimation in Markov reward processes (MRPs). We
+then observe that after appropriate relabeling of the state variables, this can also represent the problem
+of evaluating the long-term impact of actions. Most mathematical results are stated for MRPs, but
+the alternative interpretation enriches the results.
+3.1
+Value function estimation
+A trajectory in a terminating Markov reward process is a Markovian sequence
+τ = (S0, R1, S1, R2, S2, . . . , ST −1, RT , ∅),
+consisting of a sequence of states (St)t∈[T ] ⊂ S, rewards (Rt)t∈[T ] ⊂ R, and termination time T. The
+termination time is the first time at which ST = ∅, where ∅ is thought of as a special terminal state.
+Assume the distribution of Rt is independent of past rewards given the current state St.
+The law of a Markov reward process (MRP) is specified by the tuple M = (S∪{∅}, P, R, d) consisting
+of a state space, transition kernel, reward distribution, and initial state distribution. Here P is a
+transition matrix over the augmented state space S ∪∅, specifying a probability P(s′ | s) of transitioning
+from s to s′. We assume terminal sate is absorbing and reachable. That is, P(∅|∅) = 1 and for every
+state s there is some t such that the t step transition P t(∅|s) is strictly positive. The object R specifies
+the draw of rewards conditioned on a state transition as R(dr|s, s′) = P(Rt = dr | St = s, St+1 = s′).
+Throughout we use the notation r(s, s) for the mean of R(·|s, s′). We assume r(∅, ∅) = 0. The initial
+state distribution d is a probability distribution over S from which S0 is drawn.
+The value function
+V (s) = E
+� ∞
+�
+t=1
+Rt | S0 = s
+�
+= E
+� T
+�
+t=1
+r(St, St+1) | S0 = s
+�
+specifies the expected future reward earned prior to termination. It is immediate from our formulation
+that V (∅) = 0. Our formulation is the Markov reward process analogue of stochastic shortest path
+problems (Bertsekas & Tsitsiklis, 1991). Discounted problems are a special case where there is a
+constant probability of termination P(∅ | s) = 1 − γ for each non-terminal state s ∈ S. In that case,
+the horizon T follows a geometric distribution with mean 1/(1 − γ).
+A related quantity measures the value-to-go differences between states,
+A(s, s′) = V (s) − V (s′).
+We call this the advantage of s over s′, since, as revealed in the next subsection, it is closely related to
+the advantage function in RL Baird III (1993).
+We consider the problem of estimating the value-to-go at initial states. We compare methods that
+produce estimates ˆV based on n independent trajectories
+D =
+�
+τi = (S(i)
+0 , R(i)
+1 , S(i)
+1 , . . . , S(i)
+T (i)−1, R(i)
+T (i), ∅)
+�
+i=1,...,n .
+by their mean squared error E
+��
+V (s) − ˆV (s)
+�2�
+or E
+��
+A(s, s′) − ˆA(s, s′)
+�2�
+, where the expectation
+is over the randomness in D. We assume that all states have a non-zero probability of being visited in
+the dataset.
+3
+
+3.2
+Heterogenous treatment effect estimation
+By appropriate relabeling of variables, we can interpret our problem as one of evaluating the long-term
+impact of a chosen decision in a specific context.
+Here we consider a special case of our formulation where the continuing state space S = S0 ∪ SI
+is partitioned into a set of initial states S0 and a set of intermediate states SI. With probability 1,
+S0 ∈ S0, ST = ∅, and at intermediate times t ∈ {1, . . . , T − 1}, St ∈ SI.
+We give the initial states a special interpretation. We think of them as consisting of an initial context
+X0 and a decision A0 and write S0 = (X0, A0). Using the more familiar notation Q(X0, A0) ≡ V (S0),
+we have
+Q(x, a) = E
+� T
+�
+t=1
+Rt | X0 = x, A0 = a
+�
+.
+The initial distribution d0 is determined by an initial context distribution P(X0 = x) and a logging
+policy P(A0 = x | X0 = x) which determines the frequency with which actions are observed in the
+data.
+Of particular interest in this setting is the advantage
+A((x, a), (x, a′)) = Q(x, a) − Q(x, a′),
+which measures the performance difference between actions a and a′ in context x. The term ‘advantage’
+is common in RL, but in causal inference one might call this ‘heterogenous treatment effect’ estimation.
+Since policy gradient methods typically involve computing the expectation of weighted advantages,
+we expect that insights developed in this paper could apply to these methods.
+4
+Algorithms
+4.1
+Direct approach: First visit monte-Carlo (MC)
+For any candidate value function, we can evaluate its accuracy by comparing the future value it predicts
+from a given state visited in the data and the actual cumulative reward observed in the remainder of
+that trajectory. This suggests a natural direct value estimation approach: over a candidate class of
+value functions, pick one that minimizes mean squared prediction error on the dataset. This method is
+called Monte Carlo in the RL literature.
+To formally describe the algorithm, define the random time T(s) = min{t : St = s ∨ St = ∅} to be
+the first hitting time of state s if it is reached, or T otherwise. Let I(s) = {i ∈ [n] : s ∈ τi} to be set of
+trajectories that visit state s. Form a dataset
+DMC =
+�
+s∈S
+�
+i∈I(s)
+�
+�
+�
+�
+�s,
+T (i)
+�
+t=T (i)(s)+1
+R(i)
+t
+�
+�
+�
+�
+� .
+that records pairs of states and the cumulative rewards earned following the first visit to the state.
+Given a parameterized class of value functions {Vθ : θ ∈ Θ}, a direct value estimation approach is to
+solve the least squares problem
+min
+θ∈Θ
+�
+(s,v)∈DMC
+(Vθ(s) − v)2
+We focus on tabular representations, where the space of parameterized value functions spans all
+possible functions. In that case
+ˆVMC(s) = E(S,V )∼DMC [V | S = s] ,
+4
+
+where the notation (S, V ) ∼ DMC means that state/value pairs are sampled uniformly at random from
+the dataset. The value estimate at state s is simply the average reward earned after visits to state s in
+the dataset.
+What we described is called first visit Monte-carlo in the literature. It is an unbiased estimator
+because it only includes the first time a state was visited during the an episode. We focus on this
+version for analytical simplicity, but many of our key examples focus on cases where initial states are
+never revisited (See e.g. Subsection 3.2) and it coincides with an “every visit” Monte Carlo estimate.
+4.2
+Indirect approach: TD learning
+Temporal difference learning uses a reformatted dataset consisting of tuples of reward realizations and
+state transitions:
+DTD = {(S(i)
+t , R(i)
+t+1, S(i)
+t+1)t=1,...,T (i),i=1,...,n}.
+Define the temporal difference error between two candidate value functions V, V ′ to be the average
+gap in Bellman’s equation:
+ℓTD(V, V ′) = E(S,R,S′)∼DTD [V (S) − (R + V ′(S′))] ,
+where the notation (S, R, S′) ∼ DTD means that tuples are sample uniformly at random from the
+dataset. Given a parameterized class of value functions, VΘ = {Vθ : θ ∈ Θ}, batch TD algorithms
+iteratively generate parameters (θ1, θ2, . . .) by solving minθi+1 ℓ(Vθi+1, Vθi). (Online TD algorithms,
+combined with experience replay, make SGD updates instead.) For linear function approximation
+(Tsitsiklis & Van Roy, 1996), or neural networks in the ‘Neural tangent kernel’ regime (Cai et al.,
+2019), value functions are known to converge to a fixed point
+ˆVTD = arg min
+V ∈VΘ
+ℓ
+�
+V, ˆVTD
+�
+,
+which, in a sense, maximizes feasible temporal consistency.
+Again, we focus on tabular representations, where the space of parameterized value functions spans
+all possible functions. In that case, TD solves the empirical Bellman equation
+ˆVTD(s) = E(S,R,S′)∼DTD
+�
+R + ˆVTD(S′) | S = s
+�
+5
+Intuition: surrogacy and intermediate outcomes
+Figure 1: Modeling a user’s behavior
+A lot of the intuition regarding TD can be gained through the simple example in Figure 1. Imagine
+our goal is to select the website design among 100 alternatives that leads to the largest sale rate (of
+some product). Users arrive and are randomly assigned to one of the 100 pages. They either click to
+5
+
+Webpage
+Version 1
+Checkout
+page,
+Sale
+Webpage
+Version 100
+No Salepurchase and proceed to the checkout page or navigate away from the site without purchasing. Among
+those who click, only a small fraction complete the sale.
+Assume, for simplicity, that we have no access to personal information that distinguishes users
+from one another. (There is only one possible x in Section 3.2.) There are 100 possible initial states,
+corresponding to the webpage version, and the user is equally likely to start at each of those. A sale
+and a no-sale immediately precede termination. We call the checkout page an intermediate state. The
+sale state is associated with a reward of 1 and all others are associated with a reward of 0.
+What we called the Monte-Carlo estimate of the value function would directly estimate the value
+of an impression of each webpage to be the fraction who purchased among that cohort of users.
+Due to the directed nature of state transitions, TD estimation can be thought of in two steps. We
+first estimate ˆV (checkout) to be the fraction who purchased among users who visit the checkout page.
+We then estimate the value of an impression of webpage i by
+ˆV TD(webpage i) = CTR(i) × ˆV (checkout),
+the empirical click-through rate among those shown webpage i times the estimated sale rate on the
+checkout page.
+Monte-carlo estimation is unbiased, but it may be difficult to reliably estimate the efficacy of each
+webpage. If only a small fraction click initially, and among those who do only a small fraction convert
+to a sale, one would need to show each webpage to an enormous number of users. With TD, we pool
+data from across users who were shown any of the 100 webpages when estimating ˆV (checkout), greatly
+reducing variance. In this example, there is a lot of data pooling because trajectories that begin at
+distinct states quickly converge to the intermediate state. In fact, our theory reveals that certain
+intuitive measures of trajectory pooling exactly determine degree of statistical benefit TD provides.
+Another view of TD is that it uses the intermediate click/no-click outcome as a surrogate or
+proxy-metric Prentice (1989). Recognizing this, TD’s potential downsides become as transparent as a
+its benefits. If the conversion probability among users who visit the checkout page depends strongly
+on which page design they saw, the Markov property does not hold and TD is biased. We discuss this
+example again in the conclusion, mentioning how the risks and benefits interplay with the choice of
+state representation.
+6
+Empirical illustration
+6.1
+The benefits of TD
+To illustrate how much TD can improve over MC, we explore an example: we consider a layered MRP
+as describe in Figure 2. A layered MRP with width W and horizon T has W × (T − 1) states split in
+T layers. States in layer t can only transition to states in layer t + 1 and states in the last layer T − 1
+always transition to the terminal state.
+We consider a Layered MRP with width W = 5. We focus on state s(1)
+1
+and s(2)
+1
+and study the
+accuracy of the estimates of their value as we vary the horizon T of the MRP. We also study the
+accuracy of the estimate of the advantage A
+�
+s(1)
+1 , s(2)
+1
+�
+= V
+�
+s(1)
+1
+�
+− V
+�
+s(2)
+1
+�
+. Figure 3 displays the
+Mean Square Error (MSE) of the TD and MC estimates for these quantities when the dataset contains
+n = 2000 independent trajectories. MSE calculations involve 10000 Monte-Carlo replications. Alongside
+the observed MSE, we plot projected MSE based on the central limit theorem from Proposition A.4
+and Proposition A.5. There is almost perfect alignment between asymptotic and finite sample results.
+We highlight three main take-aways from this example:
+1. TD can vastly outperform MC. For the chosen states s(1)
+1
+and s(2)
+1 , the MSE is about 5
+times smaller when using TD instead of MC for a Layered MRP with width W = 5 and horizon
+T = 120.
+6
+
+s(1)
+1
+s(2)
+1
+...
+s(W )
+1
+s(1)
+2
+s(2)
+2
+...
+s(W )
+2
+s(1)
+T −1
+s(2)
+T −1
+...
+s(W )
+T −1
+∅
+Figure 2: Layered MRP with width W and horizon T. Transitions are chosen randomly and rewards
+are uniform on [r(s, s′) − 1; r(s, s′) + 1] where r(s, s′) is chosen uniformly between -1 and 1.
+0
+20
+40
+60
+80
+100
+120
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+Horizon T
+MSE
+(a) Full Y-Axis
+0
+20
+40
+60
+80
+100
+120
+0
+1 · 10−2
+2 · 10−2
+3 · 10−2
+4 · 10−2
+5 · 10−2
+Horizon T
+MSE
+(b) Truncated Y-Axis
+Value at s
+Asymptotic TD variance
+Value at s′
+Asymptotic MC variance
+Advantage
+Empirical TD Variance
+Empirical MC Variance
+Figure 3: Variance of different MC and TD estimates on Layered MRP with W = 5 and varying
+horizon H
+2. TD benefits are enhanced for advantage estimation. In this example, TD performs up
+to 100 times better than MC for the advantage estimation. This example also shows that the
+MSE of the TD estimate of the ATE is smaller than the MSE of individual estimates of the value
+of s(1)
+1
+and s(2)
+1
+when the horizon T is larger than 20. On the other hand, the MSE of the MC
+estimate of the advantage is larger than the individual MSE of the estimates of the value.
+3. TD effectively truncates the horizon. While the MSE of the MC estimate of the advantage
+scales linearly with the horizon T, the MSE of the TD estimate is constant with respect to the
+horizon T. This is all the more striking that the variance of the total reward along a trajectory
+scales linearly with the horizon.
+7
+
+6.2
+Dependence on the MRP structure
+We have seen in Section 6.1 that TD vastly outperforms MC in the case of Layered MRP. However,
+different MRP structures lead to different level of improvement of TD over MC. To illustrate this, we
+s(1)
+1
+s(2)
+1
+s(k)
+1
+s(1)
+H−1
+s(2)
+H−1
+s(k)
+H−1
+sH
+sT −1
+∅
+Figure 4: MRP with meeting horizon H
+introduce a new class of MRPs described in Figure 4. Each of the k initial states s(1)
+1 , . . . , s(k)
+1
+lead to
+disjoint trajectories for the first H − 1 steps before reaching a common state on the Hth step. We
+are interested in seeing how TD and MC perform when the crossing time H varies. Figure 5 displays
+the ratio of the MSE of the TD and MC estimates for the values of s(1)
+1 , s(2)
+1
+and for the advantage as
+the crossing time H varies. The MSE used to compute the ratios have been computed using n = 200
+independent trajectories and 1000 Monte-Carlo replications. These ratios are plotted alongside the
+asymptotic ratio from Theorem 7.2. As the crossing time H gets closer to the horizon T, the advantage
+of TD over MC vanishes until H = T, when TD and MC produce the exact same estimates. To convey
+0
+5
+10
+15
+20
+0
+0.2
+0.4
+0.6
+0.8
+1
+Meeting horizon H
+Ratio of MSE E
+�
+( ˆVT D − V )2�
+/E
+�
+( ˆVMC − V )2�
+Value at s
+value at s′
+Advantage
+Asymptotic ratio
+Empirical ratio
+Figure 5: Ratio of variance between TD and MC as a function of the meeting horizon H for T = 20
+intuition, we first focus on the two extreme cases:
+• In the case where H = 2, depicted in Figure 6a, all initial states directly transition to the same
+state. This mimics the webpage example in Figure 1. In this example, apart from the first reward,
+8
+
+trajectories do not depend of the initial state. TD pools trajectories across actions which allows
+to highly reduce the variance of the estimate of the value at s2. This low variance estimate is
+then used as a surrogate to estimate the value-to-go at initial states. On the other hand, MC
+does not leverage the structure of the MRP and produces an independent estimate for each
+initial state, using only trajectories starting at a given state to evaluate this state. In this case,
+TD will significantly improve over MC.
+• At the other extreme, consider H = T. Then, no two initial states can lead to a common state
+before the terminal state, as shown in Figure 6b. There is no opportunity to pool trajectories
+across actions so TD strictly reduces to MC in this case.
+s(1)
+1
+s(2)
+1
+s(k)
+1
+s2
+sT −1
+∅
+(a)
+s(1)
+1
+s(2)
+1
+s(k)
+1
+s(1)
+T −1
+s(2)
+T −1
+s(k)
+T −1
+∅
+(b)
+Figure 6: 6a is an instance on which TD leverages pooling to improve considerably over MC. 6b is an
+instance on which TD and MC output the same estimate.
+6.3
+Organization of the results
+In Section 7, we characterize the ratio in variance between TD and MC estimates for value estimation
+depending on the MRP structure. This characterization enables an intuitive understanding of which
+MRP structures lead to a large improvement of TD and, conversely, for which MRP structures TD
+and MC perform similarly. In Section 8, we show that the TD estimate of advantages scales with an
+effective horizon that can be much smaller than the horizon.
+7
+Value estimation and the pooling coefficient
+Recall that T(s) is the first hitting time of state s if it is reached, or T otherwise. The variables
+N(s′) =
+T
+�
+t=0
+1(St = s′), N(s → s′) =
+T
+�
+t=T (s)
+1(St = s′),
+respectively measure the total number of visits to state s′ and the number of visits to s′ which occur
+after a visit to state s.
+Define the coupling coefficient between s and s′ by
+C(s, s′) = E [N(s → s′)]
+E [N(s′)]
+,
+with C(s, s′) = 0 if E [N(s′)] = 0. Implicitly, it is understood that S0 is drawn from the MRP’s initial
+distribution d. Among all trajectories which reach state s′, the coupling coefficient measures the
+9
+
+fraction which first pass through state s. If the coupling coefficient is large, it means s and s′ are
+strongly coupled.
+The inverse trajectory pooling coefficient measures the average coupling coefficient C(s, s′) over a
+distribution of possible successor states s′. The right distribution over which to average turns out to
+be µs(·), identified in the definition below. That distribution weights highly states that are likely to be
+visited following a visit to s (high E[N(s′) | S0 = s]) and contribute heavily to to estimator variance
+(measured through the one-step variance Var (Rt+1 + V (St+1) | St = s′)).
+Definition 7.1 (Inverse trajectory pooling coefficient). For any state s ∈ S define
+C(s) = Es′∼µs [C(s, s′)] ,
+where µs(·) a probability distribution over states defined by
+µs(s′) ∝ E [N(s′) | S0 = s] × Var [Rt + V (St+1) | St = s′] .
+Again, the inverse trajectory pooling coefficient is small when there is a lot of trajectory pooling.
+The next theorem compares the asymptotic mean squared error of the value estimated under TD and
+a direct approach. The asymptotic ratio of their mean squared errors is equal to the inverse trajectory
+pooling coefficient.
+Theorem 7.2. For any s ∈ S,
+lim
+n→∞
+E
+��
+ˆVT D(s) − V (s)
+�2�
+E
+��
+ˆVMC(s) − V (s)
+�2� = C(s).
+Let us interpret this result.
+Recall that TD updates value prediction at state s using value
+predictions at successor states. The theorem shows this is helpful precisely when there is a lot of
+trajectory pooling, resulting in a small inverse trajectory pooling coefficient. When this holds, and the
+dataset D is large, there will be many trajectories which reach an important possible successor s′ of s,
+but never cross s first. TD leverages these trajectories to learn about s′ and then properly incorporates
+that knowledge to better evaluate s. Direct estimation approaches only use sub-trajectories originating
+at s to evaluate s and forego the trajectory pooling benefit. We already developed this intution by
+discussing the simple example in Figure 1. The theorem confirms that this interpretation of TD’s
+advantages is exactly the right one.
+Figures 6a describes an instance with extreme trajectory pooling. Trajectories that start in distinct
+states tend to immediately reach common successors, so TD understands value-to-go from successors
+quite well. Figure 6b is a case with no trajectory pooling at all (i.e. C(s) = 1).
+8
+Horizon truncation in advantage estimation
+Section 6 previewed two of the paper’s key insights: TD’s benefits are enhanced for advantage estimation
+and, in that setting, it effectively truncates the problem’s time horizon. Theory in this section formalizes
+these insights.
+The MSE of direct advantage estimates scales with the horizon.
+The variance of the total
+reward along a trajectory typically scales with the horizon. Therefore, it would not be surprising that
+the mean squared error of the estimate of the advantage also scales with the horizon. We show that it
+is indeed the case for MC by stating a lower bound on the mean squared error.
+10
+
+Proposition 8.1. For s, s′ ∈ S such that P [s ∈ τ ∧ s′ ∈ τ] = 0,
+lim
+n→∞ n · E
+��
+ˆAMC(s, s′) − A(s, s′)
+�2�
+≥ σ2
+min
+�E [T|S0 = s]
+P [s ∈ τ]
++ E [T|S0 = s′]
+P [s′ ∈ τ]
+�
+.
+where σ2
+min = mins∈S Var [Rt + V (St+1) | St = s]
+The condition P [s ∈ τ ∧ s′ ∈ τ] = 0 guarantees that no trajectory can visit both s and s′. It ensures
+that the MC estimate of the value at s and s′ are independent. It is verified when considering the
+heterogeneous treatment effect, described in Section 3.2, where a single action is chosen for every
+trajectory. The scaling in the inverse probability of visiting s and s′ appears because nP [s ∈ τ] and
+nP [s′ ∈ τ] are asymptotically the number of trajectories available for the Monte-Carlo estimation.
+The MSE of TD’s advantage estimates scales with a smaller trajectory crossing time.
+Rather than scale with problem’s time horizon, the mean squared error of TD’s advantage estimate is
+bounded by a smaller notion of the problem’s effective horizon. To formally capture this phenomenon,
+we introduce the trajectory crossing time H(s, s′) for two states s and s′ to be the expected time for
+trajectories starting at s and s′ to cross under the most optimistic coupling. Two trajectories always
+cross once both have terminated, as in that case both have visited the terminal state ∅, but they could
+cross much sooner. Intuition for the this definition is provided below.
+Definition 8.2. The set of distributions Ψ(s, s′) is the set of all joint distributions over trajectories
+(τ, ˜τ) such that the marginal distributions of τ and ˜τ are those of trajectories starting at s and s′,
+respectively.
+Definition 8.3 (Trajectory crossing time). The trajectory crossing time of two states s and s′ is the
+expected time for trajectories originating from s and s′ to cross under the best coupling that preserves
+the trajectories’ marginal distributions:
+H(s, s′) =
+min
+ψ∈Ψ(s,s′) E(τ,˜τ)∼ψ [inf{t|Ct(S, S′) ̸= ∅}]
+where Ct(S, S′) = {S0, . . . , St} ∩ {S′
+1, . . . , S′
+t} is the set of states visited by both trajectories at time t.
+The following theorem establishes that the mean squared error of the TD estimate of the advantage
+scales with the trajectory crossing time instead of the full horizon:
+Theorem 8.4. For s, s′ ∈ S,
+lim
+n→∞ n · E
+��
+ˆATD(s, s′) − A(s, s′)
+�2�
+≤ 2
+�
+σ2
+max
+min (P [s ∈ τ] , P [s′ ∈ τ])
+�
+· H(s, s′)
+with σ2
+max = maxs∈S Var [Rt + V (St+1) | St = s].
+Figure 3, in Section 6, provides an empirical illustration of this result.
+This result can actually understate the benefits the TD. Any trajectory pooling that happens before
+the trajectories cross further helps reducing the variance, but is not reflected in the upper bound of
+Theorem 8.4.
+Comparison with a coupling time.
+Trajectories are said to cross if one of the trajectories reaches
+a state already visited by the other one, potentially at an earlier time. It is different from the coupling
+time where trajectories have to reach a common state simultaneously. In particular, the crossing time
+is always smaller than the coupling time.The MRP defined in Figure 7 illustrates this. Trajectories
+starting at states s(1)
+1
+and s(2)
+1
+only couple when reaching the terminal state, after m + 1 timesteps.
+However, they cross in two timesteps, that is H
+�
+s(1)
+1 , s(2)
+1
+�
+= 2.
+11
+
+s(1)
+1
+s(2)
+1
+s2
+s3
+sm
+Figure 7: An example with no coupling but rapid crossing.
+Intuition for the result.
+Let us give intuition for Theorem 8.4. Under the MRP structure, two
+trajectories reaching a common state have the same future expected reward. Hence, when estimating
+the difference in expected total rewards along two trajectories, one starting at s, the other starting
+at s′, it is only useful to estimate them up until the state where trajectories cross. By computing
+estimates at every state, TD leverages this property: two trajectories reaching a common state (the
+crossing state) use the same estimate (the value at the crossing state) to update predictions along both
+trajectories. Since the same estimate is used, its value cancels out when computing the difference in
+values at s and s′. Whether the value at the crossing state is accurately estimated doesn’t affect the
+estimation of the advantage.
+9
+Open questions
+There is a subtle interplay between the choice of state representation and the benefits of imposing
+temporal consistency in value estimates. Consider again the problem in Figure 1. In that case, we chose
+to represent the ‘checkout page’ as a state, implying that the purchase probability at the checkout page
+does not depend on the initial webpage shown to the user. This makes a strong surrogacy assumption,
+which TD leverages to greatly improve data efficiency. An alternative representation of the state in the
+second period is of the form s = (website version i, checkout), retaining information about how the
+user navigated to the checkout. In this case, there is no trajectory pooling and our theory indicates
+that TD behaves as MC would. By using a very rich representation, which recalls much of the past,
+the benefits of TD disappear. Clearly, we want representations that are accurate, to avoid severe
+approximation errors. But we have shown that representations that are forgetful of aspects of the past
+offer enormous benefits; this lets value-to-go from intermediate states serve as surrogate outcomes.
+How should one balance this tradeoff while learning representations from data? This question closely
+relates to the one of understanding temporal consistency with function approximation.
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+14
+
+A
+Proofs.
+We state and prove results in the context of weighted value function which is a linear combination of
+the value function evaluated at individual states.
+Definition A.1 (Weighted value function). For a weighting over states π, the extended value function
+is defined as
+J(π) =
+�
+s∈S
+π(s)V (s)
+By setting π to be a mass point at a single state, we recover the value function at this state. When
+interpreting initial states as actions (as in Section 3.2), we recover randomized policy when using a
+distribution over actions as the weighting. In this case, the weighted value function is the expected
+value when playing according to the randomized policy. Note that our definition of weighted value
+function allows for any weighting of states, including negative weights. This will be useful for analyzing
+advantages V (s) − V (s′) by setting π(s) = 1 and π(s′) = −1.
+We also extend our definition of expected number of visits to weightings over initial states.
+Definition A.2 (Weighted expected number of visits). For a weighting over states π, we write ηπ(s)
+the weighted number of visits to s:
+ηπ(s) =
+�
+s′∈S
+π(s′)E [N(s)|S0 = s′]
+Similarly to the weighted value function, π is not enforced to be a distribution over state, allowing
+even for negative values. In the case where π is a distribution over state, we recover the probabilistic
+interpretation: ηπ(s) is the expected number of visits to state s when the initial distribution is π.
+Definition A.3 (One-step variance).
+σ2
+V (s) = Var [Rt + V (St+1) | St = s] .
+We extend trajectories into infinite horizon trajectories that stay in the terminating state and
+stop collecting rewards once the terminating state is reached: St = ∅ and Rt+1 = 0 for all t ≥ T.
+Equivalently, we define the transition P(∅ | ∅) = 1 and the reward R(∅, ∅) = 0 a.s.
+We start by stating and proving Central Limit Theorems (CLT) for the convergence of both TD
+and MC estimates. We then use these two results as building blocks to prove the main theorems.
+A.1
+Central Limit Theorems
+Proposition A.4 (Central Limit Theorem for MC). For s ∈ S,
+√n( ˆVMC(s) − V (s)) ⇒ N
+�
+0,
+1
+P [s ∈ τ]
+�
+s′∈S
+E [N(s′) | S0 = s] σ2
+V (s′)
+�
+Proof. We recall that, for tabular representation, the MC estimator takes the form
+ˆVMC(s) = E(S,V )∼DMC [V | S = s]
+=
+1
+| I(s) |
+�
+i∈I(s)
+T (i)
+�
+t=T (i)(s)
+R(i)
+t+1,
+where I(s) is the set of trajectories that visit state s and T (i)(s) is the first visit to state s in
+trajectory i. Since we consider first-visit MC, each trajectory appears at most once in the summation.
+15
+
+We start by rewriting the error ˆVMC(s) − V (s) as the product of a scaling factor and the average of
+i.i.d. random variables:
+ˆVMC(s) − V (s) =
+1
+| I(s) |
+�
+i∈I(s)
+�
+�
+T (i)
+�
+t=T (i)(s)
+R(i)
+t+1 − V (s)
+�
+�
+=
+n
+| I(s) | · 1
+n
+n
+�
+i=1
+�
+�1(s ∈ τ (i))
+�
+�
+T (i)
+�
+t=T (i)(s)
+R(i)
+t+1 − V (s)
+�
+�
+�
+�
+Recall that if s is not visited in trajectory i, T (i)(s) is defined to be T (i)(s) = T (i).
+• We start by proving a Central Limit Theorem on
+1
+n
+n
+�
+i=1
+�
+�1(s ∈ τ (i))
+T (i)
+�
+t=T (i)(s)
+R(i)
+t+1
+�
+� .
+The variables
+�
+1(s ∈ τ (i))
+��T (i)
+t=T (i)(s) R(i)
+t+1 − V (s)
+��
+i=1,...,n are n i.i.d., zero mean random
+variables. We now compute their variance.
+Var
+�
+�1(s ∈ τ)
+�
+�
+T
+�
+t=T (s)
+Rt+1 − V (s)
+�
+�
+�
+� = P [s ∈ τ] Var
+�
+�1(s ∈ τ)
+�
+�
+T
+�
+t=T (s)
+Rt+1 − V (s)
+�
+� | s ∈ τ
+�
+�
+= P [s ∈ τ] Var
+�
+�
+T
+�
+t=T (s)
+Rt+1 − V (s) | s ∈ τ
+�
+�
+Since the summation starts at the stopping time defined by the first visit to state s, the Strong
+Markov Property enables to re-index the summation in the following way:
+Var
+�
+�1(s ∈ τ)
+�
+�
+T
+�
+t=T (s)
+Rt+1 − V (s)
+�
+�
+�
+� = P [s ∈ τ] Var
+� T
+�
+t=0
+Rt+1 − V (s) | S0 = s
+�
+= P [s ∈ τ] Var
+� ∞
+�
+t=0
+Rt+1 − V (s) | S0 = s
+�
+where we allowed the sum to run to infinity since (Rt+1)t is a.s. stationary at 0 for t ≥ T.
+Similarly, we use the fact that (V (St) − V (St+1)) is a.s. stationary at 0 to write V (S0) =
+�∞
+t=0 (V (St) − V (St+1)). Plugging in the previous expression gives:
+Var
+�
+�1(s ∈ τ)
+�
+�
+T
+�
+t=T (s)
+Rt+1 − V (s)
+�
+�
+�
+� = P [s ∈ τ] Var
+� ∞
+�
+t=1
+(Rt+1 + V (St+1) − V (St)) |S0 = s
+�
+Notice that (Rt+1 + V (St+1) − V (St))t are martingale differences with respect to the filtration
+Ft = {S0, . . . St}. Using that martingale differences are uncorrelated:
+16
+
+Var
+�
+�1(s ∈ τ)
+�
+�
+T
+�
+t=T (s)
+Rt+1 − V (s)
+�
+�
+�
+� = P [s ∈ τ]
+∞
+�
+t=0
+E
+�
+(V (St+1) + Rt+1 − V (St))2|S0 = s
+�
+We then group the terms in the sum by the value of St:
+Var
+�
+�1(s ∈ τ)
+�
+�
+T
+�
+t=T (s)
+Rt+1 − V (s)
+�
+�
+�
+� = P [s ∈ τ]
+∞
+�
+t=0
+�
+s′∈S
+P [St = s′|S0 = s] E
+�
+(V (St+1) + Rt+1 − V (St))2|St = s′�
+= P [s ∈ τ]
+∞
+�
+t=0
+�
+s′∈S
+P [St = s′|S0 = s] σ2
+V (s′)
+= P [s ∈ τ]
+�
+s′∈S
+σ2
+V (s′)
+∞
+�
+t=0
+P [St = s′|S0 = s]
+= P [s ∈ τ]
+�
+s′∈S
+E [N(s′)|S0 = s] σ2
+V (s′)
+Using the Central Limit Theorem, we obtain the following convergence:
+1
+n
+n
+�
+i=1
+�
+�1(s ∈ τ (i))
+T (i)
+�
+t=T (i)(s)
+R(i)
+t+1
+�
+� ⇒ N
+�
+0, P [s ∈ τ]
+�
+s′∈S
+E [N(s′)|S0 = s] σ2
+V (s′)
+�
+.
+• The Strong Law of Large Number ensures:
+n
+| I(s) | −→
+n→∞
+1
+P [s ∈ τ]
+a.s..
+Finally, using Slutsky’s Theorem, the product converges:
+n
+| I(s) |· 1
+n
+n
+�
+i=1
+�
+�1(s ∈ τ (i))
+�
+�
+T (i)
+�
+t=T (i)(s)
+R(i)
+t+1 − V (s)
+�
+�
+�
+� ⇒ N
+�
+0,
+1
+P [s ∈ τ]
+�
+s′∈S
+E [N(s′)|S0 = s] σ2
+V (s′)
+�
+.
+Proposition A.5 (Central Limit Theorem for TD). For any weighting π,
+√n( ˆJTD(π) − J(π)) ⇒ N
+�
+0,
+�
+s′∈S
+η2
+π(s′)σ2
+V (s′)
+E [N(s′)]
+�
+.
+Corollary A.6. For s ∈ S
+√n( ˆVT D(s) − V (s)) ⇒ N
+�
+0,
+�
+s′∈S
+E [N(s′) | S0 = s]2 σ2
+V (s′)
+E [N(s′)]
+�
+.
+17
+
+A.2
+Proof of Proposition A.5
+Notations
+The idea of the proof is to use the fixed point identities verified by V and ˆVTD to obtain
+a recursive formula for the error ˆVTD − V . We then analyze the two quantities that appear when
+iterating this recursive formula: one is a sum of empirical one-step error and we show the other is of
+second order.
+We adopt a vectorial representation of the value function V = (V (s))s∈S such that J(π) = ⟨V, π⟩.
+Let T be the Bellman operator:
+T (f)(s) = ES′∼P (·|s) [R(s, S′) + f(S′)]
+The value function is the unique fixed point of the Bellman operator: V = T(V ) to verify V (∅) = 0.
+We also write the transition operator as follow:
+P(f)(s) = ES′∼P (·|s) [f(S′)] = ⟨P(· | s), f⟩.
+As discussed in Section 4.2, by trying to minimize temporal differences, TD solves the empirical
+Bellman equation:
+ˆVTD(s) = E(S,R,S′)∼DTD
+�
+R + ˆV TD(S′) | S = s
+�
+.
+Solving the empirical Bellman equation can be viewed as being a fixed point (with f(∅) = 0) of the
+empirical Bellman operator that we define as follow:
+ˆT (f)(s) = E(S,R,S′)∼DTD [R + f(S′) | S = s] .
+Similarly, we note the empirical transition operator
+ˆP(f)(s) = E(S,R,S′)∼DTD [f(S′) | S = s] .
+Finally, we introduce an explicit notation of the empirical Bellman operator that will ease proofs.
+To do so, we first need to introduce B(s), the set of all visits to state s:
+B(s) = {(i, t) | S(i)
+t
+= i}
+Using this notation, the empirical Bellman operator can be written as follow:
+ˆT (f)(s0) =
+1
+| B(s0) |
+�
+(i,t)∈B(s0)
+�
+f
+�
+S(i)
+t+1
+�
++ R(i)
+t+1
+�
+.
+Analysis
+We start by expanding the error vector ˆVTD − V into a sum of empirical one-step errors
+and a second term that we later show to be of second order.
+Lemma A.7.
+√n
+�
+ˆVTD − V
+�
+= √n
+∞
+�
+t=0
+P t( ˆT V − V ) +
+∞
+�
+t=0
+P t �√n( ˆP − P)( ˆVTD − V )
+�
+Note that, since P t(S0) = St is stationary at ∅ almost surely, the quantities summed in Lemma
+A.7 are ultimately zero almost surely.
+18
+
+Proof. By expending the difference ˆVTD − V using the fixed point identities, we obtain a recursive
+identity:
+ˆVTD − V = ˆT ˆVTD − T V
+= ( ˆT − T ) ˆVTD + T ˆVTD − T V
+= ( ˆP − P)( ˆVTD − V ) + ( ˆT − T )V + P( ˆVTD − V )
+= ( ˆP − P)( ˆVTD − V ) + ( ˆT V − V ) + P( ˆVTD − V ).
+Iterating this identity and multiplying by √n on both sides gives
+√n
+�
+ˆVTD − V
+�
+=
+∞
+�
+t=0
+P t �√n( ˆP − P)( ˆVTD − V )
+�
++ √n
+∞
+�
+t=0
+P t( ˆT V − V ).
+(1)
+We now state two lemmas that analyze the two terms that appear in equation (1). They are proved
+later.
+Lemma A.8. For all s ∈ S, as n → ∞,
+∞
+�
+t=0
+P t �√n( ˆP − P)( ˆVTD − V )
+�
+(s) → 0
+a.s.
+Lemma A.9. As n → ∞
+√n
+�
+s∈S
+π(s)
+∞
+�
+t=0
+P t( ˆT V − V )(s) ⇒ N
+�
+0,
+�
+s∈S
+ηπ(s)2
+E [N(s)]σ2
+V (s)
+�
+Finally, combining Lemma A.8 and Lemma A.9 with Slutstky Lemma concludes the proof.
+We now proceed to prove Lemma A.8 and Lemma A.9, starting with Lemma A.8
+A.2.1
+Proof of Lemma A.8
+Proof. We start by showing that, after appropriate rescaling the error in transition estimates converge
+to a Gaussian distribution. We do not calculate the variance here since it is not needed.
+Lemma A.10. For each s, √n
+�
+ˆP − P
+�
+(·|s) weakly converges to a normal distribution with mean
+zero.
+Proof. We first express ˆP(s′|s) − P(s′|s) as the product of a scaling factor and the average of n i.i.d
+random vectors:
+�
+ˆP(s′|s) − P(s′|s)
+�
+s′∈S =
+1
+| B(s) |
+�
+(i,t)∈B(s)
+�
+1(S(i)
+t+1 = s′) − P(s′|s)
+�
+s′∈S
+=
+n
+| B(s) | · 1
+n
+n
+�
+i=1
+� ∞
+�
+t=0
+1(S(i)
+t
+= s)
+�
+1(S(i)
+t+1 = s′) − P(s′|s)
+��
+s′∈S.
+We decompose this identity in two terms:
+19
+
+• Strong law of large numbers ensures that
+n
+| B(s) | −→
+n→∞
+1
+E [N(s)]
+a.s..
+• Since trajectories are independent,
+∞
+�
+t=0
+1(S(i)
+t
+= s)
+�
+1(S(i)
+t+1 = s′) − P(s′|s)
+�
+s′∈S
+are independent, identically distributed random variables with finite variance and mean 0. The
+Central Limit Theorem ensures that the rescaled average of these random variables
+√n · 1
+n
+n
+�
+i=1
+∞
+�
+t=0
+1(S(i)
+t
+= s)
+�
+1(S(i)
+t+1 = s′) − P(s′|s)
+�
+s′∈S
+weakly converges to a mean 0 normal distribution.
+Finally, the product of these two quantities also converges to a mean 0 normal distribution.
+We terminate the proof of Lemma A.8 by combining Lemma A.10 with the fact that ˆVTD − V
+converges almost surely to 0 by Slutsky’s Theorem.
+A.2.2
+Proof of Lemma A.9
+Proof. We start by expanding the transition operator P t,
+√n
+∞
+�
+t=0
+P t( ˆT V − V )(s) = √n
+∞
+�
+t=0
+E
+�
+ˆT V (St) − V (St) | S0 = s, ˆT
+�
+This term consists of a sum one-step differences of the form √n
+�
+ˆT V (S(i)
+t ) − V (S(i)
+t )
+�
+. Grouping
+S(i)
+t
+that are equal:
+√n
+∞
+�
+t=0
+P t( ˆT V − V )(s) = √n ·
+∞
+�
+t=0
+E
+��
+s′∈S
+1(St = s′)
+�
+ˆT V (s′) − V (s′)
+�
+| S0 = s, ˆT
+�
+= √n · E
+��
+s′∈S
+∞
+�
+t=0
+1(St = s′)
+�
+ˆT V (s′) − V (s′)
+�
+| S0 = s, ˆT
+�
+= √n
+�
+s′∈S
+E
+� ∞
+�
+t=0
+1(St = s′) | S0 = s′
+� �
+ˆT V (s′) − V (s′)
+�
+= √n
+�
+s′∈S
+E [N(s′)|S0 = s]
+�
+ˆT V (s′) − V (s′)
+�
+.
+To get the result for a general weighting of states π, we take the dot product of the previous identity
+with π:
+√n
+�
+s∈S
+π(s)
+∞
+�
+t=0
+P t( ˆT V − V )(s) = √n
+�
+s∈S
+ηπ(s)
+�
+ˆTV (s) − V (s)
+�
+.
+(2)
+20
+
+We are left with analyzing a linear combination of terms of the form ˆT V (s)−V (s). For an individual
+s, ˆT V (s) − V (s) is an average of i.i.d. variables and its asymptotical behavior can be controlled using
+a Central Limit Theorem. However, to find the limiting distribution of the linear combination, we need
+to prove a vectorial Central Limit Theorem for the random vector
+�
+ˆT V (s) − V (s)
+�
+s∈S. In particular,
+we show that
+�
+ˆT V (s) − V (s)
+�
+s∈S converge to independent normal distributions.
+The following lemma is key to prove the independence of the limiting distribution. It states that
+one-step differences are uncorrelated:
+Lemma A.11. For (s, t) ̸= (s′, t′):
+Cov [1(St = s) (V (St) − V (St+1) − Rt+1) , 1(St′ = s′) (V (St′) − V (St′+1) − Rt′+1)] = 0.
+Proof. The result follows from the Markovian property:
+Cov [1(St = s) (V (St) − V (St+1) − Rt+1) , 1(St′ = ˜s) (V (St′) − V (St′+1) − Rt′+1)]
+= E
+�
+Cov
+�
+1(S(i)
+t
+= s) (V (St) − V (St+1) − Rt+1) , 1(St′ = s′)
+�
+V (St′) − V (St′+1) − R(i)
+t′+1
+�
+| St, St+1, St′
+��
+= E [1(St = s) (V (St) − V (St+1) − Rt+1) 1(St′ = s′) (V (St′) − E [V (St′+1) + Rt′+1|St′])]
+= 0.
+Given the previous result, we can now state and proof the joint Central Limit Theorem of empirical
+one-step differences:
+Lemma A.12. As n → ∞,
+√n
+��
+ˆTV (s) − V (s)
+��
+s∈S ⇒ N (0, Σ)
+where Σ is a diagonal matrix with
+Σs,s =
+1
+E [N(s)]σ2
+V (s).
+Proof. We use a similar approach as in the proof of Lemma A.10: we express ˆT V (s) − V (s) as the
+product of a scaling factor that converges almost surely and an average of i.i.d. random vectors that is
+controlled by the Central Limit Theorem:
+√n ·
+�
+ˆT V (s) − V (s)
+�
+= √n ·
+1
+| B(s) |
+�
+(i,t)∈B(s)
+(V (S(i)
+t+1) + R(i)
+t+1 − V (s))
+= √n ·
+1
+| B(s) |
+n
+�
+i=1
+∞
+�
+t=0
+1(S(i)
+t
+= s)(V (S(i)
+t+1) + R(i)
+t+1 − V (S(i)
+t ))
+=
+n
+| B(s) |
+�
+√n · 1
+n
+∞
+�
+i=1
+∞
+�
+t=0
+1(S(i)
+t
+= s)(V (S(i)
+t+1) + R(i)
+t+1 − V (S(i)
+t ))
+�
+• B(s) can be rewritten as the sum of i.i.d. random variables:
+B(s) =
+n
+�
+i=1
+� ∞
+�
+t=0
+1(S(i)
+t
+= s)
+�
+.
+21
+
+The average | B(s) |
+n
+converges almost surely to E [N(s)] by the Strong Law of Large Numbers.
+Taking the inverse gives:
+n
+| B(s) | →
+1
+E [N(s)]
+a.s.
+(3)
+• The random vector
+√n · 1
+n
+n
+�
+i=1
+� ∞
+�
+t=0
+1(S(i)
+t
+= s)(V (S(i)
+t+1) + R(i)
+t+1 − V (S(i)
+t ))
+�
+s∈S
+is the re-scaled average of n i.i.d., mean 0, vectors of the form (�∞
+t=0 1(St = s)(V (St+1) + Rt+1 − V (St)))s∈S.
+The Central Limit Theorem ensures that this quantity converges to a normal distribution with
+mean 0. We now proceed to compute the variance of this distribution.
+– Lemma A.11 ensures that entries of this vector are uncorrelated:
+– The variance of a single entry is given by
+Var
+� ∞
+�
+t=0
+1(St = s)(V (St+1) + Rt+1 − V (St))
+�
+(a)
+=
+∞
+�
+t=0
+Var [1(St = s)(V (St+1) + Rt+1 − V (St))]
+=
+∞
+�
+t=0
+E
+�
+1(St = s)(V (St+1) + Rt+1 − V (St))2�
+=
+∞
+�
+t=0
+E
+�
+1(St = s)E
+�
+(V (St+1) + Rt+1 − V (St))2|St
+��
+=
+∞
+�
+t=0
+E
+�
+1(St = s)σ2
+V (St)
+�
+= E [N(s)] σ2
+V (s)
+where (a) also follows from Lemma A.11.
+From the Central Limit Theorem, we obtain:
+√n· 1
+n
+n
+�
+i=1
+� ∞
+�
+t=0
+1(S(i)
+t
+= s)(V (S(i)
+t+1) + R(i)
+t+1 − V (S(i)
+t ))
+�
+s∈S
+⇒ N
+�
+0, Diag
+�
+(E [N(s)] σ2
+V (s))s∈S
+��
+(4)
+Combining (3) and (4) gives the result.
+Finally, we just need to take the dot product of √n ·
+�
+ˆT V (s) − V (s)
+�
+s∈S with the weighted
+occupancy measure ηπ to get the result:
+√n
+�
+s∈S
+π(s)
+∞
+�
+t=0
+P t( ˆT V − V )(s) ⇒ N
+�
+0,
+�
+s∈S
+ηπ(s)2
+E [N(s)]σ2
+V (s)
+�
+22
+
+A.3
+Proof of Theorem 7.2
+The proof follows directly from Proposition A.4 and Proposition A.5.
+From, Proposition A.4, we have
+lim
+n→∞
+√n · E
+��
+ˆVMC(s) − V (s)
+�2�
+=
+1
+P [s ∈ τ]
+�
+s′∈S
+E [N(s′)|S0 = s] σ2
+V (s′) .
+Similarly, from Proposition A.5, we have
+lim
+n→∞
+√n · E
+��
+ˆVTD(s) − V (s)
+�2�
+=
+�
+s′∈S
+E [N(s′)|S0 = s]2 σ2
+V (s′)
+E [N(s′)]
+.
+Taking the ratio of these two limits, we obtain:
+lim
+n→∞
+E
+��
+ˆVTD(s) − V (s)
+�2�
+E
+��
+ˆVMC(s) − V (s)
+�2� =
+1
+�
+s′∈S E [N(s′) | S0 = s] σ2
+V (s′)
+�
+s′∈S
+E [N(s′) | S0 = s] σ2
+V (s′) E [N(s′) | S0 = s] P [s ∈ τ]
+E [N(s′)]
+= Es′∼µ(s)
+�E [N(s′) | S0 = s] P [s ∈ τ]
+E [N(s′)]
+�
+where µ(s) is defined in Definition 7.1.
+Finally,
+E [N(s′) | S0 = s] P [s ∈ τ] = E [N(s → s′)] .
+A.4
+Proof of Theorem 8.4
+We define π to be such that π(s) = 1, π(s′) = −1 and π(˜s) = 0 for all ˜s.
+That way J(π) =
+V (s) − V (s′) = A(s, s′). This allows to use Lemma A.5:
+lim
+n→∞ n · E
+��
+ˆJTD(π) − J(π)
+�2�
+=
+�
+˜s∈S
+(E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′])2 σ2
+V (˜s)
+E [N(˜s)]
+We decompose the sum into three terms that we bound separately:
+lim
+n→∞ n · E
+��
+ˆJTD(π) − J(π)
+�2�
+≤
+�
+max
+˜s∈S
+����
+E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′]
+E [N(˜s)]
+����
+�
+·
+�
+max
+˜s∈S σ2
+V (˜s)
+�
+×
+�
+˜s∈S
+|E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′]| .
+(5)
+We start by proving that if trajectories where S0 is sampled from the initial distribution visit s
+frequently often, then the occupancy measure E [N(˜s)] and E [N(˜s) | S0 = s] cannot differ too much
+(and symmetrically for s′).
+If E [N(˜s) | S0 = s] ≥ E [N(˜s) | S0 = s′] then
+����
+E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′]
+E [N(˜s)]
+���� ≤
+E [N(˜s) | S0 = s]
+P [s ∈ τ] E [N(˜s) | S0 = s] =
+1
+P [s ∈ τ].
+23
+
+When E [N(˜s) | S0 = s] < E [N(˜s) | S0 = s′], a symmetric argument ensures that
+����
+E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′]
+E [N(˜s)]
+���� ≤
+1
+P [s′ ∈ τ].
+Combining these two cases gives:
+max
+˜s∈S
+����
+E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′]
+E [N(˜s)]
+���� ≤
+1
+min (P [s ∈ τ] , P [s′ ∈ τ]).
+Plugging this result in Equation 5:
+lim
+n→∞ n·E
+��
+ˆJTD(π) − J(π)
+�2�
+≤
+max˜s∈S σ2
+V (˜s)
+min (P [s ∈ τ] , P [s′ ∈ τ])·
+�
+˜s∈S
+|E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′]| .
+(6)
+Finally, we show that the last sum scales as the crossing time. To give intuition on why this holds,
+note that we have the following identity:
+�
+˜s∈S
+E [N(˜s) | S0 = s] = E [T|S0 = s] .
+E [N(˜s) | S0 = s] is the expected number of visits to ˜s whens starting at s: summing over ˜s is the
+expected total number of states visited when starting at s (not counting the terminating state ∅) which is
+exactly the horizon. In the case of the comparison of trajectories, |E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′]|
+is how many additional times ˜s has been visited by one of the trajectories compared to the other, in
+expectation. Summing over ˜s gives the expected total number of states that have been visited by only
+one of the trajectories, which is at most twice the number of states visited before both trajectories
+reach a common state.
+Lemma A.13.
+�
+˜s∈S
+|E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′] | ≤ 2H(s, s′)
+Proof. Let τ, τ ′ = (S0, R1, . . . , ST −1, RT , ∅), (S′
+0, R′
+1, . . . , S′
+T −1, R′
+T ′, ∅) ∼ ψ where ψ is a joint distribu-
+tion such that τ and τ ′ are marginally distributed like trajectories generated with S0 = s and S′
+0 = s′,
+respectively. We define the crossing time for ψ is defined as the first time a state has been visited by
+both trajectories:
+Hψ(s, s′) = inf{t|{S0, . . . , St} ∩ { ˜S0, . . . , ˜St} ̸= ∅}.
+Since we always consider the crossing time of s and s′, we omit the s and s′ dependency and
+simply write Hψ for convenience in the rest of the proof. By definition, either S′
+Hψ ∈ {S1, . . . , SHψ}
+or SHψ ∈ {S′
+1, . . . , S′
+Hψ}. Without loss of generality, we assume S′
+Hψ ∈ {s1, . . . , sHψ} holds. Let
+Nψ = inf{t|St = S′
+Hψ}, that is SNψ = S′
+Hψ.
+We now construct a new trajectory that follows τ ′ until the crossing state SNψ = S′
+Hψ is reached
+and then follows the trajectory τ: ˆτ = (S′
+0, R′
+1, . . . , S′
+Hψ = SNψ, RN+1, SN+1, . . . , ST −1, RT , ∅). By
+Markov property, ˆτ is identically distributed as τ ′.
+We are interested in bounding the difference in occupancy measure:
+E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′] = E
+� ∞
+�
+t=0
+1(St = ˜s)
+�
+− E
+� ∞
+�
+t=1
+1(S′
+t = ˜s)
+�
+.
+24
+
+Using that ˆτ and τ ′ are identically distributed, this expression can be rewritten as
+E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′] = E
+� ∞
+�
+t=0
+1(St = ˜s)
+�
+− E
+� ∞
+�
+t=0
+1( ˆSt = ˜s)
+�
+= E
+�
+�
+Nψ−1
+�
+t=0
+1(St = ˜s)
+�
+� − E
+�
+�
+Hψ−1
+�
+t=0
+1(S′
+t = ˜s)
+�
+�
+Taking absolute values and summing over ˜s gives
+�
+˜s∈S
+|E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′]| =
+�
+˜s∈S
+E
+�
+�
+������
+Nψ−1
+�
+t=0
+1(St = ˜s) −
+Hψ−1
+�
+t=0
+1(S′
+t = ˜s)
+������
+�
+�
+≤
+�
+s′∈S
+E
+�
+�
+Nψ−1
+�
+t=0
+1(St = ˜s)
+�
+� +
+�
+˜s∈S
+E
+�
+�
+Hψ−1
+�
+t=0
+1(St = s′)
+�
+�
+= E [Nψ] + E [Hψ]
+≤ 2E [Hψ]
+Taking the infimum over all ψ ∈ Ψ(s, s′) that conserves marginal distribution gives the result.
+Finally, plugging Lemma A.13 in Equation 6 leads to the result:
+lim
+n→∞ n · E
+��
+ˆJTD(π) − J(π)
+�2�
+≤ 2
+max˜s∈S σ2
+V (˜s)
+min (P [s ∈ τ] , P [s′ ∈ τ]) · H(s, s′)
+A.4.1
+Proof of Proposition 8.1
+We now prove that the MC estimate of the advantage can scale as the full horizon, no matter how
+small the crossing time is. We focus on the advantage of s over s′ when P [s ∈ τ ∧ s′ ∈ τ] = 0, that
+is no trajectory can visit both s and s′. No trajectories visiting both s and s′ means that the MC
+value estimates at s and s′ are independent since they rely on disjoints sets of trajectories. In terms of
+variance, this implies:
+Var
+�
+ˆAMC(s, s′)
+�
+= Var
+�
+ˆVMC(s) − ˆVMC(s′)
+�
+= Var
+�
+ˆVMC(s)
+�
++ Var
+�
+ˆVMC(s′)
+�
+.
+It now suffices to prove that for any s,
+lim
+n→∞ Var
+�
+ˆVMC(s)
+�
+≥
+σ2
+min
+P [s ∈ τ]E [T|S0 = s] .
+From Proposition A.4, we know the variance of the MC estimate is
+lim
+n→∞ n · Var
+�
+ˆVMC(s)
+�
+=
+1
+P [s ∈ τ]
+�
+˜s∈S
+E [N(˜s) | S0 = s] σ2
+V (˜s) .
+Using that σ2
+V (˜s) ≥ σ2
+min for all ˜s ∈ S, this expression simplifies to:
+lim
+n→∞ n · Var
+�
+ˆVMC(s)
+�
+≥
+σ2
+min
+P [s ∈ τ]
+�
+∈S
+E [N(˜s) | S0 = s] .
+25
+
+Finally,
+�
+s′∈S
+E [N(s′) | S0 = s] =
+�
+˜s∈S
+E
+� T
+�
+t=0
+1(St = ˜s)|S0 = s
+�
+= E
+� T
+�
+t=0
+�
+˜s∈S
+1(St = ˜s)|S0 = s
+�
+= E [T|S0 = s]
+26
+
diff --git a/JtFQT4oBgHgl3EQfSTYo/content/tmp_files/load_file.txt b/JtFQT4oBgHgl3EQfSTYo/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..1d510b2bf5320c4ec4004b254b3de13cfd035d26
--- /dev/null
+++ b/JtFQT4oBgHgl3EQfSTYo/content/tmp_files/load_file.txt
@@ -0,0 +1,987 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf,len=986
+page_content='On the Statistical Benefits of Temporal Difference Learning  David Cheikhi  Daniel Russo  Columbia University  Abstract  Given a dataset on actions and resulting long-term rewards, a direct estimation approach fits  value functions that minimize prediction error on the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Temporal difference learning  (TD) methods instead fit value functions by minimizing the degree of temporal inconsistency  between estimates made at successive time-steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Focusing on finite state Markov chains, we  provide a crisp asymptotic theory of the statistical advantages of this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' First, we show that  an intuitive inverse trajectory pooling coefficient completely characterizes the percent reduction in  mean-squared error of value estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Depending on problem structure, the reduction could be  enormous or nonexistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Next, we prove that there can be dramatic improvements in estimates of  the difference in value-to-go for two states: TD’s errors are bounded in terms of a novel measure  — the problem’s trajectory crossing time — which can be much smaller than the problem’s time  horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  1  Introduction  Temporal difference learning is a distinctive approach to estimation in long-term optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Its importance to reinforcement learning is hard to overstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In their seminal book, Sutton & Barto  (2018) write: If one had to identify one idea as central and novel to reinforcement learning, it would  undoubtedly be temporal difference learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Competing with temporal difference (TD) learning is a straightforward direct-estimation approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  There, one proceeds by collecting data on past decisions and the cumulative long-term ‘reward’ that  followed them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' If actions were chosen with some experimental randomness, then regression of long-term  rewards on the draw of actions would – with enough data – correctly identify actions’ causal impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  The direct approach has two significant drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The first is delay: actions can only be evaluated  after their full long-term effects realize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The second is variance: long-term outcomes can be extremely  noisy and individual actions often have a small impact on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  TD aims to alleviate these challenges by leveraging data on intermediate outcomes – those observed  after the decision but before final outcomes are realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The availability of such data is increasingly  common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Robots collect regular sensor measurements, recommendation systems log sequential user  interactions, and digital devices can track patient’s health metrics across time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Sutton (1988) observed  that successive predictions, updated as information is gathered, should be temporally consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' He  proposed to fit maximally consistent value estimates by iteratively minimizing ‘temporal difference  errors’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' TD has since become an intellectual pillar of the reinforcement learning literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' It is used  in most successful applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Our Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  We aim to provide crisp insight into the statistical benefits of TD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' This paper  focuses on the simplest possible setting, where training data consists of a batch of trajectories sampled  independently from a finite Markov reward process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We compare the asymptotic scaling of mean  squared error under TD and direct value estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Two main findings emerge:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The relative benefits of TD are determined by a natural ‘inverse trajectory pooling coefficient.’  TD uses value-to-go at intermediate states as a surrogate Athey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Prentice (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='13289v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='LG]  30 Jan 2023  This is beneficial exactly when trajectories that originate with distinct states/actions tend to  reach common intermediate states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We present simple examples illustrating when the benefits of  TD are enormous and when they vanish entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' TD is especially beneficial for advantage estimation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' That is, when estimating the difference  in value-to-go from one state/action versus another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' While the mean squared error of direct  advantage estimation generally scales with the length of the problem horizon, we show that TD’s  errors are bounded by a smaller trajectory crossing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' This novel notion of effective horizon  can be small even in some problems with unbounded mixing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  On the Markov assumption and state representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Our focus on Markov models is standard  in the academic literature on reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' This is, in a certain technical sense, an innocuous  assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' One could always use the entire sequence of observations so far as a Markov state  Puterman (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' But practical algorithms need to use appropriate compression of the history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The  choice of representation has a subtle interplay with the benefit of TD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Indeed, we comment in section  9 that the benefits of TD can vanish when the state representation is too rich and trajectory pooling  vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  2  Related works  TD has been a central idea in RL since it was first proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' It is deceptively simple, has intriguing  connections to neuroscience Schultz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' (1997), and seems to be routed in dynamic programming  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In the 1990s, researchers gathered both limited convergence guarantees Dayan & Sejnowski  (1994);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Jaakkola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' (1993) and examples of divergence Baird (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Tsitsiklis & Van Roy (1997)  offered a clarifying theory of when TD converges, and characterized the TD fixed point it reaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Maei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' (2009);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Sutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' (2009) proposed methods to reach the TD fixed point in off-policy  settings or when nonlinear function approximation is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  While illuminating, this theory does not clarify why TD should be preferred over direct value  function estimation (dubbed ‘Monte-Carlo’ or ‘MC’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In fact, the main guarantee is convergence to an  approximate value function whose mean-squared error is larger than the one MC reaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Folklore,  intuition, and experiments suggest TD often converges to its limit at a faster rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  The literature has emphasized the distinction between online and batch TD algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The  convergence speed of batch TD methods, like LSTD Bradtke & Barto (1996), is a statistical question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  With purely online algorithms, each observation is used to make a single stochastic gradient type  update and then immediately discarded, so issues of memory, compute, and data efficiency are conflated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  The deep RL literature has adopted experience replay Andrychowicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Schaul et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' (2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' (2016), which blurs the line between batch and online implementations  by recording observations in a dataset and resampling them many times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  We give a complete and intuitive characterization of the efficiency benefits of TD in the simplest  possible setting: a batch variant applied without function approximation in a finite state Markov Reward  Process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Here it is straightforward to show TD is more efficient that MC, but more subtle to understand  when the efficiency gains are large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Grunewalder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' (2007) and Gr¨unew¨alder & Obermayer (2011)  make progress in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' They prove that LSTD is at least as statistically efficient as MC,  without quantifying the improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' They also display cases where the two procedures have the same  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Textbooks by Sutton & Barto (2018) and Szepesv´ari (2010) give illuminating examples,  but no theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  A number of papers bound the data requirements of TD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' See for example Mannor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' (2004), Lu  (2005)), Lazaric et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' (2010), Pires & Szepesv´ari (2012), Tagorti & Scherrer (2015)), Bhandari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  (2018), Pananjady & Wainwright (2020), Khamaru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' (2020), Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' (2020), or Farias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' These show certain problem instances have low data requirements, but do not clarify when  2  enforcing temporal consistency in value estimates produces large efficiency gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' To the best of our  knowledge, our insights about advantage estimation are new (See Sec 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  3  Problem Formulation  We first describe the problem of value function estimation in Markov reward processes (MRPs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We  then observe that after appropriate relabeling of the state variables, this can also represent the problem  of evaluating the long-term impact of actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Most mathematical results are stated for MRPs, but  the alternative interpretation enriches the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1  Value function estimation  A trajectory in a terminating Markov reward process is a Markovian sequence  τ = (S0, R1, S1, R2, S2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' , ST −1, RT , ∅),  consisting of a sequence of states (St)t∈[T ] ⊂ S, rewards (Rt)t∈[T ] ⊂ R, and termination time T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The  termination time is the first time at which ST = ∅, where ∅ is thought of as a special terminal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Assume the distribution of Rt is independent of past rewards given the current state St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  The law of a Markov reward process (MRP) is specified by the tuple M = (S∪{∅}, P, R, d) consisting  of a state space, transition kernel, reward distribution, and initial state distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Here P is a  transition matrix over the augmented state space S ∪∅, specifying a probability P(s′ | s) of transitioning  from s to s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We assume terminal sate is absorbing and reachable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' That is, P(∅|∅) = 1 and for every  state s there is some t such that the t step transition P t(∅|s) is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The object R specifies  the draw of rewards conditioned on a state transition as R(dr|s, s′) = P(Rt = dr | St = s, St+1 = s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Throughout we use the notation r(s, s) for the mean of R(·|s, s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We assume r(∅, ∅) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The initial  state distribution d is a probability distribution over S from which S0 is drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  The value function  V (s) = E  � ∞  �  t=1  Rt | S0 = s  �  = E  � T  �  t=1  r(St, St+1) | S0 = s  �  specifies the expected future reward earned prior to termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' It is immediate from our formulation  that V (∅) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Our formulation is the Markov reward process analogue of stochastic shortest path  problems (Bertsekas & Tsitsiklis, 1991).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Discounted problems are a special case where there is a  constant probability of termination P(∅ | s) = 1 − γ for each non-terminal state s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In that case,  the horizon T follows a geometric distribution with mean 1/(1 − γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  A related quantity measures the value-to-go differences between states,  A(s, s′) = V (s) − V (s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  We call this the advantage of s over s′, since, as revealed in the next subsection, it is closely related to  the advantage function in RL Baird III (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  We consider the problem of estimating the value-to-go at initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We compare methods that  produce estimates ˆV based on n independent trajectories  D =  �  τi = (S(i)  0 , R(i)  1 , S(i)  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' , S(i)  T (i)−1, R(i)  T (i), ∅)  �  i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=',n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  by their mean squared error E  ��  V (s) − ˆV (s)  �2�  or E  ��  A(s, s′) − ˆA(s, s′)  �2�  , where the expectation  is over the randomness in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We assume that all states have a non-zero probability of being visited in  the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  3  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='2  Heterogenous treatment effect estimation  By appropriate relabeling of variables, we can interpret our problem as one of evaluating the long-term  impact of a chosen decision in a specific context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Here we consider a special case of our formulation where the continuing state space S = S0 ∪ SI  is partitioned into a set of initial states S0 and a set of intermediate states SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' With probability 1,  S0 ∈ S0, ST = ∅, and at intermediate times t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' , T − 1}, St ∈ SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  We give the initial states a special interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We think of them as consisting of an initial context  X0 and a decision A0 and write S0 = (X0, A0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Using the more familiar notation Q(X0, A0) ≡ V (S0),  we have  Q(x, a) = E  � T  �  t=1  Rt | X0 = x, A0 = a  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  The initial distribution d0 is determined by an initial context distribution P(X0 = x) and a logging  policy P(A0 = x | X0 = x) which determines the frequency with which actions are observed in the  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Of particular interest in this setting is the advantage  A((x, a), (x, a′)) = Q(x, a) − Q(x, a′),  which measures the performance difference between actions a and a′ in context x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The term ‘advantage’  is common in RL, but in causal inference one might call this ‘heterogenous treatment effect’ estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Since policy gradient methods typically involve computing the expectation of weighted advantages,  we expect that insights developed in this paper could apply to these methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  4  Algorithms  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1  Direct approach: First visit monte-Carlo (MC)  For any candidate value function, we can evaluate its accuracy by comparing the future value it predicts  from a given state visited in the data and the actual cumulative reward observed in the remainder of  that trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' This suggests a natural direct value estimation approach: over a candidate class of  value functions, pick one that minimizes mean squared prediction error on the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' This method is  called Monte Carlo in the RL literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  To formally describe the algorithm, define the random time T(s) = min{t : St = s ∨ St = ∅} to be  the first hitting time of state s if it is reached, or T otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Let I(s) = {i ∈ [n] : s ∈ τi} to be set of  trajectories that visit state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Form a dataset  DMC =  �  s∈S  �  i∈I(s)  �  �  �  �  �s,  T (i)  �  t=T (i)(s)+1  R(i)  t  �  �  �  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  that records pairs of states and the cumulative rewards earned following the first visit to the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Given a parameterized class of value functions {Vθ : θ ∈ Θ}, a direct value estimation approach is to  solve the least squares problem  min  θ∈Θ  �  (s,v)∈DMC  (Vθ(s) − v)2  We focus on tabular representations, where the space of parameterized value functions spans all  possible functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In that case  ˆVMC(s) = E(S,V )∼DMC [V | S = s] ,  4  where the notation (S, V ) ∼ DMC means that state/value pairs are sampled uniformly at random from  the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The value estimate at state s is simply the average reward earned after visits to state s in  the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  What we described is called first visit Monte-carlo in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' It is an unbiased estimator  because it only includes the first time a state was visited during the an episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We focus on this  version for analytical simplicity, but many of our key examples focus on cases where initial states are  never revisited (See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='2) and it coincides with an “every visit” Monte Carlo estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='2  Indirect approach: TD learning  Temporal difference learning uses a reformatted dataset consisting of tuples of reward realizations and  state transitions:  DTD = {(S(i)  t , R(i)  t+1, S(i)  t+1)t=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=',T (i),i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=',n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Define the temporal difference error between two candidate value functions V, V ′ to be the average  gap in Bellman’s equation:  ℓTD(V, V ′) = E(S,R,S′)∼DTD [V (S) − (R + V ′(S′))] ,  where the notation (S, R, S′) ∼ DTD means that tuples are sample uniformly at random from the  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Given a parameterized class of value functions, VΘ = {Vθ : θ ∈ Θ}, batch TD algorithms  iteratively generate parameters (θ1, θ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=') by solving minθi+1 ℓ(Vθi+1, Vθi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' (Online TD algorithms,  combined with experience replay, make SGD updates instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=') For linear function approximation  (Tsitsiklis & Van Roy, 1996), or neural networks in the ‘Neural tangent kernel’ regime (Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=',  2019), value functions are known to converge to a fixed point  ˆVTD = arg min  V ∈VΘ  ℓ  �  V, ˆVTD  �  ,  which, in a sense, maximizes feasible temporal consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Again, we focus on tabular representations, where the space of parameterized value functions spans  all possible functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In that case, TD solves the empirical Bellman equation  ˆVTD(s) = E(S,R,S′)∼DTD  �  R + ˆVTD(S′) | S = s  �  5  Intuition: surrogacy and intermediate outcomes  Figure 1: Modeling a user’s behavior  A lot of the intuition regarding TD can be gained through the simple example in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Imagine  our goal is to select the website design among 100 alternatives that leads to the largest sale rate (of  some product).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Users arrive and are randomly assigned to one of the 100 pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' They either click to  5  Webpage  Version 1  Checkout  page,  Sale  Webpage  Version 100  No Salepurchase and proceed to the checkout page or navigate away from the site without purchasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Among  those who click, only a small fraction complete the sale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Assume, for simplicity, that we have no access to personal information that distinguishes users  from one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' (There is only one possible x in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=') There are 100 possible initial states,  corresponding to the webpage version, and the user is equally likely to start at each of those.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' A sale  and a no-sale immediately precede termination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We call the checkout page an intermediate state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The  sale state is associated with a reward of 1 and all others are associated with a reward of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  What we called the Monte-Carlo estimate of the value function would directly estimate the value  of an impression of each webpage to be the fraction who purchased among that cohort of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Due to the directed nature of state transitions, TD estimation can be thought of in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We  first estimate ˆV (checkout) to be the fraction who purchased among users who visit the checkout page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  We then estimate the value of an impression of webpage i by  ˆV TD(webpage i) = CTR(i) × ˆV (checkout),  the empirical click-through rate among those shown webpage i times the estimated sale rate on the  checkout page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Monte-carlo estimation is unbiased, but it may be difficult to reliably estimate the efficacy of each  webpage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' If only a small fraction click initially, and among those who do only a small fraction convert  to a sale, one would need to show each webpage to an enormous number of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' With TD, we pool  data from across users who were shown any of the 100 webpages when estimating ˆV (checkout), greatly  reducing variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In this example, there is a lot of data pooling because trajectories that begin at  distinct states quickly converge to the intermediate state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In fact, our theory reveals that certain  intuitive measures of trajectory pooling exactly determine degree of statistical benefit TD provides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Another view of TD is that it uses the intermediate click/no-click outcome as a surrogate or  proxy-metric Prentice (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Recognizing this, TD’s potential downsides become as transparent as a  its benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' If the conversion probability among users who visit the checkout page depends strongly  on which page design they saw, the Markov property does not hold and TD is biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We discuss this  example again in the conclusion, mentioning how the risks and benefits interplay with the choice of  state representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  6  Empirical illustration  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1  The benefits of TD  To illustrate how much TD can improve over MC, we explore an example: we consider a layered MRP  as describe in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' A layered MRP with width W and horizon T has W × (T − 1) states split in  T layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' States in layer t can only transition to states in layer t + 1 and states in the last layer T − 1  always transition to the terminal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  We consider a Layered MRP with width W = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We focus on state s(1)  1  and s(2)  1  and study the  accuracy of the estimates of their value as we vary the horizon T of the MRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We also study the  accuracy of the estimate of the advantage A  �  s(1)  1 , s(2)  1  �  = V  �  s(1)  1  �  − V  �  s(2)  1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Figure 3 displays the  Mean Square Error (MSE) of the TD and MC estimates for these quantities when the dataset contains  n = 2000 independent trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' MSE calculations involve 10000 Monte-Carlo replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Alongside  the observed MSE, we plot projected MSE based on the central limit theorem from Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='4  and Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' There is almost perfect alignment between asymptotic and finite sample results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  We highlight three main take-aways from this example:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' TD can vastly outperform MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' For the chosen states s(1)  1  and s(2)  1 , the MSE is about 5  times smaller when using TD instead of MC for a Layered MRP with width W = 5 and horizon  T = 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  6  s(1)  1  s(2)  1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  s(W )  1  s(1)  2  s(2)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  s(W )  2  s(1)  T −1  s(2)  T −1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  s(W )  T −1  ∅  Figure 2: Layered MRP with width W and horizon T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Transitions are chosen randomly and rewards  are uniform on [r(s, s′) − 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' r(s, s′) + 1] where r(s, s′) is chosen uniformly between -1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  0  20  40  60  80  100  120  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='5  Horizon T  MSE  (a) Full Y-Axis  0  20  40  60  80  100  120  0  1 · 10−2  2 · 10−2  3 · 10−2  4 · 10−2  5 · 10−2  Horizon T  MSE  (b) Truncated Y-Axis  Value at s  Asymptotic TD variance  Value at s′  Asymptotic MC variance  Advantage  Empirical TD Variance  Empirical MC Variance  Figure 3: Variance of different MC and TD estimates on Layered MRP with W = 5 and varying  horizon H  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' TD benefits are enhanced for advantage estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In this example, TD performs up  to 100 times better than MC for the advantage estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' This example also shows that the  MSE of the TD estimate of the ATE is smaller than the MSE of individual estimates of the value  of s(1)  1  and s(2)  1  when the horizon T is larger than 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' On the other hand, the MSE of the MC  estimate of the advantage is larger than the individual MSE of the estimates of the value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' TD effectively truncates the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' While the MSE of the MC estimate of the advantage  scales linearly with the horizon T, the MSE of the TD estimate is constant with respect to the  horizon T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' This is all the more striking that the variance of the total reward along a trajectory  scales linearly with the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  7  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='2  Dependence on the MRP structure  We have seen in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1 that TD vastly outperforms MC in the case of Layered MRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' However,  different MRP structures lead to different level of improvement of TD over MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' To illustrate this, we  s(1)  1  s(2)  1  s(k)  1  s(1)  H−1  s(2)  H−1  s(k)  H−1  sH  sT −1  ∅  Figure 4: MRP with meeting horizon H  introduce a new class of MRPs described in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Each of the k initial states s(1)  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' , s(k)  1  lead to  disjoint trajectories for the first H − 1 steps before reaching a common state on the Hth step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We  are interested in seeing how TD and MC perform when the crossing time H varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Figure 5 displays  the ratio of the MSE of the TD and MC estimates for the values of s(1)  1 , s(2)  1  and for the advantage as  the crossing time H varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The MSE used to compute the ratios have been computed using n = 200  independent trajectories and 1000 Monte-Carlo replications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' These ratios are plotted alongside the  asymptotic ratio from Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' As the crossing time H gets closer to the horizon T, the advantage  of TD over MC vanishes until H = T, when TD and MC produce the exact same estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' To convey  0  5  10  15  20  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='8  1  Meeting horizon H  Ratio of MSE E  �  ( ˆVT D − V )2�  /E  �  ( ˆVMC − V )2�  Value at s  value at s′  Advantage  Asymptotic ratio  Empirical ratio  Figure 5: Ratio of variance between TD and MC as a function of the meeting horizon H for T = 20  intuition, we first focus on the two extreme cases:  In the case where H = 2, depicted in Figure 6a, all initial states directly transition to the same  state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' This mimics the webpage example in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In this example, apart from the first reward,  8  trajectories do not depend of the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' TD pools trajectories across actions which allows  to highly reduce the variance of the estimate of the value at s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' This low variance estimate is  then used as a surrogate to estimate the value-to-go at initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' On the other hand, MC  does not leverage the structure of the MRP and produces an independent estimate for each  initial state, using only trajectories starting at a given state to evaluate this state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In this case,  TD will significantly improve over MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  At the other extreme, consider H = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Then, no two initial states can lead to a common state  before the terminal state, as shown in Figure 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' There is no opportunity to pool trajectories  across actions so TD strictly reduces to MC in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  s(1)  1  s(2)  1  s(k)  1  s2  sT −1  ∅  (a)  s(1)  1  s(2)  1  s(k)  1  s(1)  T −1  s(2)  T −1  s(k)  T −1  ∅  (b)  Figure 6: 6a is an instance on which TD leverages pooling to improve considerably over MC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' 6b is an  instance on which TD and MC output the same estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='3  Organization of the results  In Section 7, we characterize the ratio in variance between TD and MC estimates for value estimation  depending on the MRP structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' This characterization enables an intuitive understanding of which  MRP structures lead to a large improvement of TD and, conversely, for which MRP structures TD  and MC perform similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In Section 8, we show that the TD estimate of advantages scales with an  effective horizon that can be much smaller than the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  7  Value estimation and the pooling coefficient  Recall that T(s) is the first hitting time of state s if it is reached, or T otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The variables  N(s′) =  T  �  t=0  1(St = s′), N(s → s′) =  T  �  t=T (s)  1(St = s′),  respectively measure the total number of visits to state s′ and the number of visits to s′ which occur  after a visit to state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Define the coupling coefficient between s and s′ by  C(s, s′) = E [N(s → s′)]  E [N(s′)]  ,  with C(s, s′) = 0 if E [N(s′)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Implicitly, it is understood that S0 is drawn from the MRP’s initial  distribution d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Among all trajectories which reach state s′, the coupling coefficient measures the  9  fraction which first pass through state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' If the coupling coefficient is large, it means s and s′ are  strongly coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  The inverse trajectory pooling coefficient measures the average coupling coefficient C(s, s′) over a  distribution of possible successor states s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The right distribution over which to average turns out to  be µs(·), identified in the definition below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' That distribution weights highly states that are likely to be  visited following a visit to s (high E[N(s′) | S0 = s]) and contribute heavily to to estimator variance  (measured through the one-step variance Var (Rt+1 + V (St+1) | St = s′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1 (Inverse trajectory pooling coefficient).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' For any state s ∈ S define  C(s) = Es′∼µs [C(s, s′)] ,  where µs(·) a probability distribution over states defined by  µs(s′) ∝ E [N(s′) | S0 = s] × Var [Rt + V (St+1) | St = s′] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Again, the inverse trajectory pooling coefficient is small when there is a lot of trajectory pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  The next theorem compares the asymptotic mean squared error of the value estimated under TD and  a direct approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The asymptotic ratio of their mean squared errors is equal to the inverse trajectory  pooling coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' For any s ∈ S,  lim  n→∞  E  ��  ˆVT D(s) − V (s)  �2�  E  ��  ˆVMC(s) − V (s)  �2� = C(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Let us interpret this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Recall that TD updates value prediction at state s using value  predictions at successor states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The theorem shows this is helpful precisely when there is a lot of  trajectory pooling, resulting in a small inverse trajectory pooling coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' When this holds, and the  dataset D is large, there will be many trajectories which reach an important possible successor s′ of s,  but never cross s first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' TD leverages these trajectories to learn about s′ and then properly incorporates  that knowledge to better evaluate s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Direct estimation approaches only use sub-trajectories originating  at s to evaluate s and forego the trajectory pooling benefit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We already developed this intution by  discussing the simple example in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The theorem confirms that this interpretation of TD’s  advantages is exactly the right one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Figures 6a describes an instance with extreme trajectory pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Trajectories that start in distinct  states tend to immediately reach common successors, so TD understands value-to-go from successors  quite well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Figure 6b is a case with no trajectory pooling at all (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' C(s) = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  8  Horizon truncation in advantage estimation  Section 6 previewed two of the paper’s key insights: TD’s benefits are enhanced for advantage estimation  and, in that setting, it effectively truncates the problem’s time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Theory in this section formalizes  these insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  The MSE of direct advantage estimates scales with the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  The variance of the total  reward along a trajectory typically scales with the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Therefore, it would not be surprising that  the mean squared error of the estimate of the advantage also scales with the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We show that it  is indeed the case for MC by stating a lower bound on the mean squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  10  Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' For s, s′ ∈ S such that P [s ∈ τ ∧ s′ ∈ τ] = 0,  lim  n→∞ n · E  ��  ˆAMC(s, s′) − A(s, s′)  �2�  ≥ σ2  min  �E [T|S0 = s]  P [s ∈ τ]  + E [T|S0 = s′]  P [s′ ∈ τ]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  where σ2  min = mins∈S Var [Rt + V (St+1) | St = s]  The condition P [s ∈ τ ∧ s′ ∈ τ] = 0 guarantees that no trajectory can visit both s and s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' It ensures  that the MC estimate of the value at s and s′ are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' It is verified when considering the  heterogeneous treatment effect, described in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='2, where a single action is chosen for every  trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The scaling in the inverse probability of visiting s and s′ appears because nP [s ∈ τ] and  nP [s′ ∈ τ] are asymptotically the number of trajectories available for the Monte-Carlo estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  The MSE of TD’s advantage estimates scales with a smaller trajectory crossing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Rather than scale with problem’s time horizon, the mean squared error of TD’s advantage estimate is  bounded by a smaller notion of the problem’s effective horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' To formally capture this phenomenon,  we introduce the trajectory crossing time H(s, s′) for two states s and s′ to be the expected time for  trajectories starting at s and s′ to cross under the most optimistic coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Two trajectories always  cross once both have terminated, as in that case both have visited the terminal state ∅, but they could  cross much sooner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Intuition for the this definition is provided below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The set of distributions Ψ(s, s′) is the set of all joint distributions over trajectories  (τ, ˜τ) such that the marginal distributions of τ and ˜τ are those of trajectories starting at s and s′,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='3 (Trajectory crossing time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The trajectory crossing time of two states s and s′ is the  expected time for trajectories originating from s and s′ to cross under the best coupling that preserves  the trajectories’ marginal distributions:  H(s, s′) =  min  ψ∈Ψ(s,s′) E(τ,˜τ)∼ψ [inf{t|Ct(S, S′) ̸= ∅}]  where Ct(S, S′) = {S0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' , St} ∩ {S′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' , S′  t} is the set of states visited by both trajectories at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  The following theorem establishes that the mean squared error of the TD estimate of the advantage  scales with the trajectory crossing time instead of the full horizon:  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' For s, s′ ∈ S,  lim  n→∞ n · E  ��  ˆATD(s, s′) − A(s, s′)  �2�  ≤ 2  �  σ2  max  min (P [s ∈ τ] , P [s′ ∈ τ])  �  H(s, s′)  with σ2  max = maxs∈S Var [Rt + V (St+1) | St = s].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Figure 3, in Section 6, provides an empirical illustration of this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  This result can actually understate the benefits the TD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Any trajectory pooling that happens before  the trajectories cross further helps reducing the variance, but is not reflected in the upper bound of  Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Comparison with a coupling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Trajectories are said to cross if one of the trajectories reaches  a state already visited by the other one, potentially at an earlier time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' It is different from the coupling  time where trajectories have to reach a common state simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In particular, the crossing time  is always smaller than the coupling time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='The MRP defined in Figure 7 illustrates this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Trajectories  starting at states s(1)  1  and s(2)  1  only couple when reaching the terminal state, after m + 1 timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  However, they cross in two timesteps, that is H  �  s(1)  1 , s(2)  1  �  = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  11  s(1)  1  s(2)  1  s2  s3  sm  Figure 7: An example with no coupling but rapid crossing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Intuition for the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Let us give intuition for Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Under the MRP structure, two  trajectories reaching a common state have the same future expected reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Hence, when estimating  the difference in expected total rewards along two trajectories, one starting at s, the other starting  at s′, it is only useful to estimate them up until the state where trajectories cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' By computing  estimates at every state, TD leverages this property: two trajectories reaching a common state (the  crossing state) use the same estimate (the value at the crossing state) to update predictions along both  trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Since the same estimate is used, its value cancels out when computing the difference in  values at s and s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Whether the value at the crossing state is accurately estimated doesn’t affect the  estimation of the advantage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  9  Open questions  There is a subtle interplay between the choice of state representation and the benefits of imposing  temporal consistency in value estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Consider again the problem in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In that case, we chose  to represent the ‘checkout page’ as a state, implying that the purchase probability at the checkout page  does not depend on the initial webpage shown to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' This makes a strong surrogacy assumption,  which TD leverages to greatly improve data efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' An alternative representation of the state in the  second period is of the form s = (website version i, checkout), retaining information about how the  user navigated to the checkout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In this case, there is no trajectory pooling and our theory indicates  that TD behaves as MC would.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' By using a very rich representation, which recalls much of the past,  the benefits of TD disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Clearly, we want representations that are accurate, to avoid severe  approximation errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' But we have shown that representations that are forgetful of aspects of the past  offer enormous benefits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' this lets value-to-go from intermediate states serve as surrogate outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  How should one balance this tradeoff while learning representations from data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' This question closely  relates to the one of understanding temporal consistency with function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  References  Andrychowicz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=', Wolski, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=', Ray, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=', Schneider, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=', Fong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=', Welinder, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=', McGrew, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=', Tobin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=',  Pieter Abbeel, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=', and Zaremba, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Hindsight experience replay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Advances in neural information  processing systems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Athey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=', Chetty, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=', Imbens, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=', and Kang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The surrogate index: Combining short-term  proxies to estimate long-term treatment effects more rapidly and precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Technical report, National  Bureau of Economic Research, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Baird, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Residual algorithms: Reinforcement learning with function approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In Machine  Learning Proceedings 1995, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' 30–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Elsevier, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
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+page_content=' Sample  efficient actor-critic with experience replay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' arXiv preprint arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='01224, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  14  A  Proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  We state and prove results in the context of weighted value function which is a linear combination of  the value function evaluated at individual states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1 (Weighted value function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' For a weighting over states π, the extended value function  is defined as  J(π) =  �  s∈S  π(s)V (s)  By setting π to be a mass point at a single state, we recover the value function at this state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' When  interpreting initial states as actions (as in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='2), we recover randomized policy when using a  distribution over actions as the weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In this case, the weighted value function is the expected  value when playing according to the randomized policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Note that our definition of weighted value  function allows for any weighting of states, including negative weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' This will be useful for analyzing  advantages V (s) − V (s′) by setting π(s) = 1 and π(s′) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  We also extend our definition of expected number of visits to weightings over initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='2 (Weighted expected number of visits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' For a weighting over states π, we write ηπ(s)  the weighted number of visits to s:  ηπ(s) =  �  s′∈S  π(s′)E [N(s)|S0 = s′]  Similarly to the weighted value function, π is not enforced to be a distribution over state, allowing  even for negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In the case where π is a distribution over state, we recover the probabilistic  interpretation: ηπ(s) is the expected number of visits to state s when the initial distribution is π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='3 (One-step variance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  σ2  V (s) = Var [Rt + V (St+1) | St = s] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  We extend trajectories into infinite horizon trajectories that stay in the terminating state and  stop collecting rewards once the terminating state is reached: St = ∅ and Rt+1 = 0 for all t ≥ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Equivalently, we define the transition P(∅ | ∅) = 1 and the reward R(∅, ∅) = 0 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  We start by stating and proving Central Limit Theorems (CLT) for the convergence of both TD  and MC estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We then use these two results as building blocks to prove the main theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1  Central Limit Theorems  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='4 (Central Limit Theorem for MC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' For s ∈ S,  √n( ˆVMC(s) − V (s)) ⇒ N  �  0,  1  P [s ∈ τ]  �  s′∈S  E [N(s′) | S0 = s] σ2  V (s′)  �  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We recall that, for tabular representation, the MC estimator takes the form  ˆVMC(s) = E(S,V )∼DMC [V | S = s]  =  1  | I(s) |  �  i∈I(s)  T (i)  �  t=T (i)(s)  R(i)  t+1,  where I(s) is the set of trajectories that visit state s and T (i)(s) is the first visit to state s in  trajectory i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Since we consider first-visit MC, each trajectory appears at most once in the summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  15  We start by rewriting the error ˆVMC(s) − V (s) as the product of a scaling factor and the average of  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' random variables:  ˆVMC(s) − V (s) =  1  | I(s) |  �  i∈I(s)  �  �  T (i)  �  t=T (i)(s)  R(i)  t+1 − V (s)  �  �  =  n  | I(s) | · 1  n  n  �  i=1  �  �1(s ∈ τ (i))  �  �  T (i)  �  t=T (i)(s)  R(i)  t+1 − V (s)  �  �  �  �  Recall that if s is not visited in trajectory i, T (i)(s) is defined to be T (i)(s) = T (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  We start by proving a Central Limit Theorem on  1  n  n  �  i=1  �  �1(s ∈ τ (i))  T (i)  �  t=T (i)(s)  R(i)  t+1  �  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  The variables  �  1(s ∈ τ (i))  ��T (i)  t=T (i)(s) R(i)  t+1 − V (s)  ��  i=1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=',n are n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=', zero mean random  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We now compute their variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Var  �  �1(s ∈ τ)  �  �  T  �  t=T (s)  Rt+1 − V (s)  �  �  �  � = P [s ∈ τ] Var  �  �1(s ∈ τ)  �  �  T  �  t=T (s)  Rt+1 − V (s)  �  � | s ∈ τ  �  �  = P [s ∈ τ] Var  �  �  T  �  t=T (s)  Rt+1 − V (s) | s ∈ τ  �  �  Since the summation starts at the stopping time defined by the first visit to state s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' the Strong  Markov Property enables to re-index the summation in the following way:  Var  �  �1(s ∈ τ)  �  �  T  �  t=T (s)  Rt+1 − V (s)  �  �  �  � = P [s ∈ τ] Var  � T  �  t=0  Rt+1 − V (s) | S0 = s  �  = P [s ∈ τ] Var  � ∞  �  t=0  Rt+1 − V (s) | S0 = s  �  where we allowed the sum to run to infinity since (Rt+1)t is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' stationary at 0 for t ≥ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Similarly, we use the fact that (V (St) − V (St+1)) is a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' stationary at 0 to write V (S0) =  �∞  t=0 (V (St) − V (St+1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Plugging in the previous expression gives:  Var  �  �1(s ∈ τ)  �  �  T  �  t=T (s)  Rt+1 − V (s)  �  �  �  � = P [s ∈ τ] Var  � ∞  �  t=1  (Rt+1 + V (St+1) − V (St)) |S0 = s  �  Notice that (Rt+1 + V (St+1) − V (St))t are martingale differences with respect to the filtration  Ft = {S0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' St}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Using that martingale differences are uncorrelated:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='16  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='Var  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�1(s ∈ τ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='t=T (s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='Rt+1 − V (s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='� = P [s ∈ τ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='t=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='(V (St+1) + Rt+1 − V (St))2|S0 = s  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='We then group the terms in the sum by the value of St:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='Var  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�1(s ∈ τ)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='T  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='t=T (s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='Rt+1 − V (s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='� = P [s ∈ τ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='t=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='s′∈S  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='P [St = s′|S0 = s] E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='(V (St+1) + Rt+1 − V (St))2|St = s′�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='= P [s ∈ τ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='t=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='s′∈S  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='P [St = s′|S0 = s] σ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='V (s′)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='= P [s ∈ τ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='s′∈S  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='σ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='V (s′)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='t=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='P [St = s′|S0 = s]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='= P [s ∈ τ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='s′∈S  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='E [N(s′)|S0 = s] σ2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='V (s′)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='Using the Central Limit Theorem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' we obtain the following convergence:  1  n  n  �  i=1  �  �1(s ∈ τ (i))  T (i)  �  t=T (i)(s)  R(i)  t+1  �  � ⇒ N  �  0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' P [s ∈ τ]  �  s′∈S  E [N(s′)|S0 = s] σ2  V (s′)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  The Strong Law of Large Number ensures:  n  | I(s) | −→  n→∞  1  P [s ∈ τ]  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='.  Finally, using Slutsky’s Theorem, the product converges:  n  | I(s) |· 1  n  n  �  i=1  �  �1(s ∈ τ (i))  �  �  T (i)  �  t=T (i)(s)  R(i)  t+1 − V (s)  �  �  �  � ⇒ N  �  0,  1  P [s ∈ τ]  �  s′∈S  E [N(s′)|S0 = s] σ2  V (s′)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='5 (Central Limit Theorem for TD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' For any weighting π,  √n( ˆJTD(π) − J(π)) ⇒ N  �  0,  �  s′∈S  η2  π(s′)σ2  V (s′)  E [N(s′)]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Corollary A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' For s ∈ S  √n( ˆVT D(s) − V (s)) ⇒ N  �  0,  �  s′∈S  E [N(s′) | S0 = s]2 σ2  V (s′)  E [N(s′)]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  17  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='2  Proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='5  Notations  The idea of the proof is to use the fixed point identities verified by V and ˆVTD to obtain  a recursive formula for the error ˆVTD − V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We then analyze the two quantities that appear when  iterating this recursive formula: one is a sum of empirical one-step error and we show the other is of  second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  We adopt a vectorial representation of the value function V = (V (s))s∈S such that J(π) = ⟨V, π⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Let T be the Bellman operator:  T (f)(s) = ES′∼P (·|s) [R(s, S′) + f(S′)]  The value function is the unique fixed point of the Bellman operator: V = T(V ) to verify V (∅) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  We also write the transition operator as follow:  P(f)(s) = ES′∼P (·|s) [f(S′)] = ⟨P(· | s), f⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  As discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='2, by trying to minimize temporal differences, TD solves the empirical  Bellman equation:  ˆVTD(s) = E(S,R,S′)∼DTD  �  R + ˆV TD(S′) | S = s  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Solving the empirical Bellman equation can be viewed as being a fixed point (with f(∅) = 0) of the  empirical Bellman operator that we define as follow:  ˆT (f)(s) = E(S,R,S′)∼DTD [R + f(S′) | S = s] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Similarly, we note the empirical transition operator  ˆP(f)(s) = E(S,R,S′)∼DTD [f(S′) | S = s] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Finally, we introduce an explicit notation of the empirical Bellman operator that will ease proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  To do so, we first need to introduce B(s), the set of all visits to state s:  B(s) = {(i, t) | S(i)  t  = i}  Using this notation, the empirical Bellman operator can be written as follow:  ˆT (f)(s0) =  1  | B(s0) |  �  (i,t)∈B(s0)  �  f  �  S(i)  t+1  �  + R(i)  t+1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Analysis  We start by expanding the error vector ˆVTD − V into a sum of empirical one-step errors  and a second term that we later show to be of second order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  √n  �  ˆVTD − V  �  = √n  ∞  �  t=0  P t( ˆT V − V ) +  ∞  �  t=0  P t �√n( ˆP − P)( ˆVTD − V )  �  Note that, since P t(S0) = St is stationary at ∅ almost surely, the quantities summed in Lemma  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='7 are ultimately zero almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  18  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' By expending the difference ˆVTD − V using the fixed point identities, we obtain a recursive  identity:  ˆVTD − V = ˆT ˆVTD − T V  = ( ˆT − T ) ˆVTD + T ˆVTD − T V  = ( ˆP − P)( ˆVTD − V ) + ( ˆT − T )V + P( ˆVTD − V )  = ( ˆP − P)( ˆVTD − V ) + ( ˆT V − V ) + P( ˆVTD − V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Iterating this identity and multiplying by √n on both sides gives  √n  �  ˆVTD − V  �  =  ∞  �  t=0  P t �√n( ˆP − P)( ˆVTD − V )  �  + √n  ∞  �  t=0  P t( ˆT V − V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  (1)  We now state two lemmas that analyze the two terms that appear in equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' They are proved  later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' For all s ∈ S, as n → ∞,  ∞  �  t=0  P t �√n( ˆP − P)( ˆVTD − V )  �  (s) → 0  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' As n → ∞  √n  �  s∈S  π(s)  ∞  �  t=0  P t( ˆT V − V )(s) ⇒ N  �  0,  �  s∈S  ηπ(s)2  E [N(s)]σ2  V (s)  �  Finally, combining Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='8 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='9 with Slutstky Lemma concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  We now proceed to prove Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='8 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='9, starting with Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='8  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='8  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We start by showing that, after appropriate rescaling the error in transition estimates converge  to a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We do not calculate the variance here since it is not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' For each s, √n  �  ˆP − P  �  (·|s) weakly converges to a normal distribution with mean  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We first express ˆP(s′|s) − P(s′|s) as the product of a scaling factor and the average of n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='d  random vectors:  �  ˆP(s′|s) − P(s′|s)  �  s′∈S =  1  | B(s) |  �  (i,t)∈B(s)  �  1(S(i)  t+1 = s′) − P(s′|s)  �  s′∈S  =  n  | B(s) | · 1  n  n  �  i=1  � ∞  �  t=0  1(S(i)  t  = s)  �  1(S(i)  t+1 = s′) − P(s′|s)  ��  s′∈S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  We decompose this identity in two terms:  19  Strong law of large numbers ensures that  n  | B(s) | −→  n→∞  1  E [N(s)]  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='.  Since trajectories are independent,  ∞  �  t=0  1(S(i)  t  = s)  �  1(S(i)  t+1 = s′) − P(s′|s)  �  s′∈S  are independent, identically distributed random variables with finite variance and mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The  Central Limit Theorem ensures that the rescaled average of these random variables  √n · 1  n  n  �  i=1  ∞  �  t=0  1(S(i)  t  = s)  �  1(S(i)  t+1 = s′) − P(s′|s)  �  s′∈S  weakly converges to a mean 0 normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Finally, the product of these two quantities also converges to a mean 0 normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  We terminate the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='8 by combining Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='10 with the fact that ˆVTD − V  converges almost surely to 0 by Slutsky’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='2  Proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='9  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We start by expanding the transition operator P t,  √n  ∞  �  t=0  P t( ˆT V − V )(s) = √n  ∞  �  t=0  E  �  ˆT V (St) − V (St) | S0 = s, ˆT  �  This term consists of a sum one-step differences of the form √n  �  ˆT V (S(i)  t ) − V (S(i)  t )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Grouping  S(i)  t  that are equal:  √n  ∞  �  t=0  P t( ˆT V − V )(s) = √n ·  ∞  �  t=0  E  ��  s′∈S  1(St = s′)  �  ˆT V (s′) − V (s′)  �  | S0 = s, ˆT  �  = √n · E  ��  s′∈S  ∞  �  t=0  1(St = s′)  �  ˆT V (s′) − V (s′)  �  | S0 = s, ˆT  �  = √n  �  s′∈S  E  � ∞  �  t=0  1(St = s′) | S0 = s′  � �  ˆT V (s′) − V (s′)  �  = √n  �  s′∈S  E [N(s′)|S0 = s]  �  ˆT V (s′) − V (s′)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  To get the result for a general weighting of states π, we take the dot product of the previous identity  with π:  √n  �  s∈S  π(s)  ∞  �  t=0  P t( ˆT V − V )(s) = √n  �  s∈S  ηπ(s)  �  ˆTV (s) − V (s)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  (2)  20  We are left with analyzing a linear combination of terms of the form ˆT V (s)−V (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' For an individual  s, ˆT V (s) − V (s) is an average of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' variables and its asymptotical behavior can be controlled using  a Central Limit Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' However, to find the limiting distribution of the linear combination, we need  to prove a vectorial Central Limit Theorem for the random vector  �  ˆT V (s) − V (s)  �  s∈S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In particular,  we show that  �  ˆT V (s) − V (s)  �  s∈S converge to independent normal distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  The following lemma is key to prove the independence of the limiting distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' It states that  one-step differences are uncorrelated:  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' For (s, t) ̸= (s′, t′):  Cov [1(St = s) (V (St) − V (St+1) − Rt+1) , 1(St′ = s′) (V (St′) − V (St′+1) − Rt′+1)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' The result follows from the Markovian property:  Cov [1(St = s) (V (St) − V (St+1) − Rt+1) , 1(St′ = ˜s) (V (St′) − V (St′+1) − Rt′+1)]  = E  �  Cov  �  1(S(i)  t  = s) (V (St) − V (St+1) − Rt+1) , 1(St′ = s′)  �  V (St′) − V (St′+1) − R(i)  t′+1  �  | St, St+1, St′  ��  = E [1(St = s) (V (St) − V (St+1) − Rt+1) 1(St′ = s′) (V (St′) − E [V (St′+1) + Rt′+1|St′])]  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Given the previous result, we can now state and proof the joint Central Limit Theorem of empirical  one-step differences:  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' As n → ∞,  √n  ��  ˆTV (s) − V (s)  ��  s∈S ⇒ N (0, Σ)  where Σ is a diagonal matrix with  Σs,s =  1  E [N(s)]σ2  V (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We use a similar approach as in the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='10: we express ˆT V (s) − V (s) as the  product of a scaling factor that converges almost surely and an average of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' random vectors that is  controlled by the Central Limit Theorem:  √n ·  �  ˆT V (s) − V (s)  �  = √n ·  1  | B(s) |  �  (i,t)∈B(s)  (V (S(i)  t+1) + R(i)  t+1 − V (s))  = √n ·  1  | B(s) |  n  �  i=1  ∞  �  t=0  1(S(i)  t  = s)(V (S(i)  t+1) + R(i)  t+1 − V (S(i)  t ))  =  n  | B(s) |  �  √n · 1  n  ∞  �  i=1  ∞  �  t=0  1(S(i)  t  = s)(V (S(i)  t+1) + R(i)  t+1 − V (S(i)  t ))  �  B(s) can be rewritten as the sum of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' random variables:  B(s) =  n  �  i=1  � ∞  �  t=0  1(S(i)  t  = s)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  21  The average | B(s) |  n  converges almost surely to E [N(s)] by the Strong Law of Large Numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Taking the inverse gives:  n  | B(s) | →  1  E [N(s)]  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  (3)  The random vector  √n · 1  n  n  �  i=1  � ∞  �  t=0  1(S(i)  t  = s)(V (S(i)  t+1) + R(i)  t+1 − V (S(i)  t ))  �  s∈S  is the re-scaled average of n i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=', mean 0, vectors of the form (�∞  t=0 1(St = s)(V (St+1) + Rt+1 − V (St)))s∈S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  The Central Limit Theorem ensures that this quantity converges to a normal distribution with  mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We now proceed to compute the variance of this distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  – Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='11 ensures that entries of this vector are uncorrelated:  – The variance of a single entry is given by  Var  � ∞  �  t=0  1(St = s)(V (St+1) + Rt+1 − V (St))  �  (a)  =  ∞  �  t=0  Var [1(St = s)(V (St+1) + Rt+1 − V (St))]  =  ∞  �  t=0  E  �  1(St = s)(V (St+1) + Rt+1 − V (St))2�  =  ∞  �  t=0  E  �  1(St = s)E  �  (V (St+1) + Rt+1 − V (St))2|St  ��  =  ∞  �  t=0  E  �  1(St = s)σ2  V (St)  �  = E [N(s)] σ2  V (s)  where (a) also follows from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  From the Central Limit Theorem, we obtain:  √n· 1  n  n  �  i=1  � ∞  �  t=0  1(S(i)  t  = s)(V (S(i)  t+1) + R(i)  t+1 − V (S(i)  t ))  �  s∈S  ⇒ N  �  0, Diag  �  (E [N(s)] σ2  V (s))s∈S  ��  (4)  Combining (3) and (4) gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Finally, we just need to take the dot product of √n ·  �  ˆT V (s) − V (s)  �  s∈S with the weighted  occupancy measure ηπ to get the result:  √n  �  s∈S  π(s)  ∞  �  t=0  P t( ˆT V − V )(s) ⇒ N  �  0,  �  s∈S  ηπ(s)2  E [N(s)]σ2  V (s)  �  22  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='3  Proof of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='2  The proof follows directly from Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='4 and Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  From, Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='4, we have  lim  n→∞  √n · E  ��  ˆVMC(s) − V (s)  �2�  =  1  P [s ∈ τ]  �  s′∈S  E [N(s′)|S0 = s] σ2  V (s′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Similarly, from Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='5, we have  lim  n→∞  √n · E  ��  ˆVTD(s) − V (s)  �2�  =  �  s′∈S  E [N(s′)|S0 = s]2 σ2  V (s′)  E [N(s′)]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Taking the ratio of these two limits, we obtain:  lim  n→∞  E  ��  ˆVTD(s) − V (s)  �2�  E  ��  ˆVMC(s) − V (s)  �2� =  1  �  s′∈S E [N(s′) | S0 = s] σ2  V (s′)  �  s′∈S  E [N(s′) | S0 = s] σ2  V (s′) E [N(s′) | S0 = s] P [s ∈ τ]  E [N(s′)]  = Es′∼µ(s)  �E [N(s′) | S0 = s] P [s ∈ τ]  E [N(s′)]  �  where µ(s) is defined in Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Finally,  E [N(s′) | S0 = s] P [s ∈ τ] = E [N(s → s′)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='4  Proof of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='4  We define π to be such that π(s) = 1, π(s′) = −1 and π(˜s) = 0 for all ˜s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  That way J(π) =  V (s) − V (s′) = A(s, s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' This allows to use Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='5:  lim  n→∞ n · E  ��  ˆJTD(π) − J(π)  �2�  =  �  ˜s∈S  (E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′])2 σ2  V (˜s)  E [N(˜s)]  We decompose the sum into three terms that we bound separately:  lim  n→∞ n · E  ��  ˆJTD(π) − J(π)  �2�  ≤  �  max  ˜s∈S  ����  E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′]  E [N(˜s)]  ����  � �  max  ˜s∈S σ2  V (˜s)  �  ×  �  ˜s∈S  |E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′]| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  (5)  We start by proving that if trajectories where S0 is sampled from the initial distribution visit s  frequently often, then the occupancy measure E [N(˜s)] and E [N(˜s) | S0 = s] cannot differ too much  (and symmetrically for s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  If E [N(˜s) | S0 = s] ≥ E [N(˜s) | S0 = s′] then  ����  E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′]  E [N(˜s)]  ���� ≤  E [N(˜s) | S0 = s]  P [s ∈ τ] E [N(˜s) | S0 = s] =  1  P [s ∈ τ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  23  When E [N(˜s) | S0 = s] < E [N(˜s) | S0 = s′], a symmetric argument ensures that  ����  E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′]  E [N(˜s)]  ���� ≤  1  P [s′ ∈ τ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Combining these two cases gives:  max  ˜s∈S  ����  E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′]  E [N(˜s)]  ���� ≤  1  min (P [s ∈ τ] , P [s′ ∈ τ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Plugging this result in Equation 5:  lim  n→∞ n·E  ��  ˆJTD(π) − J(π)  �2�  ≤  max˜s∈S σ2  V (˜s)  min (P [s ∈ τ] , P [s′ ∈ τ])·  �  ˜s∈S  |E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′]| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  (6)  Finally, we show that the last sum scales as the crossing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' To give intuition on why this holds,  note that we have the following identity:  �  ˜s∈S  E [N(˜s) | S0 = s] = E [T|S0 = s] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  E [N(˜s) | S0 = s] is the expected number of visits to ˜s whens starting at s: summing over ˜s is the  expected total number of states visited when starting at s (not counting the terminating state ∅) which is  exactly the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In the case of the comparison of trajectories, |E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′]|  is how many additional times ˜s has been visited by one of the trajectories compared to the other, in  expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Summing over ˜s gives the expected total number of states that have been visited by only  one of the trajectories, which is at most twice the number of states visited before both trajectories  reach a common state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  �  ˜s∈S  |E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′] | ≤ 2H(s, s′)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Let τ, τ ′ = (S0, R1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' , ST −1, RT , ∅), (S′  0, R′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' , S′  T −1, R′  T ′, ∅) ∼ ψ where ψ is a joint distribu-  tion such that τ and τ ′ are marginally distributed like trajectories generated with S0 = s and S′  0 = s′,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We define the crossing time for ψ is defined as the first time a state has been visited by  both trajectories:  Hψ(s, s′) = inf{t|{S0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' , St} ∩ { ˜S0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' , ˜St} ̸= ∅}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Since we always consider the crossing time of s and s′, we omit the s and s′ dependency and  simply write Hψ for convenience in the rest of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' By definition, either S′  Hψ ∈ {S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' , SHψ}  or SHψ ∈ {S′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' , S′  Hψ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Without loss of generality, we assume S′  Hψ ∈ {s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' , sHψ} holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' Let  Nψ = inf{t|St = S′  Hψ}, that is SNψ = S′  Hψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  We now construct a new trajectory that follows τ ′ until the crossing state SNψ = S′  Hψ is reached  and then follows the trajectory τ: ˆτ = (S′  0, R′  1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' , S′  Hψ = SNψ, RN+1, SN+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' , ST −1, RT , ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' By  Markov property, ˆτ is identically distributed as τ ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  We are interested in bounding the difference in occupancy measure:  E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′] = E  � ∞  �  t=0  1(St = ˜s)  �  − E  � ∞  �  t=1  1(S′  t = ˜s)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  24  Using that ˆτ and τ ′ are identically distributed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' this expression can be rewritten as  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′] = E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='� ∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='t=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1(St = ˜s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='− E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='� ∞  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='t=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1( ˆSt = ˜s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='= E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='Nψ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='t=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1(St = ˜s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='� − E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='Hψ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='t=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1(S′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='t = ˜s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='Taking absolute values and summing over ˜s gives  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='˜s∈S  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='|E [N(˜s) | S0 = s] − E [N(˜s) | S0 = s′]| =  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='˜s∈S  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='Nψ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='t=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1(St = ˜s) −  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='Hψ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='t=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1(S′  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='t = ˜s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='������  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='≤  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='s′∈S  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
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+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='t=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1(St = ˜s)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='� +  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='˜s∈S  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='E  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='Hψ−1  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='t=0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1(St = s′)  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='�  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='= E [Nψ] + E [Hψ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='≤ 2E [Hψ]  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='Taking the infimum over all ψ ∈ Ψ(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' s′) that conserves marginal distribution gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Finally, plugging Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='13 in Equation 6 leads to the result:  lim  n→∞ n · E  ��  ˆJTD(π) − J(π)  �2�  ≤ 2  max˜s∈S σ2  V (˜s)  min (P [s ∈ τ] , P [s′ ∈ τ]) · H(s, s′)  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1  Proof of Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='1  We now prove that the MC estimate of the advantage can scale as the full horizon, no matter how  small the crossing time is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' We focus on the advantage of s over s′ when P [s ∈ τ ∧ s′ ∈ τ] = 0, that  is no trajectory can visit both s and s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' No trajectories visiting both s and s′ means that the MC  value estimates at s and s′ are independent since they rely on disjoints sets of trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content=' In terms of  variance, this implies:  Var  �  ˆAMC(s, s′)  �  = Var  �  ˆVMC(s) − ˆVMC(s′)  �  = Var  �  ˆVMC(s)  �  + Var  �  ˆVMC(s′)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  It now suffices to prove that for any s,  lim  n→∞ Var  �  ˆVMC(s)  �  ≥  σ2  min  P [s ∈ τ]E [T|S0 = s] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  From Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='4, we know the variance of the MC estimate is  lim  n→∞ n · Var  �  ˆVMC(s)  �  =  1  P [s ∈ τ]  �  ˜s∈S  E [N(˜s) | S0 = s] σ2  V (˜s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  Using that σ2  V (˜s) ≥ σ2  min for all ˜s ∈ S, this expression simplifies to:  lim  n→∞ n · Var  �  ˆVMC(s)  �  ≥  σ2  min  P [s ∈ τ]  �  ∈S  E [N(˜s) | S0 = s] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
+page_content='  25  Finally,  �  s′∈S  E [N(s′) | S0 = s] =  �  ˜s∈S  E  � T  �  t=0  1(St = ˜s)|S0 = s  �  = E  � T  �  t=0  �  ˜s∈S  1(St = ˜s)|S0 = s  �  = E [T|S0 = s]  26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFQT4oBgHgl3EQfSTYo/content/2301.13289v1.pdf'}
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+Boundary Aware U-Net for Glacier Segmentation
+Bibek Aryal∗1, Katie E. Miles2, Sergio A. Vargas Zesati1, and Olac Fuentes1
+1The University of Texas at El Paso, Texas, USA
+2Aberystwyth University, Aberystwyth, UK
+Abstract
+Large-scale study of glaciers improves our under-
+standing of global glacier change and is impera-
+tive for monitoring the ecological environment, pre-
+venting disasters, and studying the effects of global
+climate change.
+Glaciers in the Hindu Kush Hi-
+malaya (HKH) are particularly interesting as the
+HKH is one of the world’s most sensitive regions
+for climate change. In this work, we: (1) propose a
+modified version of the U-Net for large-scale, spa-
+tially non-overlapping, clean glacial ice, and debris-
+covered glacial ice segmentation;
+(2) introduce
+a novel self-learning boundary-aware loss to im-
+prove debris-covered glacial ice segmentation per-
+formance; and (3) propose a feature-wise saliency
+score to understand the contribution of each fea-
+ture in the multispectral Landsat 7 imagery for
+glacier mapping. Our results show that the debris-
+covered glacial ice segmentation model trained us-
+ing self-learning boundary-aware loss outperformed
+the model trained using dice loss. Furthermore, we
+conclude that red, shortwave infrared, and near-
+infrared bands have the highest contribution to-
+ward debris-covered glacial ice segmentation from
+Landsat 7 images.
+1
+Introduction
+Glacier delineation using remote sensing imagery
+has seen a growing use of deep learning in recent
+years [5, 14, 15, 34, 36].
+This can be attributed
+to factors such as the availability of large-scale re-
+mote sensing data from multiple sources, the de-
+velopment of state-of-the-art deep learning archi-
+tectures for image analysis, and the growing in-
+terest due to the impacts of climate change on
+∗Corresponding Author: baryal@miners.utep.edu
+Figure 1: a) Spatially non-overlapping regions us-
+ing fishnet grid. b) A zoomed image of one of the
+cells showing clean ice and debris glacier labels.
+glaciers in recent decades. The Himalaya is one of
+the world’s most sensitive regions to global climate
+change, with impacts manifesting at particularly
+rapid rates [17, 19]. Unsurprisingly, much research
+has been focused on mapping glaciers in the Hindu
+Kush Himalayas (HKH) [5, 6, 36]. Several studies
+have reported the performance of clean glacial ice
+and debris-covered glacial ice mapping in the HKH;
+however, most research has been focused on specific
+https://doi.org/10.7557/18.6789
+© The author(s). Licensee Septentrio Academic Publishing, Tromsø, Norway. This is an open access article distributed
+under the terms and conditions of the Creative Commons Attribution license
+(http://creativecommons.org/licenses/by/4.0/).
+1
+arXiv:2301.11454v1  [cs.CV]  26 Jan 2023
+
+70°E
+75°E
+80°E
+85°E
+90° E
+95° E
+N .09
+z
+5
+Qazaqstan
+SnltiintiTc
+oblysy
+A qm.ola
+obly sy
+z
+45°
+Ongtistik
+Qazaqslan
+ob ly sy
+40° N
+b)
+z
+3
+z
+3
+a
+250
+500
+1.000
+Kilometers
+Ganges
+Sou ces : Esri USGS, NOAA, Sources: Esri, Garmin, USGS, N
+Region of Interest
+Glacier Types
+Glaciers
+Debris
+Clean lce
+Selected ces
+A celsglacier basins within the region and not across the
+region as a whole.
+Glaciers form when snow compresses under its
+own weight and hardens over long timescales [12].
+Near its formation, glacial ice has snow or ice sur-
+face cover and is known as clean glacial ice.
+As
+glacial ice slowly moves down valleys under grav-
+ity, avalanches can deposit debris (rocks and sed-
+iment) on top of the glacier. Glacial ice having a
+significant covering of dirt/rocks/boulders is known
+as debris-covered glacial ice. Clean glacial ice and
+debris-covered glacial ice appear differently in re-
+mote sensing imagery. The challenge lies in differ-
+entiating clean glacial ice from temporary snow/ice
+cover and debris-covered glacial ice from moraines
+and the surrounding valleys. The spectral unique-
+ness of clean glacial ice compared to surrounding
+terrain makes it relatively easy to identify and lo-
+calize. However, the delineation of debris-covered
+glacial ice poses significant challenges because of its
+non-unique spectral signature.
+The earliest approaches for debris-covered glacial
+ice segmentation involving deep learning used mul-
+tilayer perceptrons to estimate the supraglacial de-
+bris loads of Himalayan glaciers using pre-defined
+glacier outlines [7, 8].
+More recent methods for
+glacier segmentation use Convolutional Neural Net-
+works (CNNs) due to their success in image-based
+applications [5, 25, 36, 39]. Most recent approaches
+to learning-based image segmentation use variants
+of the U-Net architecture [30].
+Originally intro-
+duced for biomedical image segmentation, the U-
+Net has seen successes in numerous applications
+involving satellite image segmentation [23, 29, 38].
+Moreover, different architectures based on the U-
+Net have also been used for glacier segmentation
+in recent years [5, 36]. However, unlike segmenta-
+tion for general images, the results when it comes
+to glacier segmentation are not very good, particu-
+larly for debris-covered glacial ice.
+Here we present a variant of the U-Net and train
+it using multispectral images from Landsat 7 as in-
+puts. Researchers have shown that the performance
+of deep learning models can be improved by learn-
+ing multiple objectives from a shared representa-
+tion [11]. Early approaches to learn multiple tasks
+use weighted sum of losses, where the loss weights
+are either constant or manually tuned [13, 21, 31].
+We propose a method to combine two different
+loss functions - masked dice loss [2] and boundary
+loss [9] - to simultaneously learn multiple objec-
+tives automatically during the training process for
+improved performance.
+While deep learning models have been shown to
+perform well on various tasks involving computer
+vision, the interpretability of these models is lim-
+ited.
+Deep neural networks are often considered
+black boxes, since their decision rules can not be
+described easily.
+Unlike coefficients and decision
+boundaries of simpler machine learning methods
+such as linear regression and decision trees, weights
+of neurons in deep neural networks can not be
+understood as knowledge directly.
+The develop-
+ment of transparent, understandable, and explain-
+able models is imperative for the wide-scale adop-
+tion of deep learning models. Over the years, many
+have proposed different approaches to describe deep
+leaning models [22, 37, 39]. One of the most widely
+used methods to envision which pixels in the in-
+put image affect the outputs the most is by visu-
+alizing saliency maps [33]. A saliency map is ob-
+tained by calculating the gradient of the given out-
+put class with respect to the input image by letting
+gradients backpropagate to the input. In the case
+of multispectral or hyperspectral images, spectral
+saliency [20] is used to visualize salient pixels of an
+image. Image saliency maps, computed indepen-
+dently for all channels on a multispectral image,
+can be used to visualize the contribution of each
+pixel in each channel toward the final output. We
+propose a method to quantify each channel’s con-
+tributions towards the final label in the context of
+glacier segmentation using Landsat 7 imagery.
+2
+Dataset and Methodology
+The HKH region covers an area of about 4.2 million
+km2 from about 15◦ to 39◦ N latitude and about
+60◦ to 105◦ E longitude extending across eight
+countries consisting of Afghanistan, Bangladesh,
+Bhutan, China, India, Myanmar, Nepal, and Pak-
+istan [4].
+The geographic extent of the glaciers
+within the HKH, however, ranges from about 27◦
+to 38◦ N and about 67◦ to 98◦ E (Figure 1).
+We downloaded the Landsat 7 images used for
+label creation using Google Earth Engine. Landsat
+7 contains the Enhanced Thematic Mapper Plus
+(ETM+) sensor which captures multiple spectral
+bands as shown in Table 1. The thermal infrared
+2
+
+bands were upsampled from 60 meters to 30 meters
+resulting in all bands having a spatial resolution
+of 30 meters.
+The glacier outlines (labels) [3]
+were downloaded from International Centre for
+Integrated
+Mountain
+Development
+(ICIMOD)
+Regional Database System. (http://rds.icimod.
+org/Home/DataDetail?metadataId=31029)
+The
+glacier labels contain information on clean-ice and
+debris-covered glaciers in the HKH for regions
+within Afghanistan, Bhutan, India, Nepal, and
+Pakistan. The ICIMOD glacier outline labels used
+in this research were derived using the object-
+based image classification methods separately for
+clean-ice and debris-covered glaciers and fine-tuned
+with manual intervention [4].
+Table 1: Landsat 7 bands description
+Name
+Description
+B1
+Blue
+B2
+Green
+B3
+Red
+B4
+Near Infrared
+B5
+Shortwave Infrared 1
+B6 VCID 1
+Low-gain Thermal Infrared
+B6 VCID 2
+High-gain Thermal Infrared
+B7
+Shortwave Infrared 2
+The Landsat 7 images that were used for
+delineating glacier labels [4] overlap spatially.
+To avoid spatial overlap between train and test
+regions, we created polygon features represent-
+ing a fishnet of rectangular cells for the entire
+geographical region.
+We then created a mosaic
+of all Landsat 7 images used for labeling into
+a single raster and clipped the raster mosaic to
+country boundaries for glacier labels (Figure 1) to
+avoid false negative glacier labels in the dataset.
+Finally, we discarded the rasters within the poly-
+gon cells that do not contain any glacier labels
+and downloaded clipped regions within selected
+cells. The Google Earth Engine code to replicate
+this process can be found in repository https:
+//code.earthengine.google.com/?accept_
+repo=users/bibekaryal7/get_hkh_tiff.
+The
+selected cells were then randomly split into train,
+validation, and test sets with no geospatial overlap.
+1163 out of 1364 cells were filtered out to leave
+us with 141, 20, and 40 cells in the training,
+validation, and test sets respectively.
+Each cell
+was then cropped into multiple sub-images of
+512 × 512 pixels and the sub-images with less
+than 10% of pixels as glacier labels were discarded
+to reduce class imbalance. These sub-images are
+then normalized and provided as input to the
+model. There are 333, 68, and 98 sub-images in
+the training, validation, and test sets respectively.
+Every pixel within each sub-image can have one of
+four different classes as can be seen in Figure 2.
+Figure 2: (a) Sample sub-image, (b) Corresponding
+Clean Glacial Ice, Debris Glacial Ice, Background,
+and Masked labels
+The step-by-step processing we followed to pre-
+pare input features for the model is shown in Fig-
+ure 3. The distribution of pixels for train, valida-
+tion, and test set across different classes is shown
+in Table 2 and highlights that the distribution of
+pixels across different sets is similar and labels are
+heavily imbalanced across classes.
+Table 2: Labels Distribution - Random Sampling
+split
+background
+clean
+debris
+masked
+train
+72.44%
+21.77%
+2.44%
+3.35%
+val
+68.69%
+23.22%
+3.24%
+4.85%
+test
+70.16%
+22.97%
+2.65%
+4.21%
+clean = clean glacial ice
+debris = debris-covered glacial ice
+We used a modified version of the U-Net archi-
+tecture [30] as shown in Figure 4. Each input sub-
+image is 512 × 512 pixels in size. Zero padding was
+added during each convolution operation to make
+the output labels the same size as input sub-images.
+We replaced the Rectified Linear Unit (ReLU) in
+the original U-Net architecture with Gaussian Er-
+3
+
+Figure 3: Input preprocessing
+ror Linear Units (GELU) [16]. We applied batch
+normalization after each convolution operation and
+spatial dropout [35] of 0.1 after each down-sampling
+and up-sampling block to reduce overfitting. We
+also randomly modified 15% of the training sam-
+ples by either rotating (90◦, 180◦, 270◦) or flipping
+(horizontal/vertical) the input sub-images to the
+model.
+We trained the modified U-Net architec-
+ture for 250 epochs using the Adam optimizer and
+evaluated the performance based on precision, re-
+call, and Intersection over Union (IoU).
+Figure 4: Our modified U-Net architecture has 32
+feature maps in the first convolution layer. We also
+introduce batch normalization and spatial dropout
+in the modified architecture.
+We trained two separate models, one for seg-
+menting clean glacial ice and one for debris-covered
+glacial ice, and combined the outputs to produce
+the final segmentation map.
+Definitions of what
+constitutes debris-covered glacial ice vary widely,
+however, as a glacier does not have to be fully cov-
+ered by debris to be classified as debris-covered
+glacial ice [24].
+Therefore, for the pixels where
+debris-covered glacial ice labels overlapped with
+clean glacial ice labels on the final segmentation
+map, the output label was set as debris-covered
+glacial ice. The code to replicate our process can be
+found in the GitHub repository (https://github.
+com/Aryal007/glacier_mapping.git).
+3
+Experiments
+3.1
+Self-learning
+Boundary-aware
+Loss
+The subject of this section of our work lies at the
+intersection of two branches of research, which are
+penalizing misalignment of label boundaries by us-
+ing a boundary-aware loss and learning multi-task
+weights during the training process. We propose a
+combined loss (LCombined) that is a weighted sum
+of masked dice loss (LMDice) and boundary loss
+(LBoundary), as described in Equation 1. We also
+compare the performance of our methods using the
+modified U-Net to the standard U-Net trained on
+cross entropy loss (LCE).
+LCombined = α × LMDice + (1 − α) × LBoundary
+(1)
+The value of hyperparameter α can be set manu-
+ally between 0 and 1. Having an α of 1 is equivalent
+to training the model exclusively using masked dice
+loss and an α of 0 is the same as training the model
+exclusively using boundary loss. However, tuning
+the value of α manually for the best results is an
+expensive process.
+In order to learn the weights
+for LBoundary and LMDice through backpropaga-
+tion, we initially set α to 0.5 and let the model
+find the best value of α. However, we observe that
+without any constraints on the value of α, the net-
+work updates α such that LCombined is minimized
+without necessarily having to minimize LMDice or
+LBoundary. This results in poor performance. In-
+spired by [18] for weighing two different loss func-
+4
+
+Landsat 7
+Fishnet Grid
+Glacier Labels
+Clip to country
+boundaries
+Selected cells
+Intersecting Raster
+Randomly split train,
+Rasterize Labels
+test, val
+ImputeNaN
+Filter by minimum
+Crop to 512×512
+glacier percentage
+subimage
+Normalize
+Augment feature label
+Input to modified-
+pairs
+UNet32 32
+32 32
+ftma
+64 64
+64 64
+512 × 512
+512 × 512
+256 × 256
+256 × 256
+128 128
+128 128
+Conv 3×3
+128 × 128
+128 × 128
+256 256
+256 256
+Batchnorm, GELU
+512
+SpatialDropout
+64 × 64
+64 × 64
+Max Pool 2×2
+32 × 32
+ConvTranspose2d
+2×2Table 3: Performance comparisons between standard U-Net trained using cross entropy loss and modified
+U-Net trained using combined loss and self-learning boundary-aware loss
+Loss (L)
+Lweight(s)
+clean
+debris
+Precision
+Recall
+IoU
+Precision
+Recall
+IoU
+LCE
+−
+89.39%
+84.65%
+70.16%
+0.00%
+0.00%
+0.00%
+LCombined
+0
+68.40%
+0.25%
+0.25%
+3.00%
+99.75%
+3.00%
+LCombined
+0.1
+79.82%
+79.66%
+66.31%
+50.93%
+49.30%
+33.43%
+LCombined
+0.5
+81.60%
+80.77%
+68.33%
+51.16%
+46.92%
+32.41%
+LCombined
+0.9
+81.60%
+80.77%
+68.33%
+51.16%
+46.92%
+32.41%
+LCombined
+1
+80.31%
+80.65%
+67.34%
+46.00%
+44.10%
+29.05%
+LSLBA
+Dynamic
+81.59%
+80.55%
+68.17%
+51.97%
+53.81%
+35.94%
+*clean = clean glacial ice; debris = debris-covered glacial ice
+tions, we propose Self-Learning Boundary-Aware
+loss (LSLBA) that is a combination of LMDice and
+LBoundary.
+LSLBA =
+1
+2α2
+1 × LMDice +
+1
+2α2
+2 × LBoundary + |ln (α1 × α2)|
+(2)
+In the case of LSLBA, α1 and α2 both are initially
+set to 1 and we let the model find the best value
+for α1 and α2 through backpropagation. In Table 3
+we show performance for different values of α in
+the case of LCombined and performance of LSLBA.
+One advantage of using LSLBA over LCombined is
+that there is no extra hyperparameter that requires
+fine-tuning. All experiments in Table 3 use eight
+features from Landsat 7 imagery as inputs.
+Figure 5: Masked Dice Loss weights and Boundary
+Loss weights vs. epoch for debris-covered glacial ice
+From Table 3, we see that LSLBA performs the
+best for debris-covered glacial ice segmentation and
+eliminates the need to fine-tune loss weights. We
+can also see that the model fails to converge when
+training solely on boundary loss (α = 0) and train-
+ing on glacier boundaries by incorporating bound-
+ary loss along with masked dice loss results in
+an overall improvement in performance for debris-
+covered glacial ice regardless of the weighting fac-
+tor. Figure 5 shows the weights for masked dice loss
+(
+1
+2α2
+1 ) and the weights for boundary loss (
+1
+2α2
+2 ) vs.
+epoch during training for debris-covered glacial ice.
+The optimal values for α1 and α2 are calculated to
+be 0.9569 and 1.045 for clean glacial ice segmenta-
+tion and 0.952 and 1.05 for debris-covered glacial
+ice segmentation for LSLBA.
+3.2
+Representation Analysis
+To understand the contribution of each feature in
+the multispectral image toward the final label, we
+computed a Saliency Score (SS) for each feature by
+summing all pixels in the Saliency Map (SM) for
+that feature.
+SSfeature =
+c−1
+�
+i=0
+r−1
+�
+j=0
+SM feature(i, j)∀ feature ∈ Input
+(3)
+where:
+r, c = number of rows, columns in saliency map
+Average feature saliency scores across all the im-
+ages in the training samples are shown in Fig-
+ure 6.
+The channel-wise contributions towards
+5
+
+Weights
+Masked Dice
+Boundary
+0.54
+0.52
+lue
+0.50
+0.48
+0.46
+0
+50
+100
+150
+200
+250
+EpochFigure 6: Average saliency scores for all sub-images
+in training set.
+debris-covered glacial ice segmentation in decreas-
+ing order are: red, shortwave infrared 1, near in-
+frared, green, high-gain thermal infrared, short-
+wave infrared 2, low-gain thermal infrared, and
+blue. Similarly, for clean glacial ice segmentation,
+the channel-wise contributions in decreasing order
+are: shortwave infrared 2, blue, shortwave infrared
+1, high-gain thermal infrared, red, low-gain ther-
+mal infrared, near infrared, and green. As shown
+in Figure 6, the segmentation models have different
+high contributing channels for clean glacial ice and
+debris-covered glacial ice segmentation.
+4
+Discussion
+Glaciers have been melting at an unprecedented
+rate
+in
+recent
+years
+due
+to
+global
+climate
+change [17, 19].
+Glaciers are the largest fresh-
+water reservoir on the planet [32], so it is nec-
+essary to understand the changes they undergo.
+As a result, numerous approaches to automati-
+cally delineate glacier boundaries have been pro-
+posed [5, 6, 14, 15, 34, 36]. We frequently observe
+deep learning methods outperforming traditional
+machine learning methods for glacier segmentation
+in the literature [1, 5]. However, the results have
+not been very good, particularly in the case of
+debris-covered glacial ice.
+In this work, we modify U-Net and train it using
+a novel loss function that allows the modified U-
+Net to focus on glacier boundaries. From Table 3,
+we see that standard U-Net is not able to detect
+debris-covered glacial ice in input sub-images. We
+can also see from Table 2 that only 2.44% of pix-
+els in the training set correspond to debris-covered
+glacial ice. This shows that our proposed method
+is more robust than the original U-Net to imbal-
+anced labels, which are common in remote sensing
+datasets.
+From Figure 5, we can see how the weights
+change for LSLBA while training. A higher weight
+is assigned to masked dice loss at the beginning
+and the weights for boundary loss are gradually in-
+creased during training.
+The reason behind this
+could be that for an untrained model, it may be
+easier to learn glacier instances over trying to learn
+the boundaries. However once the network learns
+to label instances, it is easier to learn the glacier
+boundaries. This also explains why the model fails
+to converge when training solely on LBoundary from
+scratch as can be seen from the results in Table 3.
+We presented methods to improve debris-covered
+glacial ice segmentation from remote sensing im-
+agery using deep learning. While we were able to
+show significant improvements over existing meth-
+ods, the IoU for debris-covered glacial ice still leaves
+much to be desired. The existing body of litera-
+ture on the topic has shown that the performance
+for debris-covered glacial ice segmentation can be
+improved by incorporating thermal signatures [28]
+and topographical information [10, 26, 27] from
+other satellites. Since debris-covered glacial ice is
+common in low-gradient areas due to how it forms
+and has cooler surface temperatures compared to
+the surrounding non-glaciated regions, we suspect
+that adding this information can further help im-
+prove the performance of debris-covered glacial ice
+segmentation. We may also be able to see an im-
+provement in performance by using images from
+the recently-launched Landsat 9 satellite, instead
+6
+
+Saliency Scares
+Debris
+1D
+0.B
+0.6
+t0
+0.2
+0.D
+BG VCID1
+BG VCID2
+B2
+相
+B5
+B3
+Clean Ice
+0.6
+0.5
+0.4
+0.3
+0.2
+0.1
+0.D
+B2
+B4B6_vCID1 B3
+B6_vCID2
+B5
+B1
+B7of the Landsat 7 images used in this work.
+The
+Operational Land Imager 2 (OLI-2) and the Ther-
+mal Infrared Sensor 2 (TIRS-2) sensors on Land-
+sat 9 provide data that is radiometrically and ge-
+ometrically superior to instruments on the previ-
+ous generation Landsat satellites. With the higher
+radiometric resolution, Landsat 9 can differentiate
+16,384 shades of a given wavelength compared to
+only 256 shades in Landsat 7.
+Meanwhile, the
+TIRS-2 in Landsat 9 enables improved atmospheric
+correction and more accurate surface temperature
+measurements. Future work includes using the im-
+ages captured through these improved sensors and
+incorporating additional information such as a dig-
+ital elevation model for improving debris-covered
+glacial ice segmentation performance.
+5
+Conclusion
+In this research study, we proposed a modified
+version of the U-Net architecture for large-scale
+debris-covered glacial ice and clean glacial ice seg-
+mentation in the HKH from Landsat 7 multispec-
+tral imagery and concluded that debris-covered
+glacial ice (IoU: 35.94%) is significantly harder
+to delineate compared to clean glacial ice (IoU:
+68.17%)(Table 3). We also introduced two differ-
+ent methods to combine commonly-used masked
+dice loss and boundary loss to incorporate label
+boundaries into the training process.
+We show
+that the performance of debris-covered glacial ice
+segmentation can be improved by encouraging the
+deep learning model to focus on label boundaries.
+The performance can be improved further by cor-
+rectly weighing loss terms. Furthermore, the rela-
+tive weights can be learned automatically from the
+data during the training process using our proposed
+loss (LSLBA). Figure 7 shows the performance of
+the models trained using LSLBA on a sample image
+from the test set. We also introduced the concept of
+feature saliency scores to quantify the contribution
+of each feature (channel) in the input image toward
+the final label and concluded that the red, short-
+wave infrared, and near infrared bands contribute
+the most towards the final label for debris-covered
+glacial ice segmentation, while shortwave infrared
+2, blue, shortwave infrared 1 bands contributed the
+most towards the final label for clean glacial ice
+segmentation.
+Figure 7: (a) Sample subimage from test set (b)
+Corresponding clean glacial ice and debris-covered
+glacial ice ground truth labels (c) True positive
+(TP), False positive (FP), False negative (FN) for
+clean glacial ice (IoU 79.17%) (d) TP, TP, FN for
+debris-covered glacial ice (IoU 59.19%)
+6
+Acknowledgements
+We would like to thank Microsoft for providing us
+with the Microsoft Azure resources through their
+AI for Earth grant program (Grant ID: AI4E-1792-
+M6P7-20121005). We also acknowledge ICIMOD
+for providing a rich dataset which this work has
+been built on. This research was supported in part
+by the Department of Computer Science at The
+University of Texas at El Paso.
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diff --git a/OtFJT4oBgHgl3EQfICxT/content/tmp_files/load_file.txt b/OtFJT4oBgHgl3EQfICxT/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf,len=665
+page_content='Boundary Aware U-Net for Glacier Segmentation  Bibek Aryal∗1, Katie E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Miles2, Sergio A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Vargas Zesati1, and Olac Fuentes1  1The University of Texas at El Paso, Texas, USA  2Aberystwyth University, Aberystwyth, UK  Abstract  Large-scale study of glaciers improves our under-  standing of global glacier change and is impera-  tive for monitoring the ecological environment, pre-  venting disasters, and studying the effects of global  climate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Glaciers in the Hindu Kush Hi-  malaya (HKH) are particularly interesting as the  HKH is one of the world’s most sensitive regions  for climate change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' In this work, we: (1) propose a  modified version of the U-Net for large-scale, spa-  tially non-overlapping, clean glacial ice, and debris-  covered glacial ice segmentation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  (2) introduce  a novel self-learning boundary-aware loss to im-  prove debris-covered glacial ice segmentation per-  formance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' and (3) propose a feature-wise saliency  score to understand the contribution of each fea-  ture in the multispectral Landsat 7 imagery for  glacier mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Our results show that the debris-  covered glacial ice segmentation model trained us-  ing self-learning boundary-aware loss outperformed  the model trained using dice loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Furthermore, we  conclude that red, shortwave infrared, and near-  infrared bands have the highest contribution to-  ward debris-covered glacial ice segmentation from  Landsat 7 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  1  Introduction  Glacier delineation using remote sensing imagery  has seen a growing use of deep learning in recent  years [5, 14, 15, 34, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  This can be attributed  to factors such as the availability of large-scale re-  mote sensing data from multiple sources, the de-  velopment of state-of-the-art deep learning archi-  tectures for image analysis, and the growing in-  terest due to the impacts of climate change on  ∗Corresponding Author: baryal@miners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='utep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='edu  Figure 1: a) Spatially non-overlapping regions us-  ing fishnet grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' b) A zoomed image of one of the  cells showing clean ice and debris glacier labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  glaciers in recent decades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' The Himalaya is one of  the world’s most sensitive regions to global climate  change, with impacts manifesting at particularly  rapid rates [17, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Unsurprisingly, much research  has been focused on mapping glaciers in the Hindu  Kush Himalayas (HKH) [5, 6, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Several studies  have reported the performance of clean glacial ice  and debris-covered glacial ice mapping in the HKH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  however, most research has been focused on specific  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
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+page_content=' Licensee Septentrio Academic Publishing, Tromsø, Norway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
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+page_content='CV]  26 Jan 2023  70°E  75°E  80°E  85°E  90° E  95° E  N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='09  z  5  Qazaqstan  SnltiintiTc  oblysy  A qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='ola  obly sy  z  45°  Ongtistik  Qazaqslan  ob ly sy  40° N  b)  z  3  z  3  a  250  500  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='000  Kilometers  Ganges  Sou ces : Esri USGS, NOAA, Sources: Esri, Garmin, USGS, N  Region of Interest  Glacier Types  Glaciers  Debris  Clean lce  Selected ces  A celsglacier basins within the region and not across the  region as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Glaciers form when snow compresses under its  own weight and hardens over long timescales [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Near its formation, glacial ice has snow or ice sur-  face cover and is known as clean glacial ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  As  glacial ice slowly moves down valleys under grav-  ity, avalanches can deposit debris (rocks and sed-  iment) on top of the glacier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Glacial ice having a  significant covering of dirt/rocks/boulders is known  as debris-covered glacial ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Clean glacial ice and  debris-covered glacial ice appear differently in re-  mote sensing imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' The challenge lies in differ-  entiating clean glacial ice from temporary snow/ice  cover and debris-covered glacial ice from moraines  and the surrounding valleys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' The spectral unique-  ness of clean glacial ice compared to surrounding  terrain makes it relatively easy to identify and lo-  calize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' However, the delineation of debris-covered  glacial ice poses significant challenges because of its  non-unique spectral signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  The earliest approaches for debris-covered glacial  ice segmentation involving deep learning used mul-  tilayer perceptrons to estimate the supraglacial de-  bris loads of Himalayan glaciers using pre-defined  glacier outlines [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  More recent methods for  glacier segmentation use Convolutional Neural Net-  works (CNNs) due to their success in image-based  applications [5, 25, 36, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Most recent approaches  to learning-based image segmentation use variants  of the U-Net architecture [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Originally intro-  duced for biomedical image segmentation, the U-  Net has seen successes in numerous applications  involving satellite image segmentation [23, 29, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Moreover, different architectures based on the U-  Net have also been used for glacier segmentation  in recent years [5, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' However, unlike segmenta-  tion for general images, the results when it comes  to glacier segmentation are not very good, particu-  larly for debris-covered glacial ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Here we present a variant of the U-Net and train  it using multispectral images from Landsat 7 as in-  puts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Researchers have shown that the performance  of deep learning models can be improved by learn-  ing multiple objectives from a shared representa-  tion [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Early approaches to learn multiple tasks  use weighted sum of losses, where the loss weights  are either constant or manually tuned [13, 21, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  We propose a method to combine two different  loss functions - masked dice loss [2] and boundary  loss [9] - to simultaneously learn multiple objec-  tives automatically during the training process for  improved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  While deep learning models have been shown to  perform well on various tasks involving computer  vision, the interpretability of these models is lim-  ited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Deep neural networks are often considered  black boxes, since their decision rules can not be  described easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Unlike coefficients and decision  boundaries of simpler machine learning methods  such as linear regression and decision trees, weights  of neurons in deep neural networks can not be  understood as knowledge directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  The develop-  ment of transparent, understandable, and explain-  able models is imperative for the wide-scale adop-  tion of deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Over the years, many  have proposed different approaches to describe deep  leaning models [22, 37, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' One of the most widely  used methods to envision which pixels in the in-  put image affect the outputs the most is by visu-  alizing saliency maps [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' A saliency map is ob-  tained by calculating the gradient of the given out-  put class with respect to the input image by letting  gradients backpropagate to the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' In the case  of multispectral or hyperspectral images, spectral  saliency [20] is used to visualize salient pixels of an  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Image saliency maps, computed indepen-  dently for all channels on a multispectral image,  can be used to visualize the contribution of each  pixel in each channel toward the final output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' We  propose a method to quantify each channel’s con-  tributions towards the final label in the context of  glacier segmentation using Landsat 7 imagery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  2  Dataset and Methodology  The HKH region covers an area of about 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='2 million  km2 from about 15◦ to 39◦ N latitude and about  60◦ to 105◦ E longitude extending across eight  countries consisting of Afghanistan, Bangladesh,  Bhutan, China, India, Myanmar, Nepal, and Pak-  istan [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  The geographic extent of the glaciers  within the HKH, however, ranges from about 27◦  to 38◦ N and about 67◦ to 98◦ E (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  We downloaded the Landsat 7 images used for  label creation using Google Earth Engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Landsat  7 contains the Enhanced Thematic Mapper Plus  (ETM+) sensor which captures multiple spectral  bands as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' The thermal infrared  2  bands were upsampled from 60 meters to 30 meters  resulting in all bands having a spatial resolution  of 30 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  The glacier outlines (labels) [3]  were downloaded from International Centre for  Integrated  Mountain  Development  (ICIMOD)  Regional Database System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' (http://rds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='icimod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  org/Home/DataDetail?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='metadataId=31029)  The  glacier labels contain information on clean-ice and  debris-covered glaciers in the HKH for regions  within Afghanistan, Bhutan, India, Nepal, and  Pakistan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' The ICIMOD glacier outline labels used  in this research were derived using the object-  based image classification methods separately for  clean-ice and debris-covered glaciers and fine-tuned  with manual intervention [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Table 1: Landsat 7 bands description  Name  Description  B1  Blue  B2  Green  B3  Red  B4  Near Infrared  B5  Shortwave Infrared 1  B6 VCID 1  Low-gain Thermal Infrared  B6 VCID 2  High-gain Thermal Infrared  B7  Shortwave Infrared 2  The Landsat 7 images that were used for  delineating glacier labels [4] overlap spatially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  To avoid spatial overlap between train and test  regions, we created polygon features represent-  ing a fishnet of rectangular cells for the entire  geographical region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  We then created a mosaic  of all Landsat 7 images used for labeling into  a single raster and clipped the raster mosaic to  country boundaries for glacier labels (Figure 1) to  avoid false negative glacier labels in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Finally, we discarded the rasters within the poly-  gon cells that do not contain any glacier labels  and downloaded clipped regions within selected  cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' The Google Earth Engine code to replicate  this process can be found in repository https:  //code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='earthengine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='google.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='com/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='accept_  repo=users/bibekaryal7/get_hkh_tiff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  The  selected cells were then randomly split into train,  validation, and test sets with no geospatial overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  1163 out of 1364 cells were filtered out to leave  us with 141, 20, and 40 cells in the training,  validation, and test sets respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Each cell  was then cropped into multiple sub-images of  512 × 512 pixels and the sub-images with less  than 10% of pixels as glacier labels were discarded  to reduce class imbalance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' These sub-images are  then normalized and provided as input to the  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' There are 333, 68, and 98 sub-images in  the training, validation, and test sets respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Every pixel within each sub-image can have one of  four different classes as can be seen in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Figure 2: (a) Sample sub-image, (b) Corresponding  Clean Glacial Ice, Debris Glacial Ice, Background,  and Masked labels  The step-by-step processing we followed to pre-  pare input features for the model is shown in Fig-  ure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' The distribution of pixels for train, valida-  tion, and test set across different classes is shown  in Table 2 and highlights that the distribution of  pixels across different sets is similar and labels are  heavily imbalanced across classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Table 2: Labels Distribution - Random Sampling  split  background  clean  debris  masked  train  72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='44%  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='77%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='44%  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='35%  val  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='69%  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='22%  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='24%  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='85%  test  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='16%  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='97%  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='65%  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='21%  clean = clean glacial ice  debris = debris-covered glacial ice  We used a modified version of the U-Net archi-  tecture [30] as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Each input sub-  image is 512 × 512 pixels in size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Zero padding was  added during each convolution operation to make  the output labels the same size as input sub-images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  We replaced the Rectified Linear Unit (ReLU) in  the original U-Net architecture with Gaussian Er-  3  Figure 3: Input preprocessing  ror Linear Units (GELU) [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' We applied batch  normalization after each convolution operation and  spatial dropout [35] of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='1 after each down-sampling  and up-sampling block to reduce overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' We  also randomly modified 15% of the training sam-  ples by either rotating (90◦, 180◦, 270◦) or flipping  (horizontal/vertical) the input sub-images to the  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  We trained the modified U-Net architec-  ture for 250 epochs using the Adam optimizer and  evaluated the performance based on precision, re-  call, and Intersection over Union (IoU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Figure 4: Our modified U-Net architecture has 32  feature maps in the first convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' We also  introduce batch normalization and spatial dropout  in the modified architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  We trained two separate models, one for seg-  menting clean glacial ice and one for debris-covered  glacial ice, and combined the outputs to produce  the final segmentation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Definitions of what  constitutes debris-covered glacial ice vary widely,  however, as a glacier does not have to be fully cov-  ered by debris to be classified as debris-covered  glacial ice [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Therefore, for the pixels where  debris-covered glacial ice labels overlapped with  clean glacial ice labels on the final segmentation  map, the output label was set as debris-covered  glacial ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' The code to replicate our process can be  found in the GitHub repository (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  com/Aryal007/glacier_mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='git).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  3  Experiments  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='1  Self-learning  Boundary-aware  Loss  The subject of this section of our work lies at the  intersection of two branches of research, which are  penalizing misalignment of label boundaries by us-  ing a boundary-aware loss and learning multi-task  weights during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' We propose a  combined loss (LCombined) that is a weighted sum  of masked dice loss (LMDice) and boundary loss  (LBoundary), as described in Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' We also  compare the performance of our methods using the  modified U-Net to the standard U-Net trained on  cross entropy loss (LCE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  LCombined = α × LMDice + (1 − α) × LBoundary  (1)  The value of hyperparameter α can be set manu-  ally between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Having an α of 1 is equivalent  to training the model exclusively using masked dice  loss and an α of 0 is the same as training the model  exclusively using boundary loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' However, tuning  the value of α manually for the best results is an  expensive process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  In order to learn the weights  for LBoundary and LMDice through backpropaga-  tion, we initially set α to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='5 and let the model  find the best value of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' However, we observe that  without any constraints on the value of α, the net-  work updates α such that LCombined is minimized  without necessarily having to minimize LMDice or  LBoundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' This results in poor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' In-  spired by [18] for weighing two different loss func-  4  Landsat 7  Fishnet Grid  Glacier Labels  Clip to country  boundaries  Selected cells  Intersecting Raster  Randomly split train,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Rasterize Labels  test,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' val  ImputeNaN  Filter by minimum  Crop to 512×512  glacier percentage  subimage  Normalize  Augment feature label  Input to modified-  pairs  UNet32 32  32 32  ftma  64 64  64 64  512 × 512  512 × 512  256 × 256  256 × 256  128 128  128 128  Conv 3×3  128 × 128  128 × 128  256 256  256 256  Batchnorm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' GELU  512  SpatialDropout  64 × 64  64 × 64  Max Pool 2×2  32 × 32  ConvTranspose2d  2×2Table 3: Performance comparisons between standard U-Net trained using cross entropy loss and modified  U-Net trained using combined loss and self-learning boundary-aware loss  Loss (L)  Lweight(s)  clean  debris  Precision  Recall  IoU  Precision  Recall  IoU  LCE  −  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='39%  84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='65%  70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
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+page_content='00%  LCombined  0  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='40%  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
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+page_content='00%  LCombined  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='1  79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
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+page_content='30%  33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='43%  LCombined  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='5  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='60%  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
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+page_content='16%  46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='92%  32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='41%  LCombined  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='9  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
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+page_content='41%  LCombined  1  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
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+page_content='05%  LSLBA  Dynamic  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='59%  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='55%  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='17%  51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='97%  53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='81%  35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='94%  clean = clean glacial ice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' debris = debris-covered glacial ice  tions, we propose Self-Learning Boundary-Aware  loss (LSLBA) that is a combination of LMDice and  LBoundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  LSLBA =  1  2α2  1 × LMDice +  1  2α2  2 × LBoundary + |ln (α1 × α2)|  (2)  In the case of LSLBA, α1 and α2 both are initially  set to 1 and we let the model find the best value  for α1 and α2 through backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' In Table 3  we show performance for different values of α in  the case of LCombined and performance of LSLBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  One advantage of using LSLBA over LCombined is  that there is no extra hyperparameter that requires  fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' All experiments in Table 3 use eight  features from Landsat 7 imagery as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Figure 5: Masked Dice Loss weights and Boundary  Loss weights vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' epoch for debris-covered glacial ice  From Table 3, we see that LSLBA performs the  best for debris-covered glacial ice segmentation and  eliminates the need to fine-tune loss weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' We  can also see that the model fails to converge when  training solely on boundary loss (α = 0) and train-  ing on glacier boundaries by incorporating bound-  ary loss along with masked dice loss results in  an overall improvement in performance for debris-  covered glacial ice regardless of the weighting fac-  tor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Figure 5 shows the weights for masked dice loss  (  1  2α2  1 ) and the weights for boundary loss (  1  2α2  2 ) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  epoch during training for debris-covered glacial ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  The optimal values for α1 and α2 are calculated to  be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='9569 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='045 for clean glacial ice segmenta-  tion and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='952 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='05 for debris-covered glacial  ice segmentation for LSLBA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='2  Representation Analysis  To understand the contribution of each feature in  the multispectral image toward the final label, we  computed a Saliency Score (SS) for each feature by  summing all pixels in the Saliency Map (SM) for  that feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  SSfeature =  c−1  �  i=0  r−1  �  j=0  SM feature(i, j)∀ feature ∈ Input  (3)  where:  r, c = number of rows, columns in saliency map  Average feature saliency scores across all the im-  ages in the training samples are shown in Fig-  ure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  The channel-wise contributions towards  5  Weights  Masked Dice  Boundary  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='54  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='52  lue  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='46  0  50  100  150  200  250  EpochFigure 6: Average saliency scores for all sub-images  in training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  debris-covered glacial ice segmentation in decreas-  ing order are: red, shortwave infrared 1, near in-  frared, green, high-gain thermal infrared, short-  wave infrared 2, low-gain thermal infrared, and  blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Similarly, for clean glacial ice segmentation,  the channel-wise contributions in decreasing order  are: shortwave infrared 2, blue, shortwave infrared  1, high-gain thermal infrared, red, low-gain ther-  mal infrared, near infrared, and green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' As shown  in Figure 6, the segmentation models have different  high contributing channels for clean glacial ice and  debris-covered glacial ice segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  4  Discussion  Glaciers have been melting at an unprecedented  rate  in  recent  years  due  to  global  climate  change [17, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Glaciers are the largest fresh-  water reservoir on the planet [32], so it is nec-  essary to understand the changes they undergo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  As a result, numerous approaches to automati-  cally delineate glacier boundaries have been pro-  posed [5, 6, 14, 15, 34, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' We frequently observe  deep learning methods outperforming traditional  machine learning methods for glacier segmentation  in the literature [1, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' However, the results have  not been very good, particularly in the case of  debris-covered glacial ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  In this work, we modify U-Net and train it using  a novel loss function that allows the modified U-  Net to focus on glacier boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' From Table 3,  we see that standard U-Net is not able to detect  debris-covered glacial ice in input sub-images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' We  can also see from Table 2 that only 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='44% of pix-  els in the training set correspond to debris-covered  glacial ice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' This shows that our proposed method  is more robust than the original U-Net to imbal-  anced labels, which are common in remote sensing  datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  From Figure 5, we can see how the weights  change for LSLBA while training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' A higher weight  is assigned to masked dice loss at the beginning  and the weights for boundary loss are gradually in-  creased during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  The reason behind this  could be that for an untrained model, it may be  easier to learn glacier instances over trying to learn  the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' However once the network learns  to label instances, it is easier to learn the glacier  boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' This also explains why the model fails  to converge when training solely on LBoundary from  scratch as can be seen from the results in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  We presented methods to improve debris-covered  glacial ice segmentation from remote sensing im-  agery using deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' While we were able to  show significant improvements over existing meth-  ods, the IoU for debris-covered glacial ice still leaves  much to be desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' The existing body of litera-  ture on the topic has shown that the performance  for debris-covered glacial ice segmentation can be  improved by incorporating thermal signatures [28]  and topographical information [10, 26, 27] from  other satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Since debris-covered glacial ice is  common in low-gradient areas due to how it forms  and has cooler surface temperatures compared to  the surrounding non-glaciated regions, we suspect  that adding this information can further help im-  prove the performance of debris-covered glacial ice  segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' We may also be able to see an im-  provement in performance by using images from  the recently-launched Landsat 9 satellite, instead  6  Saliency Scares  Debris  1D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='B  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='6  t0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='D  BG VCID1  BG VCID2  B2  相  B5  B3  Clean Ice  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='D  B2  B4B6_vCID1 B3  B6_vCID2  B5  B1  B7of the Landsat 7 images used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  The  Operational Land Imager 2 (OLI-2) and the Ther-  mal Infrared Sensor 2 (TIRS-2) sensors on Land-  sat 9 provide data that is radiometrically and ge-  ometrically superior to instruments on the previ-  ous generation Landsat satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' With the higher  radiometric resolution, Landsat 9 can differentiate  16,384 shades of a given wavelength compared to  only 256 shades in Landsat 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Meanwhile, the  TIRS-2 in Landsat 9 enables improved atmospheric  correction and more accurate surface temperature  measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Future work includes using the im-  ages captured through these improved sensors and  incorporating additional information such as a dig-  ital elevation model for improving debris-covered  glacial ice segmentation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  5  Conclusion  In this research study, we proposed a modified  version of the U-Net architecture for large-scale  debris-covered glacial ice and clean glacial ice seg-  mentation in the HKH from Landsat 7 multispec-  tral imagery and concluded that debris-covered  glacial ice (IoU: 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='94%) is significantly harder  to delineate compared to clean glacial ice (IoU:  68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='17%)(Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' We also introduced two differ-  ent methods to combine commonly-used masked  dice loss and boundary loss to incorporate label  boundaries into the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  We show  that the performance of debris-covered glacial ice  segmentation can be improved by encouraging the  deep learning model to focus on label boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  The performance can be improved further by cor-  rectly weighing loss terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Furthermore, the rela-  tive weights can be learned automatically from the  data during the training process using our proposed  loss (LSLBA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Figure 7 shows the performance of  the models trained using LSLBA on a sample image  from the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' We also introduced the concept of  feature saliency scores to quantify the contribution  of each feature (channel) in the input image toward  the final label and concluded that the red, short-  wave infrared, and near infrared bands contribute  the most towards the final label for debris-covered  glacial ice segmentation, while shortwave infrared  2, blue, shortwave infrared 1 bands contributed the  most towards the final label for clean glacial ice  segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='  Figure 7: (a) Sample subimage from test set (b)  Corresponding clean glacial ice and debris-covered  glacial ice ground truth labels (c) True positive  (TP), False positive (FP), False negative (FN) for  clean glacial ice (IoU 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='17%) (d) TP, TP, FN for  debris-covered glacial ice (IoU 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='19%)  6  Acknowledgements  We would like to thank Microsoft for providing us  with the Microsoft Azure resources through their  AI for Earth grant program (Grant ID: AI4E-1792-  M6P7-20121005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' We also acknowledge ICIMOD  for providing a rich dataset which this work has  been built on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' This research was supported in part  by the Department of Computer Science at The  University of Texas at El Paso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
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+page_content=' Sensors, 18(11), 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' ISSN 1424-  8220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
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+page_content='mdpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content='com/1424-8220/18/11/3717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
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+page_content=' Miao, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Sankaran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' Inter-  active visualization and representation anal-  ysis applied to glacier segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
+page_content=' ISPRS  International Journal of Geo-Information, 11  (8), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtFJT4oBgHgl3EQfICxT/content/2301.11454v1.pdf'}
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diff --git a/P9FJT4oBgHgl3EQf2S1v/content/tmp_files/2301.11655v1.pdf.txt b/P9FJT4oBgHgl3EQf2S1v/content/tmp_files/2301.11655v1.pdf.txt
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+Nitrogen in Silicon for Room Temperature Single Electron Tunneling Devices        
+Pooja Yadav1, Hemant Arora1, and Arup Samanta1,2* 
+1Department of Physics, Indian Institute of Technology Roorkee, Roorkee-247667, Uttarakhand, India 
+2Centre of Nanotechnology, Indian Institute of Technology Roorkee, Roorkee-247667, Uttarakhand, India  
+                                                  *Corresponding author: arup.samanta@ph.iitr.ac.in 
+Single electron transistor (SET) is an advanced tool to exploit in quantum devices. Working of such devices at room-
+temperature is essential for practical utilization. Dopant based single-electron devices are well studied at low-temperature 
+although a few devices are developed for high-temperature operation with certain limitations. Here, we propose and 
+theoretically exhibit that nitrogen (N) donor in silicon is an important candidate for effective designing of such devices. 
+Theoretical calculation of density-of-states using semi-empirical DFT method indicates that N-donor in silicon has deep 
+ground state compared to a phosphorus (P) donor. N-donor spectrum is explored in nano-silicon along with the P-donor. 
+Comparative data of Bohr radius of N-donor and P-donor is also reported. The simulated current-voltage characteristics 
+confirm that N-doped device is better suited for SET operation at room-temperature.   
+In the past two decades, a significant effort had been 
+given to develop single electron tunneling devices using dopant 
+atom for solid state-based quantum architecture by utilizing and 
+manipulating the charge and spin degrees of freedom using 
+external fields.1–4 Dopant based SET devices are key 
+components in quantum device applications e.g., quantum bit1, 
+memories5, single electron pump6, single charge sensing7, 
+single photon detectors8 etc. Extensive work has been put 
+forward on a few dopant based devices where single electron 
+tunneling effect are observed at low temperatures.9–14 However, 
+the room temperature operation is highly needed for practical 
+utilization of such devices. The reason being the low barrier 
+height and charging energy due to shallow nature of dopants 
+e.g., arsenic (As) phosphorous (P) as donor and boron (B) as 
+acceptor in silicon. Some results on high temperature operation 
+of these devices due to quantum confinement, dielectric 
+confinement, and donor’s cluster formation have also been 
+reported.15,16  
+To improve the operating temperature of such SET, 
+alternative dopants are also proposed for the utilization of 
+double donors like tellurium (Te), selenium (Se), and sulfur (S), 
+which have one order higher binding energy than shallow 
+donors.17 Recently, the spin relaxation and donor-acceptor 
+recombination of Se+ in 28Si was reported.18 However, use of 
+double donors needs control of several associated effect due to 
+the existence of multiple electrons in the donor side. Erbium 
+being a deep donor has also been studied for spin-based 
+devices.19 High temperature operations of the germanium 
+vacancy complexes based devices have recently been proposed 
+and demonstrated.20,21 Room temperature operation of deep 
+donor pair of Al-N in silicon is reported in TFET 
+configuration.22 However, the use of a single deep dopant is 
+better for quantum operation due to the minimization of 
+decoherence path.  
+Here, we propose an alternative quantum architecture 
+for such approach using nitrogen (N) donor in FinFET/SOI-
+FET configuration for high temperature operation and also 
+explored the N-donor spectrum in bulk-silicon, nano-silicon, 
+and in device configuration. Nitrogen energy level is ~190 meV 
+below the conduction band of bulk silicon, which can be a 
+suitable candidate for high temperature single electron 
+tunneling devices.23 This means the discrete states of N-donor 
+can be preserved at room temperature by suppressing the 
+crosstalk with external environment and saving the electron 
+states from thermal demons. Accounting the natural abundance 
+of Si28 isotope as a host for nitrogen donor, electron and spin 
+states can also be conserved for a longer time considering the 
+deep donor energy levels.24 In addition, this system could also 
+be an important candidate for nuclear spin qubit by analogy with 
+P-donor in silicon, since the different isotopic nuclear spin of 
+N-donor and the zero isotopic spin of the silicon host medium.1  
+In this letter, undoped, phosphorous doped and 
+nitrogen doped hydrogenated silicon nanowire (H:SiNW) in 
+normal and SET configurations along with the bulk structure are 
+designed and the comparative analyses using semi-empirical 
+density functional theory (DFT) calculation  for these 
+configurations were performed. The projected density of states 
+(PDOS), total density of states (TDOS), local density of states 
+(LDOS) along with the current-voltage (IDS-VG) characteristics 
+are studied. Single electron tunneling transport through the 
+discrete energy states are investigated for both P-doped and N-
+doped devices at low and room temperatures.  
+The numerical calculations are performed on Quantum 
+ATK software25 based on density functional theory. 
+Semiempirical extended-Hückel26 method is used for these 
+calculations at 300 K. The mesh cuttoff was set at 20 Hartree. 
+The k grid 1×1×174 is used. The atomistic structure of a silicon 
+nanowire of length 32.4 Å and diameter 10 Å is constructed 
+along [100] direction. The surface dangling bonds are 
+passivated by the hydrogen atoms. The schematic and atomistic 
+design of the structures are presented in Fig. 1(a-b). An in-built 
+optimizer is utilized for structure optimization with force 
+constant 0.008 eV-1. This structure is used to calculate the 
+TDOS and PDOS spectra of H:SiNW.   In addition, we 
+examined the density of states profile of undoped, P-doped, and 
+N-doped bulk silicon used with 4x4x4 fcc supercell (512 Si 
+lattice sites). The density mesh cut-off was set to 20 Hartree 
+with 4ⅹ4ⅹ4 k-point set proposed by Makov, Shah, and Payn.27 
+To study donor-based transistor devices, the SiNW 
+transistors having diameter of 10 Å and length of 54 Å are also 
+structured by gate all around assembly along with tunnel 
+coupling of the center dopant with source and drain reservoirs 
+
+as shown in Fig 1(c). In these devices, the gate covers the donor 
+position of the channel of the SET devices. Vacuum is used as 
+a gate dielectric for the present configurations. The source and 
+drain leads are heavily doped with n-type dopant (1×1021 cm-3). 
+IDS-VG characteristics of these devices are calculated using the 
+non-equilibrium Green’s function (NEGF) method with 
+Landauer formalism.28 
+ 
+Figure 1: (a) Quantum ATK simulated structure for (a) H:SiNW, (b) 
+substitutional P/N-doped (encircled by red dotted line) H:SiNW, and (c) P/N-
+doped H:SiNW in transistor configuration with gate terminal and source drain 
+terminals. Top Panel: schematic of (a-c) structures. SiNW radius is in x-y 
+dimension and z dimension is along the length axis. 
+ 
+Initially the TDOS spectrum of undoped, P-doped and 
+N-doped bulk system are studied in Fig. 2 (a-c). The undoped 
+bulk silicon shows a band gap of 0.6 eV as presented in Fig. 2a, 
+consistent with previous reports.29 The P-doped and N-doped 
+bulk Si systems are studied subsequently. Here, the Fermi 
+energy is set at 0 eV for all results. The donor energy state is 
+found at E-EF = 0 eV for both the cases, which are the ground 
+states of the respective donors. The separation between the 
+donor ground state (GS) and next conducting state depicts the 
+shallow and deep nature of the P-donor and N-donor, 
+respectively. In the TDOS spectrum of P-donor in Fig. 2 (b), the 
+mixed states are present at E-EF=0 eV, and donor states cannot 
+be distinctly separated due to thermal smearing. In the TDOS 
+spectrum of N-doped bulk silicon as shown in Fig. 2(c), the 
+donor state lies at 0 eV and the next conducting state at 184 
+meV. Since the donor states are merged in the conduction band 
+of bulk structure, it is difficult to separate further donor states. 
+Hence, the separation of the GS of N-donor with conduction 
+band edge (CBE) is 184 meV, consistent with experimentally 
+reported value.30  
+Now to get a full insight of the donor states and a 
+detailed analysis of donor behavior in nano-silicon devices, we 
+investigated the same analysis in H:SiNW systems. From now 
+onwards, all the calculations are performed on H:SiNW. The 
+TDOS spectra of undoped, P-doped, and N-doped H:SiNW is 
+calculated and presented in Fig. 2(d-f), respectively. In the 
+TDOS spectrum of undoped H:SiNW as shown in Fig. 2(d), the 
+separation between the conduction-band and valence-band 
+edges is very high (~1.7 eV), indicating a wide band gap due to 
+the confinement effect.   
+Modification in the TDOS features is observed in both 
+P-doped and N-doped systems compared to undoped H:SiNW 
+as presented in Fig. 2(e) and 2(f), respectively. Additional states 
+within the bandgap are observed in doped structures.   To probe 
+the TDOS spectra in details, the PDOS spectra are studied to 
+identify and locate the states. It is well documented that the 
+donor energy level in silicon splits into three energy groups due 
+to valley degeneracy: one A1 state (lowest energy state), one 
+threefold T2 state (1st excited state), and one twofold E state 
+(2nd excited state).31,32 The A1 orbital shows s-like symmetry on 
+the dopant location of nanostructure while the T2 and E orbitals 
+show a node at this position.33 The A1, T2, and E states are 
+identified for both P-donor and N-donor in this study in 
+comparison with the literature.34  
+   Figure 2: (a-c) The TDOS spectra of undoped, P-doped, and N-doped bulk silicon. (d-f) The TDOS spectra of undoped, P-doped, and N-doped H:SiNW.
+
+Silicon
+Silicon
+P/N
+Source
+Gate
+Drain
+Drain
+Sorc
+Gate
+1021cm-3
+1021cm Figure 3. (a-d) The PDOS spectra of P-doped H:SiNW for different positions along the channel starting from the P-donor location, where the states A1, T2, and E 
+are identified. (e-h) Same spectra for N-donor atom.
+We focused on the position dependent PDOS of the P-
+donor in Fig. 3(a-d). The PDOS components of s, p, and d 
+orbitals of the P-donor on itself i.e., d=0 exhibit that A1, T2, and 
+E energy levels exist at this location, where A1 state is the lowest 
+energy state and it is mainly coming from s-orbital of the P-
+donor.      
+  
+   The T2 and E energy levels are the mixture of s, p, and d orbitals. 
+The PDOS spectrum of P-donor on different silicon locations (d 
+starts from the donor location and extends towards drain 
+reservoir side) are investigated in Figs. 3(b-d). The existence of 
+the PDOS on the silicon position confirms that the hybridization 
+is present between the electronic states of P and Si atoms. We 
+mainly focus on the ground state (i.e., A1) of the P-donor, which 
+diminishes as we move away from the donor location. The 
+localization of the A1 state can be correlated with the spreading 
+of this state. Such behavior can be linked to the Bohr radius of 
+the ground state, which is the distance from where the A1 state 
+originates and where the A1 state dies out. The extinction of the 
+A1 state is the end of this state and its length from the donor 
+location is d =16.29 Å. Hence, the Bohr radius of P-donor in such 
+system is ~ 16.29 Å, which is far smaller than the bulk case (~25 
+Å).31 The reduction in the Bohr radius happens due to the 
+quantum confinement effect. We study the PDOS spectra of N-
+donor in H:SiNW as shown in Fig. 3(e-h) and found a similar 
+trend of features as we shift far from the donor location. The 
+PDOS results for A1 state reduce to negligible value at d=5.43 
+Å, which signifies the ground state is highly concentrated and 
+the Bohr radius is ~5.43 Å. We also observe that A1 state is 
+largely separated from any other states of this configuration. The 
+difference between N-donor’s ground state and the first excited 
+state (ES) is 770 meV while that for P-donor is 118 meV. This is 
+consistent with the highly localized ground state of N-donor. 
+Such highly localized and deep ground state is the pathway for 
+designing quantum devices for room temperature.  
+ 
+Figure. 4: (a-b) PDOS/TDOS spectra of a P-doped and N-doped silicon 
+nanowire, respectively. 
+To calculate the binding energy of donor electron (Eb), we 
+estimated the lowest effective conductive state of doped 
+H:SiNW following Anh et al.33 Since the donor electronic states 
+hybridize with silicon electronic states, the conductive state 
+changes in comparison with undoped system. The Eb is estimated 
+by taking the ratio of PDOS/TDOS, which signifies the relative 
+weight of donor states against the states of the whole system. 
+Figures 4(a) and (b) show the PDOS/TDOS spectra for P-donor 
+and N-donor, respectively. At lower energy range, the 
+contribution of the donor is dominated, while the contribution is 
+trivial at higher energy range (above blue dashed line) due to the 
+inclusion of more number of silicon atoms. The binding energy 
+is calculated from the point where this ratio saturates close to 0. 
+The calculated Eb values for P-donor and N-donor are 1.36 eV 
+and 2.40 eV, respectively.  
+In order to understand the behavior of undoped and doped 
+devices, LDOS spectra are simulated along with the current vs 
+gate voltage (IDS-VG) characteristics for different source to drain 
+voltage (VDS) using the transistor device parameters described in 
+Fig. 1(c). The LDOS spectrum of the undoped H:SiNW 
+transistor is obtained where no states are observed below the 
+conduction band edge as shown in Fig. 5(a). In Fig. 5(b) and 5(c), 
+extra energy states are observed in the LDOS data due to 
+substitutionally doped P-donor and N-donor, respectively.  
+
+Figure. 5: (a-c) LDOS spectra of an undoped, P-doped, and N-doped silicon nanowire transistor, respectively. (d-f) LDOS spectrum at z = 26.93 Å for undoped, P-
+doped, and N-doped silicon nanowire transistor, respectively. Inset of Fig. 5(f) indicates the zoom in spectrum, exhibiting the GS and and 1st ES of N-donor. 
+Simulated IDS-VG characteristics for (g) undoped, (h-i) P-doped, and (j-k) N-doped devices. 
+The lowest energy state in the channel region of P-doped 
+device is found at 80 meV from the next conductive state, which 
+is low compared to non-device configuration (118 meV). This 
+is most likely due to the effect of metal gate and closely spaced 
+heavily doped leads. In LDOS graph of N-doped device, a very 
+deep energy state is observed at 322 meV below the nearest 
+conductive state. To emphasize more on the donor states 
+obtained in LDOS spectra, the same information of the LDOS 
+spectra is mapped along the vertical axis in Fig. 5(d-f). The 
+LDOS spectra along the Z axis at z = 26.93 Å are presented for 
+all three device structures. No LDOS is observed below CBE 
+for undoped device as shown in Fig. 5(d). A single LDOS peak 
+in P-doped SiNW at 80 meV is observed in Fig. 5(e), indicating 
+the GS of P-donor in the device configuration. A highly 
+localized LDOS is also perceived at 322 meV from the CBE in 
+Fig. 5(f), providing the information of the GS of N-dopant in 
+the device configuration. In addition, we also observed 1st ES 
+of N-donor at 54 meV below the CBE. 
+The IDS-VG transport characteristics for all three devices 
+are simulated at the different bias voltages for T= 5 K and 300 
+K, as presented in Figs. 5(g-k). In the undoped device with VDS= 
+5 mV, the characteristics are typical FET devices, which is 
+consistent with the LDOS spectrum of this device 
+configuration. The weak modulation in IDS for T = 5 K is 
+observed at ~ VG=0.5 mV in Fig. 5(g), which is due to the 
+transport through the discrete states in the CB of the undoped 
+H:SiNW. It is also observed that the subthreshold slopes of IDS-
+VG curves at 5 K and 300 K are almost same that may be due to 
+the effect of heavy doping (1021 cm-3) of the leads along with 
+the nano-channel effect.  
+The transport characteristics for P-doped device are 
+presented in Fig. 5(h)-(i) for VDS= 1 mV and 5 mV, respectively, 
+
+LDOS (eV"
+LDOS (eV
+LDOS (eV")
+5x10
+1x104
+5x10
+j1x104
+5x10°
+1x10-5
+Undoped
+Phosphorous
+Nitrogcen
+doped
+dopedshowing a strong single electron tunneling current peak before 
+the on-set of FET current at T=5 K due to the existence of 
+localized state for P-donor within the channel window. An 
+additional weak current peak is also preserved due to the 
+transport through the next discrete conducting state. However, 
+this SET peak dies out at 300 K due to the shallow nature of 
+such dopant.  
+The IDS-VG characteristics for N-doped device are 
+presented in Fig. 5(j-k) for VDS= 1 mV and VDS = 5 mV, 
+respectively. For VDS= 1 mV, a very strong tunneling peak is 
+observed via the GS of N-donor. However, the GS and 1st ES 
+prominently participated in the transport characteristics at 
+T=300 K. The excited state features got observed at high 
+temperature due to higher tunnel rate compared to 5 K. For VDS= 
+5 mV and T=5 K, the low temperature behavior of the device is 
+similar to VDS= 1 mV with lower separation from the FET 
+current, and this current peak is also strongly sustained at 
+T=300 K. Along with the transport through the GS at 300 K, a 
+small hump corresponding to 1st excited state is also noted. 
+Observation of a single electron current peak at room 
+temperature is happened due to deep nature of N-donor. Such 
+deep donor state is capable of holding the quantum information 
+and is less fragile to environment fluctuations compared to 
+shallow donor. Silicon being one of the most promising 
+semiconductor for quantum information devices due to large 
+charge and spin coherence times,34,35,36 donor state of nitrogen 
+in silicon gives an opportunity to utilize it for better 
+performances. Hence, to design single electron transistor with a 
+single donor atom for room temperature operation, N-donor in 
+silicon could be one of the best systems. We are working in such 
+direction for the experimental realization of such devices.  
+Donor energy spectrum of nitrogen in hydrogenated-
+silicon nanowire is theoretically investigated in comparison 
+with phosphorous and undoped system. We observed that 
+binding energy of N-donor ground state is large compared to 
+the P-donor. It is also highly localized and separation of ground 
+state is also very large compared to 1st excited state.  In 
+quantum devices where perseverance of quantum states is 
+important, this advantage can be taken from a natural system 
+with deep N-donor states. Simulated LDOS and IDS-VG 
+characteristics of the devices also suggest in favor for N-dopant 
+being the potential candidate for single electron tunneling 
+devices for room temperature operation, which further infers 
+this system could be useful for designing practical quantum 
+architecture.  
+The authors are thankful to Prof. Sparsh Mittal for the computational resources 
+under the project ECR/2017/000622. The work is partially supported by DST-
+SERB (Project no: ECR/2017/001050) and IIT Roorkee (Project no: FIG-
+100778-PHY), India. P.Y. and H.A. thank to Ministry of Education and UGC, 
+India, respectively for the research scholarship.  
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+
diff --git a/P9FJT4oBgHgl3EQf2S1v/content/tmp_files/load_file.txt b/P9FJT4oBgHgl3EQf2S1v/content/tmp_files/load_file.txt
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index 0000000000000000000000000000000000000000..f27ad969298364bf019521138b659ab1d1cca8dc
--- /dev/null
+++ b/P9FJT4oBgHgl3EQf2S1v/content/tmp_files/load_file.txt
@@ -0,0 +1,635 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf,len=634
+page_content='Nitrogen in Silicon for Room Temperature Single Electron Tunneling Devices  Pooja Yadav1, Hemant Arora1, and Arup Samanta1,2*  1Department of Physics, Indian Institute of Technology Roorkee, Roorkee-247667, Uttarakhand, India  2Centre of Nanotechnology, Indian Institute of Technology Roorkee, Roorkee-247667, Uttarakhand, India  Corresponding author: arup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='samanta@ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='iitr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='in  Single electron transistor (SET) is an advanced tool to exploit in quantum devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Working of such devices at room-  temperature is essential for practical utilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Dopant based single-electron devices are well studied at low-temperature  although a few devices are developed for high-temperature operation with certain limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Here, we propose and  theoretically exhibit that nitrogen (N) donor in silicon is an important candidate for effective designing of such devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Theoretical calculation of density-of-states using semi-empirical DFT method indicates that N-donor in silicon has deep  ground state compared to a phosphorus (P) donor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' N-donor spectrum is explored in nano-silicon along with the P-donor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Comparative data of Bohr radius of N-donor and P-donor is also reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The simulated current-voltage characteristics  confirm that N-doped device is better suited for SET operation at room-temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  In the past two decades, a significant effort had been  given to develop single electron tunneling devices using dopant  atom for solid state-based quantum architecture by utilizing and  manipulating the charge and spin degrees of freedom using  external fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='1–4 Dopant based SET devices are key  components in quantum device applications e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=', quantum bit1,  memories5, single electron pump6, single charge sensing7,  single photon detectors8 etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Extensive work has been put  forward on a few dopant based devices where single electron  tunneling effect are observed at low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='9–14 However,  the room temperature operation is highly needed for practical  utilization of such devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The reason being the low barrier  height and charging energy due to shallow nature of dopants  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=', arsenic (As) phosphorous (P) as donor and boron (B) as  acceptor in silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Some results on high temperature operation  of these devices due to quantum confinement, dielectric  confinement, and donor’s cluster formation have also been  reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='15,16  To improve the operating temperature of such SET,  alternative dopants are also proposed for the utilization of  double donors like tellurium (Te), selenium (Se), and sulfur (S),  which have one order higher binding energy than shallow  donors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='17 Recently, the spin relaxation and donor-acceptor  recombination of Se+ in 28Si was reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='18 However, use of  double donors needs control of several associated effect due to  the existence of multiple electrons in the donor side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Erbium  being a deep donor has also been studied for spin-based  devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='19 High temperature operations of the germanium  vacancy complexes based devices have recently been proposed  and demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='20,21 Room temperature operation of deep  donor pair of Al-N in silicon is reported in TFET  configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='22 However, the use of a single deep dopant is  better for quantum operation due to the minimization of  decoherence path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Here, we propose an alternative quantum architecture  for such approach using nitrogen (N) donor in FinFET/SOI-  FET configuration for high temperature operation and also  explored the N-donor spectrum in bulk-silicon, nano-silicon,  and in device configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Nitrogen energy level is ~190 meV  below the conduction band of bulk silicon, which can be a  suitable candidate for high temperature single electron  tunneling devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='23 This means the discrete states of N-donor  can be preserved at room temperature by suppressing the  crosstalk with external environment and saving the electron  states from thermal demons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Accounting the natural abundance  of Si28 isotope as a host for nitrogen donor, electron and spin  states can also be conserved for a longer time considering the  deep donor energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='24 In addition, this system could also  be an important candidate for nuclear spin qubit by analogy with  P-donor in silicon, since the different isotopic nuclear spin of  N-donor and the zero isotopic spin of the silicon host medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='1  In this letter, undoped, phosphorous doped and  nitrogen doped hydrogenated silicon nanowire (H:SiNW) in  normal and SET configurations along with the bulk structure are  designed and the comparative analyses using semi-empirical  density functional theory (DFT) calculation  for these  configurations were performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The projected density of states  (PDOS), total density of states (TDOS), local density of states  (LDOS) along with the current-voltage (IDS-VG) characteristics  are studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Single electron tunneling transport through the  discrete energy states are investigated for both P-doped and N-  doped devices at low and room temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  The numerical calculations are performed on Quantum  ATK software25 based on density functional theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Semiempirical extended-Hückel26 method is used for these  calculations at 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The mesh cuttoff was set at 20 Hartree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  The k grid 1×1×174 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The atomistic structure of a silicon  nanowire of length 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='4 Å and diameter 10 Å is constructed  along [100] direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The surface dangling bonds are  passivated by the hydrogen atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The schematic and atomistic  design of the structures are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 1(a-b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' An in-built  optimizer is utilized for structure optimization with force  constant 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='008 eV-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' This structure is used to calculate the  TDOS and PDOS spectra of H:SiNW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' In addition, we  examined the density of states profile of undoped, P-doped, and  N-doped bulk silicon used with 4x4x4 fcc supercell (512 Si  lattice sites).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The density mesh cut-off was set to 20 Hartree  with 4ⅹ4ⅹ4 k-point set proposed by Makov, Shah, and Payn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='27  To study donor-based transistor devices, the SiNW  transistors having diameter of 10 Å and length of 54 Å are also  structured by gate all around assembly along with tunnel  coupling of the center dopant with source and drain reservoirs  as shown in Fig 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' In these devices, the gate covers the donor  position of the channel of the SET devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Vacuum is used as  a gate dielectric for the present configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The source and  drain leads are heavily doped with n-type dopant (1×1021 cm-3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  IDS-VG characteristics of these devices are calculated using the  non-equilibrium Green’s function (NEGF) method with  Landauer formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='28  Figure 1: (a) Quantum ATK simulated structure for (a) H:SiNW, (b)  substitutional P/N-doped (encircled by red dotted line) H:SiNW, and (c) P/N-  doped H:SiNW in transistor configuration with gate terminal and source drain  terminals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Top Panel: schematic of (a-c) structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' SiNW radius is in x-y  dimension and z dimension is along the length axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Initially the TDOS spectrum of undoped, P-doped and  N-doped bulk system are studied in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 2 (a-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The undoped  bulk silicon shows a band gap of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='6 eV as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 2a,  consistent with previous reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='29 The P-doped and N-doped  bulk Si systems are studied subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Here, the Fermi  energy is set at 0 eV for all results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The donor energy state is  found at E-EF = 0 eV for both the cases, which are the ground  states of the respective donors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The separation between the  donor ground state (GS) and next conducting state depicts the  shallow and deep nature of the P-donor and N-donor,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' In the TDOS spectrum of P-donor in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 2 (b), the  mixed states are present at E-EF=0 eV, and donor states cannot  be distinctly separated due to thermal smearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' In the TDOS  spectrum of N-doped bulk silicon as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 2(c), the  donor state lies at 0 eV and the next conducting state at 184  meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Since the donor states are merged in the conduction band  of bulk structure, it is difficult to separate further donor states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Hence, the separation of the GS of N-donor with conduction  band edge (CBE) is 184 meV, consistent with experimentally  reported value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='30  Now to get a full insight of the donor states and a  detailed analysis of donor behavior in nano-silicon devices, we  investigated the same analysis in H:SiNW systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' From now  onwards, all the calculations are performed on H:SiNW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The  TDOS spectra of undoped, P-doped, and N-doped H:SiNW is  calculated and presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 2(d-f), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' In the  TDOS spectrum of undoped H:SiNW as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 2(d), the  separation between the conduction-band and valence-band  edges is very high (~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='7 eV), indicating a wide band gap due to  the confinement effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Modification in the TDOS features is observed in both  P-doped and N-doped systems compared to undoped H:SiNW  as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 2(e) and 2(f), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Additional states  within the bandgap are observed in doped structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' To probe  the TDOS spectra in details, the PDOS spectra are studied to  identify and locate the states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' It is well documented that the  donor energy level in silicon splits into three energy groups due  to valley degeneracy: one A1 state (lowest energy state), one  threefold T2 state (1st excited state), and one twofold E state  (2nd excited state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='31,32 The A1 orbital shows s-like symmetry on  the dopant location of nanostructure while the T2 and E orbitals  show a node at this position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='33 The A1, T2, and E states are  identified for both P-donor and N-donor in this study in  comparison with the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='34  Figure 2: (a-c) The TDOS spectra of undoped, P-doped, and N-doped bulk silicon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' (d-f) The TDOS spectra of undoped, P-doped, and N-doped H:SiNW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Silicon  Silicon  P/N  Source  Gate  Drain  Drain  Sorc  Gate  1021cm-3  1021cm Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' (a-d) The PDOS spectra of P-doped H:SiNW for different positions along the channel starting from the P-donor location, where the states A1, T2, and E  are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' (e-h) Same spectra for N-donor atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  We focused on the position dependent PDOS of the P-  donor in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 3(a-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The PDOS components of s, p, and d  orbitals of the P-donor on itself i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=', d=0 exhibit that A1, T2, and  E energy levels exist at this location, where A1 state is the lowest  energy state and it is mainly coming from s-orbital of the P-  donor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  The T2 and E energy levels are the mixture of s, p, and d orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  The PDOS spectrum of P-donor on different silicon locations (d  starts from the donor location and extends towards drain  reservoir side) are investigated in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 3(b-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The existence of  the PDOS on the silicon position confirms that the hybridization  is present between the electronic states of P and Si atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' We  mainly focus on the ground state (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=', A1) of the P-donor, which  diminishes as we move away from the donor location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The  localization of the A1 state can be correlated with the spreading  of this state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Such behavior can be linked to the Bohr radius of  the ground state, which is the distance from where the A1 state  originates and where the A1 state dies out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The extinction of the  A1 state is the end of this state and its length from the donor  location is d =16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='29 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Hence, the Bohr radius of P-donor in such  system is ~ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='29 Å, which is far smaller than the bulk case (~25  Å).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='31 The reduction in the Bohr radius happens due to the  quantum confinement effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' We study the PDOS spectra of N-  donor in H:SiNW as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 3(e-h) and found a similar  trend of features as we shift far from the donor location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The  PDOS results for A1 state reduce to negligible value at d=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='43  Å, which signifies the ground state is highly concentrated and  the Bohr radius is ~5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='43 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' We also observe that A1 state is  largely separated from any other states of this configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The  difference between N-donor’s ground state and the first excited  state (ES) is 770 meV while that for P-donor is 118 meV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' This is  consistent with the highly localized ground state of N-donor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Such highly localized and deep ground state is the pathway for  designing quantum devices for room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 4: (a-b) PDOS/TDOS spectra of a P-doped and N-doped silicon  nanowire, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  To calculate the binding energy of donor electron (Eb), we  estimated the lowest effective conductive state of doped  H:SiNW following Anh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='33 Since the donor electronic states  hybridize with silicon electronic states, the conductive state  changes in comparison with undoped system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The Eb is estimated  by taking the ratio of PDOS/TDOS, which signifies the relative  weight of donor states against the states of the whole system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Figures 4(a) and (b) show the PDOS/TDOS spectra for P-donor  and N-donor, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' At lower energy range, the  contribution of the donor is dominated, while the contribution is  trivial at higher energy range (above blue dashed line) due to the  inclusion of more number of silicon atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The binding energy  is calculated from the point where this ratio saturates close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  The calculated Eb values for P-donor and N-donor are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='36 eV  and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='40 eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  In order to understand the behavior of undoped and doped  devices, LDOS spectra are simulated along with the current vs  gate voltage (IDS-VG) characteristics for different source to drain  voltage (VDS) using the transistor device parameters described in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The LDOS spectrum of the undoped H:SiNW  transistor is obtained where no states are observed below the  conduction band edge as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 5(b) and 5(c),  extra energy states are observed in the LDOS data due to  substitutionally doped P-donor and N-donor, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 5: (a-c) LDOS spectra of an undoped, P-doped, and N-doped silicon nanowire transistor, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' (d-f) LDOS spectrum at z = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='93 Å for undoped, P-  doped, and N-doped silicon nanowire transistor, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 5(f) indicates the zoom in spectrum, exhibiting the GS and and 1st ES of N-donor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Simulated IDS-VG characteristics for (g) undoped, (h-i) P-doped, and (j-k) N-doped devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  The lowest energy state in the channel region of P-doped  device is found at 80 meV from the next conductive state, which  is low compared to non-device configuration (118 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' This  is most likely due to the effect of metal gate and closely spaced  heavily doped leads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' In LDOS graph of N-doped device, a very  deep energy state is observed at 322 meV below the nearest  conductive state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' To emphasize more on the donor states  obtained in LDOS spectra, the same information of the LDOS  spectra is mapped along the vertical axis in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 5(d-f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The  LDOS spectra along the Z axis at z = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='93 Å are presented for  all three device structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' No LDOS is observed below CBE  for undoped device as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 5(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' A single LDOS peak  in P-doped SiNW at 80 meV is observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 5(e), indicating  the GS of P-donor in the device configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' A highly  localized LDOS is also perceived at 322 meV from the CBE in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 5(f), providing the information of the GS of N-dopant in  the device configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' In addition, we also observed 1st ES  of N-donor at 54 meV below the CBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  The IDS-VG transport characteristics for all three devices  are simulated at the different bias voltages for T= 5 K and 300  K, as presented in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 5(g-k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' In the undoped device with VDS=  5 mV, the characteristics are typical FET devices, which is  consistent with the LDOS spectrum of this device  configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The weak modulation in IDS for T = 5 K is  observed at ~ VG=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='5 mV in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 5(g), which is due to the  transport through the discrete states in the CB of the undoped  H:SiNW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' It is also observed that the subthreshold slopes of IDS-  VG curves at 5 K and 300 K are almost same that may be due to  the effect of heavy doping (1021 cm-3) of the leads along with  the nano-channel effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  The transport characteristics for P-doped device are  presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 5(h)-(i) for VDS= 1 mV and 5 mV, respectively,  LDOS (eV"  LDOS (eV  LDOS (eV")  5x10  1x104  5x10  j1x104  5x10°  1x10-5  Undoped  Phosphorous  Nitrogcen  doped  dopedshowing a strong single electron tunneling current peak before  the on-set of FET current at T=5 K due to the existence of  localized state for P-donor within the channel window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' An  additional weak current peak is also preserved due to the  transport through the next discrete conducting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' However,  this SET peak dies out at 300 K due to the shallow nature of  such dopant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  The IDS-VG characteristics for N-doped device are  presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 5(j-k) for VDS= 1 mV and VDS = 5 mV,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' For VDS= 1 mV, a very strong tunneling peak is  observed via the GS of N-donor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' However, the GS and 1st ES  prominently participated in the transport characteristics at  T=300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The excited state features got observed at high  temperature due to higher tunnel rate compared to 5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' For VDS=  5 mV and T=5 K, the low temperature behavior of the device is  similar to VDS= 1 mV with lower separation from the FET  current, and this current peak is also strongly sustained at  T=300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Along with the transport through the GS at 300 K, a  small hump corresponding to 1st excited state is also noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Observation of a single electron current peak at room  temperature is happened due to deep nature of N-donor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Such  deep donor state is capable of holding the quantum information  and is less fragile to environment fluctuations compared to  shallow donor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Silicon being one of the most promising  semiconductor for quantum information devices due to large  charge and spin coherence times,34,35,36 donor state of nitrogen  in silicon gives an opportunity to utilize it for better  performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Hence, to design single electron transistor with a  single donor atom for room temperature operation, N-donor in  silicon could be one of the best systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' We are working in such  direction for the experimental realization of such devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Donor energy spectrum of nitrogen in hydrogenated-  silicon nanowire is theoretically investigated in comparison  with phosphorous and undoped system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' We observed that  binding energy of N-donor ground state is large compared to  the P-donor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' It is also highly localized and separation of ground  state is also very large compared to 1st excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' In  quantum devices where perseverance of quantum states is  important, this advantage can be taken from a natural system  with deep N-donor states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Simulated LDOS and IDS-VG  characteristics of the devices also suggest in favor for N-dopant  being the potential candidate for single electron tunneling  devices for room temperature operation, which further infers  this system could be useful for designing practical quantum  architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  The authors are thankful to Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Sparsh Mittal for the computational resources  under the project ECR/2017/000622.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The work is partially supported by DST-  SERB (Project no: ECR/2017/001050) and IIT Roorkee (Project no: FIG-  100778-PHY), India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' thank to Ministry of Education and UGC,  India, respectively for the research scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  References  1 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Kane, Nature 393, 133 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  2 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Vrijen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Yablonovitch, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Jiang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Balandin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Roychowdhury, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Mor, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' DiVincenzo, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' A - At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  62, 10 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  3 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Hollenberg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Greentree, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Fowler, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Wellard, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  B - Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Matter Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 74, 1 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  4 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Hollenberg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Dzurak, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Wellard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Hamilton, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Reilly, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Milburn, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Clark, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' B - Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Matter Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 69, 2  (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  5 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Guo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Leobandung, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Chou, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 70, 850 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  6 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Lansbergen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Ono, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Fujiwara, Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 12, 763 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  7 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Pan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Zhao, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Xu, Nanoscale 13,  1048 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  8 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Astafiev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Antonov, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Kutsuwa, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Komiyama, 403, 191 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  9 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Tabe, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Moraru, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Ligowski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Anwar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Jablonski, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Ono, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Mizuno, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 105, 1 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  10 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Sellier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Lansbergen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Caro, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Rogge, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Collaert, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Ferain, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Jurczak, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Biesemans, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 97, 10 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  11 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Ono, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Khalafalla, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Nishiguchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Takashina, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Fujiwara, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Horiguchi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Inokawa, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Takahashi, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 254, 6252 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  12 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Prati, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Hori, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Guagliardo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Ferrari, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Shinada, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 7, 443 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  13 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Fuechsle, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Miwa, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Mahapatra, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Ryu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Lee, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Warschkow,  L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Hollenberg, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Klimeck, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Simmons, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 7, 242  (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  14 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Pierre, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Wacquez, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Jehl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Sanquer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Vinet, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Cueto, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 5, 133 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  15 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Samanta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Muruganathan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Hori, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Ono, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Mizuta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Tabe, and  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Moraru, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 110, (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  16 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Hamid, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Moraru, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Kuzuya, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Mizuno, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Anh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Mizuta, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Tabe, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' B - Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Matter Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 87, 3 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  17 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Calderón, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Koiller, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Das Sarma, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' B - Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Matter  Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 75, 1 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  18 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Lo Nardo, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Wolfowicz, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Simmons, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Tyryshkin, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Riemann, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Abrosimov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Becker, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Pohl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Steger, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Lyon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Thewalt,  and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Morton, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' B - Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Matter Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 92, 1 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  19 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Yin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Rancic, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' De Boo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Stavrias, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' McCallum, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Sellars,  and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Rogge, Nature 497, 91 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  20 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Achilli, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Le, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Fratesi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Manini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Onida, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Turchetti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Ferrari,  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Shinada, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Tanii, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Prati, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 31, 1 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  21 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Achilli, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Manini, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Onida, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Shinada, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
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+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Itoh, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
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+page_content=' 11, 143 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  25 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
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+page_content=' Vancraeyveld, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
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+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
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+page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Vej-Hansen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
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+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
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+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Palsgaard, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
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+page_content=' Stokbro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
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+page_content=' Matter 32, (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  26 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Stokbro, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Petersen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
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+page_content=' Blom, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Ipsen, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Kaasbjerg, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' B - Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Matter Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 82, 1 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  27 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Makov, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Shah, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Payne, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' B - Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Matter Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 53, 15513 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  28 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Epperson, Subquantum Kinetics (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  29 Group IV Elements, IV-IV and III-V Compounds Part b- Electronic,  Transport, Optical and Other Properties,” in Numerical Data and Functional  Relationships in Science and Technology, Landolt-Börnstein, New Series,  Group III, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 22, edited by O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Madelung, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Rössler, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Schulz  (Springer, Berlin, 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  30 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Guan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Siampour, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Fan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Kong, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Mesli, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Zhang, and  Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Dan, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 5, 1 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  31 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Kohn and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Luttinger, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 98, 915 (1955).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  32 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Saraiva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Baena, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' CalderÓn, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Koiller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Matter 27, (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  33 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' The Anh, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Moraru, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Manoharan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Tabe, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Mizuta, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 116, 063705 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  34 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Zhou, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Steigerwald, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Friesner, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Brus, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Hybertsen, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' B - Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Matter Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 71, 1 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  35 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Zwanenburg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Dzurak, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Morello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Simmons, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Hollenberg, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Klimeck, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Rogge, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Coppersmith, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Eriksson, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 85, 961 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  36 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Tyryshkin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Lyon, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Astashkin, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Raitsimring, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' B -  Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Matter Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' 68, 12 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  37  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Steger, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Saeedi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Thewalt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Morton, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Riemann, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content='  Abrosimov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Becker, and H- J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
+page_content=' Pohl, Science  336, 1280 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FJT4oBgHgl3EQf2S1v/content/2301.11655v1.pdf'}
diff --git a/Q9A0T4oBgHgl3EQfDf--/content/tmp_files/2301.02005v1.pdf.txt b/Q9A0T4oBgHgl3EQfDf--/content/tmp_files/2301.02005v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..5db5981229bb87326792e50c0c77689d864a01f8
--- /dev/null
+++ b/Q9A0T4oBgHgl3EQfDf--/content/tmp_files/2301.02005v1.pdf.txt
@@ -0,0 +1,836 @@
+1 
+ 
+  Rainbows in a bottle: Realizing microoptic effects by polymerizable 
+multiple emulsion particle design 
+Naresh Yandrapalliab*, Baris Kumrua, Tom Robinsonb*, Markus Antoniettia  
+ 
+a Max Planck Institute of Colloids and Interfaces, Department of Colloid Chemistry, Am Mühlenberg 1, 14424 
+Potsdam, Germany  
+b Max Planck Institute of Colloids and Interfaces, Department of Theory & Bio-Systems, Am Mühlenberg 1, 
+14424 Potsdam, Germany 
+ 
+Introduction 
+In nature, structural colour generation is based on discriminative light propagation associ-
+ated with physical structures in the range of the wavelengths of light1. These iridescent struc-
+tural colours are of immense significance2 but not easy to control experimentally and there-
+fore difficult to exploit for applications. In this work, we employ microfluidics to produce 
+polymerizable double emulsions that can not only induce the already known lensing effect3 
+but also result in the spectral separation of white light. Here, liquids of varying refractive 
+index that constitute the emulsions resulted in patterns of iridescent colours. After polymer-
+ization, the inner emulsion cores collapse and this results in curved concave surfaces on 
+these polymeric microspheres. Interestingly, the light propagation along the curved surfaces 
+undergo total internal reflection, followed by near-field interference along exit structures on 
+the polymerized microspheres4. These structured polymeric particles that are able to gener-
+ate colour dispersions can be exploited for optical devices, displays and even sensing tech-
+nologies. 
+Main 
+From the formation of glories in the sky to the spectacular vibrant colours observable on 
+various living organisms, humans have learned that light can be controlled by materials 
+
+2 
+ 
+structures/structured materials that exist and evolved in nature. We are continuously discov-
+ering that the majority of the everyday iridescent spectra, either colourful butterflies or the 
+plumage of birds, is the result of micrometre scale material structures5,6. Simple physical 
+phenomena such as absorption, reflection, and refraction, as well as diffraction and interfer-
+ence that result from light-matter interactions can combine into complex structural colour 
+generations 5. An attempt to classify the physical interaction of light with microstructures 
+resulted in identifying processes such as thin-film interference7, multi-film interference8, dif-
+fraction grating9, scattering (coherent & incoherent)10 and photonic crystal diffraction11 that 
+leads to structural colouration observed in nature12. Recently, Goodling et al. have shown 
+that light interaction with concave interfaces formed by materials with two different refractive 
+indices (η) will result in structural colouration due to the interference of total internally re-
+flected rays4. This proves that two materials (with varying refractive indices) forming a simple 
+interface with a degree of curvature can produce iridescent colours. 
+Our results show that water-in-oil-in-water (W/O/W) double emulsions produced with a 
+highly refractive oil layer (styrene (η-1.516) interfaced with water (η-1.333) on either side 
+can generate iridescent colours with spectral separation similar to that observed in glories13 
+but with more complexity and can still be exploited for their light focusing effect14. The double 
+emulsions produced here contain an aqueous inner core surrounded by a layer of styrene 
+in a continuous aqueous phase (shown in figure 1) stabilized with 0.5 wt% F108 surfactant. 
+Because the two liquids have different refractive indices, light undergos both refraction and 
+reflection based on the layering of the two liquids. The spectral dispersions or rainbows 
+observed in nature are an effect of light interacting with water drops in the air, however, in 
+the case of double emulsions (here, W/O/W), having layers of aqueous-oil-aqueous-oil-
+aqueous phase result in a similar dispersion but are followed by merging of the spectral 
+wavelengths to produce structural colouring (figure 1(b)). A z-stack of the diffused structural  
+ 
+
+3 
+ 
+ 
+Figure1: Microfluidic generation of liquid emulsions. (a) Double and multi-core emulsion production with vary-
+ing inner cores (scale bar – 200 µm) and the iridescent nature of the dispersed light at the equatorial plane, 
+off-focus plane (30 µm below the equatorial plane) of the emulsions and the intense light spots observed along 
+the z-axis of the inner cores (scale bar – 10 µm). (b) bright-field z-stacks of the light scattering from a 1-in-1 
+emulsion with each slice separated by 10 µm (top left to bottom right). (c) 2D ray-tracing of a 1-in-1 emulsion 
+with varying inner droplet sizes (diameters 5, 10, 30, 40 and 50 µm) showing the effect on the focal length and 
+TIR. (d) schematic representation of a 1-in-1 emulsion showing refraction, TIR and the effect of inner to outer 
+droplet density on the TIR (black arrows represent the direction of light with a single wavelength). S – corre-
+sponds to styrene phase and W – corresponds to aqueous phase. 
+ 
+colouring produce from a 1-in-1 W/O/W double emulsion is presented in supplementary 
+video 1,and a clear spectral separation is modelled by ray tracing (see supplementary video 
+2). As depicted schematically in figure 1d(4), the light rays from the source passing through 
+the double emulsion initially undergo refraction at the first convex interface formed by the 
+
+(a) W/o/W emulsion
+Equilateral off-focus
+Microfluidicproduction
+(b)
+1-in-1 Z-stack array (4x4)
+Z-project
+plane
+plane
+1-in-1
+2-in-1
+3-in-1
+4-in-1
+5-in-1
+(c)
+(p)
+Double
+Totalinternalreflection
+Decreasing outer droplet to inner droplet diameter ratio
+(TIR)
+TIRangle (α)= 145°
+emulsion
+1)
+W
+styrene-1.51
+non-TiR
+regime
+TIRregime
+Increasingtotal internalreflectionwithdecreasingsecondaryfocusing
+Refraction
+RefractionandTIR4 
+ 
+outer aqueous solution and the oil layer as well as through the second convex interface 
+across the aqueous inner core present inside the oil drop. Thanks to the spherical configu-
+ration of the emulsion, a single light ray can undergo a minimum of two or a maximum of 
+four refraction events as it passes out of the emulsion (1d). This results in more complex 
+colour formations than observed from single emulsions. 
+Our observations suggest that the focusing effect from these double emulsions is also prom-
+inent (see figure 1c), which was not observed in previous works on diverging light rays pass-
+ing through low refractive index inner core surrounded by a high refractive index liquid3. 
+Moreover, a point source of light passing through the emulsions such as the present case 
+resulted in dual focusing – one right below the double emulsion (resulting from the rays 
+passing through the inner aqueous core) and the other farther away (resulting from the rays 
+passing through the rest of the emulsion) (see supplementary figure S2). In agreement with 
+that, z-stack images of the double emulsion indeed show a bright spot right below the emul-
+sion due to the said focusing of the dispersed light beyond the emulsion (see supplementary 
+video 1 as well as z-project from figure 1(a)). The second focus vanishes for the double 
+emulsions whose inner aqueous core diameter is equal to or above the radius of the diam-
+eter of the W/O/W emulsion (see figure 1c). This is because the rays that are otherwise 
+contributing to the second focus are then refracted through the bigger inner aqueous core 
+converge into the primary focusing area. This is observed by the increase in the intensity of 
+the primary focusing area as the inner aqueous droplet size increases (figure 1c). 
+Previous studies have shown that refraction through equidistant alternating layers of liquid 
+or transparent material is associated with multi-layer interference8. This phenomenon is not 
+observable in the case of the double emulsions described here. The inner aqueous droplet 
+being heavier (ρ = 0.9982 g/mL at 20 ºC) than the oil phase (ρ = 0.909 g/mL at 20 ºC) sinks 
+to the bottom of the oil drop (figure 1d(1)), resulting in non-equidistant layers and a decrease 
+in peak reflectivity8. As the light is passing through a high refractive index medium to a lower 
+
+5 
+ 
+refractive index medium, followed by the curved nature of the interface4, a total internal re-
+flection (TIR) of light is possible at particular angular incidences, observable from the 2D 
+ray-tracing result (figure 1c) and presented in the scheme (figure 1d(2)). Only refracted rays 
+whose angle of incidence (α) is ~145° can undergo TIR when they pass through the inner 
+aqueous core and encounter the concave interface across styrene (high refractive index) 
+and the aqueous outer solution (low refractive index) (figure 1d(3)). The schematic produced 
+with the ray tracing data figure 1d(6) shows the density-dependent TIR with ρstyrene/ρwater less 
+than 1, while values greater than 1 result in no TIR. This suggests that a double emulsion 
+with a high refractive index outer layer and a low refractive index inner layer should require 
+inversed density values to take advantage of TIR as shown previously.4  
+Similarl to double emulsions, the emulsions with multiple inner droplets inside one W/O/W 
+double emulsion generated by the same single inlet microfluidic device (see Methods) in-
+deed showed light scattering and iridescent colours (figure 1a). Producing such multi-core 
+emulsions using a single inlet microfluidic design was made possible in this work by taking 
+advantage of the swelling property of PDMS in the presence of specific solvents like sty-
+rene.15,16 The swelling of PDMS upon the uptake of styrene resulted in narrowed channels 
+width, especially at first cross junction where water-in-oil (W/O) droplets are formed. 
+Through carefull manipulation of flow rates, when can achieve  W/O/W emulsions wih mul-
+tiple inner cores, all with the same device design (see figure 1a and Supplementary Video 
+3). In the absence of such a reliable system, multiple device designs with varied channel 
+widths would have to be fabricated. The light scattering observed in all these multi-core 
+emulsions, 2-in-1, 3-in-1, 4-in-1, and 5-in-1, are presented in the supplementary figure S1 
+as arrays of z-stack slices and in supplementary video 4, 5, and 6. At any given z-slice, the 
+light scattering observed is similar for each different type of multi-core emulsion. There is no 
+observable influence of the light scattering produced by one inner droplet on the other inner 
+droplet within such multi-core emulsions. Moreover, the iridescence within and around each 
+
+6 
+ 
+inner droplet is preserved. This is only possible if the inner droplets are in the same plane 
+(at the bottom) and are not stacked one above the other (see ray tracing data in supplemen-
+tary figure S2). Although the formation of emulsions with multiple inner droplets has been 
+presented previously, their light scattering and high iridescence properties have never been 
+explored before17,18.  
+ 
+Figure 2: Light scattering from polymerized emulsions. (a) Electron micrographs of polymerized multi-core 
+emulsion, inset showing the close-up of the curved surfaces on the microsphere (scale bar – 10 µm). (b) Bright-
+field images of the polymerized emulsions with a varying number of inner cores (scale bar – 50 µm). (c) Con-
+focal 3D reconstruction of the reflected light from the curved surfaces of two polymerized 1-in-1 emulsions 
+facing the light source. The curved dashed lines represent the surface boundary of the concave surface on the 
+microparticle . (d) Color images of the polymerized multi-core emulsions showing spectral dispersion of the 
+reflected light from the curved surfaces (scale bar – 10 µm). (e) Spectral separation of dispersed reflected light 
+
+(a)
+(c)
+Depth
+140
+120
+light
+100
+source
+80
+60
+40
+20
+0
+μm
+140
+(b)
+120
+1-in-1
+2-in-1
+3in-1
+multi-in-1
+100
+140
+80
+120
+y (μum)
+60
+100
+80
+40
+60
+cted
+z (μum)
+201
+20
+120
+80
+40
+x (um)
+(p)
+(e)
+Polymerized
+Redfilter
+Greenfilter
+Bluefilter
+Merge
+Redfilter
+Greenfilter
+Bluefilter
+Merge
+emulsions
+1-in-1
+Parallel
+100200300
+Gray Value
+2-in-1
+0
+20
+40
+20
+40
+20
+3-in-1
+Perpendicular
+multi-in-1
+0
+50
+100
+50
+100
+50
+100
+50
+100
+Distance(μm)Distance(μm)Distance(μm)
+Distance(μm)7 
+ 
+from the curved surfaces of polymerized 1-in-1 emulsion aligned 0° (parallel) and 90° (perpendicular) angle to 
+the light source (scale bar – 10 µm). 
+ 
+We then further explored the possibility to polymerize the styrene within the double and 
+multi-core emulsions, with the aim of preserving some of these effects into a more stable, 
+polymer structure. We have demonstrated that microfluidics can be used to generate uni-
+form sized hollow microspheres 19. However, it is not possible to produce structural colour-
+ation/iridescent colours with individual microspheres or hollow microspheres unless they are 
+transparent or with surface patterning20 or regularily packed to form so-called photonic 
+balls21.  
+In this work, the above produced double emulsions and multi-core emulsions with styrene 
+as the oil-phase are polymerized to generate microspheres with lensing curved surfaces. 
+Unlike for the formation of hollow polymer microspheres where the inner aqueous cores 
+have to be stabilized, microspheres with concaved dimples as shown in figure 2 are made 
+from inner aqueous cores devoid of any surfactant. The surfactant F108 used in this study 
+is a tri-block copolymer with two polypropylene hydrophobic blocks separated by a central 
+hydrophilic polyethene oxide block, which is known to form rather stable block copolymer 
+bilayers along two aqueous droplets, similar to that of lipids in a cell membrane22. However, 
+in the absence of surfactant in the inner aqueous core, there will only be a monolayer of the 
+surfactant stabilizing the emulsion interface. We hypothesize that during polymerization, de-
+stabilization of the monlayer could result in the formation of a concaved dimple on the spher-
+ical polystyrene microsphere as the inner aqueous core fuses with the outer aqueous solu-
+tion (as shown schematically in supplementary figure S3). The scanning electron micro-
+graphs of these polymeric spheres show the resulting concave surfaces (see figure 2(a)). 
+Similarly, polymerization of 2-in-1, 3-in-1, and multi-in-1 emulsions resulted in an equivalent 
+
+8 
+ 
+number of surface lenses on the respective polystyrene microspheres formed (see bright-
+field images in figure 2(b)).  
+Unlike the averaged light scattering observed in liquid phase emulsions, single polystyrene 
+microspheres can be described via the rules of the reflection of light23 and thereby show 
+optical near field effects. Like concave mirrors, the dimples on the polymeric microsphere 
+reflect and focus the light. True to this assumption, the 3D reconstruction of the confocal z-
+stack images suggests both reflection and focused light from the structured surface of the 
+polymerized 1-in-1 W/O/W emulsion (figure 2(c)). The z-stack was acquired with the curved 
+surface facing towards the source of light.  
+Similar to the liquid emulsions, our observations of their polymerized versions using a colour 
+camera have revealed the iridescence nature of these focused surface emissions. Initial 
+observations of these particles have shown their surface dimples/curves and iridescent col-
+ouring around the particles as well as bright spots from the lensing regions (figure 2(b)).  A 
+closer look reveals more spectacular details of iridescence from the lensing regions of the 
+particles (figure 2(d)). Concaved structures that are facing the light source reflect the light 
+and appear as intense bright white spots – a combination of all the spectral colours, as seen 
+in figure 2(b) and 1-in-1 polymerized emulsions of figure 2(d). Interestingly, particles with 
+concaved structures which are not facing but at an angle to the light source show a dramatic 
+colour separation. This is exhibited in figure 2(d) for 2-in-1, 3-in-1 and multi-in-1 configura-
+tions of the polymerized emulsions. In the case of multi-in-1 particles, with multiple curved 
+surfaces facing the light source at different angles (figure 2(d)), one can observe that ap-
+proximately 20 micron “rainbows” show no apparent angular dependency, within the ob-
+served upper hemisphere of the particle . Light reflected from the concave interface of pol-
+ymeric microspheres creates optical interference and the dispersion of light4. In our experi-
+ments with the results shown in figure 2, this is clearly visible that the longer wavelengths of 
+
+9 
+ 
+the reflected light are closer to the surface of the microsphere while rays of shorter wave-
+lengths are farther away from the surface (see figure 2(e)). The line profile data taken from 
+the centre of the images plotted to suggest the mixing and separation of the three main 
+wavelengths - red, green and blue. A similar patterning of light reflection and dispersion is 
+observed in polymeric microspheres, 2-in-1, 3-in-1, and multi-in-1 (figure 2(d)). This sug-
+gests that not only the liquid emulsions but also their polymerized versions can be exploited 
+for light scattering properties.  
+ 
+This study highlights some interesting optical phenoma that are enabled by manufacturing 
+controlled double and multi-core emulsions with a high refractive index differences using 
+microfluidics. We have shown that simple emulsions like W/O/W can induce spectral sepa-
+ration of white light under very local light focusing. The observed light enhancements are 
+mainly due to refraction and to a lesser extent from total internal reflection. More dramatic 
+colour patterns can be achieved by simple alteration of the double emulsions to complex 
+multi-core emulsions, also involving morphological transitions occurring under polymeriza-
+tion. While structural colouring in ordered 2D-arrays is a clear use case as a multilens array, 
+because of the focusing nature, the applications and use of the local, close-to particle light 
+fields with their colour gradients are yet to be discovered. The possibility to embed catalysts 
+within the polymerizable liquid emulsions was demonstrated earlier19 and opens applications 
+towards focused photocatalysis and photocatalytic gradients. Well ahead of applicability and 
+exploitation in multiple fields however, we want to underline that the particles presented 
+above are simple and rather effective to make, and that the observed effects are also just 
+stunningly beautiful and unexpected: little rainbows in a bottle. 
+ 
+
+10 
+ 
+References 
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+Prum RO. A Fourier Tool for the Analysis of Coherent Light Scattering by Bio-Optical 
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+11  
+Joannopoulos JD, Johnson SG, Winn JN, Meade RD. Photonic crystals: Molding the 
+flow of light. 2011http://ab-initio.mit.edu/book/ (accessed 12 Feb2021). 
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+Parker AR. 515 million years of structural colour. J. Opt. A Pure Appl. Opt. 2000; 2. 
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+Laven P. Atmospheric glories: Simulations and observations. In: Applied Optics. 
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+Calixto S, Rosete-Aguilar M, Sanchez-Marin FJ, Marañon V, Arauz-Lara JL, Olivares 
+DM et al. Optofluidic compound microlenses made by emulsion techniques. Opt 
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+Lee JN, Park C, Whitesides GM. Solvent Compatibility of Poly(dimethylsiloxane)-
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+Dangla R, Gallaire F, Baroud CN. Microchannel deformations due to solvent-induced 
+PDMS swelling. Lab Chip 2010. doi:10.1039/c003504a. 
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+Wang W, Xie R, Ju XJ, Luo T, Liu L, Weitz DA et al. Controllable microfluidic 
+production of multicomponent multiple emulsions. Lab Chip 2011. 
+doi:10.1039/c1lc20065h. 
+18  
+Vladisavljević GT, Al Nuumani R, Nabavi SA. Microfluidic production of multiple 
+emulsions. Micromachines. 2017. doi:10.3390/mi8030075. 
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+Yandrapalli N, Robinson T, Antonietti M, Kumru B. Graphitic Carbon Nitride 
+Stabilizers Meet Microfluidics: From Stable Emulsions to Photoinduced Synthesis of 
+Hollow Polymer Spheres. Small 2020; 16: 2001180. 
+20  
+Yeo SJ, Park KJ, Guo K, Yoo PJ, Lee S. Microfluidic Generation of Monodisperse 
+and Photoreconfigurable Microspheres for Floral Iridescence–Inspired Structural 
+Colorization. Adv Mater 2016; 28: 5268–5275. 
+21  
+Vogel N, Utech S, England GT, Shirman T, Phillips KR, Koay N et al. Color from 
+hierarchy: Diverse optical properties of micron-sized spherical colloidal assemblies. 
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+Proc Natl Acad Sci U S A 2015. doi:10.1073/pnas.1506272112. 
+22  
+Kulkarni C V. Lipid crystallization: From self-assembly to hierarchical and biological 
+ordering. Nanoscale 2012; 4: 5779–5791. 
+23  
+Wang M, Ye X, Wan X, Liu Y, Xie X. Brilliant white polystyrene microsphere film as a 
+diffuse back reflector for solar cells. Mater Lett 2015; 148: 122–125. 
+24  
+Yandrapalli N, Petit J, Bäumchen O, Robinson T. Surfactant-free production of 
+biomimetic giant unilamellar vesicles using PDMS-based microfluidics. Commun 
+Chem 2021 41 2021; 4: 1–10. 
+25  
+Tu R, Johnson. Ray Optics Simulation - Home. https://ricktu288.github.io/ray-optics/ 
+(accessed 11 Feb2021). 
+ 
+Supplementary information 
+Materials, supplementary Figs. 1-4 and caption for supplementary Video 1-6. 
+ 
+  
+Acknowledgments 
+The authors thank the Max Planck Society for funding. T.R. and N.Y. acknowledge support 
+from the MaxSynBio consortium, which is jointly funded by the Federal Ministry of Education 
+and Research of Germany and the Max Planck Society. 
+Author Contributions 
+N.Y., B.K., T.R., and M.A. conceptualized the project. N.Y. performed the experiments. 
+N.Y., T.R., and M.A. analyzed the data and wrote the manuscript. 
+Author Information 
+Naresh Yandrapalli 
+Department of Colloid Chemistry, Max Planck Institute of Colloids and InterfacesAm Müh-
+lenberg 1, Potsdam 14424, Germany 
+Email: naresh.yandrapalli@mpikg.mpg.de 
+ 
+Baris Kumru 
+Department of Colloid Chemistry, Max Planck Institute of Colloids and InterfacesAm Müh-
+lenberg 1, Potsdam 14424, Germany 
+Email: baris.kumru@mpikg.mpg.de 
+ 
+Tom Robinson  
+Department of Theory & Bio-Systems, Max Planck Institute of Colloids and Interfaces, Am 
+Mühlenberg 1, Potsdam 14424, Germany 
+E-mail: tom.robinson@mpikg.mpg.de 
+
+12 
+ 
+ 
+Markus Antonietti 
+Department of Colloid Chemistry, Max Planck Institute of Colloids and InterfacesAm Müh-
+lenberg 1, Potsdam 14424, Germany 
+E-mail: markus.antonietti@mpikg.mpg.de 
+ 
+Data Availability 
+All data generated or analysed during this study are included in the published article and 
+supplementary information, and are available from the corresponding authors upon rea-
+sonable request. 
+Competing Interest Declaration 
+No competing interests to declare. 
+Corresponding Author 
+Correspondence to Naresh Yandrapalli and Tom Robinson 
+ 
+Figure Legends 
+Figure1: Microfluidic generation of liquid emulsions. (a) Double and multi-core emulsion production with vary-
+ing inner cores (scale bar – 200 µm) and the iridescent nature of the dispersed light at the equatorial plane, 
+off-focus plane (30 µm below the equatorial plane) of the emulsions and the intense light spots observed along 
+the z-axis of the inner cores (scale bar – 10 µm). (b) bright-field z-stacks of the light scattering from a 1-in-1 
+emulsion with each slice separated by 10 µm (top left to bottom right). (c) 2D ray-tracing of a 1-in-1 emulsion 
+with varying inner droplet sizes (diameters 5, 10, 30, 40 and 50 µm) showing the effect on the focal length and 
+TIR. (d) schematic representation of a 1-in-1 emulsion showing refraction, TIR and the effect of inner to outer 
+droplet density on the TIR (black arrows represent the direction of light with a single wavelength). S – corre-
+sponds to styrene phase and W – corresponds to aqueous phase. 
+ 
+Figure 2: Light scattering from polymerized emulsions. (a) Electron micrographs of polymerized multi-core 
+emulsion, inset showing the close-up of the curved surfaces on the microsphere (scale bar – 10 µm). (b) Bright-
+field images of the polymerized emulsions with a varying number of inner cores (scale bar – 50 µm). (c) Con-
+focal 3D reconstruction of the reflected light from the curved surfaces of two polymerized 1-in-1 emulsions 
+
+13 
+ 
+facing the light source. The curved dashed lines represent the surface boundary of the concave surface on the 
+microparticle . (d) Color images of the polymerized multi-core emulsions showing spectral dispersion of the 
+reflected light from the curved surfaces (scale bar – 10 µm). (e) Spectral separation of dispersed reflected light 
+from the curved surfaces of polymerized 1-in-1 emulsion aligned 0° (parallel) and 90° (perpendicular) angle to 
+the light source (scale bar – 10 µm). 
+ 
+ 
+Methods 
+Device Fabrication 
+ 
+Master mould for the microfluidic device was created using UV-based photolithography. Ini-
+tially, a 4 inch silicon wafer was pre-heated for 30 min at 200 °C and 80 µm thick layer of 
+photoresist (SU8 2025)  was spin-coated on top (model no. WS-650MZ-23NPPB, Laurell 
+Tech. Corp) as per the specifications provided by the manufacturer at 23 °C. After the coating 
+step, the wafer is heated at 65 °C for 3 min and 95 °C for 9 min before UV exposure. After 
+8 sec of UV exposure through a specific design (see the supplementary figure S4) using 
+kloe photolithographic instrument (model no. UV-KUB 2), the wafer was post-baked at 65 °C 
+for 2 min and 95 °C for 7 min. The device design was revealed on the wafer after the washing 
+steps with a developer solution and isopropanol. Finally, the wafer was baked at 200 °C for 
+2 h before performing overnight surface passivation with 50 µL of 1H,1H,2H,2H-per-
+fluorodecyltrichlorosilane in a dessicator.  
+ 
+To produce the microfluidic chips, a PDMS:curing agent (10:1) mixture was thoroughly mixed 
+and degassed for 30 min in a desiccator connected to low pressure (150 millibars). The 
+degassed mixture was poured on top of the surface passivated wafer and cured at 90 °C for 
+3 h. Crosslinked PDMS was pealed from the master mould and diced into small pieces. The 
+inlets and outlets were created using 1 mm biopsy puncher (Kai Europe GmbH). Finally, 
+plasma cleaned (at 600 mbar for 1 min) (Plasma Cleaner PDC-002-CE, Harrick Plasma) 
+glass coverslips and diced PDMS chips with the desired design were bonded to form the 
+microfluidic chip. These chips were further heated at 60 °C for 2h to complete the bonding 
+process and retention of hydrophobic surface.  
+ 
+Device Surface Passivation 
+ 
+Surface passivation of the double emulsion microfluidic design is necessary for the formation 
+of stable double emulsions24. A series of solutions are flown through the outlet to the outer 
+aqueous (OA) solution inlet to render the hydrophobic PDMS surface hydrophilic (see the 
+supplementary figure S4(a)). To achieve this, initially, a 2:1 mixture of H2O2-HCl solution was 
+flushed for 30 sec. This was followed by flushing of 10 wt% of PDADMAC solution for 2 min 
+and later by 5 wt% of PSS solution for another 2 min. After every step, MilliQ® water was 
+flushed for 30 sec to remove excess material. Thus, flushed solutions form a hydrophilic 
+
+14 
+ 
+polyelectrolyte layer on top of the hydrophobic PDMS surface along the OA solution inlet to 
+the outlet.     
+ 
+Production of Double and Multi-core emulsions 
+ 
+Briefly, the inner aqueous solution (IA) is flushed through the first cross-junction to form a 
+water-in-oil (W/O) emulsion, followed by a second shearing step at the second cross-junc-
+tion. This results in the formation of water-in-oil-in-water (W/O/W) double emulsion with a 
+single inner aqueous core surrounded by styrene (S) which is stabilized through F108 (0.5 
+wt%) containing outer aqueous solution (OA) (see the supplementary figure S4(b)). Further-
+more, to form multi-core emulsions, the swelling property of the PDMS in the presence of 
+styrene is exploited to reproducibly narrow the channels at the first junction (at its lowest 
+dimension) where water-in-oil (W/O) droplets are created. A careful alteration of the flow 
+pressures resulted in controlling the number of aqueous droplets that get encased inside 
+the final double emulsion. Using this property, double emulsions with double, triple, quadru-
+ple, and quintuple cores are produced. Since styrene is used as the oil phase, thus produced 
+droplets can be polymerized to yield polymeric styrene microspheres. For fluid flow control, 
+four-channel pressure devices are used (MFCS-EZ, Fluigent Inc.). 
+ 
+Emulsion polymerization 
+ 
+Double and multi-core emulsions prepared with styrene:octanol (95:5 %) oil phase dissolved 
+with 0.5 wt% BAPO, were placed under UV illumination (395-400 nm, custom made device, 
+50W LED chips (Foxpic High Power 50 W LED Chip Bulb Light DIY White 3800LM 6500 K) 
+and 30 W UV chip (Fdit, 395-400 nm UV LED chip) were connected to a self-made circuit 
+and cooling system.) for 4 hours for complete polymerization of styrene. Produced micro-
+particles were washed via centirifugation before imaging. Freeze drying was performed be-
+fore scanning electron microscopy analysis. 
+ 
+Microscopy 
+ 
+Microfluidic production of multi-core emulsions is recorded with a MicroLab 310 camera at 
+full-frame and ~3000 fps (Vision Research Inc.) that is connected to wide-field Olympus IX73 
+microscope using a x5 objective in bright-field transmission mode. Monochrome confocal 
+images are acquired with Leica TCS SP8 (Leica Microsystems Inc.) confocal microscope. 
+Colour images with Nikon DS-Fi3 high definition camera fitted to a wide-field Olympus IX73 
+microscope using a x40 objective in bright-field transmission mode. Additionally, red (580/30 
+nm), green (510/30 nm) and blue (420/30 nm) filters are also used to image the spectral 
+separation. In all the cases, the 50 µL of emulsion suspensions were pipetted on to a 0.17 
+mm glass coverslip fitted with imaging spacers (SecurteSealTM) for imaging. General image 
+processing is performed using ImageJ/Fiji, z-stack arrays with Huygens Professional (Sci-
+entific Volume Imaging Inc.), and 3D rendering of confocal z-stacks with LAS X Core (Leica) 
+module. 
+ 
+Ray tracing 
+
+15 
+ 
+ 
+Ray tracing simulations are performed using Ray-Optics Simulation software25 and COM-
+SOL Multiphysics® (COMSOL AB, Stockholm, Sweden). Dimensions of the droplets and 
+emulsions simulated were obtained from the microscopic images of the emulsions. Further-
+more, the COMSOL ray tracing box setup with dimensions is visualized in the supplementary 
+figure S5.  
+  
+ 
+ 
+  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Supplementary Information for 
+Rainbows in a bottle: Realizing microoptic effects by polymerizable multiple emulsion 
+particle design  
+Naresh Yandrapalliab*, Baris Kumrua, Tom Robinsonb*, Markus Antoniettia  
+ 
+a Max Planck Institute of Colloids and Interfaces, Department of Colloid Chemistry, Am Mühlenberg 1, 
+14424 Potsdam, Germany  
+b Max Planck Institute of Colloids and Interfaces, Department of Theory & Bio-Systems, Am Mühlenberg 
+1, 14424 Potsdam, Germany 
+ 
+ 
+ 
+Materials  
+ 
+All materials were used as purchased unless noted otherwise. 1-octanol (99 %, Sigma 
+Aldrich), phenylbis(2,4,6-trimethylbenzoyl)phosphine oxide (BAPO initiator, 97%, Sigma 
+Aldrich), Synperonic® F 108 surfactant (Sigma Aldrich). Polydi-methylsiloxane (PDMS) 
+and curing agent were obtained as SYLGARD® 184 silicone elastomer kit from Dow 
+Corning. 1H,1H,2H,2H-Perfluorodecyltrichlorosilane was purchased from abcr GmbH. 
+Poly(diallyldimethylammonium chloride (PDADMAC) and poly(sodium 4-styrenesulfonate 
+(PSS) were obtained from Sigma Aldrich. SU8 2025 (Microchem Inc.), Silicon wafer 
+(Siegert Wafers), SU8 developer solution (Microchem Inc.) Styrene (99 %, Sigma Aldrich) 
+was passed through alumina column to remove inhibitor before use.  
+ 
+ 
+ 
+ 
+ 
+
+ 
+ 
+Figure S1: Bright-field z-stack of the spectral iridescence from 2-in-1, 3-in-1, 4-in-1, and 5-in-1 multi-core 
+emulsions with each slice separated by 10 µm. 
+ 
+ 
+
+ 
+ 
+ 
+Figure S2: Comparative 2D ray-tracing of multiple emulsions. Ray tracing data reveals the formation of 
+multiple focal points from single light source. 1 represents primary focusing point and 2- secondary focusing 
+point. S – Corresponds to Styrene pphase and W – corresponds to aqueous phase. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+1-in-1
+2-in-1
+3-in-1
+4-in-1
+5-in-1
+s
+s
+S
+S
+S
+A
+W
+W
+W
+W
+Direction of light
+X - axis 
+ 
+ 
+Figure S3: Mechanism for the formation of curved microspheres. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+3
+4
+5
+2
+2
+interface collapse andfusion
+to form curved surface.
+UV-induced polymerization
+Styrene
+Aquoeus solution
+F108surfactant
+Polystyrene 
+ 
+Figure S4: Microfluidic chip design with flow directions. (a) Inlet and outlets for successful surface 
+passivation of the microfluidic chip with solutions flowing from outlet to the OA solution inlet and (b) 
+showing multiple inlets and outlet with respective solutions and their flow direction to produce emulsions.   
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+(a)
+Coating
+Pressure
+solutions
+connection
+goes in here
+goes here
+(0.05 bars)
+(q)
+2nd
+Junction
+Outlet
+Outer Aqueous (OA)
+Inner Aqueous (IA)
+solution
+solution inlet
+Middle
+inlet
+styrene
+inlet 
+ 
+Figure S5: 3D COMSOL raytracing setup with example double emulsion with two inner cores (2-in-1) 
+surrounded by water box topped with a glass coverslip toward the direction of the point light source 
+positioned at (-165,0,0).  
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+μm
+50
+-100
+0
+-50
+-50
+μm
+-100
+0
+-150
+-100
+50
+wn
+-50
+0
+50
+100
+100Supplementary Video 1 
+ 
+Z-stack video of 1-in-1 W/O/W double emulsion produced using a microfluidic device. 
+Scale bar corresponds to 10 µm. 
+ 
+Supplementary Video 2 
+ 
+Comsol Multiphysics® ray-tracing simulation of ray-path (1000 rays) and dispersion of 
+light along 1-in-1 W/O/W double emulsion. Colour legend depicts the wavelength of 
+dispersed light. 
+ 
+Supplementary Video 3 
+ 
+Microfluidic production of 2-in-1 (IA - 66 mbar, Styrene – 102 mbar, OA – 112 mbar), 3-in-
+1 (IA - 66 mbar, Styrene – 106 mbar, OA – 105 mbar) , 4-in-1 ( IA - 66 mbar, Styrene – 
+104 mbar, OA – 94 mbar ) and 5-in-1 (IA - 66 mbar, Styrene – 105 mbar, OA – 86 mbar) 
+multi-emulsions containing aqueous inner and outer solution and middle styrene solution. 
+Image sequence was acquired using high speed camera at ~3000 fps. 
+ 
+Supplementary Video 4 
+ 
+Z-stack video of 2-in-1 W/O/W double emulsion produced using a microfluidic device. 
+Scale bar corresponds to 10 µm. 
+ 
+Supplementary Video 5 
+ 
+Z-stack video of 3-in-1 & 4-in-1 W/O/W double emulsion produced using a microfluidic 
+device. Scale bar corresponds to 10 µm. 
+ 
+Supplementary Video 6 
+ 
+Z-stack video of 5-in-1 W/O/W double emulsion produced using a microfluidic device. 
+Scale bar corresponds to 10 µm. 
+ 
+
+ 
+
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+arXiv:2301.05584v1  [math.CV]  13 Jan 2023
+A NOTE ON CYCLIC VECTORS IN DIRICHLET-TYPE
+SPACES IN THE UNIT BALL OF Cn
+DIMITRIOS VAVITSAS
+Abstract. We characterize model polynomials that are cyclic in
+Dirichlet-type spaces in the unit ball of Cn, and we give a sufficient
+capacity condition in order to identify non-cyclic vectors.
+1. Introduction
+Studying Dirichlet-type spaces in the unit ball of Cn we can draw
+conclusions for classical Hilbert spaces of holomorphic functions such
+as the Hardy, Bergman and Dirichlet spaces. General introduction to
+this theory can be found in [18], [22].
+The purpose of this note is to characterize model polynomials and to
+study special families of functions that are cyclic for the shift operators
+on these spaces. Moreover, we give a sufficient capacity condition in
+order to identify non-cyclic functions. Norm comparisons, sharp decay
+of norms for special subspaces, capacity conditions studied in [3], [4],
+[6], [21] are the main motivation for this work.
+The cyclicity of a
+function f in a space of holomorphic functions is connected also with
+the problem of approximating 1/f, see [19], [20] for the study of this
+subject.
+Full characterization of polynomials in more than two variables looks
+like a hard problem either in the unit ball or the polydisc. The cyclicity
+problem of polynomials for the bidisk was solved in [5] and shortly after
+extended in [13]. The corresponding problem in the setting of the unit
+ball of C2 was solved in [14].
+1.1. Dirichlet-type spaces in the unit ball. Denote the unit ball
+by
+Bn = {z ∈ Cn : ||z|| < 1},
+2020 Mathematics Subject Classification. 31C25, 32A37, 47A15.
+Key words and phrases. Dirichlet-type spaces,
+cyclic vectors,
+anisotropic
+capacities.
+Partially supported by NCN grant SONATA BIS no. 2017/26/E/ST1/00723 of
+the National Science Centre, Poland.
+1
+
+2
+DIMITRIOS VAVITSAS
+and its boundary, the unit sphere by
+Sn = {z ∈ Cn : ||z|| = 1},
+where ||z|| =
+�
+|z1|2 + ... + |zn|2 is the associated norm of the usual
+Euclidean inner product ⟨z, w⟩ = z1 ¯w1 + ... + zn ¯wn. Denote the class
+of holomorphic functions in Bn by Hol(Bn). Any function f ∈ Hol(Bn)
+has a power series expansion
+(1)
+f(z) =
+∞
+�
+k=0
+akzk =
+∞
+�
+k1=0
+...
+∞
+�
+kn=0
+ak1,...,knzk1
+1 · · · zkn
+n ,
+z ∈ Bn,
+where k = (k1, ..., kn) is a n-tuple index of non-negative integers, k! =
+k1! · · ·kn! and zk = zk1
+1 · · · zkn
+n . The power series in (1) exist, con-
+verges normal in Bn and it is unique since the unit ball is a connected
+Reinhardt domain containing the origin, i.e. (z1, ..., zn) ∈ Bn implies
+(eiθ1z1, ..., eiθnzn) ∈ Bn for arbitrary real θ1, ..., θn, (see [11]).
+To simplify the notation we may write (1) as follows:
+(2)
+f(z) =
+∞
+�
+m=0
+∞
+�
+|k|=m
+akzk =
+∞
+�
+|k|=0
+akzk,
+z ∈ Bn,
+where |k| = k1 + ... + kn.
+Let f ∈ Hol(Bn). We say that f belongs to the Dirichlet-type space
+Dα(Bn), where α ∈ R is a fixed parameter, if
+(3)
+||f||2
+α :=
+∞
+�
+|k|=0
+(n + |k|)α
+(n − 1)!k!
+(n − 1 + |k|)!|ak|2 < ∞.
+General introduction to the theory of Dirichlet-type spaces in the
+unit ball of Cn can be found in [1], [2], [15], [16], [19], [21], [22]. One
+variable Dirichlet-type spaces are discussed in the textbook [12]. The
+weights in the norm in (3) are chosen in such a way that D0(Bn) and
+D−1(Bn) coincide with the Hardy and Bergman spaces of the ball, re-
+spectively. The Dirichlet space having M¨obius invariant norm corre-
+sponds to the parameter choice α = n.
+By the definition, Dα(Bn) ⊂ Dβ(Bn), when α ≥ β. Polynomials are
+dense in the spaces Dα(Bn), α ∈ R, and zi · f ∈ Dα(Bn), i = 1, ..., n
+whenever f ∈ Dα(Bn).
+A multiplier in Dα(Bn) is a holomorphic function φ : Bn → C that
+satisfies φ · f ∈ Dα(Bn) for all f ∈ Dα(Bn). Polynomials, as well as
+holomorphic functions in a neighbourhood of the closed unit ball, are
+multipliers in every space Dα(Bn).
+
+CYCLIC VECTORS IN DIRICHLET-TYPE SPACES
+3
+1.2. Shift operators and cyclic vectors. Consider the bounded lin-
+ear operators S1, ..., Sn : Dα(Bn) → Dα(Bn) defined by Si : f �→ zi · f.
+We say that f ∈ Dα(Bn) is a cyclic vector if the closed invariant sub-
+space, i.e.
+[f] := clos span{zk1
+1 · · · zkn
+n f : k1, ..., kn = 0, 1, ...}
+coincides with Dα(Bn) (the closure is taken with respect to the Dα(Bn)
+norm). An equivalent definition is that f is cyclic if and only if 1 ∈ [f].
+Since Dα(Bn) enjoys the bounded point evaluation property a func-
+tion that is cyclic cannot vanish inside the unit ball. Thus, we focus
+on functions non-vanishing in the domain.
+Also, non-zero constant
+functions are cyclic in every space Dα(Bn). More information regard-
+ing cyclic vectors in Dirichlet-type spaces over the disk, the polydisc
+and the unit ball can be found in [3], [4], [5], [6], [8], [12], [13], [14],
+[20], [21].
+Just as in the settings of the bidisk and the unit ball of two variables,
+the cyclicity of a function f ∈ Dα(Bn) is inextricably linked with its
+zero set
+Z(f) = {z ∈ Cn : f(z) = 0}.
+The zeros of a function lying on the sphere are called the boundary
+zeros.
+1.3. Plan of the paper. Section 2 studies Dirichlet-type spaces. In
+particular, we give a crucial relation among them.
+Using fractional
+radial derivatives and the Cauchy formula of functions lying in the
+ball algebra A(Bn) which contains functions that are continuous on the
+closed unit ball and holomorphic in its interior, we give an equivalent
+norm of Dirichlet-type spaces for a wide range of parameters α.
+Section 3 studies diagonal subspaces. In particular, we extend re-
+sult from [21]. It makes sense to define functions f ∈ Hol(Bn) using
+functions ˜f ∈ Hol(D(µ)) for a proper µ > 0. Geometrically speaking,
+we are looking at a disk embedded in the ball but not in a coordinate
+plane. Thus, we may switch the problem of cyclicity from the ball to
+spaces of holomorphic functions of one variable that are well known.
+Then we use optimal approximants in order to identify cyclicity.
+Moreover, we prove cyclicity for model polynomials for proper pa-
+rameters.
+In the setting of the unit ball of two variables, see [21],
+the model polynomials are the following: 1 − z1 which vanishes in
+the closed unit ball on a singleton, i.e. Z(1 − z1) ∩ S2 = {(1, 0)}, and
+1−2z1z2 which vanishes along an analytic curve, i.e. Z(1−2z1z2)∩S2 =
+{(eiθ/
+√
+2, e−iθ/
+√
+2) : θ ∈ R}. In our case, the corresponding candidates
+
+4
+DIMITRIOS VAVITSAS
+are the following:
+p(z) = 1 − mm/2z1 · · ·zm,
+1 ≤ m ≤ n.
+They vanish in the closed unit ball along the following analytic sets:
+Z(p) ∩ Sn = {1/√m(eiθ1, .., eiθm−1, e−i(θ1+...+θm−1), 0, .., 0) : θi ∈ R}.
+These polynomials are also studied with respect to the Drury-Arveson
+space in [19].
+In two variables, 1 − z1 is cyclic in Dα(B2) precisely when α ≤ 2,
+and 1 − 2z1z2 is cyclic in Dα(B2) precisely when α ≤ 3/2. Here, there
+are more than two fixed parameters. The characterization of cyclicity
+of these two polynomials was crucial in [14].
+Section 4 studies the radial dilation of a polynomial. Using the equiv-
+alent norm of Section 2, we identify cyclicity for the model polynomials
+via the powerful radial dilation method. In particular, we show that if
+p/pr → 1 weakly, where pr(z) = p(rz) is a radial dilation of p, then p
+is cyclic, (see [13] for the bidisk settings and [14] for the unit ball in
+two variables). This method is quite interesting since it can be applied
+to an arbitrary polynomial. Note that in [13], [14] the radial dilation
+method is one of the main tools of solving cyclicity problem for poly-
+nomials. The main result of this section verifies the arguments made
+about polynomials in Section 3.
+Section 5 studies non-cyclic vectors. We use the notion of Riesz α-
+capacity in order to identify non-cyclic functions. Moreover, we study
+Cauchy transforms of Borel measures supported on zero sets of the
+radial limits of a given function f ∈ Dα(Bn) and we give asymptotic
+expansions of their norms. Then employing a standard scheme due to
+Brown and Shields, see [8], we prove the main result. Note that this
+sufficient capacity condition for non-cyclicity in Dirichlet-type spaces
+in the unit ball of two variables was proved by A. Sola in [21].
+Standard tools
+Let us give some standard tools which will be useful in the sequel.
+The binomial series:
+1
+(1 − x)α =
+∞
+�
+k=0
+Γ(k + α)
+Γ(α)k! xk,
+where |x| < 1 is a complex number and α is a non-negative real number.
+The asymptotic behaviour of the Γ-function is the following: Γ(k +
+α) ≍ (k − 1)!kα, where the symbol ≍ denotes that the ratio of the two
+quantities either tends to a constant as k tends to infinity or it is rather
+two sides bound by constants.
+
+CYCLIC VECTORS IN DIRICHLET-TYPE SPACES
+5
+The multinomial formula:
+(x1 + ... + xn)k =
+�
+|j|=k
+k!
+j!xj1
+1 · · · xjn
+n ,
+where j = (j1, ..., jn) is a n-tuple index of non-negative integers and xi
+are complex numbers.
+The Stirling formula that describes the asymptotic behaviour of the
+gamma function:
+k! ≍ k1/2kk/ek.
+Denote the normalized area measure on Cn = R2n by du(z) and the
+normalized rotation-invariant positive Borel measure on Sn by dσ(ζ),
+(see [18], [22]). The measures du(z) and dσ(ζ) are related by the for-
+mula
+�
+Cn f(z)du(z) = 2n
+� ∞
+0
+�
+Sn
+ǫ2n−1f(ǫζ)dσ(ζ)dǫ.
+The holomorphic monomials are orthogonal to each other in L2(σ),
+that is, if k and l are multi-indices such that k ̸= l, then
+�
+Sn
+ζk¯ζldσ(ζ) = 0.
+Moreover,
+�
+Sn
+|ζk|2dσ(ζ) =
+(n − 1)!k!
+(n − 1 + |k|)!
+and
+�
+Bn
+|zk|2du(z) =
+n!k!
+(n + |k|)!.
+2. Relation among Dirichlet-type spaces and equivalent
+norms
+We study the structure of Dirichlet-type spaces. Note that
+R(f)(z) = z1∂z1f(z) + ... + zn∂znf(z)
+is the radial derivative of a function f. The radial derivative plays a key
+role in the function theory of the unit ball. A crucial relation among
+these spaces is the following.
+Proposition 1. Let f ∈ Hol(Bn) and α ∈ R be fixed. Then
+f ∈ Dα(Bn)
+if and only if
+nqf + Rq(f) + q
+q−1
+�
+i=1
+niRq−i(f) ∈ Dυ(Bn),
+where α = 2q + υ, q ∈ N and Rq is the q-image of the operator R.
+
+6
+DIMITRIOS VAVITSAS
+Proof. Indeed, it is enough to check that
+||nf +R(f)||2
+α−2 =
+∞
+�
+|k|=0
+(n+|k|)α−2
+(n − 1)!k!
+(n − 1 + |k|)!(n+|k|)2|ak|2 = ||f||2
+α.
+□
+We continue by giving an equivalent characterization of Dirichlet-
+type norms. In Dirichlet-type spaces in the unit ball, one of the integral
+representations of the norm is achieved in a limited range of parameters.
+Lemma 2 (see[16]). If α ∈ (−1, 1), then ||f||2
+α is equivalent to
+|f|2
+α :=
+�
+Bn
+||∇(f)(z)||2 − |R(f)(z)|2
+(1 − ||z||2)α
+du(z).
+Above, ∇(f)(z) = (∂z1f(z), ..., ∂znf(z)) denotes the holomorphic
+gradient of a holomorphic function f. Note that Proposition 1 allows
+us to use Lemma 2 whenever υ ∈ (−1, 1). . Let γ, t ∈ R be such that
+neither n + γ nor n + γ + t is a negative integer. If f = �∞
+|k|=0 akzk
+is the homogeneous expansion of a function f ∈ Hol(Bn), then we
+may define an invertible continuous linear operator with respect to the
+topology of uniform convergence on compact subsets of Bn, denoted by
+Rγ,t : Hol(Bn) → Hol(Bn) and having expression
+Rγ,tf(z) =
+∞
+�
+|k|=0
+C(γ, t, k)akzk,
+z ∈ Bn,
+where
+(4)
+C(γ, t, k) = Γ(n + 1 + γ)Γ(n + 1 + |k| + γ + t)
+Γ(n + 1 + γ + t)Γ(n + 1 + |k| + γ) ≍ |k|t.
+See [22] for more information regarding these fractional radial deriva-
+tives.
+Lemma 3. Let t ∈ R be such that n − 1 + t ≥ 0. If f ∈ A(Bn), then
+R−1,tf(z) =
+�
+Sn
+f(ζ)
+(1 − ⟨z, ζ⟩)n+tdσ(ζ),
+z ∈ Bn.
+Proof. The continuous linear operator Rγ,t, see [22], satisfies
+Rγ,t�
+1
+(1 − ⟨z, w⟩)n+1+γ
+�
+=
+1
+(1 − ⟨z, w⟩)n+1+γ+t
+for all w ∈ Bn. Next, define fǫ for ǫ ∈ (0, 1) by
+fǫ(z) =
+�
+Sn
+f(ζ)
+(1 − ⟨z, ǫζ⟩)ndσ(ζ),
+z ∈ Bn.
+
+CYCLIC VECTORS IN DIRICHLET-TYPE SPACES
+7
+The Cauchy formula holds for f ∈ A(Bn) and hence f = limǫ→1− fǫ. It
+follows that
+R−1,tf(z) = R−1,t�
+lim
+ǫ→1−
+�
+Sn
+f(ζ)
+(1 − ⟨z, ǫζ⟩)ndσ(ζ)
+�
+= lim
+ǫ→1− R−1,t� �
+Sn
+f(ζ)
+(1 − ⟨z, ǫζ⟩)ndσ(ζ)
+�
+= lim
+ǫ→1−
+�
+Sn
+f(ζ)R−1,t�
+1
+(1 − ⟨z, ǫζ⟩)n
+�
+dσ(ζ)
+= lim
+ǫ→1−
+�
+Sn
+f(ζ)
+(1 − ⟨z, ǫζ⟩)n+tdσ(ζ)
+=
+�
+Sn
+f(ζ)
+(1 − ⟨z, ζ⟩)n+tdσ(ζ)
+and the assertion follows.
+□
+Theorem 4. Let α ∈ R be such that n − 1 + α/2 ≥ 0 and f ∈ A(Bn).
+Then f ∈ Dα(Bn) if and only if
+�
+Bn
+(1 − ||z||2)
+���
+�
+Sn
+f(ζ)¯ζp
+(1 − ⟨z, ζ⟩)n+α/2+1dσ(ζ)
+���
+2
+du(z) < ∞
+and
+�
+Bn
+���
+�
+Sn
+(zpζq − zqζp)f(ζ)
+(1 − ⟨z, ζ⟩)n+α/2+1dσ(ζ)
+���
+2
+du(z) < ∞,
+where p, q = 1, ..., n.
+Proof. Choose t so that α = 2t. Note that n, t are fixed and hence
+||f||2
+α ≍
+∞
+�
+|k|=0
+(n − 1)!k!
+(n − 1 + |k|)!||k|tak|2.
+Thus, (4) implies that ||R−1,tf||0 ≍ ||f||α. One can apply then the
+integral representation of Dirichlet-type norms to R−1,tf ∈ Hol(Bn),
+i.e. ||R−1,tf||0 is equivalent to |R−1,tf|0. According to Lemma 3 we get
+that
+∂zp(R−1,tf)(z) =
+�
+Sn
+f(ζ)¯ζp
+(1 − ⟨z, ζ⟩)n+t+1dσ(ζ),
+z ∈ Bn,
+where p = 1, ..., n. Expand the term ||∇(f)||2 − |R(f)|2 as follows:
+||∇(f)||2 − |R(f)|2 = (1 − ||z||2)||∇(f)||2 +
+�
+p,q
+|¯zp∂zqf − ¯zq∂zpf|2.
+The assertion follows by Lemma 2.
+□
+
+8
+DIMITRIOS VAVITSAS
+3. Diagonal subspaces
+In [3], a method of construction of optimal approximants via deter-
+minants in Dirichlet-type spaces in the unit disk is provided. Similarly,
+we may define optimal approximants in several variables, (see [19]).
+Fix N ∈ N. We define the space of polynomials p ∈ C[z1, ..., zn] with
+degree at most nN as follows:
+P n
+N := {p(z) =
+N
+�
+k1=0
+...
+N
+�
+kn=0
+ak1,...,knzk1
+1 · · · zkn
+n }.
+Remark 5. Let (X, || · ||) be a normed space and fix x ∈ X, C ⊂ X.
+The distance between x and the set C is the following:
+distX(x, C) := inf{||x − c|| : c ∈ C}.
+It is well known that if X is a Hilbert space and C ⊂ X a convex
+closed subset, then for any x ∈ X, there exists a unique y ∈ C such
+that ||x − y|| = distX(x, C). Let f ∈ Dα(Bn) be non-zero constant. We
+deduce that for any N ∈ N, there exists exactly one pN ∈ P n
+N satisfying
+||pNf − 1||α = distDα(Bn)(1, f · P n
+N).
+Let f ∈ Dα(Bn). We say that a polynomial pN ∈ P n
+N is an optimal
+approximant of order N to 1/f if pN minimizes ||pf − 1||α among all
+polynomials p ∈ P n
+N. We call ||pNf − 1||α the optimal norm of order N
+associated with f.
+Let M = (M1, ..., Mn) be a multi-index, where Mi are non-negative
+integers, and m ∈ {1, ..., n}. Setting
+µ(m) := (M1 + ... + Mm)M1+...+Mm
+MM1
+1
+· · · MMm
+m
+,
+we see that
+(5)
+µ(m)1/2|z1|M1 · · · |zm|Mm ≤ 1,
+z ∈ Bn.
+Using (5) we may construct polynomials that vanish in the closed
+unit ball along analytic subsets of the unit sphere.
+Remark 6. Let ˜f ∈ Hol(D(µ(m)−1/4)), where
+D(µ) = {z ∈ C : |z| < µ},
+µ > 0.
+According to (5) we define the following function:
+f(z) = f(z1, ..., zn) = ˜f(µ(m)1/4zM1
+1
+· · · zMm
+m ),
+z ∈ Bn.
+
+CYCLIC VECTORS IN DIRICHLET-TYPE SPACES
+9
+Then f ∈ Hol(Bn) and it depends on m variables. Note that we may
+change the variables z1, ..., zm by any other m variables. For conve-
+nience, we choose the m first variables. The power 1/4 will be conve-
+nient in the sequel.
+Thus, the question that arises out is if we may define closed subspaces
+of Dα(Bn) passing through one variable functions. We shall see that
+these subspaces are called diagonal subspaces due to the nature of the
+power series expansion of their elements.
+Instead of the classical one variable Dirichlet-type spaces of the unit
+disk, we may consider spaces dβ, β ∈ R, consisting of holomorphic func-
+tions ˜f ∈ Hol(D(µ−1/4)). Moreover, such functions with power series
+expansion ˜f(z) = �∞
+l=0 alzl are said to belong to dβ if
+|| ˜f||2
+dβ :=
+∞
+�
+l=0
+µ−l/2(l + 1)β|al|2 < ∞.
+There is a natural identification between the function theories of
+Dβ(D): one variable Dirichlet-type spaces of the unit disk, and dβ, and
+one verifies that the results in [3] are valid for dβ.
+We are ready to define diagonal closed subspaces. Set
+β(α) := α − n + m + 1
+2
+.
+Let α, M, m be as above. The diagonal closed subspace of Dα(Bn)
+is the following:
+Jα,M,m := {f ∈ Dα(Bn) : ∃ ˜f ∈ dβ(α), f(z) = ˜f(µ(m)1/4zM1
+1
+· · · zMm
+m )}.
+The existence of a holomorphic function ˜f is unique by identity prin-
+ciple and hence there is no any amiss in the definition. Any function
+f ∈ Jα,M,m has an expansion of the form
+f(z) =
+∞
+�
+l=0
+al(zM1
+1
+· · · zMm
+m )l.
+The relation of norms between one variable and diagonal subspaces
+follows.
+Proposition 7. If f ∈ Jα,M,m, then ||f||α ≍ || ˜f||dβ(α).
+Proof. If f ∈ Jα,M,m, then
+||f||2
+α ≍
+∞
+�
+l=0
+(l + 1)α
+(M1l)! · · · (Mml)!
+(n − 1 + (M1 + · · · Mm)l)!|al|2.
+
+10
+DIMITRIOS VAVITSAS
+By Stirling’s formula, we obtain
+||f||2
+α ≍
+∞
+�
+l=0
+(l + 1)α−n+m/2+1/2µ(m)−l|al|2.
+On the other hand, define the function f ′(z) = �∞
+l=0 µ(m)−l/4alzl. Then
+f ′(µ(m)1/4zM1
+1
+· · · zMm
+m ) = f(z1, ..., zn) and
+||f ′||2
+dβ(α) ≍
+�
+l=0
+(l + 1)α−n+m/2+1/2µ(m)−l|al|2.
+The assertion follows since f ′ coincides with ˜f.
+□
+The corresponding Lemma 3.4 of [4] in our case is the following.
+Lemma 8. Let f ∈ Jα,M,m, where α, M, m be as above. Let rN ∈ P n
+N
+with expansion
+rN(z) =
+N
+�
+k1=0
+· · ·
+N
+�
+kn=0
+ak1,...,knzk1
+1 ...zkn
+n ,
+and consider its projection onto Jα,M,m
+π(rN)(z) =
+�
+{l:M1l,...,Mml≤N}
+cM1l,...,Mml,0,...,0zM1l
+1
+· · · zMml
+m
+.
+Then
+||rNf − 1||α ≥ ||π(rN)f − 1||α.
+Moreover, just as in Proposition 7, there is a relation of optimal
+approximants between one variable and diagonal subspaces.
+Proposition 9. If f ∈ Jα,M,m, then
+distDα(Bn)(1, f · P n
+N) ≍ distdβ(α)(1, ˜f · P 1
+N).
+Proof. Let rN, π(rN) be as in Lemma 8. Then π(rN)f − 1 ∈ Jα,M,m. It
+follows that
+||rNf − 1||α ≥ ||π(rN)f − 1||α ≍ ||˜π(rN) ˜f − 1||dβ(α) ≥ distdβ(α)(1, ˜f · P 1
+N),
+since ˜π(rN) ∈ P 1
+N. On the other hand, let
+distdβ(α)(1, ˜f · P 1
+N) = ||qN ˜f − 1||dβ(α),
+qN(z) =
+N
+�
+l=0
+alzl.
+Then, the polynomial
+q′
+N(z1, ..., zn) =
+N
+�
+l=0
+µ(m)−l/4alzM1l
+1
+· · · zMml
+m
+
+CYCLIC VECTORS IN DIRICHLET-TYPE SPACES
+11
+satisfies q′
+N ∈ Jα,M,m ∩ P n
+N and q′
+Nf − 1 ∈ Jα,M,m. Thus,
+||qN ˜f − 1||dβ(α) = ||˜q′
+N ˜f − 1||dβ(α) ≍ ||q′
+Nf − 1||α ≥ distDα(Bn)(1, f · P n
+N)
+and the assertion follows.
+□
+Define the function φβ : [0, ∞) → [0, ∞) by
+φβ(t) =
+�
+t1−β,
+β < 1
+log+(t),
+β = 1 ,
+where log+(t) := max{log t, 0}. We have the following.
+Theorem 10. Let α ∈ R be such that β(α) ≤ 1. Let f ∈ Jα,M,m
+be as above and suppose the corresponding ˜f has no zeros inside its
+domain, has at least one zero on the boundary, and admit an analytic
+continuation to a strictly bigger domain. Then f is cyclic in Dα(Bn)
+whenever α ≤ 2n−m+1
+2
+and
+dist2
+Dα(Bn)(1, f · P n
+N) ≍ φβ(α)(N + 1)−1.
+Proof. It is an immediate consequence of the identification between
+Dβ(D) and dβ and previous lemmas and propositions.
+□
+If we focus on polynomials, then the following is true.
+Theorem 11. Consider the polynomial p(z) = 1−mm/2z1 · · · zm, where
+1 ≤ m ≤ n. Then p is cyclic in Dα(Bn) whenever α ≤ 2n+1−m
+2
+.
+Note that the Theorem11 is not a characterization. We shall study
+the case α > 2n+1−m
+2
+.
+4. Cyclicity for model polynomials via radial dilation
+The radial dilation of a function f : Bn → C is defined for r ∈ (0, 1)
+by fr(z) = f(rz). To prove Theorem 11, it is enough to prove the
+following lemma.
+Lemma 12. Consider the polynomial p(z) = 1 − mm/2z1 · · · zm, where
+1 ≤ m ≤ n. Then ||p/pr||α < ∞ as r → 1− whenever α ≤ 2n+1−m
+2
+.
+We follow the arguments of [14], [13]. Indeed, if Lemma 12 holds,
+then φr · p → 1 weakly, where φr := 1/pr. This is a consequence of a
+crucial property of Dirichlet-type spaces: if {fn} ⊂ Dα(Bn), then fn →
+0 weakly if and only if fn → 0 pointwise and supn{||fn||α} < ∞. Since
+φr extends holomorphically past the closed unit ball, φr are multipliers,
+and hence, φr · p ∈ [p]. Finally, 1 is weak limit of φr · p and [p] is closed
+and convex or, equivalently, weakly closed. It is clear that 1 ∈ [p], and
+hence, p is cyclic.
+
+12
+DIMITRIOS VAVITSAS
+Moreover, it is enough to prove that ||p/pr||α < ∞, as r → 1−,
+for α0 =
+2n+1−m
+2
+. Then the case α < α0 follows since the inclusion
+Dα0(Bn) ֒→ Dα(Bn) is a compact linear map and weak convergence in
+Dα0(Bn) gives weak convergence in Dα(Bn).
+Proof of Lemma 12. By Theorem 4 it is enough to show the following:
+Ip :=
+�
+Bn
+(1 − ||z||2)
+���
+�
+Sn
+(1 − λζ1 · · · ζm)¯ζp
+(1 − rmλζ1 · · · ζm)(1 − ⟨z, ζ⟩)β dσ(ζ)
+���
+2
+du(z)
+and
+Ip,q :=
+�
+Bn
+���
+�
+Sn
+(zpζq − zqζp)(1 − λζ1 · · · ζm)
+(1 − rmλζ1 · · ·ζm)(1 − ⟨z, ζ⟩)β dσ(ζ)
+���
+2
+du(z)
+are finite, as r → 1−, where β = n + t + 1, t = 2n+1−m
+4
+, and λ = mm/2.
+Denote
+Sp :=
+�
+Sn
+(1 − λζ1 · · · ζm)¯ζp
+(1 − rmλζ1 · · · ζm)(1 − ⟨z, ζ⟩)β dσ(ζ),
+where the last integral is equal to
+1
+2π
+�
+Sn
+� 2π
+0
+(1 − λeimθζ1 · · · ζm)e−iθ ¯ζp
+(1 − rmλeimθζ1 · · · ζm)(1 − e−iθ⟨z, ζ⟩)β dθdσ(ζ).
+Let z, ζ be fixed. Then
+� 2π
+0
+e−iθ
+(1 − e−iθ⟨z, ζ⟩)β dθ = 0.
+Thus, replacing p(eiθζ)/p(reiθζ) by p(eiθζ)/p(reiθζ) − 1 we obtain
+Sp = λ(rm − 1)
+2π
+�
+Sn
+� 2π
+0
+¯ζpζ1 · · · ζmei(m−1)θ
+(1 − rmλeimθζ1 · · ·ζm)(1 − e−iθ⟨z, ζ⟩)β dθdσ(ζ).
+Next, expand the binomials
+� 2π
+0
+ei(m−1)θ
+(1 − rmλeimθζ1 · · · ζm)(1 − e−iθ⟨z, ζ⟩)β dθ
+=
+∞
+�
+k=0
+∞
+�
+l=0
+Γ(k + β)
+Γ(β)k! (rmλζ1 · · · ζm)l⟨z, ζ⟩k
+� 2π
+0
+ei(m(l+1)−k−1)θdθ
+= 2π
+∞
+�
+k=0
+Γ(m(k + 1) − 1 + β)
+Γ(β)(m(k + 1) − 1)! (rmλζ1 · · ·ζm)k⟨z, ζ⟩m(k+1)−1
+= 2π
+∞
+�
+k=0
+�
+|j|=m(k+1)−1
+Γ(m(k + 1) − 1 + β)
+Γ(β)j!
+(rmλζ1 · · · ζm)kzj ¯ζj.
+
+CYCLIC VECTORS IN DIRICHLET-TYPE SPACES
+13
+Therefore,
+Sp = λ(rm − 1)
+∞
+�
+k=0
+�
+|j|=m(k+1)−1
+Γ(m(k + 1) − 1 + β)
+Γ(β)j!
+(rmλ)kzjc(k),
+where c(k) =
+�
+Sn ζα(k)¯ζb(k)dσ(ζ), α(k) = (k+1, .., k+1(m-comp.), 0, .., 0)
+and b(k) = (j1, ..., jp−1, jp + 1, jp+1, ..., jn). Whence, 1 ≤ p ≤ m. Since
+the holomorphic monomials are orthogonal to each other in L2(σ) we
+get that
+|Sp| ≍ (1 − rm)
+���z′
+p
+∞
+�
+k=0
+(k + 1)β−n(rmλz1 · · · zm)k���,
+where z′
+p = z1 · · · zp−1zp+1 · · · zm. Hence we obtain
+Ip ≍ (1−rm)2
+∞
+�
+k=0
+(k+1)2(β−n)(rmλ)2k
+�
+Bn
+(1−||z||2)|z′
+p|2|z1 · · · zm|2kdu(z),
+where has been used again the orthogonality of the holomorphic mono-
+mials in L2(σ). To handle the integral above we use polar coordinates
+�
+Bn
+(1 − ||z||2)|z′
+p|2|z1 · · · zm|2kdu(z)
+≍
+� 1
+0
+�
+Sn
+ǫ2n−1(1 − ǫ2)ǫ2km+2m−2|ζ′
+p|2|ζ1 · · · ζm|2kdσ(ζ)dǫ
+≍
+[(k + 1)!]m−1k!
+(n + m(k + 1) − 2)! ·
+1
+(k + 1)2.
+If we recall that β = n + t + 1, t =
+2n+1−m
+4
+and λ2k = mmk, then
+applying the Stirling formula more than one time we see that
+Ip ≍ (1 − rm)2
+∞
+�
+k=0
+(k + 1)r2mk.
+This proves the assertion made about Ip.
+It remains to estimate the following term:
+Ip,q =
+�
+Bn
+���
+�
+Sn
+(zpζq − zqζp)(1 − λζ1 · · ·ζm)
+(1 − rmλζ1 · · · ζm)(1 − ⟨z, ζ⟩)β dσ(ζ)
+���
+2
+du(z).
+We shall show that Ip,q ≍ Ip. Denote again the inner integral by Sp,q
+which is convenient to expand it as Sp,q = ¯zpSq − ¯zqSp. Recall that
+z′
+p = z1 · · · zp−1zp+1 · · · zm. Similar calculations to the one above lead to
+|Sp,q| ≍ (1 − rm)|¯zpz′
+q − ¯zqz′
+p|
+���
+∞
+�
+k=0
+(k + 1)β−n(rmλz1 · · · zm)k���.
+
+14
+DIMITRIOS VAVITSAS
+Moreover, the orthogonality of the holomorphic monomials in L2(σ)
+gives the following estimation:
+Ip,q ≍ (1−rm)2
+∞
+�
+k=0
+(k+1)2β−2n(rmλ)2k
+�
+Bn
+|¯zpz′
+q−¯zqz′
+p|2|z1 · · · zm|2kdu(z).
+It is easy to see that |¯zpz′
+q − ¯zqz′
+p|2 = |zp|2|z′
+q|2 +|zq|2|z′
+p|2 −2|z1 · · · zm|2.
+Let us estimate the integral
+�
+Bn
+(|zp|2|z′
+q|2 − |z1 · · · zm|2)|z1 · · · zm|2kdu(z).
+Passing through polar coordinates we get, for p ̸= q, that
+�
+Bn
+|zp|2|z′
+q|2|z1 · · · zm|2kdu(z)
+≍ 2n(n − 1)!
+[(k + 1)!]m−1k!
+(mk + n + m − 1)!
+k + 2
+2km + 2n + 2m,
+and
+�
+Bn
+|z1 · · ·zm|2(k+1)du(z)
+= 2n(n − 1)!
+[(k + 1)!]m−1k!
+(mk + n + m − 1)!
+k + 1
+2km + 2n + 2m.
+Hence we obtain
+�
+Bn
+(|zp|2|z′
+q|2 − |z1 · · · zm|2)|z1 · · · zm|2kdu(z)
+≍
+[(k + 1)!]m−1k!
+(mk + n + m − 2)!(k + 1)2.
+Again, applying the Stirling formula to the one above estimates we
+obtain
+Ip,q ≍ (1 − rm)2
+∞
+�
+k=0
+(k + 1)r2mk.
+This proves the assertion made about Ip,q.
+□
+5. Sufficient conditions for non-cyclicity via Cauchy
+transforms and α-capacities
+We consider the Cauchy transform of a complex Borel measure µ on
+the unit sphere by
+C[µ](z) =
+�
+Sn
+1
+(1 − ⟨z, ¯ζ⟩)ndµ(ζ),
+z ∈ Bn.
+Note that this definition differs from the classical one.
+
+CYCLIC VECTORS IN DIRICHLET-TYPE SPACES
+15
+Let f ∈ Dα(Bn) and put a measure µ on Z(f ∗): the zero set in
+the sphere of the radial limits of f. The results in [21] about Cauchy
+transforms and non-cyclicity are valid in our settings. We deduce that
+[f] ̸= Dα(Bn), and hence non-cyclicity, whenever C[µ] ∈ D−α(Bn).
+Thus, it is important to compute the Dirichlet-type norm of the Cauchy
+transform.
+Let µ be a Borel measure on Sn and set µ∗(j) =
+�
+Sn ζjdµ(ζ), ¯µ∗(j) =
+�
+Sn ¯ζjdµ(ζ). We have the following.
+Lemma 13. Let µ be a Borel measure on Sn. Then
+||C[µ]||2
+−α ≍
+∞
+�
+k=0
+�
+|j|=k
+(k + 1)n−1−αk!
+j!
+|¯µ∗(j)|2.
+Proof. Our Cauchy integral of µ on Bn has the following expansion
+C[µ](z) =
+∞
+�
+k=0
+�
+|j|=k
+Γ(k + n)
+Γ(n)j! ¯µ∗(j)zj.
+Therefore, one can compute the norm of C[µ] in the space D−α(Bn).
+The assertion follows.
+□
+The following lemma is crucial in the sequel. It is probably known,
+but we were not able to locate it in the literature, and hence we include
+its proof.
+Lemma 14. Let j1, ..., jn, k be non-negative integers satisfying j1+...+
+jn = nk. Then
+j1! · · · jn! ≥ (k!)n.
+Proof. The Γ-function is logarithmically convex, and hence, we may
+apply the Jensen inequality to it:
+log Γ
+�x1
+n + ... + xn
+n
+�
+≤ log Γ(x1)
+n
++ ... + log Γ(xn)
+n
+.
+Set xi := ji + 1, i = 1, ..., n. Since j1 + ... + jn = nk, the assertion
+follows.
+□
+We may identify non-cyclicity for model polynomials via Cauchy
+transforms.
+Lemma 15. Consider the polynomial p(z) = 1 − mm/2z1 · · · zm, where
+1 ≤ m ≤ n. Then p is not cyclic in Dα(Bn) whenever α > 2n+1−m
+2
+.
+Proof. Recall that the model polynomials vanish in the closed unit ball
+along analytic sets of the form:
+Z(p) ∩ Sn = {1/√m(eiθ1, .., eiθm−1, e−i(θ1+...+θm−1), 0, .., 0) : θi ∈ R}.
+
+16
+DIMITRIOS VAVITSAS
+It is easy to see that for a proper measure µ, µ∗(j) is non-zero when
+mjm = k and µ∗(j) ≍ m−k/2. By Stirling’s formula and Lemma 14 we
+get that
+||C[µ]||2
+−α ≤ C
+∞
+�
+k=0
+(mk + 1)n−1−α(mk)!
+(k!)mmmk
+≍
+∞
+�
+k=0
+(k + 1)1/2(2n−m−1)−α.
+Thus, p is not cyclic in Dα(Bn) for α > 2n+1−m
+2
+.
+□
+We consider Riesz α-capacity for a fixed parameter α ∈ (0, n) with
+respect to the anisotropic distance in Sn given by
+d(ζ, η) = |1 − ⟨ζ, η⟩|1/2
+and the non-negative kernel Kα : (0, ∞) → [0, ∞) given by
+Kα(t) =
+�
+tα−n,
+α ∈ (0, n)
+log(e/t),
+α = n
+.
+Note that we may extend the definition of K to 0 by defining K(0) :=
+limt→0+ K(t).
+Let µ be any Borel probability measure supported on some Borel set
+E ⊂ Sn. Then the Riesz α-energy of µ is given by
+Iα[µ] =
+��
+Sn
+Kα(|1 − ⟨ζ, η⟩|)dµ(ζ)dµ(η)
+and the Riesz α-capacity of E by
+capα(E) = inf{Iα[µ] : µ ∈ P(E)}−1,
+where P(E) is the set of all Borel probability measures supported on
+E. Note that if capα(E) > 0, then there exist at least one probability
+measure supported on E having finite Riesz α-energy. Moreover, any
+f ∈ Dα(Bn) has finite radial limits f ∗ on Sn, except possibly, on a
+set E having capα(E) = 0. Theory regarding to the above standard
+construction in potential theory can be found in [1], [9], [12], [17].
+The relation between non-cyclicity of a function and the Riesz α-
+capacity of the zeros of its radial limits follows.
+Theorem 16. Fix α ∈ (0, n] and let f ∈ Dα(Bn). If capα(Z(f ∗)) > 0,
+then f is not cyclic in Dα(Bn).
+
+CYCLIC VECTORS IN DIRICHLET-TYPE SPACES
+17
+Proof. Let µ be a probability measure supported in Z(f ∗), with finite
+Riesz n-energy. If r ∈ (0, 1), then
+log
+e
+|1 − r⟨ζ, η⟩| = 1 + Re
+�
+log
+1
+1 − r⟨ζ, η⟩
+�
+= 1 + Re
+∞
+�
+k=1
+�
+|j|=k
+rkk!
+kj! ζjηj.
+Note that µ is a probability measure and hence
+��
+Sn
+log
+e
+|1 − r⟨ζ, η⟩|dµ(ζ)dµ(η) = 1 +
+∞
+�
+k=1
+�
+|j|=k
+rkk!
+kj! |µ∗(j)|2.
+Since |1 − w|/|1 − rw| ≤ 2 for r ∈ (0, 1) and w ∈ D, the dominated
+convergence theorem and Lemma 13 give
+||C[µ]||2
+−n ≍
+∞
+�
+k=1
+�
+|j|=k
+k!
+kj!|µ∗(j)|2 < ∞.
+The assertion follows.
+We continue setting a probability measure µ, supported in Z(f ∗),
+with finite Riesz α-energy, where α ∈ (0, n). If r ∈ (0, 1), then
+1
+(1 − r⟨ζ, η⟩)n−α =
+∞
+�
+k=0
+�
+|j|=k
+Γ(k + n − α)k!rk
+k!Γ(n − α)j!
+ζjηj.
+Similar arguments to the one above show that
+Iα[µ] ≥
+���
+��
+Sn
+Re
+�
+1
+(1 − r⟨ζ, η⟩)n−α
+�
+dµ(ζ)dµ(η)
+���
+=
+���
+∞
+�
+k=0
+�
+|j|=k
+Γ(k + n − α)k!rk
+k!Γ(n − α)j!
+��
+Sn
+ζjηjdµ(ζ)dµ(η)
+���
+≍
+∞
+�
+k=0
+�
+|j|=k
+(k + 1)n−1−αk!
+j!
+rk|µ∗(j)|2.
+Again, letting r → 1− by Lemma 13 we obtain that C[µ] ∈ D−α(Bn).
+The assertion follows.
+□
+Remark 17. According to [14] one can expect that the cyclicity problem
+of polynomials in the unit ball of Cn depends on the real dimension of
+their zero set restricted on the unit sphere: dimR(Z(p) ∩ Sn).
+Let us point out the nature of the boundary zeros of a polynomial
+non-vanishing in the ball. See [14] for the two dimensional case where
+had been used the Curve Selection Lemma of [10].
+
+18
+DIMITRIOS VAVITSAS
+Let p ∈ C[z1, ..., zn] be a polynomial non-vanishing in the ball. Look-
+ing at Z(p) ∩ Sn as at a semi-algebraic set, we conclude that it is the
+disjoint union of a finite number of Nash manifolds Mi, each Nash
+diffeomorphic to an open hypercube (0, 1)dim(Mi). Note that the Nash
+diffeomorphisms over the closed field of the real numbers satisfy some
+additional properties (see [7], Proposition 2.9.10).
+One can expect then that the characterization of cyclicity and the
+nature of the boundary zeros of the model polynomials, as well as,
+the unitary invariance of the Dirichlet norm and the sufficient capacity
+condition, will be crucial in the characterization of cyclic polynomials
+in arbitrary dimension.
+Acknowledgments.
+I would like to thank �L.
+Kosi´nski for the
+helpful conversations during the preparation of the present work.
+I
+would like to thank also the anonymous referee for numerous remarks
+that substantially improved the shape of the paper.
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+applications to exceptional and boundary interpolation sets, Michigan Math. J.
+42 (1995), 79–97.
+[10] Z. Denkowska and M.P. Denkowski, A long and winding road to definable sets,
+Journal of Singularities, 13 (2015), 57–86.
+[11] L. H¨ormander, An introduction to complex analysis in several variables, 3rd
+ed., North-Holland Mathematical library 7, Elsevier Science Publishers B.V.,
+Amsterdam, 1990.
+
+CYCLIC VECTORS IN DIRICHLET-TYPE SPACES
+19
+[12] O. El-Fallah, K. Kellay, J. Mashreghi, and T. Ransford, A primer on the
+Dirichlet space, Cambridge Tracts in Mathematics 203, Cambridge University
+Press, 2014.
+[13] G. Knese, �L. Kosi´nski, T.J. Ransford, and A.A. Sola, Cyclic polynomials in
+anisotropic Dirichlet spaces, J. Anal. Math. 138 (2019), 23–47.
+[14] �L. Kosi´nski, D. Vavitsas, Cyclic polynomials in Dirichlet-type spaces in the
+unit ball of C2, (2022), to appear in Constructive Approximation Journal.
+[15] S. Li, Some new characterizations of Dirichlet type spaces on the unit ball of
+Cn, J. Math. Anal. Appl. 324 (2006), 1073–1083.
+[16] M. Michalska, On Dirichlet type spaces in the unit ball of Cn, Ann. Univ.
+Mariae Curie-Sk�lodowska Sect. A 65 (2011), 75–86.
+[17] D. Pestana and J.M. Rodr´ıguez, Capacity distortion by inner functions in the
+unit ball of Cn, Michigan Math. J. 44 (1997), 125–137.
+[18] W. Rudin, Function theory in the unit ball of Cn, Grundlehren der mathema-
+tischen Wissenschaften 241, Springer, New York, 1980.
+[19] M. Sargent, A.A. Sola, Optimal approximants and orthogonal polynomials in
+several variables, Canad. J. Math. 74 (2022), 428–256.
+[20] M. Sargent, A.A. Sola, Optimal approximants and orthogonal polynomials in
+several variables II: Families of polynomials in the unit ball, Proc. Amer. Math.
+Soc. 149 (2021), 5321–5330.
+[21] A. Sola, A note on Dirichlet-type spaces and cyclic vectors in the unit ball of
+C2, Arch. Math. 104 (2015), 247–257.
+[22] K. Zhu, Spaces of holomorphic functions in the unit ball, Graduate Texts in
+Mathematics 226, Springer, New York, 2005.
+Email address: dimitris.vavitsas@doctoral.uj.edu.pl
+Institute of Mathematics, Faculty of Mathematics and Computer
+Science, Jagiellonian University, �Lojasiewicza 6, 30-348 Krak´ow, Poland
+
diff --git a/RtE5T4oBgHgl3EQfZw_r/content/tmp_files/load_file.txt b/RtE5T4oBgHgl3EQfZw_r/content/tmp_files/load_file.txt
new file mode 100644
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@@ -0,0 +1,629 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf,len=628
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='05584v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='CV]  13 Jan 2023  A NOTE ON CYCLIC VECTORS IN DIRICHLET-TYPE  SPACES IN THE UNIT BALL OF Cn  DIMITRIOS VAVITSAS  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' We characterize model polynomials that are cyclic in  Dirichlet-type spaces in the unit ball of Cn, and we give a sufficient  capacity condition in order to identify non-cyclic vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Introduction  Studying Dirichlet-type spaces in the unit ball of Cn we can draw  conclusions for classical Hilbert spaces of holomorphic functions such  as the Hardy, Bergman and Dirichlet spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' General introduction to  this theory can be found in [18], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  The purpose of this note is to characterize model polynomials and to  study special families of functions that are cyclic for the shift operators  on these spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Moreover, we give a sufficient capacity condition in  order to identify non-cyclic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Norm comparisons, sharp decay  of norms for special subspaces, capacity conditions studied in [3], [4],  [6], [21] are the main motivation for this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  The cyclicity of a  function f in a space of holomorphic functions is connected also with  the problem of approximating 1/f, see [19], [20] for the study of this  subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Full characterization of polynomials in more than two variables looks  like a hard problem either in the unit ball or the polydisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' The cyclicity  problem of polynomials for the bidisk was solved in [5] and shortly after  extended in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' The corresponding problem in the setting of the unit  ball of C2 was solved in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Dirichlet-type spaces in the unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Denote the unit ball  by  Bn = {z ∈ Cn : ||z|| < 1},  2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' 31C25, 32A37, 47A15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Dirichlet-type spaces,  cyclic vectors,  anisotropic  capacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Partially supported by NCN grant SONATA BIS no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' 2017/26/E/ST1/00723 of  the National Science Centre, Poland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  1  2  DIMITRIOS VAVITSAS  and its boundary, the unit sphere by  Sn = {z ∈ Cn : ||z|| = 1},  where ||z|| =  �  |z1|2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' + |zn|2 is the associated norm of the usual  Euclidean inner product ⟨z, w⟩ = z1 ¯w1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' + zn ¯wn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Denote the class  of holomorphic functions in Bn by Hol(Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Any function f ∈ Hol(Bn)  has a power series expansion  (1)  f(z) =  ∞  �  k=0  akzk =  ∞  �  k1=0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  ∞  �  kn=0  ak1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=',knzk1  1 · · · zkn  n ,  z ∈ Bn,  where k = (k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', kn) is a n-tuple index of non-negative integers, k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' =  k1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' · · ·kn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' and zk = zk1  1 · · · zkn  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' The power series in (1) exist, con-  verges normal in Bn and it is unique since the unit ball is a connected  Reinhardt domain containing the origin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', zn) ∈ Bn implies  (eiθ1z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', eiθnzn) ∈ Bn for arbitrary real θ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', θn, (see [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  To simplify the notation we may write (1) as follows:  (2)  f(z) =  ∞  �  m=0  ∞  �  |k|=m  akzk =  ∞  �  |k|=0  akzk,  z ∈ Bn,  where |k| = k1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' + kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Let f ∈ Hol(Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' We say that f belongs to the Dirichlet-type space  Dα(Bn), where α ∈ R is a fixed parameter, if  (3)  ||f||2  α :=  ∞  �  |k|=0  (n + |k|)α  (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  (n − 1 + |k|)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='|ak|2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  General introduction to the theory of Dirichlet-type spaces in the  unit ball of Cn can be found in [1], [2], [15], [16], [19], [21], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' One  variable Dirichlet-type spaces are discussed in the textbook [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' The  weights in the norm in (3) are chosen in such a way that D0(Bn) and  D−1(Bn) coincide with the Hardy and Bergman spaces of the ball, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' The Dirichlet space having M¨obius invariant norm corre-  sponds to the parameter choice α = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  By the definition, Dα(Bn) ⊂ Dβ(Bn), when α ≥ β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Polynomials are  dense in the spaces Dα(Bn), α ∈ R, and zi · f ∈ Dα(Bn), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', n  whenever f ∈ Dα(Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  A multiplier in Dα(Bn) is a holomorphic function φ : Bn → C that  satisfies φ · f ∈ Dα(Bn) for all f ∈ Dα(Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Polynomials, as well as  holomorphic functions in a neighbourhood of the closed unit ball, are  multipliers in every space Dα(Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  CYCLIC VECTORS IN DIRICHLET-TYPE SPACES  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Shift operators and cyclic vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Consider the bounded lin-  ear operators S1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', Sn : Dα(Bn) → Dα(Bn) defined by Si : f �→ zi · f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  We say that f ∈ Dα(Bn) is a cyclic vector if the closed invariant sub-  space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  [f] := clos span{zk1  1 · · · zkn  n f : k1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', kn = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='}  coincides with Dα(Bn) (the closure is taken with respect to the Dα(Bn)  norm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' An equivalent definition is that f is cyclic if and only if 1 ∈ [f].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Since Dα(Bn) enjoys the bounded point evaluation property a func-  tion that is cyclic cannot vanish inside the unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Thus, we focus  on functions non-vanishing in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Also, non-zero constant  functions are cyclic in every space Dα(Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' More information regard-  ing cyclic vectors in Dirichlet-type spaces over the disk, the polydisc  and the unit ball can be found in [3], [4], [5], [6], [8], [12], [13], [14],  [20], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Just as in the settings of the bidisk and the unit ball of two variables,  the cyclicity of a function f ∈ Dα(Bn) is inextricably linked with its  zero set  Z(f) = {z ∈ Cn : f(z) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  The zeros of a function lying on the sphere are called the boundary  zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Plan of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Section 2 studies Dirichlet-type spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' In  particular, we give a crucial relation among them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Using fractional  radial derivatives and the Cauchy formula of functions lying in the  ball algebra A(Bn) which contains functions that are continuous on the  closed unit ball and holomorphic in its interior, we give an equivalent  norm of Dirichlet-type spaces for a wide range of parameters α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Section 3 studies diagonal subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' In particular, we extend re-  sult from [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' It makes sense to define functions f ∈ Hol(Bn) using  functions ˜f ∈ Hol(D(µ)) for a proper µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Geometrically speaking,  we are looking at a disk embedded in the ball but not in a coordinate  plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Thus, we may switch the problem of cyclicity from the ball to  spaces of holomorphic functions of one variable that are well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Then we use optimal approximants in order to identify cyclicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Moreover, we prove cyclicity for model polynomials for proper pa-  rameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  In the setting of the unit ball of two variables, see [21],  the model polynomials are the following: 1 − z1 which vanishes in  the closed unit ball on a singleton, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Z(1 − z1) ∩ S2 = {(1, 0)}, and  1−2z1z2 which vanishes along an analytic curve, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Z(1−2z1z2)∩S2 =  {(eiθ/  √  2, e−iθ/  √  2) : θ ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' In our case, the corresponding candidates  4  DIMITRIOS VAVITSAS  are the following:  p(z) = 1 − mm/2z1 · · ·zm,  1 ≤ m ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  They vanish in the closed unit ball along the following analytic sets:  Z(p) ∩ Sn = {1/√m(eiθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='., eiθm−1, e−i(θ1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='+θm−1), 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='., 0) : θi ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  These polynomials are also studied with respect to the Drury-Arveson  space in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  In two variables, 1 − z1 is cyclic in Dα(B2) precisely when α ≤ 2,  and 1 − 2z1z2 is cyclic in Dα(B2) precisely when α ≤ 3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Here, there  are more than two fixed parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' The characterization of cyclicity  of these two polynomials was crucial in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Section 4 studies the radial dilation of a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Using the equiv-  alent norm of Section 2, we identify cyclicity for the model polynomials  via the powerful radial dilation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' In particular, we show that if  p/pr → 1 weakly, where pr(z) = p(rz) is a radial dilation of p, then p  is cyclic, (see [13] for the bidisk settings and [14] for the unit ball in  two variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' This method is quite interesting since it can be applied  to an arbitrary polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Note that in [13], [14] the radial dilation  method is one of the main tools of solving cyclicity problem for poly-  nomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' The main result of this section verifies the arguments made  about polynomials in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Section 5 studies non-cyclic vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' We use the notion of Riesz α-  capacity in order to identify non-cyclic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Moreover, we study  Cauchy transforms of Borel measures supported on zero sets of the  radial limits of a given function f ∈ Dα(Bn) and we give asymptotic  expansions of their norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Then employing a standard scheme due to  Brown and Shields, see [8], we prove the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Note that this  sufficient capacity condition for non-cyclicity in Dirichlet-type spaces  in the unit ball of two variables was proved by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Sola in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Standard tools  Let us give some standard tools which will be useful in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  The binomial series:  1  (1 − x)α =  ∞  �  k=0  Γ(k + α)  Γ(α)k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' xk,  where |x| < 1 is a complex number and α is a non-negative real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  The asymptotic behaviour of the Γ-function is the following: Γ(k +  α) ≍ (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='kα, where the symbol ≍ denotes that the ratio of the two  quantities either tends to a constant as k tends to infinity or it is rather  two sides bound by constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  CYCLIC VECTORS IN DIRICHLET-TYPE SPACES  5  The multinomial formula:  (x1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' + xn)k =  �  |j|=k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='xj1  1 · · · xjn  n ,  where j = (j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', jn) is a n-tuple index of non-negative integers and xi  are complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  The Stirling formula that describes the asymptotic behaviour of the  gamma function:  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' ≍ k1/2kk/ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Denote the normalized area measure on Cn = R2n by du(z) and the  normalized rotation-invariant positive Borel measure on Sn by dσ(ζ),  (see [18], [22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' The measures du(z) and dσ(ζ) are related by the for-  mula  �  Cn f(z)du(z) = 2n  � ∞  0  �  Sn  ǫ2n−1f(ǫζ)dσ(ζ)dǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  The holomorphic monomials are orthogonal to each other in L2(σ),  that is, if k and l are multi-indices such that k ̸= l, then  �  Sn  ζk¯ζldσ(ζ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Moreover,  �  Sn  |ζk|2dσ(ζ) =  (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  (n − 1 + |k|)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  and  �  Bn  |zk|2du(z) =  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  (n + |k|)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='.  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Relation among Dirichlet-type spaces and equivalent  norms  We study the structure of Dirichlet-type spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Note that  R(f)(z) = z1∂z1f(z) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' + zn∂znf(z)  is the radial derivative of a function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' The radial derivative plays a key  role in the function theory of the unit ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' A crucial relation among  these spaces is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Let f ∈ Hol(Bn) and α ∈ R be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Then  f ∈ Dα(Bn)  if and only if  nqf + Rq(f) + q  q−1  �  i=1  niRq−i(f) ∈ Dυ(Bn),  where α = 2q + υ, q ∈ N and Rq is the q-image of the operator R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  6  DIMITRIOS VAVITSAS  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Indeed, it is enough to check that  ||nf +R(f)||2  α−2 =  ∞  �  |k|=0  (n+|k|)α−2  (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  (n − 1 + |k|)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' (n+|k|)2|ak|2 = ||f||2  α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  □  We continue by giving an equivalent characterization of Dirichlet-  type norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' In Dirichlet-type spaces in the unit ball, one of the integral  representations of the norm is achieved in a limited range of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Lemma 2 (see[16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' If α ∈ (−1, 1), then ||f||2  α is equivalent to  |f|2  α :=  �  Bn  ||∇(f)(z)||2 − |R(f)(z)|2  (1 − ||z||2)α  du(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Above, ∇(f)(z) = (∂z1f(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', ∂znf(z)) denotes the holomorphic  gradient of a holomorphic function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Note that Proposition 1 allows  us to use Lemma 2 whenever υ ∈ (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Let γ, t ∈ R be such that  neither n + γ nor n + γ + t is a negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' If f = �∞  |k|=0 akzk  is the homogeneous expansion of a function f ∈ Hol(Bn), then we  may define an invertible continuous linear operator with respect to the  topology of uniform convergence on compact subsets of Bn, denoted by  Rγ,t : Hol(Bn) → Hol(Bn) and having expression  Rγ,tf(z) =  ∞  �  |k|=0  C(γ, t, k)akzk,  z ∈ Bn,  where  (4)  C(γ, t, k) = Γ(n + 1 + γ)Γ(n + 1 + |k| + γ + t)  Γ(n + 1 + γ + t)Γ(n + 1 + |k| + γ) ≍ |k|t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  See [22] for more information regarding these fractional radial deriva-  tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Let t ∈ R be such that n − 1 + t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' If f ∈ A(Bn), then  R−1,tf(z) =  �  Sn  f(ζ)  (1 − ⟨z, ζ⟩)n+tdσ(ζ),  z ∈ Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' The continuous linear operator Rγ,t, see [22], satisfies  Rγ,t�  1  (1 − ⟨z, w⟩)n+1+γ  �  =  1  (1 − ⟨z, w⟩)n+1+γ+t  for all w ∈ Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Next, define fǫ for ǫ ∈ (0, 1) by  fǫ(z) =  �  Sn  f(ζ)  (1 − ⟨z, ǫζ⟩)ndσ(ζ),  z ∈ Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  CYCLIC VECTORS IN DIRICHLET-TYPE SPACES  7  The Cauchy formula holds for f ∈ A(Bn) and hence f = limǫ→1− fǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' It  follows that  R−1,tf(z) = R−1,t�  lim  ǫ→1−  �  Sn  f(ζ)  (1 − ⟨z, ǫζ⟩)ndσ(ζ)  �  = lim  ǫ→1− R−1,t� �  Sn  f(ζ)  (1 − ⟨z, ǫζ⟩)ndσ(ζ)  �  = lim  ǫ→1−  �  Sn  f(ζ)R−1,t�  1  (1 − ⟨z, ǫζ⟩)n  �  dσ(ζ)  = lim  ǫ→1−  �  Sn  f(ζ)  (1 − ⟨z, ǫζ⟩)n+tdσ(ζ)  =  �  Sn  f(ζ)  (1 − ⟨z, ζ⟩)n+tdσ(ζ)  and the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  □  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Let α ∈ R be such that n − 1 + α/2 ≥ 0 and f ∈ A(Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Then f ∈ Dα(Bn) if and only if  �  Bn  (1 − ||z||2)  ���  �  Sn  f(ζ)¯ζp  (1 − ⟨z, ζ⟩)n+α/2+1dσ(ζ)  ���  2  du(z) < ∞  and  �  Bn  ���  �  Sn  (zpζq − zqζp)f(ζ)  (1 − ⟨z, ζ⟩)n+α/2+1dσ(ζ)  ���  2  du(z) < ∞,  where p, q = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Choose t so that α = 2t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Note that n, t are fixed and hence  ||f||2  α ≍  ∞  �  |k|=0  (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  (n − 1 + |k|)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='||k|tak|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Thus, (4) implies that ||R−1,tf||0 ≍ ||f||α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' One can apply then the  integral representation of Dirichlet-type norms to R−1,tf ∈ Hol(Bn),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' ||R−1,tf||0 is equivalent to |R−1,tf|0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' According to Lemma 3 we get  that  ∂zp(R−1,tf)(z) =  �  Sn  f(ζ)¯ζp  (1 − ⟨z, ζ⟩)n+t+1dσ(ζ),  z ∈ Bn,  where p = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Expand the term ||∇(f)||2 − |R(f)|2 as follows:  ||∇(f)||2 − |R(f)|2 = (1 − ||z||2)||∇(f)||2 +  �  p,q  |¯zp∂zqf − ¯zq∂zpf|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  The assertion follows by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  □  8  DIMITRIOS VAVITSAS  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Diagonal subspaces  In [3], a method of construction of optimal approximants via deter-  minants in Dirichlet-type spaces in the unit disk is provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Similarly,  we may define optimal approximants in several variables, (see [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Fix N ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' We define the space of polynomials p ∈ C[z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', zn] with  degree at most nN as follows:  P n  N := {p(z) =  N  �  k1=0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  N  �  kn=0  ak1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=',knzk1  1 · · · zkn  n }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Let (X, || · ||) be a normed space and fix x ∈ X, C ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  The distance between x and the set C is the following:  distX(x, C) := inf{||x − c|| : c ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  It is well known that if X is a Hilbert space and C ⊂ X a convex  closed subset, then for any x ∈ X, there exists a unique y ∈ C such  that ||x − y|| = distX(x, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Let f ∈ Dα(Bn) be non-zero constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' We  deduce that for any N ∈ N, there exists exactly one pN ∈ P n  N satisfying  ||pNf − 1||α = distDα(Bn)(1, f · P n  N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Let f ∈ Dα(Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' We say that a polynomial pN ∈ P n  N is an optimal  approximant of order N to 1/f if pN minimizes ||pf − 1||α among all  polynomials p ∈ P n  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' We call ||pNf − 1||α the optimal norm of order N  associated with f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Let M = (M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', Mn) be a multi-index, where Mi are non-negative  integers, and m ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Setting  µ(m) := (M1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' + Mm)M1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='+Mm  MM1  1  · · MMm  m  ,  we see that  (5)  µ(m)1/2|z1|M1 · · · |zm|Mm ≤ 1,  z ∈ Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Using (5) we may construct polynomials that vanish in the closed  unit ball along analytic subsets of the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Let ˜f ∈ Hol(D(µ(m)−1/4)), where  D(µ) = {z ∈ C : |z| < µ},  µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  According to (5) we define the following function:  f(z) = f(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', zn) = ˜f(µ(m)1/4zM1  1  · · zMm  m ),  z ∈ Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  CYCLIC VECTORS IN DIRICHLET-TYPE SPACES  9  Then f ∈ Hol(Bn) and it depends on m variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Note that we may  change the variables z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', zm by any other m variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' For conve-  nience, we choose the m first variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' The power 1/4 will be conve-  nient in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Thus, the question that arises out is if we may define closed subspaces  of Dα(Bn) passing through one variable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' We shall see that  these subspaces are called diagonal subspaces due to the nature of the  power series expansion of their elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Instead of the classical one variable Dirichlet-type spaces of the unit  disk, we may consider spaces dβ, β ∈ R, consisting of holomorphic func-  tions ˜f ∈ Hol(D(µ−1/4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Moreover, such functions with power series  expansion ˜f(z) = �∞  l=0 alzl are said to belong to dβ if  || ˜f||2  dβ :=  ∞  �  l=0  µ−l/2(l + 1)β|al|2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  There is a natural identification between the function theories of  Dβ(D): one variable Dirichlet-type spaces of the unit disk, and dβ, and  one verifies that the results in [3] are valid for dβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  We are ready to define diagonal closed subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Set  β(α) := α − n + m + 1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Let α, M, m be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' The diagonal closed subspace of Dα(Bn)  is the following:  Jα,M,m := {f ∈ Dα(Bn) : ∃ ˜f ∈ dβ(α), f(z) = ˜f(µ(m)1/4zM1  1  · · zMm  m )}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  The existence of a holomorphic function ˜f is unique by identity prin-  ciple and hence there is no any amiss in the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Any function  f ∈ Jα,M,m has an expansion of the form  f(z) =  ∞  �  l=0  al(zM1  1  · · zMm  m )l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  The relation of norms between one variable and diagonal subspaces  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' If f ∈ Jα,M,m, then ||f||α ≍ || ˜f||dβ(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' If f ∈ Jα,M,m, then  ||f||2  α ≍  ∞  �  l=0  (l + 1)α  (M1l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' · · · (Mml)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  (n − 1 + (M1 + · · · Mm)l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='|al|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  10  DIMITRIOS VAVITSAS  By Stirling’s formula, we obtain  ||f||2  α ≍  ∞  �  l=0  (l + 1)α−n+m/2+1/2µ(m)−l|al|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  On the other hand, define the function f ′(z) = �∞  l=0 µ(m)−l/4alzl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Then  f ′(µ(m)1/4zM1  1  · · zMm  m ) = f(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', zn) and  ||f ′||2  dβ(α) ≍  �  l=0  (l + 1)α−n+m/2+1/2µ(m)−l|al|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  The assertion follows since f ′ coincides with ˜f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  □  The corresponding Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='4 of [4] in our case is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Let f ∈ Jα,M,m, where α, M, m be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Let rN ∈ P n  N  with expansion  rN(z) =  N  �  k1=0  · ·  N  �  kn=0  ak1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=',knzk1  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='zkn  n ,  and consider its projection onto Jα,M,m  π(rN)(z) =  �  {l:M1l,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=',Mml≤N}  cM1l,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=',Mml,0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=',0zM1l  1  · · zMml  m  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Then  ||rNf − 1||α ≥ ||π(rN)f − 1||α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Moreover, just as in Proposition 7, there is a relation of optimal  approximants between one variable and diagonal subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' If f ∈ Jα,M,m, then  distDα(Bn)(1, f · P n  N) ≍ distdβ(α)(1, ˜f · P 1  N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Let rN, π(rN) be as in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Then π(rN)f − 1 ∈ Jα,M,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' It  follows that  ||rNf − 1||α ≥ ||π(rN)f − 1||α ≍ ||˜π(rN) ˜f − 1||dβ(α) ≥ distdβ(α)(1, ˜f · P 1  N),  since ˜π(rN) ∈ P 1  N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' On the other hand, let  distdβ(α)(1, ˜f · P 1  N) = ||qN ˜f − 1||dβ(α),  qN(z) =  N  �  l=0  alzl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Then, the polynomial  q′  N(z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', zn) =  N  �  l=0  µ(m)−l/4alzM1l  1  · · zMml  m  CYCLIC VECTORS IN DIRICHLET-TYPE SPACES  11  satisfies q′  N ∈ Jα,M,m ∩ P n  N and q′  Nf − 1 ∈ Jα,M,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Thus,  ||qN ˜f − 1||dβ(α) = ||˜q′  N ˜f − 1||dβ(α) ≍ ||q′  Nf − 1||α ≥ distDα(Bn)(1, f · P n  N)  and the assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  □  Define the function φβ : [0, ∞) → [0, ∞) by  φβ(t) =  �  t1−β,  β < 1  log+(t),  β = 1 ,  where log+(t) := max{log t, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' We have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Let α ∈ R be such that β(α) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Let f ∈ Jα,M,m  be as above and suppose the corresponding ˜f has no zeros inside its  domain, has at least one zero on the boundary, and admit an analytic  continuation to a strictly bigger domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Then f is cyclic in Dα(Bn)  whenever α ≤ 2n−m+1  2  and  dist2  Dα(Bn)(1, f · P n  N) ≍ φβ(α)(N + 1)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' It is an immediate consequence of the identification between  Dβ(D) and dβ and previous lemmas and propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  □  If we focus on polynomials, then the following is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Consider the polynomial p(z) = 1−mm/2z1 · · · zm, where  1 ≤ m ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Then p is cyclic in Dα(Bn) whenever α ≤ 2n+1−m  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Note that the Theorem11 is not a characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' We shall study  the case α > 2n+1−m  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Cyclicity for model polynomials via radial dilation  The radial dilation of a function f : Bn → C is defined for r ∈ (0, 1)  by fr(z) = f(rz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' To prove Theorem 11, it is enough to prove the  following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Consider the polynomial p(z) = 1 − mm/2z1 · · · zm, where  1 ≤ m ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Then ||p/pr||α < ∞ as r → 1− whenever α ≤ 2n+1−m  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  We follow the arguments of [14], [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Indeed, if Lemma 12 holds,  then φr · p → 1 weakly, where φr := 1/pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' This is a consequence of a  crucial property of Dirichlet-type spaces: if {fn} ⊂ Dα(Bn), then fn →  0 weakly if and only if fn → 0 pointwise and supn{||fn||α} < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Since  φr extends holomorphically past the closed unit ball, φr are multipliers,  and hence, φr · p ∈ [p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Finally, 1 is weak limit of φr · p and [p] is closed  and convex or, equivalently, weakly closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' It is clear that 1 ∈ [p], and  hence, p is cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  12  DIMITRIOS VAVITSAS  Moreover, it is enough to prove that ||p/pr||α < ∞, as r → 1−,  for α0 =  2n+1−m  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Then the case α < α0 follows since the inclusion  Dα0(Bn) ֒→ Dα(Bn) is a compact linear map and weak convergence in  Dα0(Bn) gives weak convergence in Dα(Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Proof of Lemma 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' By Theorem 4 it is enough to show the following:  Ip :=  �  Bn  (1 − ||z||2)  ���  �  Sn  (1 − λζ1 · · · ζm)¯ζp  (1 − rmλζ1 · · · ζm)(1 − ⟨z, ζ⟩)β dσ(ζ)  ���  2  du(z)  and  Ip,q :=  �  Bn  ���  �  Sn  (zpζq − zqζp)(1 − λζ1 · · · ζm)  (1 − rmλζ1 · · ·ζm)(1 − ⟨z, ζ⟩)β dσ(ζ)  ���  2  du(z)  are finite, as r → 1−, where β = n + t + 1, t = 2n+1−m  4  , and λ = mm/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Denote  Sp :=  �  Sn  (1 − λζ1 · · · ζm)¯ζp  (1 − rmλζ1 · · · ζm)(1 − ⟨z, ζ⟩)β dσ(ζ),  where the last integral is equal to  1  2π  �  Sn  � 2π  0  (1 − λeimθζ1 · · · ζm)e−iθ ¯ζp  (1 − rmλeimθζ1 · · · ζm)(1 − e−iθ⟨z, ζ⟩)β dθdσ(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Let z, ζ be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Then  � 2π  0  e−iθ  (1 − e−iθ⟨z, ζ⟩)β dθ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Thus, replacing p(eiθζ)/p(reiθζ) by p(eiθζ)/p(reiθζ) − 1 we obtain  Sp = λ(rm − 1)  2π  �  Sn  � 2π  0  ¯ζpζ1 · · · ζmei(m−1)θ  (1 − rmλeimθζ1 · · ·ζm)(1 − e−iθ⟨z, ζ⟩)β dθdσ(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Next, expand the binomials  � 2π  0  ei(m−1)θ  (1 − rmλeimθζ1 · · · ζm)(1 − e−iθ⟨z, ζ⟩)β dθ  =  ∞  �  k=0  ∞  �  l=0  Γ(k + β)  Γ(β)k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' (rmλζ1 · · · ζm)l⟨z, ζ⟩k  � 2π  0  ei(m(l+1)−k−1)θdθ  = 2π  ∞  �  k=0  Γ(m(k + 1) − 1 + β)  Γ(β)(m(k + 1) − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' (rmλζ1 · · ·ζm)k⟨z, ζ⟩m(k+1)−1  = 2π  ∞  �  k=0  �  |j|=m(k+1)−1  Γ(m(k + 1) − 1 + β)  Γ(β)j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  (rmλζ1 · · · ζm)kzj ¯ζj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  CYCLIC VECTORS IN DIRICHLET-TYPE SPACES  13  Therefore,  Sp = λ(rm − 1)  ∞  �  k=0  �  |j|=m(k+1)−1  Γ(m(k + 1) − 1 + β)  Γ(β)j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  (rmλ)kzjc(k),  where c(k) =  �  Sn ζα(k)¯ζb(k)dσ(ζ), α(k) = (k+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='., k+1(m-comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' ), 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='., 0)  and b(k) = (j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', jp−1, jp + 1, jp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', jn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Whence, 1 ≤ p ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Since  the holomorphic monomials are orthogonal to each other in L2(σ) we  get that  |Sp| ≍ (1 − rm)  ���z′  p  ∞  �  k=0  (k + 1)β−n(rmλz1 · · · zm)k���,  where z′  p = z1 · · · zp−1zp+1 · · · zm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Hence we obtain  Ip ≍ (1−rm)2  ∞  �  k=0  (k+1)2(β−n)(rmλ)2k  �  Bn  (1−||z||2)|z′  p|2|z1 · · · zm|2kdu(z),  where has been used again the orthogonality of the holomorphic mono-  mials in L2(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' To handle the integral above we use polar coordinates  �  Bn  (1 − ||z||2)|z′  p|2|z1 · · · zm|2kdu(z)  ≍  � 1  0  �  Sn  ǫ2n−1(1 − ǫ2)ǫ2km+2m−2|ζ′  p|2|ζ1 · · · ζm|2kdσ(ζ)dǫ  ≍  [(k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' ]m−1k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  (n + m(k + 1) − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' ·  1  (k + 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  If we recall that β = n + t + 1, t =  2n+1−m  4  and λ2k = mmk, then  applying the Stirling formula more than one time we see that  Ip ≍ (1 − rm)2  ∞  �  k=0  (k + 1)r2mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  This proves the assertion made about Ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  It remains to estimate the following term:  Ip,q =  �  Bn  ���  �  Sn  (zpζq − zqζp)(1 − λζ1 · · ·ζm)  (1 − rmλζ1 · · · ζm)(1 − ⟨z, ζ⟩)β dσ(ζ)  ���  2  du(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  We shall show that Ip,q ≍ Ip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Denote again the inner integral by Sp,q  which is convenient to expand it as Sp,q = ¯zpSq − ¯zqSp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Recall that  z′  p = z1 · · · zp−1zp+1 · · · zm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Similar calculations to the one above lead to  |Sp,q| ≍ (1 − rm)|¯zpz′  q − ¯zqz′  p|  ���  ∞  �  k=0  (k + 1)β−n(rmλz1 · · · zm)k���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  14  DIMITRIOS VAVITSAS  Moreover, the orthogonality of the holomorphic monomials in L2(σ)  gives the following estimation:  Ip,q ≍ (1−rm)2  ∞  �  k=0  (k+1)2β−2n(rmλ)2k  �  Bn  |¯zpz′  q−¯zqz′  p|2|z1 · · · zm|2kdu(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  It is easy to see that |¯zpz′  q − ¯zqz′  p|2 = |zp|2|z′  q|2 +|zq|2|z′  p|2 −2|z1 · · · zm|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Let us estimate the integral  �  Bn  (|zp|2|z′  q|2 − |z1 · · · zm|2)|z1 · · · zm|2kdu(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Passing through polar coordinates we get, for p ̸= q, that  �  Bn  |zp|2|z′  q|2|z1 · · · zm|2kdu(z)  ≍ 2n(n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  [(k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' ]m−1k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  (mk + n + m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  k + 2  2km + 2n + 2m,  and  �  Bn  |z1 · · ·zm|2(k+1)du(z)  = 2n(n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  [(k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' ]m−1k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  (mk + n + m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  k + 1  2km + 2n + 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Hence we obtain  �  Bn  (|zp|2|z′  q|2 − |z1 · · · zm|2)|z1 · · · zm|2kdu(z)  ≍  [(k + 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' ]m−1k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  (mk + n + m − 2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' (k + 1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Again, applying the Stirling formula to the one above estimates we  obtain  Ip,q ≍ (1 − rm)2  ∞  �  k=0  (k + 1)r2mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  This proves the assertion made about Ip,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  □  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Sufficient conditions for non-cyclicity via Cauchy  transforms and α-capacities  We consider the Cauchy transform of a complex Borel measure µ on  the unit sphere by  C[µ](z) =  �  Sn  1  (1 − ⟨z, ¯ζ⟩)ndµ(ζ),  z ∈ Bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Note that this definition differs from the classical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  CYCLIC VECTORS IN DIRICHLET-TYPE SPACES  15  Let f ∈ Dα(Bn) and put a measure µ on Z(f ∗): the zero set in  the sphere of the radial limits of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' The results in [21] about Cauchy  transforms and non-cyclicity are valid in our settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' We deduce that  [f] ̸= Dα(Bn), and hence non-cyclicity, whenever C[µ] ∈ D−α(Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Thus, it is important to compute the Dirichlet-type norm of the Cauchy  transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Let µ be a Borel measure on Sn and set µ∗(j) =  �  Sn ζjdµ(ζ), ¯µ∗(j) =  �  Sn ¯ζjdµ(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' We have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Lemma 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Let µ be a Borel measure on Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Then  ||C[µ]||2  −α ≍  ∞  �  k=0  �  |j|=k  (k + 1)n−1−αk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  |¯µ∗(j)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Our Cauchy integral of µ on Bn has the following expansion  C[µ](z) =  ∞  �  k=0  �  |j|=k  Γ(k + n)  Γ(n)j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' ¯µ∗(j)zj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Therefore, one can compute the norm of C[µ] in the space D−α(Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  The assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  □  The following lemma is crucial in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' It is probably known,  but we were not able to locate it in the literature, and hence we include  its proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Lemma 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Let j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', jn, k be non-negative integers satisfying j1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='+  jn = nk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Then  j1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' · · · jn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' ≥ (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' The Γ-function is logarithmically convex, and hence, we may  apply the Jensen inequality to it:  log Γ  �x1  n + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' + xn  n  �  ≤ log Γ(x1)  n  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' + log Γ(xn)  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Set xi := ji + 1, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Since j1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' + jn = nk, the assertion  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  □  We may identify non-cyclicity for model polynomials via Cauchy  transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Lemma 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Consider the polynomial p(z) = 1 − mm/2z1 · · · zm, where  1 ≤ m ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Then p is not cyclic in Dα(Bn) whenever α > 2n+1−m  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Recall that the model polynomials vanish in the closed unit ball  along analytic sets of the form:  Z(p) ∩ Sn = {1/√m(eiθ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='., eiθm−1, e−i(θ1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='+θm−1), 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='., 0) : θi ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  16  DIMITRIOS VAVITSAS  It is easy to see that for a proper measure µ, µ∗(j) is non-zero when  mjm = k and µ∗(j) ≍ m−k/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' By Stirling’s formula and Lemma 14 we  get that  ||C[µ]||2  −α ≤ C  ∞  �  k=0  (mk + 1)n−1−α(mk)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' )mmmk  ≍  ∞  �  k=0  (k + 1)1/2(2n−m−1)−α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Thus, p is not cyclic in Dα(Bn) for α > 2n+1−m  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  □  We consider Riesz α-capacity for a fixed parameter α ∈ (0, n) with  respect to the anisotropic distance in Sn given by  d(ζ, η) = |1 − ⟨ζ, η⟩|1/2  and the non-negative kernel Kα : (0, ∞) → [0, ∞) given by  Kα(t) =  �  tα−n,  α ∈ (0, n)  log(e/t),  α = n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Note that we may extend the definition of K to 0 by defining K(0) :=  limt→0+ K(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Let µ be any Borel probability measure supported on some Borel set  E ⊂ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Then the Riesz α-energy of µ is given by  Iα[µ] =  ��  Sn  Kα(|1 − ⟨ζ, η⟩|)dµ(ζ)dµ(η)  and the Riesz α-capacity of E by  capα(E) = inf{Iα[µ] : µ ∈ P(E)}−1,  where P(E) is the set of all Borel probability measures supported on  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Note that if capα(E) > 0, then there exist at least one probability  measure supported on E having finite Riesz α-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Moreover, any  f ∈ Dα(Bn) has finite radial limits f ∗ on Sn, except possibly, on a  set E having capα(E) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Theory regarding to the above standard  construction in potential theory can be found in [1], [9], [12], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  The relation between non-cyclicity of a function and the Riesz α-  capacity of the zeros of its radial limits follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Theorem 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Fix α ∈ (0, n] and let f ∈ Dα(Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' If capα(Z(f ∗)) > 0,  then f is not cyclic in Dα(Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  CYCLIC VECTORS IN DIRICHLET-TYPE SPACES  17  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Let µ be a probability measure supported in Z(f ∗), with finite  Riesz n-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' If r ∈ (0, 1), then  log  e  |1 − r⟨ζ, η⟩| = 1 + Re  �  log  1  1 − r⟨ζ, η⟩  �  = 1 + Re  ∞  �  k=1  �  |j|=k  rkk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  kj!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' ζjηj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Note that µ is a probability measure and hence  ��  Sn  log  e  |1 − r⟨ζ, η⟩|dµ(ζ)dµ(η) = 1 +  ∞  �  k=1  �  |j|=k  rkk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  kj!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' |µ∗(j)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Since |1 − w|/|1 − rw| ≤ 2 for r ∈ (0, 1) and w ∈ D, the dominated  convergence theorem and Lemma 13 give  ||C[µ]||2  −n ≍  ∞  �  k=1  �  |j|=k  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  kj!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='|µ∗(j)|2 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  The assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  We continue setting a probability measure µ, supported in Z(f ∗),  with finite Riesz α-energy, where α ∈ (0, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' If r ∈ (0, 1), then  1  (1 − r⟨ζ, η⟩)n−α =  ∞  �  k=0  �  |j|=k  Γ(k + n − α)k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='rk  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='Γ(n − α)j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  ζjηj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Similar arguments to the one above show that  Iα[µ] ≥  ���  ��  Sn  Re  �  1  (1 − r⟨ζ, η⟩)n−α  �  dµ(ζ)dµ(η)  ���  =  ���  ∞  �  k=0  �  |j|=k  Γ(k + n − α)k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='rk  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='Γ(n − α)j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  ��  Sn  ζjηjdµ(ζ)dµ(η)  ���  ≍  ∞  �  k=0  �  |j|=k  (k + 1)n−1−αk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  rk|µ∗(j)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Again, letting r → 1− by Lemma 13 we obtain that C[µ] ∈ D−α(Bn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  The assertion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  □  Remark 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' According to [14] one can expect that the cyclicity problem  of polynomials in the unit ball of Cn depends on the real dimension of  their zero set restricted on the unit sphere: dimR(Z(p) ∩ Sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Let us point out the nature of the boundary zeros of a polynomial  non-vanishing in the ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' See [14] for the two dimensional case where  had been used the Curve Selection Lemma of [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  18  DIMITRIOS VAVITSAS  Let p ∈ C[z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=', zn] be a polynomial non-vanishing in the ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Look-  ing at Z(p) ∩ Sn as at a semi-algebraic set, we conclude that it is the  disjoint union of a finite number of Nash manifolds Mi, each Nash  diffeomorphic to an open hypercube (0, 1)dim(Mi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content=' Note that the Nash  diffeomorphisms over the closed field of the real numbers satisfy some  additional properties (see [7], Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  One can expect then that the characterization of cyclicity and the  nature of the boundary zeros of the model polynomials, as well as,  the unitary invariance of the Dirichlet norm and the sufficient capacity  condition, will be crucial in the characterization of cyclic polynomials  in arbitrary dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  I would like to thank �L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  Kosi´nski for the  helpful conversations during the preparation of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
+page_content='  I  would like to thank also the anonymous referee for numerous remarks  that substantially improved the shape of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/RtE5T4oBgHgl3EQfZw_r/content/2301.05584v1.pdf'}
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+arXiv:2301.04734v1  [cond-mat.str-el]  11 Jan 2023
+Construction of Wannier functions from the spectral moments of correlated electron
+systems
+Frank Freimuth1,2,∗ Stefan Bl¨ugel1, and Yuriy Mokrousov1,2
+1Peter Gr¨unberg Institut and Institute for Advanced Simulation,
+Forschungszentrum J¨ulich and JARA, 52425 J¨ulich, Germany and
+2 Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany
+When the first four spectral moments are considered, spectral features missing in standard Kohn-
+Sham (KS) density-functional theory (DFT), such as upper and lower Hubbard bands, as well as
+spectral satellite peaks, can be described, and the bandwidths can be corrected.
+Therefore, we
+have devised a moment-functional based spectral density functional theory (MFbSDFT) recently.
+However, many computational tools in theoretical solid state physics, such as the construction of
+maximally localized Wannier functions (MLWFs), have been developed for KS-DFT and require
+modifications if they are supposed to be used in MFbSDFT. Here, we show how generalized Wan-
+nier functions may be constructed from the first four spectral moment matrices. We call these func-
+tions maximally localized spectral moment Wannier functions (MLSMWFs). We demonstrate how
+MLSMWFs may be used to compute the anomalous Hall effect (AHE) in fcc Ni by Wannier interpo-
+lation. More generally, MLSMWFs may be computed from the first 2P moments (P = 1, 2, 3, . . . ).
+Using more than 4 moments opens the perspective of reproducing all spectral features accurately in
+MFbSDFT.
+I.
+INTRODUCTION
+Maximally localized Wannier functions (MLWFs) have
+become a widely-applied tool in computational solid state
+physics.
+Recent reviews [1, 2] give a comprehensive
+overview of their applications. Using Wannier interpola-
+tion [3] one may significantly reduce the computing time
+requirements of calculations of response functions such
+as the anomalous Hall effect (AHE) [4], thermoelectric
+coefficients [5], and the spin-orbit torque (SOT) [6, 7].
+The Wannier interpolation of the AHE does not suffer
+from band truncation errors, because it directly interpo-
+lates the Berry curvature.
+In order to interpolate the
+SOT and the Dzyaloshinskii-Moriya interaction without
+band truncation errors one may use higher-dimensional
+Wannier functions [8] in order to interpolate the mixed
+Berry curvature [9, 10]. Response functions that cannot
+be expressed in terms of these geometric properties of the
+electronic bands cannot be treated within the conven-
+tional Wannier interpolation method without any band
+truncation error. However, for such response functions
+Wannier function perturbation theory has been devel-
+oped recently [11].
+All these MLWF-based interpolation methods have
+been devised essentially in the context of KS-DFT.
+While impressive progress has been achieved in devel-
+oping exchange-correlation functionals for KS-DFT that
+describe ground state properties of solids with high pre-
+cision [12], experimental spectra are often not repro-
+duced well by standard KS-DFT [13]. In Ref. [14] and
+in Ref. [15] we have explained how the spectral func-
+tion may be constructed from the first four spectral mo-
+ment matrices. These moment matrices may be obtained
+either by computing several correlation functions self-
+consistently [14], or by evaluating suitable moment func-
+tionals [15]. We expect that the moment potentials re-
+quired for the latter approach, which we call MFbSDFT,
+are universal functionals of the spin density.
+Indeed, we have shown that parameter-free moment
+functionals can be found that improve the spectra of
+Na and SrVO3 significantly [16] in comparison to stan-
+dard KS-DFT with LDA. In order to formulate these
+parameter-free moment functionals we have used an ex-
+isting model of the second moment of the uniform elec-
+tron gas (UEG) [17] as well as models of the momentum
+distribution function nk of the UEG [18–20]. However,
+formulating universal moment functionals that are gen-
+erally applicable is still a long way. Notably, the correct
+inclusion of spin-polarization and the extension by gra-
+dient corrections are important necessary developments
+left for future work.
+Nevertheless, even while the universal moment func-
+tionals are not yet available, one may generally im-
+prove spectra by optimizing the parameters in suitable
+parameterized moment functionals.
+Adjusting in this
+way e.g. the bandwidths and gaps in order to repro-
+duce experimental data will increase the accuracy of re-
+sponse function calculations, in particular those of op-
+tical responses, such as laser-induced currents [21] and
+torques [22], which are expected to be generally sensi-
+tive to gaps, bandwidths, and band positions, because
+they are not Fermi-surface effects like the AHE [23–25]
+for example.
+We have demonstrated that MFbSDFT can be used
+to improve the description of the electronic structure of
+fcc Ni significantly in comparison to LDA, because it
+yields bandwidth, exchange splitting, and satellite peak
+positions in good agreement with the experimental spec-
+trum [15, 16].
+The satellite peak in Ni roughly 6 eV
+below the Fermi energy is a correlation effect [26–28],
+
+2
+which is missing in the KS-spectrum. Since the method
+of spectral moments captures the splitting of bands into
+lower and upper Hubbard bands, it cannot be mapped
+onto a non-interacting effective KS Hamiltonian in gen-
+eral without changing the spectrum. The question there-
+fore poses itself of how to obtain generalized Wannier
+functions from the spectral moment matrices.
+The method of spectral moments yields state energies,
+state wavefunctions, and corresponding spectral weights,
+in contrast to KS-DFT, where the spectral weights are
+by definition unity. Moreover, the state wavefunctions
+corresponding to different energies are not guaranteed
+to be orthogonal. While many-body generalizations of
+Wannier functions have been considered before [29], these
+generalizations are not optimized to construct localized
+Wannier functions from the first four spectral moment
+matrices. For example, Ref. [29] does not take the spec-
+tral weights of the quasiparticles into account and does
+not consider the possibility that the quasiparticle wave-
+functions do not necessarily form a set of mutually or-
+thogonal functions.
+However, the standard method of
+constructing MLWFs assumes the Bloch functions of dif-
+ferent bands to be orthogonal [30, 31]. Here, we will show
+that spectral weights and non-orthogonality of quasipar-
+ticle wavefunctions can be taken into account by gener-
+alizing the MLWFs concept for the method of spectral
+moments.
+We demonstrate the construction of MLSMWFs for
+fcc Ni and use them to compute the AHE by Wannier
+interpolation. We choose fcc Ni because it is known that
+standard KS-DFT overestimates the bandwidth and the
+exchange splitting in this material [27] and it predicts
+the AHE to be significantly larger than the experimental
+value [32–34]. Moreover, even the sign of the magnetic
+anisotropy energy (MAE) is not predicted correctly by
+standard KS-DFT, i.e., LDA does not predict the cor-
+rect easy axis [35].
+Previously, we have demonstrated
+that MFbSDFT can be used to reproduce the experi-
+mental values of the exchange splitting, the bandwidth,
+and the position of the satellite peaks [15, 16]. Here, we
+demonstrate that also the AHE is predicted to be close
+to the experimental value if MFbSDFT is used.
+The rest of this paper is structured as follows. In Sec. II
+we explain how we construct MLSMWFs from the first
+four spectral moment matrices. Practical issues, such as
+the use of the wannier90 code [2] for the generation of
+MLSMWFs are described in Sec. II E and in Sec. II F.
+MLSMWFs with spin-orbit interaction (SOI) are dis-
+cussed in Sec. II G. In Sec. II H we explain that ab-initio
+programs which can compute MLWFs can be extended
+easily to compute additionally MLSMWFs. In Sec. II I
+we describe Wannier interpolation based on MLSMWFs
+using the example of the AHE. The interpolation of addi-
+tional matrix elements such as spin and torque operators
+is discussed in Sec. II J. In Sec. III we explain how the
+method of Sec. II may be generalized to include the first
+2P moments (P = 3, 4, . . .). In Sec. IV we present appli-
+cations of our method to fcc Ni. This paper ends with a
+summary in Sec. V.
+II.
+THEORY
+Before discussing the construction of MLSMWFs from
+the first four spectral moment matrices in Sec. II D we
+first revisit the generation of MLWFs in Sec. II A as well
+as the calculation of the spectral function in Sec. II B.
+This will help us to explain the necessary modifications
+of the MLWFs formalism in Sec. II D, when MLSMWFs
+are constructed from the spectral moment matrices.
+A.
+MLWFs and Wannier interpolation
+The Bloch functions |ψkm⟩ are eigenfunctions of the
+KS-Hamiltonian H with eigenenergies Ekm:
+H|ψkm⟩ = Ekm|ψkm⟩,
+(1)
+where k is the k-point and m is the band index. The ML-
+WFs |WRn⟩ are constructed from these Bloch functions
+by the transformation [30]
+|WRn⟩ = 1
+N
+�
+k
+NB
+�
+m=1
+U (k)
+mne−ik·R|ψkm⟩,
+(2)
+where N is the number of k points. The matrix U(k) is
+a rectangular matrix when the number of bands NB is
+larger than the number of MLWFs NW, otherwise it is a
+square matrix, which may occur for example when ML-
+WFs are constructed from isolated groups of bands [31].
+The matrix U(k) is determined by the condition that the
+MLWFs minimize the spatial spread
+Ω =
+NW
+�
+n=1
+�
+⟨WRn|r2|WRn⟩ − (⟨WRn|r|WRn⟩)2�
+.
+(3)
+Due to the localization of the MLWFs in real-space
+the matrix elements of the Hamiltonian H decay rapidly
+when the distance between the MLWFs increases:
+lim
+R→∞⟨W0m|H|WRn⟩ → 0.
+(4)
+This localization property is important for Wannier inter-
+polation, because it implies that in order to interpolate
+the electronic structure at any desired k point it is suffi-
+cient to provide the matrix elements ⟨W0m|H|WRn⟩ for
+a finite and small set of R vectors. The reason for this is
+that the electronic structure at any desired k point may
+be computed by performing a Fourier transformation of
+⟨W0m|H|WRn⟩ and that the computational time for this
+
+3
+is small if the set of R vectors is small. Explicitly, the
+matrix elements of the Hamiltonian may be written as
+⟨W0m|H|WRn⟩ =
+= 1
+N
+�
+k
+NB
+�
+m′=1
+e−ik·REkm′
+�
+U (k)
+m′m
+�∗
+U (k)
+m′n.
+(5)
+In order to describe the AHE in magnetically collinear
+ferromagnets, the spin-orbit interaction (SOI) has to be
+taken into account. Wannier interpolation is very effi-
+cient in computing the AHE [4]. In the presence of SOI,
+the Bloch functions are spinors
+⟨r|ψkm⟩ = ψkm(r) =
+�
+χkm↑(r)
+χkm↓(r)
+�
+,
+(6)
+and the MLWFs are spinors as well:
+⟨r|WRn⟩ =
+�WRn↑(r)
+WRn↓(r)
+�
+.
+(7)
+For the interpolation of the AHE in bcc Fe [4] one first
+constructs MLWFs using a coarse k mesh, e.g. an 8 ×
+8 × 8 mesh.
+Next, one computes the matrix elements
+of the Hamiltonian according to Eq. (5).
+Finally, one
+may Fourier transform these matrix elements for all k
+points in the fine interpolation mesh, e.g. an 800 × 800 ×
+800 mesh. Using this interpolated Hamiltonian, one may
+compute the AHE numerically efficiently [4].
+B.
+Construction of the spectral function from the
+first four spectral moment matrices
+In Ref. [15] we describe an algorithm to construct the
+spectral function from the first four spectral moments.
+In the following we assume that the spectral moments
+are so expressed in a basis set of NS orthonormalized
+functions φkn(r) that they are given by NS×NS matrices.
+Due to the orthonormalization of the basis functions, the
+zeroth moment M (0)
+k
+is simply the unit matrix. When
+the spectral function is determined approximately from
+the first four spectral moments (M (0)
+k , M (1)
+k , M (2)
+k , and
+M (3)
+k ), the poles of the single-particle spectral function
+are given by the eigenenergies of the hermitean 2NS×2NS
+matrix [15]
+Hk =
+�
+M (1)
+k
+B1k
+B†
+1k
+D1k
+�
+,
+(8)
+where M (1)
+k
+is the first moment, B1k = U k
+√Dk, B2k =
+[M (3)
+k
+− M (2)
+k M (1)
+k ][B†
+1k]−1, and D1k = B−1
+1k [B2k −
+M (1)
+k B1k]. Here, Dk is a diagonal matrix, and Uk is
+a unitary matrix so that UkDkU †
+k = M (2)
+k
+−M (1)
+k M (1)
+k .
+The eigenvectors of Hk, Eq. (8), may be written as
+Ψkn =
+�ψkn→
+ψkn←
+�
+,
+(9)
+where ψkn→ and ψkn← are both column vectors with
+NS components, while Ψkn is a column vector with 2NS
+entries. We denote the eigenvalues of Hk by Ekn, i.e.,
+HkΨkn = EknΨkn.
+(10)
+Within MFbSDFT, the charge density is computed only
+from the upper part ψkn→ of the state vector Ψkn
+(Eq. (9)), while the lower part plays the role of an aux-
+iliary component.
+Note that while the eigenfunctions
+|ψkm⟩ of the KS-Hamiltonian (Eq. (1)) are orthonormal,
+i.e.,
+⟨ψkm|ψk′n⟩ = δnmδkk′,
+(11)
+the upper parts ψkn→ of the state vectors Ψkn (Eq. (9))
+are not even orthogonal, i.e.,
+[ψkm→]† ψkn→ ̸∝ δnm,
+(12)
+while the complete state vectors are orthonormal:
+[Ψkm]† Ψkn = δnm.
+(13)
+We may obtain the spectral weight of the state Ψkn
+from
+akn = [ψkn→]† ψkn→ =
+NS
+�
+m′=1
+[ψknm′→]∗ψknm′→.
+(14)
+Here, ψknm′→ is the m′-th entry in the column vector
+ψkn→. These spectral weights are useful to quantify the
+relative importance of a given state with energy Ekn. For
+example, it may occur that the spectral function has a
+pole at Ekn with a spectral weight akn ≪ 1. Due to the
+small spectral weight this pole might not be observable
+in the experimental spectrum. Therefore, both the poles
+Ekn and the spectral weights akn are generally necessary
+to discuss the spectrum.
+In order to construct MLSMWFs from the state vec-
+tors Ψkn, we need their real-space representation Ψkn(r).
+Clearly, ψkn→ is given by
+ψkn→(r) =
+NS
+�
+m=1
+φkm(r)ψknm→,
+(15)
+in real-space, where φkm(r) is the m-th function in the
+orthonormal set used to express the spectral moments at
+k.
+In MFbSDFT the functions ψkn→(r) replace the KS
+wavefunctions from standard KS-DFT [15]: The charge
+density and the DOS may be obtained from ψkn→(r),
+while the auxiliary vector ψkn← is only needed to solve
+
+4
+Eq. (10), and may often be discarded afterwards. How-
+ever, as will become clear in Sec. II D, we need the lower
+part ψkn←(r) for the construction of MLSMWFs. The
+matrices B1k and B2k describe linear maps from the
+space of eigenfunctions of M (2)
+k −M (1)
+k M (1)
+k
+to the space
+of orthonormal basis functions φkn(r). Consequently, the
+matrix D1k describes a linear map from the space of
+eigenfunctions of M (2)
+k
+− M (1)
+k M (1)
+k
+to itself. Therefore,
+the components of ψkn← refer to the space of eigenfunc-
+tions of M (2)
+k
+− M (1)
+k M (1)
+k
+and an additional unitary
+transformation to the space of orthonormal basis func-
+tions φkn(r) is necessary to obtain the real-space repre-
+sentation of ψkn←:
+ψkn←(r) =
+NS
+�
+m,m′=1
+Ukmm′φkm(r)ψknm′←.
+(16)
+Another approach leading to Eq. (16) considers the
+unitary transformation
+�k =
+�1
+0
+0 U k
+�
+,
+(17)
+where 1 is the NS×NS unit matrix, while 0 is the NS×NS
+zero matrix. When this transformation is applied to Hk
+it does not change its eigenvalues Ekn nor the upper part
+ψkn→ of the eigenvectors. Only the lower part ψkn← of
+the eigenvectors is changed so that Eq. (10) turns into
+¯Hk
+� ψkn→
+U kψkn←
+�
+= Ekn
+� ψkn→
+U kψkn←
+�
+,
+(18)
+where
+¯Hk = �kHk� †
+k
+(19)
+is the transformed Hamiltonian.
+¯Hk describes a map
+Vk×Vk → Vk×Vk, where we denote the space of orthog-
+onal basis functions φkn(r) by Vk. Since the lower com-
+ponents of the eigenvectors of ¯Hk are given by U kψkn←
+according to Eq. (18), it is clear that the real-space rep-
+resentation of ψkn← is given by Eq. (16).
+C.
+Choice of the moment functionals
+The spectral moment matrices M (I)
+k
+(I = 1, 2, 3, . . .)
+may be obtained either by computing several correlation
+functions self-consistently [14], or by evaluating suitable
+moment functionals [15, 16].
+In the latter approach,
+which we call MFbSDFT, the I-th moment is decom-
+posed into the I-th power of the first moment plus the
+additional contribution M (I+)
+k
+[15, 16]:
+M (I)
+k
+=
+�
+M (1)
+k
+�I
++ M (I+)
+k
+.
+(20)
+The first moment, M (1)
+k , may be obtained easily within
+the standard KS framework: It is simply given by the KS
+Hamiltonian, if instead of the full exchange-correlation
+potential only the first-order exchange is used. The addi-
+tional contributions M (I+)
+k
+may be computed from suit-
+able potentials V(I+)(r) [15, 16]:
+M (I+)
+knm =
+�
+d3rV(I+)(r)φ∗
+kn(r)φkm(r).
+(21)
+We expect that the V(I+)(r) depend only on the elec-
+tron density n(r), i.e., there are universal functionals of
+n(r), from which V(I+)(r) may be obtained. This expec-
+tation is corroborated by our finding [16] that parameter-
+free expressions for the moment potentials can be found
+that improve the spectra of Na and of SrVO3 significantly
+in comparison to standard KS-DFT with LDA. However,
+general and accurate expressions for V(I+)(r) are cur-
+rently not yet available. Therefore, we proposed several
+parameterizations of V(I+)(r), which can be used to re-
+produce spectral features such as satellite peaks and to
+correct the band width e.g. in Ni [15, 16].
+Defining the dimensionless density parameter
+rs(r) = 1
+aB
+�
+3
+4πn(r)
+� 1
+3
+,
+(22)
+where aB is Bohr’s radius, we may express V(I+)(r)
+through [15]
+V(I+)(r) =
+c(I+)
+[rs(r)]I + . . .
+(23)
+in the low-density limit, i.e., when rs(r) is large. Alter-
+natively, one may use [15, 16]
+V(I+)(r) = d(I+)[Vc(rs)]I + . . . ,
+(24)
+where
+Vc = d(ǫcn)
+dn
+(25)
+is the correlation potential, and ǫc is the correlation en-
+ergy.
+In these parameterized expressions of V(I+) one may
+treat e.g. d(2+) and d(3+) as independent parameters and
+optimize both in order to match the experimental spec-
+tra as well as possible. Alternatively, one may compute
+V(3+) for a given V(2+) by enforcing the constraint of the
+momentum distribution function of the UEG [16].
+D.
+Construction of MLSMWFs from the first four
+spectral moment matrices
+In order to compute MLSMWFs from the first four
+spectral moment matrices, we need to use the states
+
+5
+Eq. (9) instead of the usual Bloch functions. An obvious
+generalization of Eq. (2) based on these state vectors is
+�⟨r|WRn→⟩
+⟨r|WRn←⟩
+�
+= 1
+N
+�
+k
+2NS
+�
+m=1
+U (k)
+mne−ik·R
+�ψkm→(r)
+ψkm←(r)
+�
+(26)
+where ψkm→(r) and ψkm←(r) are given in Eq. (15) and
+Eq. (16), respectively, and the 2NS × NW matrix U(k) is
+so chosen that the spread
+Ω =
+NW
+�
+n=1
+�
+⟨WRn→|r2|WRn→⟩ − (⟨WRn→|r|WRn→⟩)2�
++
+NW
+�
+n=1
+�
+⟨WRn←|r2|WRn←⟩ − (⟨WRn←|r|WRn←⟩)2�
+(27)
+is minimized.
+As a consequence of the spatial localization, the ma-
+trix elements ⟨W0m|H|WRn⟩ decay rapidly in real-space
+similar to Eq. (4):
+lim
+R→∞⟨W0m|H|WRn⟩ → 0.
+(28)
+Explicitly, these matrix elements are given by
+HRmn = ⟨W0m|H|WRn⟩ =
+= 1
+N
+�
+k
+2NS
+�
+m′=1
+e−ik·REkm′
+�
+U (k)
+m′m
+�∗
+U (k)
+m′n.
+(29)
+The derivation of Eq. (29) shows clearly that both
+ψkn→(r) and ψkn←(r) need to be taken into account in
+the construction of MLSMWFs: Only when both compo-
+nents, ψkn→ and ψkn←, are considered, Ψkn is an eigen-
+vector of Hk. Moreover, it is clear that both components,
+⟨r|WRn→⟩ and ⟨r|WRn←⟩, have to be localized together
+to minimize Eq. (27), because otherwise Eq. (28) is not
+valid and the Fourier transformation below in Eq. (30)
+cannot be performed numerically efficiently.
+In order to obtain the interpolated band structure, we
+first carry out the Fourier transformation
+˜Hk =
+�
+R
+eik·RHR,
+(30)
+where HR is the matrix with the components HRmn de-
+fined in Eq. (29). Next, we diagonalize ˜Hk:
+[X k]† ˜HkX k = ˜�k.
+(31)
+Here, X k is a unitary matrix and ˜�k is a diagonal matrix
+holding the interpolated energies:
+˜Eknm = ˜Eknδnm.
+(32)
+Often, we would like to interpolate not only the band
+energies but also the spectral weights akn, Eq. (14). For
+this purpose, we first need to compute the matrix ele-
+ments
+sknm = (ψ†
+kn→, ψ†
+kn←)
+�1 0
+0 0
+� �ψkm→
+ψkm←
+�
+= ψ†
+kn→ψkm→
+(33)
+for all k points in the coarse k mesh that are used in the
+construction of the MLSMWFs. Here, 0 is the NS × NS
+zero matrix and 1 is the NS×NS unit matrix. Next, these
+matrix elements need to be expressed in the MLSMWF
+basis:
+sRnm = 1
+N
+�
+k
+2NS
+�
+n′,m′=1
+skn′m′
+�
+U(k)
+n′n
+�∗
+U(k)
+m′me−ik·R. (34)
+After carrying out these preparations before the actual
+Wannier interpolation step, one may interpolate sknm to
+a given k point in the fine interpolation mesh by per-
+forming the Fourier transformation
+˜sknm =
+�
+R
+sRnmeik·R
+(35)
+in the course of the Wannier interpolation. Finally, ˜sknm
+needs to be transformed into the eigenbasis in order to
+obtain the interpolated spectral weights:
+˜akn =
+�
+n′m′
+˜skn′m′Xkm′n [Xkn′n]∗ .
+(36)
+While the applications shown below in Sec. IV use the
+MFbSDFT approach of Ref. [15] in order to obtain the
+spectral moments, the theory for the construction of the
+MLSMWFs from the spectral moment matrices that we
+present here can also be used when the spectral moments
+are obtained by computing several correlation functions
+self-consistently as in Ref. [14].
+E.
+Wavefunction overlaps
+The wannier90 code [2] computes the spread Eq. (3)
+from the overlaps between the lattice periodic parts
+ukm(r) = e−ik·rψkm(r) of the Bloch functions at the
+nearest-neighbor k-points k and k + b. Therefore, the
+matrix elements
+M (k,b)
+mn
+= ⟨ukm|uk+b,n⟩
+(37)
+need to be provided to wannier90 in order to determine
+the MLWFs through the matrix U (k)
+mn in Eq. (2), which
+minimizes the spread Eq. (3).
+In order to find the matrix U (k)
+mn that defines the
+MLSMWFs in Eq. (26) one may use the wannier90
+code [2] as well. In this case one needs to provide the
+matrix elements
+M (k,b)
+mn
+= ⟨ukm→|uk+b,n→⟩ + ⟨ukm←|uk+b,n←⟩
+(38)
+to wannier90, which ensures that all contributions to the
+spread in Eq. (27) are taken into account.
+
+6
+F.
+Initial projections
+In order to obtain a good starting point for the iterative
+minimization of the spreads, Eq. (3) (for MLWFs) and
+Eq. (27) (for MLSMWFs), one may define first guesses
+|gn⟩ for these Wannier functions [30, 31]. In the case of
+MLWFs the matrix elements
+A(k)
+mn = ⟨ψkm|gn⟩
+(39)
+may be computed and provided to the wannier90 code [2]
+for this purpose. In order to provide the first guesses in
+the case of MLSMWFs, one may generalize Eq. (39) as
+follows:
+A(k)
+mn = ⟨ψkm→|gn→⟩ + ⟨ψkm←|gn←⟩.
+(40)
+When one computes MLWFs of bulk transition metals
+such as bcc Fe, fcc Ni, fcc Pt, and fcc Pd, one typically
+constructs 9 MLWFs per spin in order to obtain Wan-
+nier functions that describe the valence bands and the
+first few conduction bands. In this case suitable initial
+projections are one s, three p, and five d states, which
+are 9 states in total. Alternatively, one may use 6 sp3d2
+hybrid states plus dxy, dyz, and dzx. From Sec. II D it
+follows that the number of MLSMWFs is typically cho-
+sen twice as large as the number of MLWFs would be
+chosen in the same material. If we assume that for half
+of the MLSMWFs the →-component is larger than the
+←-component, while for the remaining other half of the
+MLSMWFs the ←-component is more dominant than the
+→-component, an obvious choice for the initial projec-
+tions is to use states that are purely ← or purely →.
+G.
+Construction of MLSMWFs in systems with
+SOI
+In magnetically collinear systems without SOI, one
+typically constructs MLWFs separately for spin-up and
+spin-down, i.e., one constructs two sets of MLWFs. In the
+presence of SOI this is not possible, because the Hamil-
+tonian couples the spin-up and spin-down bands. Conse-
+quently, only a single set of MLWFs is constructed. For
+example, in ferromagnetic fcc Ni one computes 9 spin-up
+MLWFs and 9 spin-down MLWFs when SOI is not taken
+into account, while one constructs 18 spinor-MLWFs (see
+Eq. (7)) when SOI is considered.
+Analogously,
+only a single set of MLSMWFs is
+constructed in systems with SOI. In this case every
+MLSMWF has four components:
+⟨r|WRn⟩ =
+
+
+
+
+⟨r|WRn→↑⟩
+⟨r|WRn→↓⟩
+⟨r|WRn←↑⟩
+⟨r|WRn←↓⟩
+
+
+
+ .
+(41)
+Similarly, the eigenvectors Ψkn in Eq. (10) have four
+components:
+Ψkn =
+�ψkn→
+ψkn←
+�
+=
+
+
+
+
+ψkn→↑
+ψkn→↓
+ψkn←↑
+ψkn←↓
+
+
+
+ .
+(42)
+Consequently, the matrix elements
+M (k,b)
+mn
+=
+�
+p=→,←
+�
+σ=↑,←
+⟨ukmpσ|uk+b,npσ⟩
+(43)
+and
+A(k)
+mn =
+�
+p=→,←
+�
+σ=↑,←
+⟨ψkmpσ|gnpσ⟩
+(44)
+need to be provided to the wannier90 code in this case
+in order to determine the MLSMWFs.
+H.
+Implementation within the FLAPW method
+In Ref. [36] we describe in detail how the matrix
+elements M (k,b)
+mn
+and A(k)
+mn required by wannier90 for
+the calculation of the MLWFs may be implemented
+within the full-potential linearized augmented plane-
+wave method (FLAPW). For the construction of the
+MLSMWFs we need to compute these matrix elements
+according to the prescriptions of Eq. (38) and Eq. (40)
+(when SOI is included in the calculations Eq. (43) and
+Eq. (44) should be used instead). It is straightforward
+to extend the implementation described in Ref. [36] by
+adding the additional loop over the MFbSDFT indices
+→ and ←.
+I.
+Wannier interpolation of response functions
+In Ref. [14] we have described how the AHE con-
+ductivity may be computed within the method of
+spectral moments using correlation functions such as
+⟨[[c†
+kαckβ, H]−, c†
+kγckδ]−⟩ (see e.g. Eq. (34), Eq. (C1), and
+Eq. (C2) in Ref. [14]). However, we have also reported in
+Ref. [14] that in the case of the Hubbard-Rashba model
+the AHE is well-approximated by
+σxy = e2ℏ
+V N
+�
+k
+NW
+�
+n,n′=1
+[fkn − fkn′]×
+× Im [⟨ψkn→|vx|ψkn′→⟩⟨ψkn′→|vy|ψkn→⟩]
+(Ekn′ − Ekn)2 + 0+
+,
+(45)
+which does not require us to compute correlation func-
+tions such as ⟨[[c†
+kαckβ, H]−, c†
+kγckδ]−⟩. Here, we assume
+that Eq. (45) can also be used to compute the AHE of
+
+7
+realistic materials approximately within the spectral mo-
+ment approach. We leave if for future work to test this
+approximation by computing the AHE also from the cor-
+relation functions ⟨[[c†
+kαckβ, H]−, c†
+kγckδ]−⟩, and focus on
+the evaluation of Eq. (45) in order to provide an example
+of Wannier interpolation with MLSMWFs.
+We may obtain ⟨ψkn→|vx|ψkn′→⟩ from Wannier inter-
+polation by computing first the matrix elements
+hknm =
+NS
+�
+n′,m′=1
+M (1)
+n′m′ψkmm′→ [ψknn′→]∗
+(46)
+of the first moment for all k points in the coarse k mesh
+that are used in the construction of the MLSMWFs.
+Subsequently, we compute the corresponding matrix ele-
+ments in the MLSMWFs basis:
+hRnm = 1
+N
+�
+k
+2NS
+�
+n′,m′=1
+hkn′m′
+�
+U(k)
+n′n
+�∗
+U(k)
+m′me−ik·R.
+(47)
+These are preparatory steps that are carried out before
+the actual Wannier interpolation. In order to interpolate
+hknm to a given k point in the fine interpolation mesh
+we first carry out the Fourier-transformation
+˜hknm =
+�
+R
+hRnmeik·R.
+(48)
+The velocity operator matrix elements are obtained from
+the k derivative:
+˜vknm = 1
+ℏ
+�
+R
+iReik·RhRnm.
+(49)
+Finally, we need to transform these matrix elements into
+the eigenbasis, which we obtain from Eq. (31):
+⟨ψkn→|v|ψkm→⟩ =
+�
+n′m′
+˜vkn′m′Xkm′m [Xkn′n]∗ ,
+(50)
+where Xkm′m are the elements of the unitary matrix de-
+fined in Eq. (31). Now, the matrix elements Eq. (50) may
+be used together with the eigenvalues Ekn and the Fermi
+factors fkn = f(Ekn) (where f is the Fermi function) to
+evaluate Eq. (45).
+This interpolation approach suffers from a band trun-
+cation error, because only Wannier interpolated states
+are used to evaluate Eq. (45). However, the band trun-
+cation error has been shown to be small in the case of
+AHE [4] and also in the case of SHE [11].
+J.
+Wannier interpolation of the spin and torque
+operators
+Within MFbSDFT, the matrix elements of the spin
+operator are defined by
+Sknm = ℏ
+2(ψ†
+kn→, ψ†
+kn←)
+�σ 0
+0 0
+� �ψkm→
+ψkm←
+�
+= ℏ
+2ψ†
+kn→σψkm→
+= ℏ
+2(ψ†
+kn→↑, ψ†
+kn→↓)σ
+�ψkm→↑
+ψkm→↓
+�
+.
+(51)
+In order to compute for example spin photocurrents [21]
+from Wannier interpolation within MFbSDFT, these ma-
+trix elements need to be interpolated.
+We obtain the
+interpolated ˜Sknm similarly to Eq. (34) and Eq. (35) (re-
+place sknm → Sknm, sRnm → SRnm, and ˜sknm → ˜Sknm
+in these equations). Finally, the interpolated ˜Sknm may
+be transformed into the eigenbasis similarly to Eq. (50):
+⟨ψkn→|S|ψkm→⟩ =
+�
+n′m′
+˜Skn′m′Xkm′m [Xkn′n]∗ .
+(52)
+The torque operator T is needed for the calculation of
+the SOT [6, 7]. At first glance, it is tempting to define
+the torque operator by
+T knm = −µB
+�
+d3r[ψkn→(r)]†σψkm→(r) × Ωxc(r)
+(53)
+within MFbSDFT, where µB is the Bohr magneton, and
+Ωxc(r) is the exchange field. However, the moment po-
+tentials V(2+)
+σ
+(r) and V(3+)
+σ
+(r) (see Eq. (21)) may be spin-
+polarized in general, similarly to the exchange potential
+in the first moment. There is no convincing argument
+that one may substitute the exchange potential of the
+first moment for Ωxc(r) in Eq. (53).
+Instead, we ex-
+pect that a suitable expression for Ωxc(r) may be de-
+rived within the MFbSDFT framework, and that it will
+depend on the potentials of the first, second, and third
+moments.
+Therefore, we consider the alternative expression for
+the torque operator
+T knm = − i
+2
+�
+d3r[ψkn→(r)]†[HSOI(r), σ]ψkm→(r),
+(54)
+where HSOI(r) is the SOI. The torque operator may be
+interpolated analogously to the interpolation of the spin
+operator discussed above.
+The torque operator may also be used to compute the
+MAE [6, 37]. Within MFbSDFT, the torque due to the
+magnetic anisotropy is given by
+T mae = − 1
+N
+�
+kn
+fkn⟨ψkn→|T |ψkn→⟩.
+(55)
+
+8
+III.
+EXTENSION TO MORE MOMENTS
+In Ref. [16] we present an efficient algorithm to con-
+struct the spectral function from the first 2P spectral
+moment matrices, where P = 1, 2, 3, . . .. The algorithm
+described in Ref. [15], which we revisit briefly in Sec. II B,
+is the special case with P = 2 of this more general algo-
+rithm. We expect that the accuracy of the MFbSDFT
+approach can be enhanced by increasing P. For example,
+in Ref. [16] we explain that it easy to reproduce the jump
+of the momentum distribution function nk of the UEG
+at kF when P ≥ 3, while this is difficult to achieve with
+P = 2.
+In Sec. II D we describe the generation of MLSMWFs
+when P = 2. The extension to P > 2 is straightforward.
+As an example, consider the case P = 3, i.e., assume
+that we construct the spectral function from the first 6
+spectral moment matrices using the algorithm described
+in Ref. [16]. In this case the poles of the spectral function
+are the eigenvalues of a 3NS × 3NS matrix Hk.
+The
+eigenvectors of Hk have 3NS components in this case
+and they may be written in the form
+Ψkn =
+
+
+ψkn→
+ψknտ
+ψknւ
+
+ ,
+(56)
+where ψkn→, ψknտ, and ψknւ are NS-component vec-
+tors. ψkn→ is the physical component from which the
+charge density, the DOS, the spectral weights, and the
+expectation values of operators can be computed. ψknտ,
+and ψknւ are auxiliary components, which may be dis-
+carded in a standard MFbSDFT selfconsistency loop af-
+ter diagonalizing Hk. However, like in Sec. II D, these
+auxiliary components need to be included into the gen-
+eration of the MLSMWFs. Therefore, we construct the
+MLSMWFs from
+
+
+⟨r|WRn→⟩
+⟨r|WRnտ⟩
+⟨r|WRnւ⟩
+
+ = 1
+N
+�
+k
+3NS
+�
+m=1
+U (k)
+mne−ik·R
+
+
+ψkm→(r)
+ψkmտ(r)
+ψkmւ(r)
+
+
+(57)
+in this case. Here, U(k) is a 3NS × NW matrix.
+IV.
+APPLICATIONS
+When the magnetization is along the [001] direction,
+GGA predicts the intrinsic AHE in Ni to be -2200 S/cm,
+which is significantly larger than the experimental value
+of -646 S/cm [34].
+Using GGA+U with U = 1.9 eV,
+one obtains the intrinsic AHE of -1066 S/cm [34]. The
+remaining discrepancy between experiment and theory
+is 420 S/cm. This discrepancy can be explained by the
+side-jump AHE [32].
+MFbSDFT may be used to reproduce the experimental
+values of the exchange splitting, of the band width, and
+15
+16
+17
+18
+19
+20
+d
+(2+)
+-3000
+-2500
+-2000
+-1500
+-1000
+-500
+0
+AHE σxy [S/cm]
+FIG. 1. AHE conductivity σxy vs. the prefactor d(2+) of the
+second moment potential.
+of the valence band satellite position in fcc Ni [15, 16].
+To compute the AHE in Ni from MLSMWFs, we first
+perform self-consistent MFbSDFT calculations with SOI.
+We perform these calculations with various different d(2+)
+parameters in the range 15-20 to investigate the depen-
+dence of the AHE on d(2+), i.e. we use Eq. (24), but
+we set d(3+) = 0.
+In order to keep the magnetic mo-
+ment fixed at around 0.6 µB, which is the value measured
+in experiments, we need to spin-polarize V(2+). We use
+V(2+)
+σ
+= ζt
+σV(2+), where ζσ = (1 − σ(n↑ − n↓)/n), and t
+is determined at every value of d(2+) to match the exper-
+imental magnetic moment. Next, we compute the ma-
+trix elements M (k,b)
+mn
+and A(k)
+mn as discussed in Sec. II E,
+Sec. II F, and Sec. II G. We generate MLSMWFs using
+the wannier90 code [2] and disentanglement, where we
+set the lower bound of the frozen window at around 80 eV
+below the Fermi energy and the upper bound at around
+4 eV above the Fermi energy. We construct 36 spinor
+MLSMWFs from 72 MFbSDFT bands.
+In Fig. 1 we plot the AHE obtained from MLSMWFs
+as explained in Sec. II I as a function of the prefactor
+d(2+) used in the potential of the second moment. With
+increasing d(2+) the magnitude of σxy decreases.
+At
+d(2+) = 20.0 the intrinsic AHE is -1000 S/cm.
+If we
+assume that the side-jump contribution to the AHE is
+around 400 S/cm [32], this is in good agreement with the
+experimental value of -646 S/cm.
+In Ref. [15] we used d(2+) = 15.0 in order to reproduce
+the experimental bandwidth, exchange splitting, and po-
+sition of the satellite peak. However, using d(2+) = 20.0
+instead reproduces these experimental features also quite
+well, which we show in Fig. 2. In Ref. [16] we have found
+that the valence band satellite is in much better agree-
+ment with DMFT calculations and with experiment if
+the third moment potential is computed from the second
+
+9
+-8
+-6
+-4
+-2
+0
+2
+E-EF [eV]
+0
+1
+2
+3
+4
+5
+DOS [States/(u.c. eV)]
+Majority
+Minority
+FIG. 2. DOS in fcc Ni obtained from MFbSDFT.
+moment potential using the constraint of the momen-
+tum distribution function of the UEG. However, since
+we have currently developed this procedure only for the
+UEG without spin-polarization we needed to apply a sim-
+ilar spin-polarization factor ζt
+σ like in the present calcu-
+lations. As a result, the spectral density of Ni in Ref. [16]
+matches experiment concerning the spin-polarization of
+the satellite peak, and the band width of the main band.
+However, it suffers from a similar overestimation of the
+exchange splitting as standard KS-DFT with LDA. In
+contrast, the present calculation yields the exchange-
+splitting close to experiments. Since the AHE depends
+strongly on the Fermi surface [23–25] we therefore use
+here the simpler approach of Eq. (24) instead of the im-
+proved approach of Ref. [16].
+V.
+SUMMARY
+We describe the construction of Wannier functions
+from the first 4 spectral moment matrices.
+We show
+that these MLSMWFs can be used for the efficient in-
+terpolation of material property tensors such as the AHE
+within MFbSDFT. This paves the way for the application
+of MFbSDFT to compute response properties of materi-
+als. We demonstrate that MFbSDFT is able to reproduce
+the experimentally measured AHE in fcc Ni, similarly to
+LDA+U. Finally, we discuss that MLSMWFs may be
+computed also from the first 6 moments, and generally
+from the first 2P moments. This opens the perspective
+of using as many moments as necessary to reproduce all
+spectral features accurately in MFbSDFT.
+ACKNOWLEDGMENTS
+The project is funded by the Deutsche Forschungs-
+gemeinschaft (DFG, German Research Foundation) −
+TRR 288 − 422213477 (project B06), CRC 1238, Control
+and Dynamics of Quantum Materials: Spin orbit cou-
+pling, correlations, and topology (Project No. C01), SPP
+2137 “Skyrmionics”, and Sino-German research project
+DISTOMAT (DFG project MO 1731/10-1).
+We also
+acknowledge financial support from the European Re-
+search Council (ERC) under the European Union’s Hori-
+zon 2020 research and innovation program (Grant No.
+856538, project “3D MAGiC”) and computing resources
+granted by the J¨ulich Supercomputing Centre under
+project No. jiff40.
+∗ Corresp. author: f.freimuth@fz-juelich.de
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf,len=765
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='04734v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='str-el]  11 Jan 2023  Construction of Wannier functions from the spectral moments of correlated electron  systems  Frank Freimuth1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='∗ Stefan Bl¨ugel1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' and Yuriy Mokrousov1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='2  1Peter Gr¨unberg Institut and Institute for Advanced Simulation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Forschungszentrum J¨ulich and JARA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' 52425 J¨ulich,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Germany and  2 Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Johannes Gutenberg University Mainz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' 55099 Mainz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Germany  When the first four spectral moments are considered,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' spectral features missing in standard Kohn-  Sham (KS) density-functional theory (DFT),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' such as upper and lower Hubbard bands,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' as well as  spectral satellite peaks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' can be described,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' and the bandwidths can be corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Therefore, we  have devised a moment-functional based spectral density functional theory (MFbSDFT) recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  However, many computational tools in theoretical solid state physics, such as the construction of  maximally localized Wannier functions (MLWFs), have been developed for KS-DFT and require  modifications if they are supposed to be used in MFbSDFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Here, we show how generalized Wan-  nier functions may be constructed from the first four spectral moment matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' We call these func-  tions maximally localized spectral moment Wannier functions (MLSMWFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' We demonstrate how  MLSMWFs may be used to compute the anomalous Hall effect (AHE) in fcc Ni by Wannier interpo-  lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' More generally, MLSMWFs may be computed from the first 2P moments (P = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Using more than 4 moments opens the perspective of reproducing all spectral features accurately in  MFbSDFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  INTRODUCTION  Maximally localized Wannier functions (MLWFs) have  become a widely-applied tool in computational solid state  physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Recent reviews [1, 2] give a comprehensive  overview of their applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Using Wannier interpola-  tion [3] one may significantly reduce the computing time  requirements of calculations of response functions such  as the anomalous Hall effect (AHE) [4], thermoelectric  coefficients [5], and the spin-orbit torque (SOT) [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  The Wannier interpolation of the AHE does not suffer  from band truncation errors, because it directly interpo-  lates the Berry curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  In order to interpolate the  SOT and the Dzyaloshinskii-Moriya interaction without  band truncation errors one may use higher-dimensional  Wannier functions [8] in order to interpolate the mixed  Berry curvature [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Response functions that cannot  be expressed in terms of these geometric properties of the  electronic bands cannot be treated within the conven-  tional Wannier interpolation method without any band  truncation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' However, for such response functions  Wannier function perturbation theory has been devel-  oped recently [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  All these MLWF-based interpolation methods have  been devised essentially in the context of KS-DFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  While impressive progress has been achieved in devel-  oping exchange-correlation functionals for KS-DFT that  describe ground state properties of solids with high pre-  cision [12], experimental spectra are often not repro-  duced well by standard KS-DFT [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' [14] and  in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' [15] we have explained how the spectral func-  tion may be constructed from the first four spectral mo-  ment matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' These moment matrices may be obtained  either by computing several correlation functions self-  consistently [14], or by evaluating suitable moment func-  tionals [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' We expect that the moment potentials re-  quired for the latter approach, which we call MFbSDFT,  are universal functionals of the spin density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Indeed, we have shown that parameter-free moment  functionals can be found that improve the spectra of  Na and SrVO3 significantly [16] in comparison to stan-  dard KS-DFT with LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' In order to formulate these  parameter-free moment functionals we have used an ex-  isting model of the second moment of the uniform elec-  tron gas (UEG) [17] as well as models of the momentum  distribution function nk of the UEG [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' However,  formulating universal moment functionals that are gen-  erally applicable is still a long way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Notably, the correct  inclusion of spin-polarization and the extension by gra-  dient corrections are important necessary developments  left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Nevertheless, even while the universal moment func-  tionals are not yet available, one may generally im-  prove spectra by optimizing the parameters in suitable  parameterized moment functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Adjusting in this  way e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' the bandwidths and gaps in order to repro-  duce experimental data will increase the accuracy of re-  sponse function calculations, in particular those of op-  tical responses, such as laser-induced currents [21] and  torques [22], which are expected to be generally sensi-  tive to gaps, bandwidths, and band positions, because  they are not Fermi-surface effects like the AHE [23–25]  for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  We have demonstrated that MFbSDFT can be used  to improve the description of the electronic structure of  fcc Ni significantly in comparison to LDA, because it  yields bandwidth, exchange splitting, and satellite peak  positions in good agreement with the experimental spec-  trum [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  The satellite peak in Ni roughly 6 eV  below the Fermi energy is a correlation effect [26–28],  2  which is missing in the KS-spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Since the method  of spectral moments captures the splitting of bands into  lower and upper Hubbard bands, it cannot be mapped  onto a non-interacting effective KS Hamiltonian in gen-  eral without changing the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' The question there-  fore poses itself of how to obtain generalized Wannier  functions from the spectral moment matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  The method of spectral moments yields state energies,  state wavefunctions, and corresponding spectral weights,  in contrast to KS-DFT, where the spectral weights are  by definition unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Moreover, the state wavefunctions  corresponding to different energies are not guaranteed  to be orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' While many-body generalizations of  Wannier functions have been considered before [29], these  generalizations are not optimized to construct localized  Wannier functions from the first four spectral moment  matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' For example, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' [29] does not take the spec-  tral weights of the quasiparticles into account and does  not consider the possibility that the quasiparticle wave-  functions do not necessarily form a set of mutually or-  thogonal functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  However, the standard method of  constructing MLWFs assumes the Bloch functions of dif-  ferent bands to be orthogonal [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Here, we will show  that spectral weights and non-orthogonality of quasipar-  ticle wavefunctions can be taken into account by gener-  alizing the MLWFs concept for the method of spectral  moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  We demonstrate the construction of MLSMWFs for  fcc Ni and use them to compute the AHE by Wannier  interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' We choose fcc Ni because it is known that  standard KS-DFT overestimates the bandwidth and the  exchange splitting in this material [27] and it predicts  the AHE to be significantly larger than the experimental  value [32–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Moreover, even the sign of the magnetic  anisotropy energy (MAE) is not predicted correctly by  standard KS-DFT, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=', LDA does not predict the cor-  rect easy axis [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Previously, we have demonstrated  that MFbSDFT can be used to reproduce the experi-  mental values of the exchange splitting, the bandwidth,  and the position of the satellite peaks [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Here, we  demonstrate that also the AHE is predicted to be close  to the experimental value if MFbSDFT is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  The rest of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II  we explain how we construct MLSMWFs from the first  four spectral moment matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Practical issues, such as  the use of the wannier90 code [2] for the generation of  MLSMWFs are described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II E and in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  MLSMWFs with spin-orbit interaction (SOI) are dis-  cussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II H we explain that ab-initio  programs which can compute MLWFs can be extended  easily to compute additionally MLSMWFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II I  we describe Wannier interpolation based on MLSMWFs  using the example of the AHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' The interpolation of addi-  tional matrix elements such as spin and torque operators  is discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' III we explain how the  method of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II may be generalized to include the first  2P moments (P = 3, 4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' IV we present appli-  cations of our method to fcc Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' This paper ends with a  summary in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  THEORY  Before discussing the construction of MLSMWFs from  the first four spectral moment matrices in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II D we  first revisit the generation of MLWFs in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II A as well  as the calculation of the spectral function in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  This will help us to explain the necessary modifications  of the MLWFs formalism in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II D, when MLSMWFs  are constructed from the spectral moment matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  MLWFs and Wannier interpolation  The Bloch functions |ψkm⟩ are eigenfunctions of the  KS-Hamiltonian H with eigenenergies Ekm:  H|ψkm⟩ = Ekm|ψkm⟩,  (1)  where k is the k-point and m is the band index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' The ML-  WFs |WRn⟩ are constructed from these Bloch functions  by the transformation [30]  |WRn⟩ = 1  N  �  k  NB  �  m=1  U (k)  mne−ik·R|ψkm⟩,  (2)  where N is the number of k points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' The matrix U(k) is  a rectangular matrix when the number of bands NB is  larger than the number of MLWFs NW, otherwise it is a  square matrix, which may occur for example when ML-  WFs are constructed from isolated groups of bands [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  The matrix U(k) is determined by the condition that the  MLWFs minimize the spatial spread  Ω =  NW  �  n=1  �  ⟨WRn|r2|WRn⟩ − (⟨WRn|r|WRn⟩)2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (3)  Due to the localization of the MLWFs in real-space  the matrix elements of the Hamiltonian H decay rapidly  when the distance between the MLWFs increases:  lim  R→∞⟨W0m|H|WRn⟩ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (4)  This localization property is important for Wannier inter-  polation, because it implies that in order to interpolate  the electronic structure at any desired k point it is suffi-  cient to provide the matrix elements ⟨W0m|H|WRn⟩ for  a finite and small set of R vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' The reason for this is  that the electronic structure at any desired k point may  be computed by performing a Fourier transformation of  ⟨W0m|H|WRn⟩ and that the computational time for this  3  is small if the set of R vectors is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Explicitly, the  matrix elements of the Hamiltonian may be written as  ⟨W0m|H|WRn⟩ =  = 1  N  �  k  NB  �  m′=1  e−ik·REkm′  �  U (k)  m′m  �∗  U (k)  m′n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (5)  In order to describe the AHE in magnetically collinear  ferromagnets, the spin-orbit interaction (SOI) has to be  taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Wannier interpolation is very effi-  cient in computing the AHE [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' In the presence of SOI,  the Bloch functions are spinors  ⟨r|ψkm⟩ = ψkm(r) =  �  χkm↑(r)  χkm↓(r)  �  ,  (6)  and the MLWFs are spinors as well:  ⟨r|WRn⟩ =  �WRn↑(r)  WRn↓(r)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (7)  For the interpolation of the AHE in bcc Fe [4] one first  constructs MLWFs using a coarse k mesh, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' an 8 ×  8 × 8 mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Next, one computes the matrix elements  of the Hamiltonian according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Finally, one  may Fourier transform these matrix elements for all k  points in the fine interpolation mesh, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' an 800 × 800 ×  800 mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Using this interpolated Hamiltonian, one may  compute the AHE numerically efficiently [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Construction of the spectral function from the  first four spectral moment matrices  In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' [15] we describe an algorithm to construct the  spectral function from the first four spectral moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  In the following we assume that the spectral moments  are so expressed in a basis set of NS orthonormalized  functions φkn(r) that they are given by NS×NS matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Due to the orthonormalization of the basis functions, the  zeroth moment M (0)  k  is simply the unit matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' When  the spectral function is determined approximately from  the first four spectral moments (M (0)  k , M (1)  k , M (2)  k , and  M (3)  k ), the poles of the single-particle spectral function  are given by the eigenenergies of the hermitean 2NS×2NS  matrix [15]  Hk =  �  M (1)  k  B1k  B†  1k  D1k  �  ,  (8)  where M (1)  k  is the first moment, B1k = U k  √Dk, B2k =  [M (3)  k  − M (2)  k M (1)  k ][B†  1k]−1, and D1k = B−1  1k [B2k −  M (1)  k B1k].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Here, Dk is a diagonal matrix, and Uk is  a unitary matrix so that UkDkU †  k = M (2)  k  −M (1)  k M (1)  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  The eigenvectors of Hk, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (8), may be written as  Ψkn =  �ψkn→  ψkn←  �  ,  (9)  where ψkn→ and ψkn← are both column vectors with  NS components, while Ψkn is a column vector with 2NS  entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' We denote the eigenvalues of Hk by Ekn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=',  HkΨkn = EknΨkn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (10)  Within MFbSDFT, the charge density is computed only  from the upper part ψkn→ of the state vector Ψkn  (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (9)), while the lower part plays the role of an aux-  iliary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Note that while the eigenfunctions  |ψkm⟩ of the KS-Hamiltonian (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (1)) are orthonormal,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=',  ⟨ψkm|ψk′n⟩ = δnmδkk′,  (11)  the upper parts ψkn→ of the state vectors Ψkn (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (9))  are not even orthogonal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=',  [ψkm→]† ψkn→ ̸∝ δnm,  (12)  while the complete state vectors are orthonormal:  [Ψkm]† Ψkn = δnm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (13)  We may obtain the spectral weight of the state Ψkn  from  akn = [ψkn→]† ψkn→ =  NS  �  m′=1  [ψknm′→]∗ψknm′→.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (14)  Here, ψknm′→ is the m′-th entry in the column vector  ψkn→.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' These spectral weights are useful to quantify the  relative importance of a given state with energy Ekn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' For  example, it may occur that the spectral function has a  pole at Ekn with a spectral weight akn ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Due to the  small spectral weight this pole might not be observable  in the experimental spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Therefore, both the poles  Ekn and the spectral weights akn are generally necessary  to discuss the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  In order to construct MLSMWFs from the state vec-  tors Ψkn, we need their real-space representation Ψkn(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Clearly, ψkn→ is given by  ψkn→(r) =  NS  �  m=1  φkm(r)ψknm→,  (15)  in real-space, where φkm(r) is the m-th function in the  orthonormal set used to express the spectral moments at  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  In MFbSDFT the functions ψkn→(r) replace the KS  wavefunctions from standard KS-DFT [15]: The charge  density and the DOS may be obtained from ψkn→(r),  while the auxiliary vector ψkn← is only needed to solve  4  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (10), and may often be discarded afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' How-  ever, as will become clear in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II D, we need the lower  part ψkn←(r) for the construction of MLSMWFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' The  matrices B1k and B2k describe linear maps from the  space of eigenfunctions of M (2)  k −M (1)  k M (1)  k  to the space  of orthonormal basis functions φkn(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Consequently, the  matrix D1k describes a linear map from the space of  eigenfunctions of M (2)  k  − M (1)  k M (1)  k  to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Therefore,  the components of ψkn← refer to the space of eigenfunc-  tions of M (2)  k  − M (1)  k M (1)  k  and an additional unitary  transformation to the space of orthonormal basis func-  tions φkn(r) is necessary to obtain the real-space repre-  sentation of ψkn←:  ψkn←(r) =  NS  �  m,m′=1  Ukmm′φkm(r)ψknm′←.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (16)  Another approach leading to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (16) considers the  unitary transformation  �k =  �1  0  0 U k  �  ,  (17)  where 1 is the NS×NS unit matrix, while 0 is the NS×NS  zero matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' When this transformation is applied to Hk  it does not change its eigenvalues Ekn nor the upper part  ψkn→ of the eigenvectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Only the lower part ψkn← of  the eigenvectors is changed so that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (10) turns into  ¯Hk  � ψkn→  U kψkn←  �  = Ekn  � ψkn→  U kψkn←  �  ,  (18)  where  ¯Hk = �kHk� †  k  (19)  is the transformed Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  ¯Hk describes a map  Vk×Vk → Vk×Vk, where we denote the space of orthog-  onal basis functions φkn(r) by Vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Since the lower com-  ponents of the eigenvectors of ¯Hk are given by U kψkn←  according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (18), it is clear that the real-space rep-  resentation of ψkn← is given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Choice of the moment functionals  The spectral moment matrices M (I)  k  (I = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=')  may be obtained either by computing several correlation  functions self-consistently [14], or by evaluating suitable  moment functionals [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  In the latter approach,  which we call MFbSDFT, the I-th moment is decom-  posed into the I-th power of the first moment plus the  additional contribution M (I+)  k  [15, 16]:  M (I)  k  =  �  M (1)  k  �I  + M (I+)  k  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (20)  The first moment, M (1)  k , may be obtained easily within  the standard KS framework: It is simply given by the KS  Hamiltonian, if instead of the full exchange-correlation  potential only the first-order exchange is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' The addi-  tional contributions M (I+)  k  may be computed from suit-  able potentials V(I+)(r) [15, 16]:  M (I+)  knm =  �  d3rV(I+)(r)φ∗  kn(r)φkm(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (21)  We expect that the V(I+)(r) depend only on the elec-  tron density n(r), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=', there are universal functionals of  n(r), from which V(I+)(r) may be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' This expec-  tation is corroborated by our finding [16] that parameter-  free expressions for the moment potentials can be found  that improve the spectra of Na and of SrVO3 significantly  in comparison to standard KS-DFT with LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' However,  general and accurate expressions for V(I+)(r) are cur-  rently not yet available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Therefore, we proposed several  parameterizations of V(I+)(r), which can be used to re-  produce spectral features such as satellite peaks and to  correct the band width e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' in Ni [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Defining the dimensionless density parameter  rs(r) = 1  aB  �  3  4πn(r)  � 1  3  ,  (22)  where aB is Bohr’s radius, we may express V(I+)(r)  through [15]  V(I+)(r) =  c(I+)  [rs(r)]I + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (23)  in the low-density limit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=', when rs(r) is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Alter-  natively, one may use [15, 16]  V(I+)(r) = d(I+)[Vc(rs)]I + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' ,  (24)  where  Vc = d(ǫcn)  dn  (25)  is the correlation potential, and ǫc is the correlation en-  ergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  In these parameterized expressions of V(I+) one may  treat e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' d(2+) and d(3+) as independent parameters and  optimize both in order to match the experimental spec-  tra as well as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Alternatively, one may compute  V(3+) for a given V(2+) by enforcing the constraint of the  momentum distribution function of the UEG [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Construction of MLSMWFs from the first four  spectral moment matrices  In order to compute MLSMWFs from the first four  spectral moment matrices, we need to use the states  5  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (9) instead of the usual Bloch functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' An obvious  generalization of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (2) based on these state vectors is  �⟨r|WRn→⟩  ⟨r|WRn←⟩  �  = 1  N  �  k  2NS  �  m=1  U (k)  mne−ik·R  �ψkm→(r)  ψkm←(r)  �  (26)  where ψkm→(r) and ψkm←(r) are given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (15) and  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (16), respectively, and the 2NS × NW matrix U(k) is  so chosen that the spread  Ω =  NW  �  n=1  �  ⟨WRn→|r2|WRn→⟩ − (⟨WRn→|r|WRn→⟩)2�  +  NW  �  n=1  �  ⟨WRn←|r2|WRn←⟩ − (⟨WRn←|r|WRn←⟩)2�  (27)  is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  As a consequence of the spatial localization, the ma-  trix elements ⟨W0m|H|WRn⟩ decay rapidly in real-space  similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (4):  lim  R→∞⟨W0m|H|WRn⟩ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (28)  Explicitly, these matrix elements are given by  HRmn = ⟨W0m|H|WRn⟩ =  = 1  N  �  k  2NS  �  m′=1  e−ik·REkm′  �  U (k)  m′m  �∗  U (k)  m′n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (29)  The derivation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (29) shows clearly that both  ψkn→(r) and ψkn←(r) need to be taken into account in  the construction of MLSMWFs: Only when both compo-  nents, ψkn→ and ψkn←, are considered, Ψkn is an eigen-  vector of Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Moreover, it is clear that both components,  ⟨r|WRn→⟩ and ⟨r|WRn←⟩, have to be localized together  to minimize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (27), because otherwise Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (28) is not  valid and the Fourier transformation below in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (30)  cannot be performed numerically efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  In order to obtain the interpolated band structure, we  first carry out the Fourier transformation  ˜Hk =  �  R  eik·RHR,  (30)  where HR is the matrix with the components HRmn de-  fined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Next, we diagonalize ˜Hk:  [X k]† ˜HkX k = ˜�k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (31)  Here, X k is a unitary matrix and ˜�k is a diagonal matrix  holding the interpolated energies:  ˜Eknm = ˜Eknδnm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (32)  Often, we would like to interpolate not only the band  energies but also the spectral weights akn, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' For  this purpose, we first need to compute the matrix ele-  ments  sknm = (ψ†  kn→, ψ†  kn←)  �1 0  0 0  � �ψkm→  ψkm←  �  = ψ†  kn→ψkm→  (33)  for all k points in the coarse k mesh that are used in the  construction of the MLSMWFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Here, 0 is the NS × NS  zero matrix and 1 is the NS×NS unit matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Next, these  matrix elements need to be expressed in the MLSMWF  basis:  sRnm = 1  N  �  k  2NS  �  n′,m′=1  skn′m′  �  U(k)  n′n  �∗  U(k)  m′me−ik·R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (34)  After carrying out these preparations before the actual  Wannier interpolation step, one may interpolate sknm to  a given k point in the fine interpolation mesh by per-  forming the Fourier transformation  ˜sknm =  �  R  sRnmeik·R  (35)  in the course of the Wannier interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Finally, ˜sknm  needs to be transformed into the eigenbasis in order to  obtain the interpolated spectral weights:  ˜akn =  �  n′m′  ˜skn′m′Xkm′n [Xkn′n]∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (36)  While the applications shown below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' IV use the  MFbSDFT approach of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' [15] in order to obtain the  spectral moments, the theory for the construction of the  MLSMWFs from the spectral moment matrices that we  present here can also be used when the spectral moments  are obtained by computing several correlation functions  self-consistently as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Wavefunction overlaps  The wannier90 code [2] computes the spread Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (3)  from the overlaps between the lattice periodic parts  ukm(r) = e−ik·rψkm(r) of the Bloch functions at the  nearest-neighbor k-points k and k + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Therefore, the  matrix elements  M (k,b)  mn  = ⟨ukm|uk+b,n⟩  (37)  need to be provided to wannier90 in order to determine  the MLWFs through the matrix U (k)  mn in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (2), which  minimizes the spread Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  In order to find the matrix U (k)  mn that defines the  MLSMWFs in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (26) one may use the wannier90  code [2] as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' In this case one needs to provide the  matrix elements  M (k,b)  mn  = ⟨ukm→|uk+b,n→⟩ + ⟨ukm←|uk+b,n←⟩  (38)  to wannier90, which ensures that all contributions to the  spread in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (27) are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  6  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Initial projections  In order to obtain a good starting point for the iterative  minimization of the spreads, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (3) (for MLWFs) and  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (27) (for MLSMWFs), one may define first guesses  |gn⟩ for these Wannier functions [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' In the case of  MLWFs the matrix elements  A(k)  mn = ⟨ψkm|gn⟩  (39)  may be computed and provided to the wannier90 code [2]  for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' In order to provide the first guesses in  the case of MLSMWFs, one may generalize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (39) as  follows:  A(k)  mn = ⟨ψkm→|gn→⟩ + ⟨ψkm←|gn←⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (40)  When one computes MLWFs of bulk transition metals  such as bcc Fe, fcc Ni, fcc Pt, and fcc Pd, one typically  constructs 9 MLWFs per spin in order to obtain Wan-  nier functions that describe the valence bands and the  first few conduction bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' In this case suitable initial  projections are one s, three p, and five d states, which  are 9 states in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Alternatively, one may use 6 sp3d2  hybrid states plus dxy, dyz, and dzx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' From Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II D it  follows that the number of MLSMWFs is typically cho-  sen twice as large as the number of MLWFs would be  chosen in the same material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' If we assume that for half  of the MLSMWFs the →-component is larger than the  ←-component, while for the remaining other half of the  MLSMWFs the ←-component is more dominant than the  →-component, an obvious choice for the initial projec-  tions is to use states that are purely ← or purely →.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Construction of MLSMWFs in systems with  SOI  In magnetically collinear systems without SOI, one  typically constructs MLWFs separately for spin-up and  spin-down, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=', one constructs two sets of MLWFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' In the  presence of SOI this is not possible, because the Hamil-  tonian couples the spin-up and spin-down bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Conse-  quently, only a single set of MLWFs is constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' For  example, in ferromagnetic fcc Ni one computes 9 spin-up  MLWFs and 9 spin-down MLWFs when SOI is not taken  into account, while one constructs 18 spinor-MLWFs (see  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (7)) when SOI is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Analogously,  only a single set of MLSMWFs is  constructed in systems with SOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' In this case every  MLSMWF has four components:  ⟨r|WRn⟩ =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  ⟨r|WRn→↑⟩  ⟨r|WRn→↓⟩  ⟨r|WRn←↑⟩  ⟨r|WRn←↓⟩  \uf8f6  \uf8f7  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (41)  Similarly, the eigenvectors Ψkn in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (10) have four  components:  Ψkn =  �ψkn→  ψkn←  �  =  \uf8eb  \uf8ec  \uf8ec  \uf8ed  ψkn→↑  ψkn→↓  ψkn←↑  ψkn←↓  \uf8f6  \uf8f7  \uf8f7  \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (42)  Consequently, the matrix elements  M (k,b)  mn  =  �  p=→,←  �  σ=↑,←  ⟨ukmpσ|uk+b,npσ⟩  (43)  and  A(k)  mn =  �  p=→,←  �  σ=↑,←  ⟨ψkmpσ|gnpσ⟩  (44)  need to be provided to the wannier90 code in this case  in order to determine the MLSMWFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Implementation within the FLAPW method  In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' [36] we describe in detail how the matrix  elements M (k,b)  mn  and A(k)  mn required by wannier90 for  the calculation of the MLWFs may be implemented  within the full-potential linearized augmented plane-  wave method (FLAPW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' For the construction of the  MLSMWFs we need to compute these matrix elements  according to the prescriptions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (38) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (40)  (when SOI is included in the calculations Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (43) and  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (44) should be used instead).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' It is straightforward  to extend the implementation described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' [36] by  adding the additional loop over the MFbSDFT indices  → and ←.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Wannier interpolation of response functions  In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' [14] we have described how the AHE con-  ductivity may be computed within the method of  spectral moments using correlation functions such as  ⟨[[c†  kαckβ, H]−, c†  kγckδ]−⟩ (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (34), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (C1), and  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (C2) in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' However, we have also reported in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' [14] that in the case of the Hubbard-Rashba model  the AHE is well-approximated by  σxy = e2ℏ  V N  �  k  NW  �  n,n′=1  [fkn − fkn′]×  × Im [⟨ψkn→|vx|ψkn′→⟩⟨ψkn′→|vy|ψkn→⟩]  (Ekn′ − Ekn)2 + 0+  ,  (45)  which does not require us to compute correlation func-  tions such as ⟨[[c†  kαckβ, H]−, c†  kγckδ]−⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Here, we assume  that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (45) can also be used to compute the AHE of  7  realistic materials approximately within the spectral mo-  ment approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' We leave if for future work to test this  approximation by computing the AHE also from the cor-  relation functions ⟨[[c†  kαckβ, H]−, c†  kγckδ]−⟩, and focus on  the evaluation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (45) in order to provide an example  of Wannier interpolation with MLSMWFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  We may obtain ⟨ψkn→|vx|ψkn′→⟩ from Wannier inter-  polation by computing first the matrix elements  hknm =  NS  �  n′,m′=1  M (1)  n′m′ψkmm′→ [ψknn′→]∗  (46)  of the first moment for all k points in the coarse k mesh  that are used in the construction of the MLSMWFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Subsequently, we compute the corresponding matrix ele-  ments in the MLSMWFs basis:  hRnm = 1  N  �  k  2NS  �  n′,m′=1  hkn′m′  �  U(k)  n′n  �∗  U(k)  m′me−ik·R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (47)  These are preparatory steps that are carried out before  the actual Wannier interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' In order to interpolate  hknm to a given k point in the fine interpolation mesh  we first carry out the Fourier-transformation  ˜hknm =  �  R  hRnmeik·R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (48)  The velocity operator matrix elements are obtained from  the k derivative:  ˜vknm = 1  ℏ  �  R  iReik·RhRnm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (49)  Finally, we need to transform these matrix elements into  the eigenbasis, which we obtain from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (31):  ⟨ψkn→|v|ψkm→⟩ =  �  n′m′  ˜vkn′m′Xkm′m [Xkn′n]∗ ,  (50)  where Xkm′m are the elements of the unitary matrix de-  fined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Now, the matrix elements Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (50) may  be used together with the eigenvalues Ekn and the Fermi  factors fkn = f(Ekn) (where f is the Fermi function) to  evaluate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  This interpolation approach suffers from a band trun-  cation error, because only Wannier interpolated states  are used to evaluate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' However, the band trun-  cation error has been shown to be small in the case of  AHE [4] and also in the case of SHE [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Wannier interpolation of the spin and torque  operators  Within MFbSDFT, the matrix elements of the spin  operator are defined by  Sknm = ℏ  2(ψ†  kn→, ψ†  kn←)  �σ 0  0 0  � �ψkm→  ψkm←  �  = ℏ  2ψ†  kn→σψkm→  = ℏ  2(ψ†  kn→↑, ψ†  kn→↓)σ  �ψkm→↑  ψkm→↓  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (51)  In order to compute for example spin photocurrents [21]  from Wannier interpolation within MFbSDFT, these ma-  trix elements need to be interpolated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  We obtain the  interpolated ˜Sknm similarly to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (34) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (35) (re-  place sknm → Sknm, sRnm → SRnm, and ˜sknm → ˜Sknm  in these equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Finally, the interpolated ˜Sknm may  be transformed into the eigenbasis similarly to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (50):  ⟨ψkn→|S|ψkm→⟩ =  �  n′m′  ˜Skn′m′Xkm′m [Xkn′n]∗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (52)  The torque operator T is needed for the calculation of  the SOT [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' At first glance, it is tempting to define  the torque operator by  T knm = −µB  �  d3r[ψkn→(r)]†σψkm→(r) × Ωxc(r)  (53)  within MFbSDFT, where µB is the Bohr magneton, and  Ωxc(r) is the exchange field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' However, the moment po-  tentials V(2+)  σ  (r) and V(3+)  σ  (r) (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (21)) may be spin-  polarized in general, similarly to the exchange potential  in the first moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' There is no convincing argument  that one may substitute the exchange potential of the  first moment for Ωxc(r) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Instead, we ex-  pect that a suitable expression for Ωxc(r) may be de-  rived within the MFbSDFT framework, and that it will  depend on the potentials of the first, second, and third  moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Therefore, we consider the alternative expression for  the torque operator  T knm = − i  2  �  d3r[ψkn→(r)]†[HSOI(r), σ]ψkm→(r),  (54)  where HSOI(r) is the SOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' The torque operator may be  interpolated analogously to the interpolation of the spin  operator discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  The torque operator may also be used to compute the  MAE [6, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Within MFbSDFT, the torque due to the  magnetic anisotropy is given by  T mae = − 1  N  �  kn  fkn⟨ψkn→|T |ψkn→⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  (55)  8  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  EXTENSION TO MORE MOMENTS  In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' [16] we present an efficient algorithm to con-  struct the spectral function from the first 2P spectral  moment matrices, where P = 1, 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='. The algorithm  described in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' [15], which we revisit briefly in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II B,  is the special case with P = 2 of this more general algo-  rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' We expect that the accuracy of the MFbSDFT  approach can be enhanced by increasing P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' For example,  in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' [16] we explain that it easy to reproduce the jump  of the momentum distribution function nk of the UEG  at kF when P ≥ 3, while this is difficult to achieve with  P = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II D we describe the generation of MLSMWFs  when P = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' The extension to P > 2 is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  As an example, consider the case P = 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=', assume  that we construct the spectral function from the first 6  spectral moment matrices using the algorithm described  in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' In this case the poles of the spectral function  are the eigenvalues of a 3NS × 3NS matrix Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  The  eigenvectors of Hk have 3NS components in this case  and they may be written in the form  Ψkn =  \uf8eb  \uf8ed  ψkn→  ψknտ  ψknւ  \uf8f6  \uf8f8 ,  (56)  where ψkn→, ψknտ, and ψknւ are NS-component vec-  tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' ψkn→ is the physical component from which the  charge density, the DOS, the spectral weights, and the  expectation values of operators can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' ψknտ,  and ψknւ are auxiliary components, which may be dis-  carded in a standard MFbSDFT selfconsistency loop af-  ter diagonalizing Hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' However, like in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II D, these  auxiliary components need to be included into the gen-  eration of the MLSMWFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Therefore, we construct the  MLSMWFs from  \uf8eb  \uf8ed  ⟨r|WRn→⟩  ⟨r|WRnտ⟩  ⟨r|WRnւ⟩  \uf8f6  \uf8f8 = 1  N  �  k  3NS  �  m=1  U (k)  mne−ik·R  \uf8eb  \uf8ed  ψkm→(r)  ψkmտ(r)  ψkmւ(r)  \uf8f6  \uf8f8  (57)  in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Here, U(k) is a 3NS × NW matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  APPLICATIONS  When the magnetization is along the [001] direction,  GGA predicts the intrinsic AHE in Ni to be -2200 S/cm,  which is significantly larger than the experimental value  of -646 S/cm [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  Using GGA+U with U = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='9 eV,  one obtains the intrinsic AHE of -1066 S/cm [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' The  remaining discrepancy between experiment and theory  is 420 S/cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' This discrepancy can be explained by the  side-jump AHE [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  MFbSDFT may be used to reproduce the experimental  values of the exchange splitting, of the band width, and  15  16  17  18  19  20  d  (2+)  3000  2500  2000  1500  1000  500  0  AHE σxy [S/cm]  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' AHE conductivity σxy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' the prefactor d(2+) of the  second moment potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  of the valence band satellite position in fcc Ni [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  To compute the AHE in Ni from MLSMWFs, we first  perform self-consistent MFbSDFT calculations with SOI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  We perform these calculations with various different d(2+)  parameters in the range 15-20 to investigate the depen-  dence of the AHE on d(2+), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (24), but  we set d(3+) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  In order to keep the magnetic mo-  ment fixed at around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='6 µB, which is the value measured  in experiments, we need to spin-polarize V(2+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' We use  V(2+)  σ  = ζt  σV(2+), where ζσ = (1 − σ(n↑ − n↓)/n), and t  is determined at every value of d(2+) to match the exper-  imental magnetic moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Next, we compute the ma-  trix elements M (k,b)  mn  and A(k)  mn as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II E,  Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II F, and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' We generate MLSMWFs using  the wannier90 code [2] and disentanglement, where we  set the lower bound of the frozen window at around 80 eV  below the Fermi energy and the upper bound at around  4 eV above the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' We construct 36 spinor  MLSMWFs from 72 MFbSDFT bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' 1 we plot the AHE obtained from MLSMWFs  as explained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' II I as a function of the prefactor  d(2+) used in the potential of the second moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' With  increasing d(2+) the magnitude of σxy decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  At  d(2+) = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='0 the intrinsic AHE is -1000 S/cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  If we  assume that the side-jump contribution to the AHE is  around 400 S/cm [32], this is in good agreement with the  experimental value of -646 S/cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' [15] we used d(2+) = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='0 in order to reproduce  the experimental bandwidth, exchange splitting, and po-  sition of the satellite peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' However, using d(2+) = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='0  instead reproduces these experimental features also quite  well, which we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' [16] we have found  that the valence band satellite is in much better agree-  ment with DMFT calculations and with experiment if  the third moment potential is computed from the second  9  8  6  4  2  0  2  E-EF [eV]  0  1  2  3  4  5  DOS [States/(u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' eV)]  Majority  Minority  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' DOS in fcc Ni obtained from MFbSDFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  moment potential using the constraint of the momen-  tum distribution function of the UEG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' However, since  we have currently developed this procedure only for the  UEG without spin-polarization we needed to apply a sim-  ilar spin-polarization factor ζt  σ like in the present calcu-  lations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' As a result, the spectral density of Ni in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' [16]  matches experiment concerning the spin-polarization of  the satellite peak, and the band width of the main band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  However, it suffers from a similar overestimation of the  exchange splitting as standard KS-DFT with LDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' In  contrast, the present calculation yields the exchange-  splitting close to experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Since the AHE depends  strongly on the Fermi surface [23–25] we therefore use  here the simpler approach of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' (24) instead of the im-  proved approach of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  SUMMARY  We describe the construction of Wannier functions  from the first 4 spectral moment matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  We show  that these MLSMWFs can be used for the efficient in-  terpolation of material property tensors such as the AHE  within MFbSDFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' This paves the way for the application  of MFbSDFT to compute response properties of materi-  als.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' We demonstrate that MFbSDFT is able to reproduce  the experimentally measured AHE in fcc Ni, similarly to  LDA+U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' Finally, we discuss that MLSMWFs may be  computed also from the first 6 moments, and generally  from the first 2P moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' This opens the perspective  of using as many moments as necessary to reproduce all  spectral features accurately in MFbSDFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  ACKNOWLEDGMENTS  The project is funded by the Deutsche Forschungs-  gemeinschaft (DFG, German Research Foundation) −  TRR 288 − 422213477 (project B06), CRC 1238, Control  and Dynamics of Quantum Materials: Spin orbit cou-  pling, correlations, and topology (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' C01), SPP  2137 “Skyrmionics”, and Sino-German research project  DISTOMAT (DFG project MO 1731/10-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  We also  acknowledge financial support from the European Re-  search Council (ERC) under the European Union’s Hori-  zon 2020 research and innovation program (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  856538, project “3D MAGiC”) and computing resources  granted by the J¨ulich Supercomputing Centre under  project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' jiff40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='  ∗ Corresp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content=' author: f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
+page_content='freimuth@fz-juelich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
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+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
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+page_content=' Souza, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/UtE3T4oBgHgl3EQf0Au1/content/2301.04734v1.pdf'}
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diff --git a/V9FPT4oBgHgl3EQfqjWj/content/tmp_files/2301.13142v1.pdf.txt b/V9FPT4oBgHgl3EQfqjWj/content/tmp_files/2301.13142v1.pdf.txt
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@@ -0,0 +1,496 @@
+Self-Compressing Neural Networks
+Szabolcs Cséfalvay 
+szabolcs.csefalvay@imgtec.com 
+James Imber 
+james.imber@imgtec.com
+ 
+ 
+Abstract 
+This work focuses on reducing neural network size, which is 
+a major driver of neural network execution time, power con-
+sumption, bandwidth, and memory footprint. A key challenge 
+is to reduce size in a manner that can be exploited readily for 
+efficient training and inference without the need for special-
+ized hardware. We propose Self-Compression: a simple, gen-
+eral method that simultaneously achieves two goals: (1) re-
+moving redundant weights, and (2) reducing the number of 
+bits required to represent the remaining weights. This is 
+achieved using a generalized loss function to minimize over-
+all network size. In our experiments we demonstrate floating 
+point accuracy with as few as 3% of the bits and 18% of the 
+weights remaining in the network. 
+ Introduction 
+The ongoing revolution in the capabilities of machine learn-
+ing models can in large part be attributed to their increasing 
+size. For example, the exceptional capabilities of recent 
+state-of-the-art language models (Brown et al. 2020) have 
+only been achieved at the expense of immense network size, 
+slow training and execution, and high energy/carbon con-
+sumption (Lacoste et al. 2019). However, performance opti-
+mization, particularly for power- and area-efficient infer-
+ence on dedicated accelerators, has been relatively ne-
+glected, which limits the deployment of powerful models on 
+resource-limited devices (Demirci and Ferhatosmanoglu, 
+2021). 
+In this work our objective is threefold: (1) to compress 
+networks during training to realize benefits in training time; 
+(2) to reduce the size of weight and activation tensors by 
+eliminating redundant channels; and (3) to reduce the num-
+ber of bits required to represent weights. The second and 
+third points produce a smaller network expected to execute 
+more efficiently on devices supporting variable bit depth 
+weight formats (Lee et al. 2019). Despite being conceptually 
+simple, the approach we take is effective and we demon-
+strate high compression rates on an example classification 
+network. We achieve the following advantages: 
+• Fewer weights in the final network. 
+• Fewer bits in the remaining parameters (depending on the 
+target device). 
+• Reduced training and execution time. 
+• Freeing the network designer from manually optimizing 
+architectural hyperparameters such as layer widths and bit 
+depths. 
+• No requirement for special hardware to take advantage of 
+most optimizations (e.g., no need for sparse matrix multi-
+plication (Le Cun et al. 1989) or support for hash func-
+tions (Han et al. 2016)). 
+We achieve this by means of a novel quantization-aware 
+training (QAT) scheme in which the quantization nodes are 
+differentiable in their exponents and number of bits. This al-
+lows bit depths to be reduced simultaneously with maxim-
+izing accuracy on the task being trained for. Redundant 
+channels are automatically detected when they reach zero 
+bits and periodically eliminated, leading to a speedup in both 
+training and inference due to reduced bandwidth and com-
+pute requirements. 
+Related Work 
+Our proposed solution bridges multiple active research ar-
+eas: low bit depth neural networks, QAT, and induced spar-
+sity (particularly channel pruning). 
+ Early contributions in the field of low bit depth neural net-
+works showed that it is possible to achieve reasonable accu-
+racy at very low bit depths with specialized operators 
+(Rastegari et al. 2016, Li et al. 2016). Where specialized op-
+erators are needed, specialized inference hardware may also 
+be required (Wang et al. 2019). The present work is de-
+signed to yield networks that may be deployed efficiently on  
+low-bit-depth integer pipelines, as are available in many 
+GPUs and neural network accelerators. 
+ There exist many methods for performing QAT for net-
+work parameters. One important advance is the Straight-
+Through Estimator (STE) for rounding (Bengio et al. 2013), 
+which allows gradient updates to be propagated to weights 
+through a rounding operation during training. Other meth-
+ods smooth the rounding function, using stochastic rounding 
+(Défossez et al. 2022) or explicit smoothing (Gong et al. 
+2019). Importantly, Défossez et al. (2022) take QAT a step 
+further by also learning bit depths. 
+ The literature on induced network sparsity started with Le 
+Cun et al. (1989). Recent related developments include 
+methods for efficient inference of sparse networks (Demirci 
+
+and Ferhatosmanoglu 2021), and induced structured sparsity 
+such as channel pruning (He et al. 2017). 
+In our experiments we compare with the related method 
+of Défossez et al. (2022), as described in more detail in the 
+Experiments section below. The following differences with 
+our method should be noted: 
+1. We allow bit depths to reduce to zero, eliminating 
+some weights, instead of limiting minimum com-
+pression to 1 bit. 
+2. We define the quantization function in such a way 
+that it is fully differentiable with respect to all pa-
+rameters, including the number format parameters 
+(scale/exponent and bit depth (Jacob et al. 2017)). 
+Importantly, this turns all number format parame-
+ters into network parameters that can be trained di-
+rectly as if they were weights. 
+3. We use the basic STE for all training instead of us-
+ing pseudo-quantization noise. 
+4. We use a coarser grouping of weights: instead of 
+using groups of 4, 8 or 16 weights, we group all 
+weights in a channel, achieving greater stability 
+and less forgetting during training. This also allows 
+for a significant reduction in compute requirements 
+without requiring specialized hardware by a com-
+plete elimination of channels. 
+Self-Compression and Differentiable Quanti-
+zation 
+In this paper, our experiments use a differentiable number 
+format (eq. 1) that is shared by a group of weights, repre-
+sented as signed integers with floating point exponents 𝑒 and 
+bit depths 𝑏 (however, this is fully expected to generalize to 
+other formats such as Q8A). Our quantization function is as 
+follows: 
+𝑞(𝑥, 𝑏, 𝑒) = 2𝑒⌊min(max(2−𝑒𝑥, −2𝑏−1) , 2𝑏−1 − 1)⌉ (1) 
+Where ⌊⋅⌉ is the rounding function which rounds to nearest 
+integer with ties to nearest even. Since this formula is only 
+valid for non-negative values of 𝑏, we constrain the range of 
+𝑏 to be greater than or equal to zero. Use of the STE to re-
+define the derivative of the rounding function makes it pos-
+sible to optimize an objective function with respect to the 
+quantization parameters b and e. 
+The choice of rounding mode is important: when 𝑏 = 0 
+the output of the 𝑞 function is always zero. Therefore, when 
+a weight is represented with zero bits, it makes no contribu-
+tion to the output of the network, and may be removed with-
+out changing the result. By sharing the quantization param-
+eters across entire channels, it becomes possible to remove 
+(prune) zero bit channels without impacting the network’s 
+output. This has the effect both of reducing the size of 
+weight and activation tensors in the network (Figure 1), but 
+also accelerating training over time (Figure 2) without af-
+fecting the accuracy of the final network. 
+Reducing a network’s size by removing channels has the 
+advantage of not requiring specialized hardware to handle 
+the reduced network. Our proposed method therefore pro-
+ceeds as follows: 
+1. Quantizing each output channel of the weights with 
+a single quantization parameter pair of bit depth 
+and exponent (𝑏,𝑒). 
+2. Training the network using a loss function that 
+maximizes accuracy on the original task whilst pe-
+nalizing the number of bits used. 
+3. Removing network parameters (i.e. weight output 
+channels) when the corresponding bit depths reach 
+zero. This is also propagated to subsequent ops that 
+consumed the removed output channel, resulting in 
+a reduction in the size of following layers, and the 
+removal of the corresponding input channel of a 
+following convolution, where present. 
+Although the method described in this work learns to 
+compress and eliminate channels, it is expected to generalize 
+to other hardware-exploitable learned sparsity patterns. 
+Figure 1: Using the proposed method, network size (num-
+ber of bits) shrinks quickly early in training, with further 
+reductions becoming progressively more gradual. 
+Figure 2: Training time accelerates as parameters are re-
+moved from the network. 
+
+Training Time by Epoch
+Seconds
+EpochTime
+16
+Best Fit
+14
+Time in
+12
+Training
+10
+8
+0
+200
+400
+600
+800
+EpochWhen removing parts of a network during training the op-
+timiser state must also be modified by removing the corre-
+sponding meta-parameters (e.g. momentum vectors) of the 
+removed parameters. 
+Optimization Objective 
+In this work, it is shown that an optimization objective may 
+be defined that improves one or more aspects of neural net-
+work performance in addition to the usual objective of re-
+ducing error on the training dataset. These aspects could in-
+clude the network’s size, total bandwidth consumed, num-
+ber of hardware operations, power consumption, energy per 
+inference, performance on a specific target hardware, etc. 
+All of the above can be minimised by using bit depths as a 
+proxy. In this work we therefore chose to minimize the num-
+ber of bits, which additionally makes direct use of our pro-
+posed differentiable number format (1) for learning quanti-
+zation parameters. We do this by including a new term γ𝑄 
+in the optimisation objective: 
+𝛬(𝑥) = 𝛬0(𝑥) + γ𝑄
+(2) 
+Where 𝛬0 is the original loss of the network, γ is the com-
+pression factor (a larger 𝛾 produces a smaller, less accurate 
+network), and 𝑄 is the average bit depth. 𝑄 is defined as the 
+sum of the sizes 𝑧𝑙 of all layers 𝑙, divided by the total number 
+of weights 𝑁 in the starting network: 
+𝑄 = 1
+𝑁 ∑
+𝑧𝑙
+𝐿
+𝑙=1
+(3) 
+The size of a layer can be expressed as the total number 
+of bits used to represent its output channels: 
+𝑧𝑙 = 𝐼𝑙𝐻𝑙𝑊𝑙 ∑
+𝑏𝑙
+𝑖
+𝑂𝑙
+𝑖=1
+(4) 
+Where 𝑂𝑙, 𝐼𝑙, 𝐻𝑙 and 𝑊𝑙 are the output, input, height, and 
+width dimensions of the weight tensor of layer 𝑙 respec-
+tively, and 𝑏𝑙
+𝑖 is the bit depth of output channel 𝑖 of layer 𝑙. 
+When this metric is minimized, some 𝑏𝑖
+𝑙 can reach zero. 
+When this happens the corresponding output channel can of-
+ten be removed from the network without losing accuracy. 
+In addition, if the output of layer 𝑙′ is directly used by a 
+layer 𝑙, the corresponding input channel of the next layer 𝑙 
+also becomes redundant. Therefore, the compression loss 
+may be improved by including this relationship: 
+𝑧𝑙 = 𝐻𝑤𝑊𝑤 ∑
+𝟏𝑏𝑙′
+𝑗 >0
+𝐼
+𝑗=1
+∑
+𝑏𝑙
+𝑖
+𝑂
+𝑖=1
++𝐻𝑤𝑊𝑤 ∑
+𝟏𝑏𝑙
+𝑖>0 
+𝑂
+𝑖=1
+∑
+𝑏𝑙′
+𝑗
+𝐼
+𝑗=1
+ 
+(5)
+ 
+Where 𝑏𝑙′ is the vector of bit depths used to encode the pre-
+vious convolution layer’s output (where present). 
+Once a channel can be compressed to zero bits it becomes 
+a candidate for removal. However, removing a channel only 
+outputting zeros could significantly change the network’s 
+output if a bias was to be added to that channel. A sudden 
+change to the network’s output can irreversibly disrupt the 
+training, so to handle this, an 𝐿1 loss is applied to biases op-
+erating on zero-bit channels to reduce them to zero. Only 
+when the biases are reduced to zero are these output chan-
+nels (and corresponding input channels from the next layer) 
+removed, since at this point removing such a channel does 
+not change the network’s output. 
+A sudden change of quantisation parameters can also ir-
+reversibly degrade the network during training, which is a 
+problem described in the next section. 
+Irreversible Forgetting 
+Compressing networks in this way can be challenging. We 
+conjecture that the network is continuously trying to remove 
+(forget) channels (or more generally groups of weights 
+quantised by a common bit depth parameter) that are not 
+necessary to produce a low error at that moment in training. 
+However, this process could erroneously remove parts of a 
+network that are useful, albeit not heavily used during pro-
+cessing of recent minibatches. For example, one might con-
+sider a network channel in the first layer trained to match 
+horizontal lines. If multiple subsequent training batches con-
+tain no horizontal lines affecting the output, the training 
+might determine that horizontal lines are not necessary and 
+reduce the channel’s quantization bit depth too much, and 
+possibly to a point whereat the training can no longer re-
+learn the feature if needed by recovering the corresponding 
+bit depth. We will call this irreversible forgetting. 
+This phenomenon is more likely to occur deeper in the 
+network in wider layers where more abstract (and less often-
+needed) features are located. We have identified ways to 
+mitigate irreversible forgetting, including: 
+1. Having more weights share the same quantization 
+parameters. Even if some of the weights in a group 
+seem unnecessary, their encoding bit depth will 
+stay high if other weights in the group are being 
+used. 
+2. Use the Adam optimizer that adapts the learning 
+rate when a gradient is noisy, with relatively high 
+epsilon parameter to reduce the “acceleration” of 
+bit depth parameters during the early phase of 
+training. 
+Another factor that might affect the compression rate of the 
+network is the error function’s smoothness, but exploring 
+this aspect is left for future work. 
+ 
+
+ 
+ 
+Figure 3: Top-1 accuracy on CIFAR-10 for different 
+choices of compression factor γ. Also shown are results 
+from the method of Défossez et al. (2022). Bit depths are 
+determined using eq. (3) and (4) after channels of zeroes 
+have been removed. 
+Experiments 
+To demonstrate the proposed method, a fast-training classi-
+fication network was chosen (Page 2019). This is important 
+for being able to iterate algorithm development quickly, and 
+to explore the tradeoff space between training time (Figure 
+2), network size (Figure 3), and accuracy in reasonable time. 
+Experiments were conducted on the CIFAR-10 dataset 
+using the following data augmentation methods, applied in 
+the order: (1) 4 pixel padding; (2) PyTorch AutoAugment 
+policy for CIFAR-10; (3) random horizontal flip; (4) 32x32 
+random crop; (5) random erasing; and (6) normalization. 
+The optimizer used was Adam. For training the quantiza-
+tion parameters and weights we use a learning rate of 0.5 
+and 10−3 respectively, and an ϵ parameter of 10−3 and 10−5 
+respectively. A 𝐿2 decay of 5 × 10−4 was applied only to 
+the weights. Training was run for 850 iterations, then the 
+network was allowed to “anneal” to a final state by using 
+PyTorch's ReduceLROnPlateau scheduler until conver-
+gence. 
+ The same training method was used when implementing 
+the method of Défossez et al. (2022) for fair comparison 
+with our method.  
+Results 
+A major advantage of Self-Compression is a parameterized 
+trade-off between size and accuracy, in our case governed 
+by the parameter 𝛾 (Equation 2). The network was trained 
+with 𝛾 log-uniformly sampled from the interval [10-3, 10-0.5]. 
+As can be seen in Figure 3, this forms a locus in a plot of 
+accuracy against final network size, wherein high values of 
+𝛾 correspond to higher compression/lower accuracy. Base-
+line 32-bit float accuracy on this network is 95.69 ± 0.22, 
+which we can match down to as few as 3% of the network 
+weight bits (18% of weights) remaining. 
+Also shown in Figure 3 are results for the method of Dé-
+fossez et al. (2022) which also learns bit depths simultane-
+ously with optimization of accuracy. Their method typically 
+achieves floating point accuracy when the final size is above 
+~8% of the original number of bits. However, our proposed 
+method maintains high accuracy at lower numbers of bits. 
+We also note that the locus of their method is considerably 
+noisier, which may be due to their use of a smaller weight 
+granularity and stochastic rounding. One key difference be-
+tween our proposed method and that of Défossez et al. is the 
+form of the quantization function (STE vs. stochastic round-
+ing). For this reason, we also include results in Figure 3 for 
+their method using the STE instead of stochastic rounding, 
+which results in a modest improvement in accuracy. 
+Figure 4 shows the number of channels before and after 
+Self-Compression is applied with 𝛾 = 0.015. The boxes 
+represent convolution blocks, comprising convolutions with 
+optional batch norm and bias. The numbers on the arrows 
+indicate number of activation channels, and the numbers on 
+the convolution blocks represent the number of output chan-
+nels. Where a summation has been performed, the number 
+of input channels is instead noted. 
+Conclusion 
+We have introduced Self-Compression: an efficient, con-
+ceptually simple means of learning the bit depths used to 
+represent a network’s parameters simultaneously with learn-
+ing its weights, so that during training the network size is 
+reduced simultaneously with maximizing accuracy on its 
+task. Results on the CIFAR-10 classification task indicate 
+that accuracy close to 32-bit floating point can be achieved 
+with as few as 1-3% of the original bits remaining. Im-
+portantly, performance improvements are realisable on typ-
+ical hardware for accelerating neural networks including 
+Figure 4: An overview of the number of weight channels in 
+the example classification network before (top) and after 
+(bottom) applying Self-Compression. 
+
+NetworkSizevs.Accuracy(CIFAR-1o)
+96
+X-
+95
+x x
+94
+Accuracy
++/
++
++
++
++
+93
+Top-1
+1+
+92
++
+Our Method
+Best Fit
+Défossez et.al.with STE
+91
+Best Fit
+Défossez et.al.
+Best Fit
+90
+0%
+2%
+4%
+6%
+8%
+10%
+12%
+Percentage of Original Network Bits Kept (vs.32bit float)CPUs, GPUs, and neural network accelerators, without the 
+need for specialized hardware or execution algorithms. 
+Acknowledgments 
+Our special thanks go to Timothy Gale and Gunduz Vehbi 
+Demirci. We would also like to thank our other colleagues 
+at Imagination Technologies who supported this work. 
+References  
+Bengio, Y.; Léonard N.; and Courville A. 2013. Estimating or 
+Propagating Gradients through Stochastic Neurons for Conditional 
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+
diff --git a/V9FPT4oBgHgl3EQfqjWj/content/tmp_files/load_file.txt b/V9FPT4oBgHgl3EQfqjWj/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf,len=406
+page_content='Self-Compressing Neural Networks  Szabolcs Cséfalvay  szabolcs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='csefalvay@imgtec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='com  James Imber  james.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='imber@imgtec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='com  Abstract  This work focuses on reducing neural network size, which is  a major driver of neural network execution time, power con-  sumption, bandwidth, and memory footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' A key challenge  is to reduce size in a manner that can be exploited readily for  efficient training and inference without the need for special-  ized hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' We propose Self-Compression: a simple, gen-  eral method that simultaneously achieves two goals: (1) re-  moving redundant weights, and (2) reducing the number of  bits required to represent the remaining weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' This is  achieved using a generalized loss function to minimize over-  all network size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' In our experiments we demonstrate floating  point accuracy with as few as 3% of the bits and 18% of the  weights remaining in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Introduction  The ongoing revolution in the capabilities of machine learn-  ing models can in large part be attributed to their increasing  size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' For example, the exceptional capabilities of recent  state-of-the-art language models (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' 2020) have  only been achieved at the expense of immense network size,  slow training and execution, and high energy/carbon con-  sumption (Lacoste et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' However, performance opti-  mization, particularly for power- and area-efficient infer-  ence on dedicated accelerators, has been relatively ne-  glected, which limits the deployment of powerful models on  resource-limited devices (Demirci and Ferhatosmanoglu,  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  In this work our objective is threefold: (1) to compress  networks during training to realize benefits in training time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  (2) to reduce the size of weight and activation tensors by  eliminating redundant channels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' and (3) to reduce the num-  ber of bits required to represent weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' The second and  third points produce a smaller network expected to execute  more efficiently on devices supporting variable bit depth  weight formats (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Despite being conceptually  simple, the approach we take is effective and we demon-  strate high compression rates on an example classification  network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' We achieve the following advantages:  Fewer weights in the final network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Fewer bits in the remaining parameters (depending on the  target device).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Reduced training and execution time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Freeing the network designer from manually optimizing  architectural hyperparameters such as layer widths and bit  depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  No requirement for special hardware to take advantage of  most optimizations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=', no need for sparse matrix multi-  plication (Le Cun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' 1989) or support for hash func-  tions (Han et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' 2016)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  We achieve this by means of a novel quantization-aware  training (QAT) scheme in which the quantization nodes are  differentiable in their exponents and number of bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' This al-  lows bit depths to be reduced simultaneously with maxim-  izing accuracy on the task being trained for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Redundant  channels are automatically detected when they reach zero  bits and periodically eliminated, leading to a speedup in both  training and inference due to reduced bandwidth and com-  pute requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Related Work  Our proposed solution bridges multiple active research ar-  eas: low bit depth neural networks, QAT, and induced spar-  sity (particularly channel pruning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Early contributions in the field of low bit depth neural net-  works showed that it is possible to achieve reasonable accu-  racy at very low bit depths with specialized operators  (Rastegari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' 2016, Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Where specialized op-  erators are needed, specialized inference hardware may also  be required (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' The present work is de-  signed to yield networks that may be deployed efficiently on  low-bit-depth integer pipelines, as are available in many  GPUs and neural network accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  There exist many methods for performing QAT for net-  work parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' One important advance is the Straight-  Through Estimator (STE) for rounding (Bengio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' 2013),  which allows gradient updates to be propagated to weights  through a rounding operation during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Other meth-  ods smooth the rounding function, using stochastic rounding  (Défossez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' 2022) or explicit smoothing (Gong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Importantly, Défossez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' (2022) take QAT a step  further by also learning bit depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  The literature on induced network sparsity started with Le  Cun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Recent related developments include  methods for efficient inference of sparse networks (Demirci  and Ferhatosmanoglu 2021), and induced structured sparsity  such as channel pruning (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  In our experiments we compare with the related method  of Défossez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' (2022), as described in more detail in the  Experiments section below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' The following differences with  our method should be noted:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' We allow bit depths to reduce to zero, eliminating  some weights, instead of limiting minimum com-  pression to 1 bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' We define the quantization function in such a way  that it is fully differentiable with respect to all pa-  rameters, including the number format parameters  (scale/exponent and bit depth (Jacob et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' 2017)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Importantly, this turns all number format parame-  ters into network parameters that can be trained di-  rectly as if they were weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' We use the basic STE for all training instead of us-  ing pseudo-quantization noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' We use a coarser grouping of weights: instead of  using groups of 4, 8 or 16 weights, we group all  weights in a channel, achieving greater stability  and less forgetting during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' This also allows  for a significant reduction in compute requirements  without requiring specialized hardware by a com-  plete elimination of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Self-Compression and Differentiable Quanti-  zation  In this paper, our experiments use a differentiable number  format (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' 1) that is shared by a group of weights, repre-  sented as signed integers with floating point exponents 𝑒 and  bit depths 𝑏 (however, this is fully expected to generalize to  other formats such as Q8A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Our quantization function is as  follows:  𝑞(𝑥, 𝑏, 𝑒) = 2𝑒⌊min(max(2−𝑒𝑥, −2𝑏−1) , 2𝑏−1 − 1)⌉ (1)  Where ⌊⋅⌉ is the rounding function which rounds to nearest  integer with ties to nearest even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Since this formula is only  valid for non-negative values of 𝑏, we constrain the range of  𝑏 to be greater than or equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Use of the STE to re-  define the derivative of the rounding function makes it pos-  sible to optimize an objective function with respect to the  quantization parameters b and e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  The choice of rounding mode is important: when 𝑏 = 0  the output of the 𝑞 function is always zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Therefore, when  a weight is represented with zero bits, it makes no contribu-  tion to the output of the network, and may be removed with-  out changing the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' By sharing the quantization param-  eters across entire channels, it becomes possible to remove  (prune) zero bit channels without impacting the network’s  output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' This has the effect both of reducing the size of  weight and activation tensors in the network (Figure 1), but  also accelerating training over time (Figure 2) without af-  fecting the accuracy of the final network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Reducing a network’s size by removing channels has the  advantage of not requiring specialized hardware to handle  the reduced network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Our proposed method therefore pro-  ceeds as follows:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Quantizing each output channel of the weights with  a single quantization parameter pair of bit depth  and exponent (𝑏,𝑒).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Training the network using a loss function that  maximizes accuracy on the original task whilst pe-  nalizing the number of bits used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Removing network parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' weight output  channels) when the corresponding bit depths reach  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' This is also propagated to subsequent ops that  consumed the removed output channel, resulting in  a reduction in the size of following layers, and the  removal of the corresponding input channel of a  following convolution, where present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Although the method described in this work learns to  compress and eliminate channels, it is expected to generalize  to other hardware-exploitable learned sparsity patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Figure 1: Using the proposed method, network size (num-  ber of bits) shrinks quickly early in training, with further  reductions becoming progressively more gradual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Figure 2: Training time accelerates as parameters are re-  moved from the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Training Time by Epoch  Seconds  EpochTime  16  Best Fit  14  Time in  12  Training  10  8  0  200  400  600  800  EpochWhen removing parts of a network during training the op-  timiser state must also be modified by removing the corre-  sponding meta-parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' momentum vectors) of the  removed parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Optimization Objective  In this work, it is shown that an optimization objective may  be defined that improves one or more aspects of neural net-  work performance in addition to the usual objective of re-  ducing error on the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' These aspects could in-  clude the network’s size, total bandwidth consumed, num-  ber of hardware operations, power consumption, energy per  inference, performance on a specific target hardware, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  All of the above can be minimised by using bit depths as a  proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' In this work we therefore chose to minimize the num-  ber of bits, which additionally makes direct use of our pro-  posed differentiable number format (1) for learning quanti-  zation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' We do this by including a new term γ𝑄  in the optimisation objective:  𝛬(𝑥) = 𝛬0(𝑥) + γ𝑄  (2)  Where 𝛬0 is the original loss of the network, γ is the com-  pression factor (a larger 𝛾 produces a smaller, less accurate  network), and 𝑄 is the average bit depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' 𝑄 is defined as the  sum of the sizes 𝑧𝑙 of all layers 𝑙, divided by the total number  of weights 𝑁 in the starting network:  𝑄 = 1  𝑁 ∑  𝑧𝑙  𝐿  𝑙=1  (3)  The size of a layer can be expressed as the total number  of bits used to represent its output channels:  𝑧𝑙 = 𝐼𝑙𝐻𝑙𝑊𝑙 ∑  𝑏𝑙  𝑖  𝑂𝑙  𝑖=1  (4)  Where 𝑂𝑙, 𝐼𝑙, 𝐻𝑙 and 𝑊𝑙 are the output, input, height, and  width dimensions of the weight tensor of layer 𝑙 respec-  tively, and 𝑏𝑙  𝑖 is the bit depth of output channel 𝑖 of layer 𝑙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  When this metric is minimized, some 𝑏𝑖  𝑙 can reach zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  When this happens the corresponding output channel can of-  ten be removed from the network without losing accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  In addition, if the output of layer 𝑙′ is directly used by a  layer 𝑙, the corresponding input channel of the next layer 𝑙  also becomes redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Therefore, the compression loss  may be improved by including this relationship:  𝑧𝑙 = 𝐻𝑤𝑊𝑤 ∑  𝟏𝑏𝑙′  𝑗 >0  𝐼  𝑗=1  ∑  𝑏𝑙  𝑖  𝑂  𝑖=1  +𝐻𝑤𝑊𝑤 ∑  𝟏𝑏𝑙  𝑖>0  𝑂  𝑖=1  ∑  𝑏𝑙′  𝑗  𝐼  𝑗=1  (5)  Where 𝑏𝑙′ is the vector of bit depths used to encode the pre-  vious convolution layer’s output (where present).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Once a channel can be compressed to zero bits it becomes  a candidate for removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' However, removing a channel only  outputting zeros could significantly change the network’s  output if a bias was to be added to that channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' A sudden  change to the network’s output can irreversibly disrupt the  training, so to handle this, an 𝐿1 loss is applied to biases op-  erating on zero-bit channels to reduce them to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Only  when the biases are reduced to zero are these output chan-  nels (and corresponding input channels from the next layer)  removed, since at this point removing such a channel does  not change the network’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  A sudden change of quantisation parameters can also ir-  reversibly degrade the network during training, which is a  problem described in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Irreversible Forgetting  Compressing networks in this way can be challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' We  conjecture that the network is continuously trying to remove  (forget) channels (or more generally groups of weights  quantised by a common bit depth parameter) that are not  necessary to produce a low error at that moment in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  However, this process could erroneously remove parts of a  network that are useful, albeit not heavily used during pro-  cessing of recent minibatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' For example, one might con-  sider a network channel in the first layer trained to match  horizontal lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' If multiple subsequent training batches con-  tain no horizontal lines affecting the output, the training  might determine that horizontal lines are not necessary and  reduce the channel’s quantization bit depth too much, and  possibly to a point whereat the training can no longer re-  learn the feature if needed by recovering the corresponding  bit depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' We will call this irreversible forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  This phenomenon is more likely to occur deeper in the  network in wider layers where more abstract (and less often-  needed) features are located.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' We have identified ways to  mitigate irreversible forgetting, including:  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Having more weights share the same quantization  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Even if some of the weights in a group  seem unnecessary, their encoding bit depth will  stay high if other weights in the group are being  used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Use the Adam optimizer that adapts the learning  rate when a gradient is noisy, with relatively high  epsilon parameter to reduce the “acceleration” of  bit depth parameters during the early phase of  training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Another factor that might affect the compression rate of the  network is the error function’s smoothness, but exploring  this aspect is left for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Figure 3: Top-1 accuracy on CIFAR-10 for different  choices of compression factor γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Also shown are results  from the method of Défossez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Bit depths are  determined using eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' (3) and (4) after channels of zeroes  have been removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Experiments  To demonstrate the proposed method, a fast-training classi-  fication network was chosen (Page 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' This is important  for being able to iterate algorithm development quickly, and  to explore the tradeoff space between training time (Figure  2), network size (Figure 3), and accuracy in reasonable time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Experiments were conducted on the CIFAR-10 dataset  using the following data augmentation methods, applied in  the order: (1) 4 pixel padding;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' (2) PyTorch AutoAugment  policy for CIFAR-10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' (3) random horizontal flip;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' (4) 32x32  random crop;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' (5) random erasing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' and (6) normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  The optimizer used was Adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' For training the quantiza-  tion parameters and weights we use a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='5  and 10−3 respectively, and an ϵ parameter of 10−3 and 10−5  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' A 𝐿2 decay of 5 × 10−4 was applied only to  the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=" Training was run for 850 iterations, then the  network was allowed to “anneal” to a final state by using  PyTorch's ReduceLROnPlateau scheduler until conver-  gence." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  The same training method was used when implementing  the method of Défossez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' (2022) for fair comparison  with our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Results  A major advantage of Self-Compression is a parameterized  trade-off between size and accuracy, in our case governed  by the parameter 𝛾 (Equation 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' The network was trained  with 𝛾 log-uniformly sampled from the interval [10-3, 10-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  As can be seen in Figure 3, this forms a locus in a plot of  accuracy against final network size, wherein high values of  𝛾 correspond to higher compression/lower accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Base-  line 32-bit float accuracy on this network is 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='22,  which we can match down to as few as 3% of the network  weight bits (18% of weights) remaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Also shown in Figure 3 are results for the method of Dé-  fossez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' (2022) which also learns bit depths simultane-  ously with optimization of accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Their method typically  achieves floating point accuracy when the final size is above  ~8% of the original number of bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' However, our proposed  method maintains high accuracy at lower numbers of bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  We also note that the locus of their method is considerably  noisier, which may be due to their use of a smaller weight  granularity and stochastic rounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' One key difference be-  tween our proposed method and that of Défossez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' is the  form of the quantization function (STE vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' stochastic round-  ing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' For this reason, we also include results in Figure 3 for  their method using the STE instead of stochastic rounding,  which results in a modest improvement in accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Figure 4 shows the number of channels before and after  Self-Compression is applied with 𝛾 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' The boxes  represent convolution blocks, comprising convolutions with  optional batch norm and bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' The numbers on the arrows  indicate number of activation channels, and the numbers on  the convolution blocks represent the number of output chan-  nels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Where a summation has been performed, the number  of input channels is instead noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Conclusion  We have introduced Self-Compression: an efficient, con-  ceptually simple means of learning the bit depths used to  represent a network’s parameters simultaneously with learn-  ing its weights, so that during training the network size is  reduced simultaneously with maximizing accuracy on its  task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Results on the CIFAR-10 classification task indicate  that accuracy close to 32-bit floating point can be achieved  with as few as 1-3% of the original bits remaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Im-  portantly, performance improvements are realisable on typ-  ical hardware for accelerating neural networks including  Figure 4: An overview of the number of weight channels in  the example classification network before (top) and after  (bottom) applying Self-Compression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  NetworkSizevs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='Accuracy(CIFAR-1o)  96  X-  95  x x  94  Accuracy  +/  +  +  +  +  93  Top-1  1+  92  +  Our Method  Best Fit  Défossez et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='with STE  91  Best Fit  Défossez et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Best Fit  90  0%  2%  4%  6%  8%  10%  12%  Percentage of Original Network Bits Kept (vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='32bit float)CPUs, GPUs, and neural network accelerators, without the  need for specialized hardware or execution algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Acknowledgments  Our special thanks go to Timothy Gale and Gunduz Vehbi  Demirci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' We would also like to thank our other colleagues  at Imagination Technologies who supported this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
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+page_content=' Dhari-  wal, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Neelakantan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Shyam, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Sastry, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Askell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Agarwal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Herbert-Voss, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Krueger, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Henighan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Child,  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Ramesh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Ziegler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Winter, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Hesse, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Chen,  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Sigler, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Litwin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Gray, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Chess, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Clark, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Berner,  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' McCandlish, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Radford, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Sutskever, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' and Amodei, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Language Models are Few-Shot Learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' and Liu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Ternary Weight Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  arXiv preprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' arXiv:1605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='04711v2 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='CV].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Ithaca, NY: Cornell  University Library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Page,  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  cifar-10-fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='com/da-  vidcpage/cifar10-fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Accessed: 2022-11-03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Rastegari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Ordonez, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Redmon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' and Farhadi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  XNOR-Net: ImageNet Classification using Binary Convolutional  Neural Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' In Proceedings of the 14th European Conference  on Computer Vision (ECCV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  Wang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Davis, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' Cheung, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' and Constantinides, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='  LUTNet: Learning FPGA Configurations for Highly Efficient  Neural Network Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' In IEEE Transactions on Computers  69: 1795-1808.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='1109/TC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
+page_content='2978817' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9FPT4oBgHgl3EQfqjWj/content/2301.13142v1.pdf'}
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+Thermodynamic Properties of the Mott Insulator-Metal Transition
+in a Triangular Lattice System Without Magnetic Order
+Emre Yesil1, Shusaku Imajo2,∗ Satoshi Yamashita1, Hiroki Akutsu1, Yohei
+Saito3, Andrej Pustogow4, Atsushi Kawamoto5, and Yasuhiro Nakazawa1†
+1Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
+2Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan
+3Institute of Physics, Goethe-University Frankfurt, 60438 Frankfurt (M), Germany
+4Institute of Solid State Physics, TU Wien, 1040 Vienna, Austria
+5Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan
+(Dated: January 12, 2023)
+The organic system, κ-[(BEDT-TTF)1−x(BEDT-STF)x]2Cu2(CN)3, showing the Mott transition
+between a nonmagnetic Mott insulating (NMI) state and a Fermi liquid (FL), is systematically
+studied by calorimetric measurements. An increase of the electronic heat capacity at the transition
+from the NMI state to the FL state which keeps the triangular dimer lattice demonstrates that
+the charge sector lost in the Mott insulating state is recovered in the FL state. We observed that
+the remaining low-energy spin excitations in the Mott insulating state show unique temperature
+dependence, and that the NMI state has a larger lattice entropy originating from the frustrated
+lattice, which leads to the Pomeranchuk-like effect on the electron localization.
+Near the Mott
+boundary, an unexpected enhancement and magnetic-field dependence of heat capacity are observed.
+This anomalous heat capacity is different from the behavior in the typical first-order Mott transition
+and shows similarities with quantum critical behavior.
+To reconcile our results with previously
+reported scenarios about a spin gap and the first-order Mott transition, further studies are desired.
+I.
+INTRODUCTION
+The dimer-Mott compounds with the chemical for-
+mula
+of
+κ-(BEDT-TTF)2X,
+where
+BEDT-TTF
+is
+bis(ethylenedithio)tetrathiafulvalene and X is a mono-
+valent counter anion, provide extensive possibilities for
+understanding physical phenomena induced by electron
+correlations of π-electrons[1, 2]. Electrons in them form
+relatively narrow electron bands governed by overlaps of
+molecular orbitals, and the spin, charge, and lattice de-
+grees of freedom appear in various manners in them. The
+electronic states of the dimer-Mott system can be de-
+scribed in the frame of the Mott-Hubbard physics with
+on-site Coulomb repulsion U and bandwidth W (pro-
+portional to transfer integral t)[1–5]. Additionally, the
+dimer lattice of the κ-type molecular arrangement has
+geometrical frustration depending on the ratio of t and
+t′, nearest-neighbor and second-nearest-neighbor trans-
+fer integrals, as shown in Fig. 1(a). Using the two pa-
+rameters U/t vs.
+t′/t, the electronic phase diagram
+has been understood, as shown in Fig. 1(b)[6, 7]. For
+the less-frustrated salts (t′/t<1), the discontinuity and
+hysteresis in the electrical transport indicate that the
+superconductivity-antiferromagnetic insulator (SC-AFI)
+transition dominated by a change in U/t (the blue ar-
+row in Fig. 1(b)) is first-order[8–12]. As schematically
+described in Fig. 1(c), the 1st-order Mott boundary dis-
+appears at a critical endpoint of ∼35 K, and the nature of
+the Mott physics around the endpoint has been discussed
+∗ imajo@issp.u-tokyo.ac.jp
+† nakazawa@chem.sci.osaka-u.ac.jp
+in terms of high-energy criticality caused by the competi-
+tion between the large U and W >1000 K[10–12]. From
+the AF magnetic order induced by antiferromagnetic in-
+teractions, the SC with relatively high-T c has been ex-
+tensively discussed in terms of unconventional pairing re-
+lated to antiferromagnetic spin fluctuations in κ-(BEDT-
+TTF)2X and also in β′-, λ-type compounds[13–17]. The
+variation in physical parameters near the Mott bound-
+ary has been studied by various measurements across the
+boundary[18–22]. Based on the variation in the electronic
+heat capacity coefficient γ of the normal state shown in
+Fig. 1(c)[14, 23–26], the low-temperature FL state can
+be understood by the electron-mass enhancement with
+increasing electron correlations (the green arrow) and
+the decrease in the metallic portion due to the growth
+of phase coexistence near the Mott boundary (the or-
+ange arrow). Although slight percolative superconduc-
+tivity is left in the AF Mott insulating salts very near
+the boundary, its γ is almost zero because the volume
+fraction of the FL is negligible. It should be noted that
+the information contains the magnetic entropy change re-
+lated to the AFI ground state of π-electrons and that the
+change is not a genuine feature expected in the Hubbard
+model because no symmetry breaking is assumed in this
+framework[27–31]. When t′/t=1, the AFI ground state
+should be destabilized by the geometrical frustration, and
+non-ordered states may be stable even at low temper-
+atures.
+Indeed, κ-(BEDT-TTF)2Cu2(CN)3, which has
+been considered a prime candidate showing the quantum
+spin liquid (QSL) state, does not show long-range mag-
+netic orders down to extremely low temperatures because
+t′/t is almost unity[32, 33]. However, recently, Miksch et
+al. suggested that its ground state might be a gapped
+valence bond solid (VBS) with a spin gap from observa-
+arXiv:2301.04310v1  [cond-mat.str-el]  11 Jan 2023
+
+2
+γ (mJK-2mol-1)
+(a)
+(b)
+AFI
+SC
+Mott
+insulator
+Fermi
+liquid
+cross
+over
+10
+100
+T (K)
+U/t
+P
+①
+②
+③
+④
+⑤
+X =
+① : Cu[N(CN)2]Cl
+② : Cu[N(CN)2]Br
+dn : n = number of 
+deuterium in BEDT-TTF
+③ : Cu(NCS)2
+④ : Ag(CN)2H2O
+⑤ : I3
+κ-(BEDT-TTF)2X
+d8
+d6
+d4
+d2
+d0
+Mott
+transition
+mass
+enhancement
+ρ = AT2
+Cele = γT
+χ
+= χP
+(c)
+nonmagnetic
+Mott
+insulator
+(NMI)
+dT/dx>0
+quantum
+critical
+regime
+Fermi
+liquid
+(FL)
+crossover
+(d)
+1
+10
+100
+T (K)
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+x
+FIG. 1. (Color online) (a) Molecular arrangement in the conducting plane of the present system and BEDT-TTF and BEDT-
+STF molecules. t and t′ indicate nearest-neighbor and second-nearest-neighbor transfer integrals in the dimer lattice, respec-
+tively. (b) Electronic ground states of dimer-Mott system with parameters of electron correlations U/t (U/W) and frustration
+factor t′/t[6, 7]. The red arrow indicates the route controlled by substitution in the present system, whereas the light blue
+arrow represents the numerous previous studies on the less-frustrated dimer-Mott system. (c) Electronic phase diagram of
+the less-frustrated κ-type salts with the shown counter anions. The AFI and FL phases are divided by the first-order Mott
+transition, which terminates at a critical endpoint ∼35 K. Inside the FL, SC with relatively high-T c∼10 K occurs near the
+boundary. The lower panel shows the variation in γ of the normal state depending on electron correlations U/t and chemical
+pressure of the counter anion P. (d) Schematic phase diagram of κ-[(BEDT-TTF)1−x(BEDT-STF)x]2Cu2(CN)3 deduced from
+Refs. [36–39]. An increase in the mixing ratio x corresponds to a decrease in U/t shown by the red arrow in the phase diagram
+(b). The NMI and electronic FL states exist across the quantum critical regime, which is located around x=0.2. The dashed
+curve indicates the phase boundary between the NMI and FL phases, its slope dT/dx is positive at low temperature. The
+black dots in the NMI region represent the x dependence of the so-called 6 K anomaly, in which error bars are determined by
+the present heat capacity measurements.
+tion of a drop of spin susceptibility below 6 K[31, 34].
+Although the controversy still persists because of the re-
+maining discrepancy with the gapless spin excitations in
+heat capacity[35], we hereafter use NMI for describing
+the Mott insulating state.
+Recently, Saito et al.
+reported that a donor al-
+loying
+system
+of
+(BEDT-TTF)1−x(BEDT-STF)x,
+where
+BEDT-TTF
+and
+BEDT-STF
+are
+the
+ab-
+breviations
+of
+bis(ethylenedithio)tetrathiafulvalene
+and
+bis(ethylenedithio)diselenadithiafulvalene,
+with
+Cu2(CN)3− exhibits continuous tuning of U/W with
+keeping the triangularity of the dimer lattice[36–39].
+The Se substitution into the BEDT-TTF molecule
+shown in Fig. 1(a) results in a larger overlap of the
+wave function with the neighboring molecules.
+Since
+the increase in x is considered to work as positive
+chemical pressure without inducing large change in
+average t′/t, the insulating state is altered into the FL
+state via the genuine Mott transition at x=0.1-0.2[36–
+39], as indicated by the red arrow in Fig. 1(b).
+This
+variation is similar to the tuning by external pressure
+to κ-(BEDT-TTF)2Cu2(CN)3 where the ground state
+without magnetic order shifts to a FL across the Mott
+insulator-metal transition[8, 9]. Namely, this variation
+provides profound information on the Mott transition
+genuinely dominated by the itinerancy/localization of
+the charge degrees of freedom. The T-x electronic phase
+diagram of the present alloying system is predicted
+from the results in Refs. [36–39], as shown in Fig. 1(d).
+High-resolution
+thermodynamic
+measurements
+under
+pressure are typically challenging; however, using the
+present chemically tunable system, thermodynamic and
+entropic information near the metal-insulator boundary
+can be obtained by ambient-pressure heat capacity
+measurements.
+In this study, we systematically inves-
+tigated
+κ-[(BEDT-TTF)1−x(BEDT-STF)x]2Cu2(CN)3
+by calorimetry to unveil thermodynamics features of
+the Mott transition between the potential QSL and FL
+states.
+II.
+EXPERIMENTAL
+Single crystals of the alloying compounds κ-[(BEDT-
+TTF)1−x(BEDT-STF)x]2Cu2(CN)3 are grown by elec-
+trochemical oxidation method[37]. As shown in Table I,
+the crystal structural parameters were characterized by
+x-ray diffraction analyses, and the macroscopic homo-
+geneity of the alloying crystals was confirmed. To evalu-
+ate the change in t′/t with mixing BEDT-STF molecules,
+
+SEDTLSTF10
+AF
+sc
+5
+F
+0
+0.5
+103
+TABLE I. Crystallographic data. Fw represents the formula
+weight for each sample. V shows the cell volume. Z denotes
+the number of formula units in the unit cell divided by the
+number of independent general positions. d′/d is the ratio of
+average dimer-dimer distance along the t′ and t directions.
+x
+0.04
+0.10
+0.12
+0.19
+0.28
+0.44
+Fw
+982.04
+993.30
+997.05 1010.18 1027.06 1057.07
+Space group P21/c
+P21/c
+P21/c
+P21/c
+P21/c
+P21/c
+a (˚A)
+16.1080 16.1054 16.1136 16.1569 16.1580 16.1761
+b (˚A)
+8.5861
+8.5816
+8.5874
+8.6000
+8.6017
+8.5982
+c (˚A)
+13.3591 13.3751 13.3550 13.3663 13.3979 13.4037
+α (◦)
+90
+90
+90
+90
+90
+90
+β (◦)
+113.691 113.565 113.66 113.519 113.551 113.208
+γ (◦)
+90
+90
+90
+90
+90
+90
+V (˚A3)
+1691.9
+1694.4
+1692.6
+1703.0
+1707.0
+1713.4
+Z
+2
+2
+2
+2
+2
+2
+d′/d
+0.925
+0.926
+0.925
+0.924
+0.925
+0.926
+we here introduce d′/d, the ratio of average dimer-dimer
+distance along the t′ and t directions. The small changes
+in the lattice parameters within 0.5% were observed.
+Heat capacity measurements were carried out by a typ-
+ical relaxation technique using a home-made thermal-
+relaxation-type calorimeter in a 3He refrigerator with a
+15 T superconducting magnet. The temperature range
+of these measurements is about 0.6-10 K. Magnetic fields
+were applied perpendicular to the conducting plane. We
+measured the background data with a small amount of
+Apiezon N grease before mounting samples. These mea-
+surements were performed with single crystalline samples
+weighing about 80-300 µg. The details of the calorimeter
+and experimental setup are reported in Ref. [40].
+III.
+RESULTS
+In Fig. 2, we present the temperature dependences of
+the heat capacity of the alloying system in the CpT −1
+vs. T 2 plot. The data in the temperature range up to
+10 K are displayed in Fig. 2(a). There is only a subtle
+difference of about 6% at 10 K, mainly originating from
+the decrease in the Debye temperature induced by the Se
+substitution[41]. This means that the STF substitution
+induces only a small change of about a few percent in
+phonon contributions. In Fig. 2(b), the entropy S as a
+function of temperature is shown as a logarithmic plot.
+The entropy is calculated by integration of the exper-
+imentally obtained CpT −1 and the extrapolation down
+to 0 K estimated by polynomial fittings, and thus, the
+calculated S includes the electronic and phonon contri-
+butions together. At higher temperatures, the x depen-
+dence of the entropy is small because the main portion of
+the total entropy is the phonon contribution. In the lower
+temperature region, the x=0.19 salt shows the larger en-
+tropy compared to the others. To shed light on the low-
+temperature region, the enlarged plots of CpT −1 below
+about 3.2 K (=10 K2) are shown in Fig. 2(c) and (d). The
+datasets for x<0.15 are shown in Fig. 2(c) while those for
+x>0.15 are in Fig. 2(d) because the Mott insulating char-
+acter at the low-x region changes into the metallic one
+across the boundary region at x=0.1-0.2 according to the
+previous reports[36, 38, 39]. The small change in lattice
+heat capacity indicates that the origin of the change ob-
+served in the low-temperature region should mainly come
+from the electronic contribution. In the case of typical
+metals having Fermi surfaces composed of itinerant elec-
+trons, CpT −1 at low temperatures obeys CpT −1=γ+βT 2,
+where γ and β represent the Sommerfeld coefficient of
+electronic heat capacity and the Debye coefficient of
+lattice heat capacity.
+Indeed, the x=0.44 salt, which
+is deep inside the metallic FL region, shows the lin-
+ear behavior below 2 K with γ=24.1 mJK−2mol−1 and
+β=15.0 mJK−4mol−1, which are comparable with those
+of typical BEDT-TTF-based metallic salts[14, 16, 42].
+Above 2 K, the behavior is gradually deviated by ex-
+cess heat capacity that may originate from librational op-
+tical modes Copt∼R(TE/T)exp(TE/T)/[exp(TE/T)−1]2,
+where TE represents the Einstein temperature, as sug-
+gested for the other organic charge-transfer complexes
+with various structures[43].
+On the other hand, the insulating salts shown in
+Fig. 2(d) do not share this behavior. At first glance, it ap-
+pears to follow the linear behavior below 2 K, as indicated
+by the black dotted line. Also, the analysis of the data
+for x=0.04 using the typical CpT −1=γ+βT 2 relation
+leads to γ=12.6 mJK−2mol−1 and β=21.2 mJK−4mol−1,
+which are comparable with the previously reported
+γ=12 mJK−2mol−1 and β=21 mJK−4mol−1 for x=0[35].
+However, above 2 K, the CpT −1 is lower than this lin-
+ear dependence. Since higher-order terms of the Debye
+model appears only at higher temperatures, this behav-
+ior indicates that the Mott insulating state cannot be ex-
+plained by the framework of the typical FL states. Nev-
+ertheless, the large low-temperature heat capacity in the
+insulating state proves the presence of low-energy spin
+excitations, which have been discussed as the finite γ
+and/or the relatively large β specific to the organic QSL
+state in the previous works[35, 44]. As a rough estimate,
+we show the lattice heat capacity ClatT −1, which is sim-
+ply obtained by subtracting the γ term from the x=0.44
+data (the thin black line), CpT −1=γ+βT 2+Copt−1, as
+is shown in Fig. 2(d). The difference from this estimate
+(Cp−Clat)T −1, which corresponds to the contribution of
+the spin excitations, is displayed as a CeleT −1 vs.
+T
+plot in Fig. 2(e). This component does not appear to
+be a simple γ term.
+ESR results[34] suggest that the
+ground state is a gapped VBS state with a relatively large
+∆/kB∼12 K. However, the red curve, exp(−∆/kBT) be-
+havior for ∆/kB=12 K, does not describe the present
+results. Even if we assume that ∆/kB is a variable pa-
+rameter, it is difficult to reproduce the temperature de-
+pendence and ∆/kB must be extremely tiny. Including
+the present result, heat capacity measurements, sensi-
+tive to low-energy excitations, indicate the presence of
+low-energy spin excitations, which is puzzling in view of
+
+4
+10
+1
+10
+2
+10
+3
+10
+4
+S (mJK
+-1mol
+-1)
+5
+6
+7 8 9
+1
+2
+3
+4
+5
+6
+7 8 9
+10
+T (K)
+ x=0.04
+ x=0.10
+ x=0.12
+ x=0.19
+ x=0.28
+ x=0.44
+~6%
+ClatT-1
+(Cp−Clat)T-1
+(a)
+βT2
+γ
+CoptT-1
+γ + βT2 + CoptT-1
+ClatT-1
+γ
+superconducting
+transition
+(c)
+(d)
+(e)
+(f)
+(b)
+x
+~exp(−Δ/kBT)
+Δ/kB=12 K
+200
+150
+100
+50
+0
+Cp/T (mJK
+-2mol
+-1)
+8
+6
+4
+2
+0
+T
+2 (K
+2)
+ x=0.04
+ x=0.10
+ x=0.12
+80
+60
+40
+20
+0
+Cele/T (mJK
+-2mol
+-1)
+6
+4
+2
+0
+T (K)
+ x=0.04
+ x=0.12
+4
+3
+2
+1
+0
+T (K)
+ x=0.19
+10
+8
+6
+4
+2
+0
+T
+2 (K
+2)
+ x=0.19
+ x=0.28
+ x=0.44
+1500
+1000
+500
+0
+Cp/T (mJK
+-2mol
+-1)
+100
+80
+60
+40
+20
+0
+T
+2 (K
+2)
+ x=0.04
+ x=0.10
+ x=0.12
+ x=0.19
+ x=0.28
+ x=0.44
+Schottky
+anomaly
+~lnT
+FIG. 2. (Color online) (a) CpT −1 vs. T 2 below 10 K for x=0.04-0.44. (b) Logarithmis plot of S as a function of temperature.
+(c),(d) Enlarged plots of the low-temperature region below T<3.2 K for x<0.15 (c) and x>0.15 (d). The dotted line in (c) is
+a fit to CpT −1=γ+βT 2 below 2 K for x=0.12. The thin black curve in (c) is a rough estimate of lattice heat capacity ClatT −1
+obtained from the FL salt (x=0.44) by subtracting the γ term, which highlights the contribution of the low-energy excitations
+in the NMI state (shaded area). The black curves in (d) show the respective components of the fit of CpT −1=γ+βT 2+CoptT −1
+to the x=0.44 data. (e),(f) Electronic heat capacity CeleT −1 obtained by subtracting the lattice heat capacity ClatT −1 from the
+total CpT −1 for x=0.04, 0.12 (e) and 0.19 (f). The red curve in (e) shows activation-type gapped behavior when ∆/kB=12 K.
+The red curve in (f) indicates a fit to −lnT while the black and green curves represent the typical superconducting and Schottky
+anomalies, respectively.
+a spin gap concluded from other measurements[34, 45].
+To clarify this point, experiments at lower temperatures
+seem necessary. Although the drop of the magnetic sus-
+ceptibility below 6 K is observed, an exact zero suscepti-
+bility in a low-temperature limit has not been reported in
+these works[34]. To reconcile these arguments based on
+the temperature range of the measurements (ESR mea-
+surement above 2 K), one possibility is that the ground
+state has an incomplete spin gap, yielding some low-
+energy excitations, even below the putative transition
+at 6 K. Alternatively, an extrinsic origin, such as impu-
+rity spins or domain walls, was suggested to describe the
+low-temperature magnetic behavior[31, 46]. However, it
+is unclear how to model the present temperature depen-
+dence with the suggested local orphan spins and local
+domain wall fluctuations, which may give the Schottky-
+type heat capacity and glass-like γT heat capacity, re-
+spectively.
+For the x=0.19 salt, located in the intermediate
+region[36, 39], the temperature dependence (Fig. 2(d))
+does not obey the CpT −1=γ+βT 2 relation due to the
+gradual upward deviation below ∼2.5 K. This behav-
+ior is more clear in the plot of (Cp-Clat)T −1, as shown
+in Fig. 2(f).
+Even though remnants of superconduc-
+tive components are observed near the 1st-order Mott
+boundary of several κ-type salts, including some STF
+compounds[47], this behavior is completely distinct from
+the superconducting transition (the black curve). Fur-
+thermore, this deviation cannot be reproduced by ex-
+trinsic Schottky anomaly arising from magnetic impu-
+rities (the green curve).
+In the case of the AFI-FL
+Mott transition, such behavior is absent, and the sim-
+ple CpT −1=γ+βT 2 relation is observed even very near
+the first-order Mott boundary[23, 24, 26].
+The low-
+temperature gradual divergence is reminiscent of quan-
+tum critical behavior near a quantum critical point
+(QCP) because of the −lnT-like behavior (the red curve).
+Indeed, the present alloying system does not show signif-
+icant first-order-like discontinuous behavior in our heat
+capacity data and resistivity data[36, 38], albeit a dielec-
+tric catastrophe suggestive of phase inhomogeneity has
+been reported[39].
+The first-order Mott transition ob-
+served in other κ-type salts is less obvious in κ-(BEDT-
+TTF)2Cu2(CN)3; nevertheless, the weak first-order Mott
+transition with the critical endpoint located at 15-20 K
+has been observed in transport, NMR, and dielectric
+
+5
+measurements[8, 9, 39]. A small difference between the
+alloying system and κ-(BEDT-TTF)2Cu2(CN)3 may fur-
+ther lead to suppression of the remaining first-order na-
+ture. Randomness effects should be also taken into ac-
+count, as a disorder can lower the temperature of the
+critical endpoint[48]. Regardless of the origin of the sup-
+pression of the first-order nature, the low-temperature
+diverging heat capacity indicates that quantum fluctua-
+tions are developed in this temperature region (<2.5 K).
+Since this behavior is significant in x=0.19 and smaller
+in x=0.28, the QCP should be located close to x=0.2,
+which is not far from the reported position of the metal-
+insulator transition[36, 38, 39].
+The magnetic field dependences of the heat capac-
+ity for x=0.04, 0.12, 0.19, 0.28, and 0.44 are shown in
+Fig. 3. The upper panels show the low-temperature re-
+gion below 10 K2 while the lower ones display the data
+up to 120 K2. The fields are applied perpendicularly to
+the two-dimensional plane. For the NMI (x=0.04) salt,
+the magnetic-field dependence is not significant even at
+high magnetic fields. This fact indicates the robustness
+of these low-energy excitations against fields. However,
+the response to the magnetic field for the x=0.19 sam-
+ple, located in the quantum critical region (QCR), is
+distinct from those of the other salts. The upturn ob-
+served at 0 T disappears with increasing magnetic field,
+while a broad hump structure in the temperature de-
+pendence of CpT −1 appears at relatively high magnetic
+fields of 5-6 T. Since this behavior indicates that the
+low-temperature entropy shifts to higher temperatures
+in magnetic fields, the origin of this field dependence
+cannot be attributed to the 6 K anomaly and percola-
+tive superconductivity which is often observed near the
+1st-order Mott transition. Considering the magnetic field
+dependence of the Mott boundary[23] and the bent quan-
+tum phase boundary[38, 39] (Fig. 1(b)), the origin of the
+hump structure is also attributed to the critical behav-
+ior. By further increasing fields, the broad hump is also
+suppressed, and the field dependence is diminished. This
+behavior suggests that the high-field electronic state at
+low temperatures is out of the critical regime and can be
+regarded as the FL state.
+IV.
+DISCUSSION
+To deepen the understanding of the variations in the
+low-energy excitations around the Mott transition, we
+here show the low-temperature heat capacity at 1 K,
+Cp(1 K), as a function of x in Fig. 4(a).
+In order to
+highlight the area near the Mott transition, each region
+is color-coded in a different color in the figure. Based on
+the variation in γ depending on U/t (Fig. 1(c)), Cp(1 K)
+should vary like the blue broken curve if the Mott tran-
+sition is between the AFI and FL states. Namely, the
+deviation from the blue broken curve is the peculiar-
+ity of the NMI-FL Mott transition.
+For the x=0.04
+salt, the value of 33.8 mJK−2mol−1, much larger than
+β=15 mJK−4mol−1 for the x=0.44 salt, suggests that
+the heat capacity involves finite low-energy excitations of
+the spin sector (the light blue arrow). With approaching
+Mott transition, the Cp(1 K) increases from the constant
+value in the NMI region. This behavior deviates from
+the blue broken curve because γ is constantly zero inside
+the AFI Mott phase. Once crossing the boundary and
+entering the FL regime, the Cp(1 K) asymmetrically de-
+creases and reaches 39.1 mJK−1mol−1 at x=0.44, which
+is comparable with the typical value of Cp(T=1 K)=γ+β
+for the BEDT-TTF-based metallic salts with γ=20-
+25 mJK−2mol−1 and β=10-15 mJK−4mol−1. The dif-
+ference between the NMI and FL regions, shown by the
+pink arrow, should correspond to the contribution of the
+charge sectors of the π-electrons, which is absent in the
+NMI state. If the inhomogeneity appearing near the first-
+order Mott transition develops around x=0.2, the strong
+enhancement of Cp(1 K) inside the FL region should not
+be observed near the boundary because the inhomogene-
+ity significantly reduces the electronic heat capacity, as
+shown in Fig. 1(c).
+Here, we examine the slope of the phase boundary be-
+tween the NMI and FL states on the electronic phase
+diagram, dT/dx in Fig. 1(d).
+The positive slope in-
+dicates that the localization of electrons in the NMI
+state gives a larger entropy than the itinerancy of elec-
+trons in the FL state. This unusual behavior is reminis-
+cent of the Pomeranchuk effect observed in 3He, melting
+solid 3He with lowering the temperature through spin-
+lattice coupling[49, 50].
+In Fig. 4(b), we present the
+x-dependence of the entropy S at 1 K (left axis) and
+5 K (right axis). At 1 K, the entropy of the NMI state
+is lower than the entropy of the FL state. At 5 K, it
+is the opposite. As suggested by a theory[51], it is ex-
+pected that there is only small energy difference between
+the FL and Mott states because the gain in kinetic en-
+ergy of electrons in the FL state is compensated by the
+loss in potential energy.
+In particular, when the frus-
+tration parameter t′/t is close to unity, the slope of the
+Mott boundary dU/dT is almost zero or a small negative
+value[52], and thus, the energy difference between the two
+states should be very small. This delicate energy balance
+makes the Mott transition winding on the phase diagram
+shown in Fig. 1(c). In real materials involving a variety
+of degrees of freedom, we must consider what contribu-
+tion is an eventual factor determining how large or small
+the entropy is. The similar x dependence of S(1 K) and
+Cp(1 K) demonstrates that the low-temperature behavior
+can be explained by the electronic part and that the en-
+tropy of the NMI state is smaller at lower temperatures.
+However, at higher temperatures above 5 K, the lattice
+part must also be considered because the lattice com-
+ponents account for a large portion (>90%) of the total
+entropy in the soft organic crystal. To explain the rever-
+sal of the entropy appearing with elevating temperature,
+we need to discuss entropy originating from the phonons
+as well as the low-energy spin excitations. The charac-
+teristic of the present system is the confinement of the
+
+6
+100
+50
+0
+T
+2 (K
+2)
+x = 0.44
+    FL
+100
+50
+0
+T
+2 (K
+2)
+ 0 T
+ 4 T
+ 8 T
+x = 0.28
+    FL
+100
+50
+0
+T
+2 (K
+2)
+x = 0.12
+   NMI
+8
+6
+4
+2
+0
+x = 0.19
+  QCR
+10
+8
+6
+4
+2
+0
+x = 0.44
+    FL
+8
+6
+4
+2
+0
+x = 0.28
+    FL
+100
+50
+0
+T
+2 (K
+2)
+x = 0.19
+  QCR
+200
+150
+100
+50
+0
+Cp/T (mJK
+-2mol
+-1)
+8
+6
+4
+2
+0
+x = 0.04
+   NMI
+1.5
+1.0
+0.5
+0.0
+Cp/T (JK
+-2mol
+-1)
+100
+50
+0
+T
+2 (K
+2)
+x = 0.04
+   NMI
+8
+6
+4
+2
+0
+x = 0.12
+   NMI
+1.5
+1.0
+0.5
+0.0
+CpT
+-1/ JK
+-2mol
+-1
+100
+50
+0
+T
+2/ K
+2
+ 0 T
+ 
+ 2 T
+ 
+ 4 T
+ 
+ 5 T
+ 
+ 6 T
+ 
+ 8 T
+ 
+ 10 T
+ 
+ 12 T
+x = 0.04
+   NMI
+FIG. 3. (Color online) CpT −1 vs. T 2 at various magnetic fields for x=0.04, 0.12 at the NMI, 0.19 at the quantum critical
+region (QCR), and 0.28, 0.44 at the FL. The upper panels show the data below 10 K2 while the lower ones show the data up
+to 120 K2. The arrow for the x=0.19 data indicates a hump observed at fields of 5-6 T.
+electrons in the triangular lattice making the antiferro-
+magnetically interacting spins frustrated and disordered,
+which should result in lattice softening through the spin-
+lattice coupling. The lattice softening entails shifting the
+phonon density of states down to a lower-temperature
+region, as is observed in the NMI state.
+Low-energy
+phonon excitations in the non-ordered dimer-Mott tri-
+angular lattice system have been discussed by thermal
+conductivity measurements[45], and therefore, the soft-
+ening of phonons can be a possible reason to explain the
+larger entropy in the NMI state. Nevertheless, another
+possibility is that the spin excitations explain the evolu-
+tion of entropy with temperature in the NMI state. When
+dU/dT is negative, the entropy of the NMI state can be-
+come larger than that of the FL state. The formation
+of the possible non-magnetic VBS ordered state[31, 34]
+suggests a rapid increase in spin entropy with an en-
+hancement of heat capacity near the transition temper-
+ature. This gap-closing behavior around 5 K may also
+relate to the reversal of the entropy. Although the spin
+entropy in the present system is not large, the relation
+may not be reversed even with the gain in the lattice en-
+tropy without the spin entropy. For the NMI system, the
+low-temperature Pomeranchuk-like phase boundary[53]
+is probably related to both contributions, namely the
+phonon softening effect and spin contributions. To dis-
+cuss these in more detail, the temperature dependence of
+entropy up to higher temperatures is necessary.
+We emphasize that the low-temperature heat ca-
+pacity Cp(1 K) should reflect the variation of the
+ground state driven by quantum fluctuations predom-
+inantly.
+The gradual increase in Cp(1 K) with ap-
+proaching the Mott boundary in the NMI phase sug-
+gests the continuous change in the low-energy excita-
+tions as the possible VBS state is suppressed near the
+Mott boundary.
+According to the correspondence be-
+tween the chemical pressure characterized by x and
+physical pressure (∆P=1.5 kbar roughly corresponds to
+∆x=0.1)[39] and the slope of the metal-insulator bound-
+ary dx/dT∼2*10−3 K−1 (at T=5 K), the Clausius–
+Clapeyron relation dP/dT=∆S/∆V leads to the volume
+change ∆V ∼2*10−8 m3mol−1 with the entropy differ-
+ence ∆S(5 K)∼50 mJK−1mol−1. Despite the rough es-
+timation, the obtained ∆V is one order of magnitude
+smaller than the difference of ∆V between x=0 and 0.1,
+∆V ∼2*10−7 m3mol−1[36]. Thus, even if the boundary
+is a first-order transition, its discontinuity must be al-
+most negligible. Near the QCP where the charge gap is
+just 0 K, quantum fluctuations related to the instabil-
+ity of the charge itinerancy are enhanced and destabi-
+lize the quasiparticles characterizing the FL. It should
+be noticed that the quantum critical behavior is ap-
+parent only in the low-temperature region.
+It is wor-
+thy to note that this energy scale is completely differ-
+ent from that of the high-temperature critical behav-
+ior induced by U and W, which is commonly observed
+
+7
+FL
+NMI
+β
+γ+β
+QCR
+(a)
+(b)
+60
+40
+20
+0
+Cp(T=1 K) (mJK
+-1mol
+-1)
+60
+40
+20
+0
+S(T=1 K) (mJK
+-1mol
+-1)
+0.5
+0.4
+0.3
+0.2
+0.1
+0
+x
+1000
+900
+800
+S(T=5 K) (mJK
+-1mol
+-1)
+FIG. 4. (Color online) Heat capacity Cp at 1 K (a) and en-
+tropy S at 1 K (left) and 5 K (right) (b) as a function of
+x. The blue broken curve is the behavior expected based on
+the variation in γ for the Mott transition between the AFI
+and FL states (Fig. 1(c)). The light blue and pink arrows in
+(a) highlight the contribution of the spin and charge sectors in
+the electronic heat capacity of the FL state, respectively. The
+violet region represents the quantum critical region (QCR).
+The thick translucent curve superimposed on the data points
+in (b) is a visual guide to make the x-dependence of S(T=5 K)
+clearer.
+in all dimer-Mott systems irrespective of the geometri-
+cal frustration[54].
+Indeed, the low-temperature criti-
+cal behavior is absent in the less-frustrated system κ-
+(d[n,n]-BEDT-TTF)2Cu[N(CN)2]Br, which can access
+the 1st-order Mott transition between the AFI and the
+FL states[23, 24]. As the peculiarity of the NMI salt is
+the persistence of the low-energy excitations related to
+the spin part, the present critical behavior may be in-
+duced by the instability of the fractionalization of the
+electron into the spin and charge sectors. The present
+result and scenario agree with the discussion of the re-
+cent transport experiment under pressure[9] and the ther-
+modynamic investigation of κ-[(BEDSe-TTF)x(BEDT-
+TTF)1−x]2Cu[N(CN)2]Br[55], which is also another can-
+didate hosting the genuine Mott transition between the
+NMI and FL state, as well as theoretical works[29, 30, 56].
+Finally, we briefly discuss the so-called “6K anomaly”
+for the NMI sample, which has been discussed in the
+pristine x=0 sample[31, 35, 57].
+Although the recent
+studies with high-quality samples and various sensitive
+measurements[34, 58] have allowed us to get closer to the
+details of this anomaly, the detail is still unclear because
+of some unresolved questions, such as the presence of the
+gapless excitations discussed above.
+Since the pristine
+salts reported in the previous work[35] are synthesized by
+other methods, their sample quality may differ from that
+of the present alloying series and quantitative comparison
+may be challenging. Nevertheless, the systematic change
+in the physical parameters shown in Fig. 4 allows us to
+qualitatively compare our data with the results reported
+in the earlier work[35]. The data in Fig. 2(e) indicates
+that the anomaly seems to be broadened and suppressed
+down to 3-4 K for the x=0.04 and 0.12 samples compared
+to that of the pristine sample. The black dots shown in
+Fig. 1(d) represent the x dependence of the peak tem-
+perature of the anomaly. Considering the relation to the
+charge disproportionation[59], it seems reasonable that
+the anomaly is smeared out by the suppression of the
+electron localization with approaching the Mott bound-
+ary.
+In the lower panels in Fig. 3, the magnetic field
+dependence of the anomaly is very small. This feature
+robust against the magnetic field is consistent with the
+estimation of a critical field of order 60 T for the pris-
+tine salt[31]. To elucidate this enigmatic anomaly, more
+detailed investigations are desired in future studies.
+V.
+CONCLUSIONS
+In summary, we report the low-temperature thermo-
+dynamic properties for the chemical pressure tuning sys-
+tem of the dimer-Mott compounds that show no long-
+range ordering even at low temperatures. The present
+result also provides evidence that the NMI state sup-
+ports some gapless spin excitations.
+However, we also
+found that this low-energy excitations do not seem to
+be described by a simple FL-like γ term. The system-
+atic change in the heat capacity depending on x revealed
+that the genuine Mott transition is potentially continu-
+ous via the QCP, which hosts the low-energy quantum
+fluctuations. In the NMI state, the lattice softening orig-
+inating from the geometrical frustrated lattice gives the
+larger heat capacity in total, although the opening of the
+charge gap reduces the electronic heat capacity. Based
+on the entropy, the balance of these degrees of freedom
+makes the Pomeranchuk-like unique electronic phase di-
+agram. The coupling of these degrees of freedom makes
+the low-temperature phase competition between the NMI
+and FL states, leading to the low-energy quantum criti-
+cal behavior that may be related to the instability of the
+fractionalization of the electron into the spin and charge
+sectors.
+This work was partially supported by JSPS KAKENHI
+Grant No.19K22169 and 20H01862.
+
+8
+[1] H. Kino and H. Fukuyama, Phase Diagram of Two-
+Dimensional Organic Conductors: (BEDT-TTF)2X. J.
+Phys. Soc. Jpn. 67, 2158-2169 (1996).
+[2] K. Kanoda, Metal-Insulator Transition in κ-(ET)2X and
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+Enzyme-enriched condensates show self-propulsion, positioning, and coexistence
+Leonardo Demarchi,1, ∗ Andriy Goychuk,1, † Ivan Maryshev,1 and Erwin Frey1, 2, ‡
+1Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience,
+Department of Physics, Ludwig-Maximilians-Universit¨at M¨unchen,
+Theresienstraße 37, D-80333 M¨unchen, Germany
+2Max Planck School Matter to Life, Hofgartenstraße 8, D-80539 M¨unchen, Germany
+Enzyme-enriched condensates can organize the spatial distribution of their substrates by catalyz-
+ing non-equilibrium reactions. Conversely, an inhomogeneous substrate distribution induces enzyme
+fluxes through substrate-enzyme interactions. We find that condensates move towards the center
+of a confining domain when this feedback is weak. Above a feedback threshold, they exhibit self-
+propulsion, leading to oscillatory dynamics. Moreover, catalysis-driven enzyme fluxes can lead to
+interrupted coarsening, resulting in equidistant condensate positioning, and to condensate division.
+Liquid-liquid phase separation in living cells can lead
+to the formation of biomolecular condensates that aid
+intracellular organization [1–7]. These condensates have
+different functions such as compartmentalization of re-
+actions [4], buffering of molecules [8], and midcell local-
+ization during cell division [9]. However, in a thermal
+equilibrium system, the liquids will completely segregate
+through a coarsening process (Ostwald ripening) [10–13].
+To arrest this process, the system must be brought out
+of equilibrium by supplying energy, e.g., via fuel-driven
+chemical reactions. This has been shown to lead to ‘ac-
+tive droplet’ systems that exhibit a wealth of novel phe-
+nomena not encountered in thermal equilibrium [3, 6, 14].
+Previous studies have considered systems with a con-
+tinuous turnover of condensate (droplet) material by
+chemical reactions [14–24]. The resulting material fluxes
+lead to multi-droplet coexistence [18, 19, 22–24] and
+droplet division [21]. Here, we study a different class of
+systems where conserved enzymes spontaneously phase
+separate, or localize to an existing condensate [25]. These
+enzymes then regulate reactions among other molecules,
+by transiently binding substrate and catalyzing its con-
+version into product via a lower activation barrier.
+For example, in the bacterium Myxococcus xanthus, a
+PomXY cluster (moving on the nucleoid) regulates the
+cycling of PomZ between two conformations [9, 26–28].
+We show that such substrate turnover and the result-
+ing enzyme fluxes lead to condensate self-propulsion, po-
+sitioning, interrupted coarsening, and condensate divi-
+sion. Interestingly, previous studies have shown that liq-
+uid droplets can self-propel on a surface through active
+stresses [29–34], altering their wetting properties [35, 36],
+or in viscous fluids through Marangoni flows [37]. In con-
+trast, in our case, condensate motion is driven by the
+bulk interactions between the various chemical species
+and does not require surfaces or hydrodynamic coupling.
+While condensates might consist of several compo-
+nents, here we focus on the enzyme concentration c(x, t).
+To describe the dynamics of liquid-liquid phase sepa-
+ration, we take the Cahn-Hilliard (CH) equation as a
+starting point with the following chemical potential [38]:
+µ0(c) = −r(c − ˜c) + u(c − ˜c)3 − κ∇2c.
+This chemical
+potential µ0(c) = δF[c]/δc corresponds to the Ginzburg-
+Landau free energy functional F[c] for a symmetric bi-
+nary mixture with the critical density ˜c and phenomeno-
+logical parameters r, u, and κ; in particular, the con-
+trol parameter r measures the distance from the critical
+point [13].
+The enzymes interact with substrates and
+products, which are present at concentrations s(x, t) and
+p(x, t). These couplings, quantified by the Flory-Huggins
+(FH) parameters χs and χp, modify the local chemical
+potential of enzymes. Assuming that the particle cur-
+rents are proportional to gradients in the chemical po-
+tential [39, 40], the enzyme dynamics is given by
+∂tc(x, t) = ∇ ·
+�
+M c ∇
+�
+µ0(c) + χss + χpp
+��
+,
+(1)
+where M denotes a mobility and the term in the square
+brackets is the enzyme flux j(x, t). This gradient dynam-
+ics leads to a gradual minimization of the free energy
+functional from which it is derived [25], a hallmark of
+systems close to thermal equilibrium. Analogously, the
+thermodynamic fluxes of substrates and products can be
+derived from the same free energy functional [25].
+Active systems, however, exhibit processes that break
+detailed balance in protein reaction networks [41, 42]. For
+example, in the conversion of nucleoside triphosphatases
+(NTPases) between an NDP-bound (“product”) and a
+less stable NTP-bound (“substrate”) state, abundance
+of NTP (“fuel”) in solution may shift the equilibrium
+toward the latter [43] and replenish substrate with the
+net rate k2 p. Transient binding of an enzyme (NTPase-
+activating protein, NAP) to substrate can, by lowering
+the activation barrier of hydrolysis, kinetically select the
+fuel-independent reaction pathway and replenish product
+with the net rate k1c s, which follows from the law of
+mass action.
+We assume that these separate reaction
+pathways are far from their respective equilibria, so that
+we can disregard thermodynamic constraints [39, 44] and
+treat the rate constants k1,2 as independent parameters.
+Hence, we write for the dynamics of the substrate and
+the product:
+∂ts = ∇ · (D ∇s + Λ s χs∇c) − k1 c s + k2 p ,
+(2a)
+∂tp = ∇ · (D ∇p + Λ p χp∇c) + k1 c s − k2 p .
+(2b)
+arXiv:2301.00392v1  [physics.bio-ph]  1 Jan 2023
+
+2
+Catalytic substrate turnover is preceded by enzyme
+binding, suggesting effective pairwise attraction, χs < 0.
+Converting substrate into product reduces its affinity for
+the enzymes, χs < χp, leading to unbinding. Note that in
+Eq. (2) we have taken the liberty of formally decoupling
+the diffusion coefficient D from the mobility Λ, thus intro-
+ducing a further source of far from equilibrium dynamics
+by breaking the fluctuation-dissipation relation valid for
+thermal equilibrium systems. In the present context, this
+is the Einstein-Smoluchowsky relation D = Λ kBT [45].
+With the aim of simplifying the analysis, in the present
+work we consider Λ = 0, an approximation that is valid
+for weak FH parameters χs,p and will be addressed else-
+where [46].
+For our initial exploration of the dynamics, we con-
+sider a finite-sized domain [−L, L] in a one-dimensional
+(1d) geometry with no-flux boundary conditions at
+x = ±L.
+A droplet then corresponds to a plateau
+with a high enzyme concentration, surrounded by an
+enzyme-poor phase [Fig. 1].
+If the width of the inter-
+face between these phases, w =
+�
+2κ/r, is much smaller
+than all other length scales, one can use a sharp inter-
+face approximation with piecewise constant concentra-
+tion c. For our analysis, we consider weak interactions,
+|χss| + |χpp| ≪ r (c+ − c−), and therefore approximate
+the enzyme concentrations in the two phases by their
+equilibrium values, c± = ˜c ±
+�
+r/u.
+First, consider a stationary droplet, where a closed an-
+alytic solution of Eqs. (1) and (2) can be obtained [25].
+The stationary state is maintained by a balance of re-
+active and diffusive fluxes [Fig. 1]: In the droplet, the
+enzymes catalyze the conversion of substrate to prod-
+uct, consuming the former and accumulating the lat-
+enzyme 𝑐
+substrate 𝑠
+product 𝑝
+−𝐷𝜕𝑥𝑠
+−𝐷𝜕𝑥𝑝
+𝑘1𝑐𝑠
+𝑘2𝑝
+concentration profiles
+−𝐿
+−𝑅
++𝐿
++𝑅
+FIG. 1.
+Steady state concentration profiles for a 1d
+droplet with no-flux boundary conditions at x = ±L.
+Ar-
+rows indicate reactive (vertical) and diffusive (horizontal)
+fluxes.
+The analytical solutions in the sharp interface ap-
+proximation (lines) match our simulations (dots).
+We use
+c+ as reference concentration and define the characteris-
+tic time τ0 := k−1
+2 , diffusion length in the absence of en-
+zymes l0 :=
+�
+D/k2, and reference energy ϵ0 := rc+. The re-
+maining parameters, c− = 0.1c+, w = 0.1l0, R = l0, L = 5l0,
+M = 100D/ϵ0, k1 = k2/c+, χs = −0.05r, χp = −0.01r, and
+s + p = c+, are fixed for all figures unless stated otherwise.
+−5
+concentration
+5
+10
+−0.2
+0.2
+−10
+𝑣
+𝑐
+𝑠
+𝑝
+𝑐+
+𝑐−
+𝑣 [𝑙0/𝜏0]
+space 𝑥 [𝑙0]
+Δ𝑠 [𝑐+]
+Theory
+1d
+2d
+3d
+5
+10
+1000
+500
+5
+10
+5
+10
+5
+10
+0
+5
+𝑣 [𝑙0/𝜏0]
+𝑀𝜖0/𝐷
+𝑘1𝑐+/𝑘2
+(b)
+(a)
+(c)
+FIG. 2.
+Self-propulsion instability. (a) Analytical profiles
+for a droplet moving with velocity v = 2 l0/τ0. Lighter col-
+ors indicate earlier times. (b) Graphical analysis of the self-
+consistency relation (3) for Mϵ0/D = 1000. The solid curve
+indicates the substrate imbalance ∆s(v), while the slope of
+the dashed line corresponds to the right-hand side of inequal-
+ity (4). Stationary droplets correspond to unstable solutions
+(empty circle), while self-propelling droplets are stable (filled
+circles). (c) Theoretical prediction and simulation results for
+the self-propulsion velocity (color scale) with M and k1 as
+free parameters.
+The solid black lines indicate the critical
+mobility M ∗ (for the explicit closed expression see [25]).
+ter.
+Diffusive fluxes in turn replenish substrate in the
+droplet while expelling product, resulting in concentra-
+tion gradients over the characteristic diffusion lengths
+l± =
+�
+D/(k1c± + k2) inside and outside the droplet, re-
+spectively.
+This leads to cyclic diffusive and reactive
+fluxes such that time-reversal symmetry is broken and
+one has a reaction-driven non-equilibrium steady state.
+If there is an appreciable difference in substrate or
+product concentration between the two droplet inter-
+faces (henceforth referred to as “imbalance”), this gen-
+erally results in a chemical potential gradient that can
+drive droplet motion through a net flux of enzymes
+[Eq. (1)].
+Indeed, using finite element (FEM) simula-
+tions of Eqs. (1) and (2), we find a broad parameter
+regime with ballistic droplet motion [Fig. 2c, Video 1].
+To analytically determine the conditions for the onset of
+this self-propulsion instability, we next study the sharp-
+interface limit of a single 1d droplet in an infinite domain.
+Similar to the analysis of Fisher waves [47], we solve
+Eq. (2) in the reference frame of a moving droplet to
+obtain the concentration profiles of substrates and prod-
+ucts, and then use Eq. (1) to derive a self-consistency
+relation for the droplet velocity v. Specifically, the conti-
+nuity equation (1) implies that the moving steady-state
+enzyme profile, c(z) with z := x − vt and constant ve-
+locity v, can only be maintained if ∂zj(z) = v ∂zc(z)
+holds at all times. The local flux of enzymes is there-
+fore given by j(z) = v [c(z) − c−], and vanishes in the far
+
+3
+field where all concentrations become homogeneous [25].
+The concentration of enzymes in the droplet is en-
+riched by ∆c = c+ − c− with respect to the far-field value
+c(±∞) = c−. While the enzyme flux is driven by the local
+chemical potential and concentration gradients [Eq. (1)],
+integrating the flux over the droplet domain [−R, R]
+yields an expression that depends only on the values
+at the droplet boundaries. In the sharp-interface limit,
+the µ0-term becomes mirror-symmetric relative to the
+droplet center and hence doesn’t contribute to the self-
+consistency relation for the droplet velocity:
+2R ∆c v = −Mc+
+�
+χs∆s(v) + χp∆p(v)
+�
+.
+(3)
+Here, ∆s(v) = s(R) − s(−R) is the substrate concentra-
+tion imbalance between the two opposite sides of the
+droplet, with an analogous expression ∆p(v) for the prod-
+uct. This result quantifies how asymmetric substrate and
+product concentration profiles drive droplet motion.
+Using the closed analytic expressions for the substrate
+concentration profiles [25], shown in Fig. 2a, one can
+graphically solve the self-consistency relation (3) for the
+droplet velocity v [Fig. 2b]. Specifically, for Λ = 0, where
+the total concentration of substrates and products is con-
+stant [Eqs. (2)], the right-hand side of Eq. (3) simplifies
+to Mc+(χp − χs)∆s(v). Based on the graphical form of
+∆s(v), a non-vanishing solution to the self-consistency
+relation for the velocity v exists if [Fig. 2b]:
+∂
+∂v ∆s(v)
+���
+v=0 >
+2R∆c
+Mc+(χp − χs) .
+(4)
+Thus, traveling wave solutions emerge if, for example, the
+mobility M or the difference between the FH parameters
+χp − χs are sufficiently large. Then, enzymes are pulled
+more towards substrates than products, so that enzy-
+matic substrate depletion can induce a chemophoretic
+effect. This theoretical analysis quantitatively explains
+the onset of the self-propulsion instability that we ob-
+served in our simulations; see Fig. 2c for a comparison.
+Droplet movement is driven by asymmetries in the con-
+centration profiles of substrates and products. Droplets
+induce such asymmetries autonomously during self-
+propulsion, but also near impermeable domain bound-
+aries. Figure 3a and Video 2 show the results of FEM
+simulations in a closed domain for four characteristic val-
+ues of the mobility M. Below the self-propulsion thresh-
+old M ∗, the droplet exhibits an overdamped relaxation
+toward the domain center (red), where the concentration
+profiles become symmetric. Thus, the impermeable do-
+main boundaries effectively repel the droplet, due to sub-
+strate depletion and product enrichment within a range
+l−. When increasing M ∗, there is a transition from over-
+damped to underdamped oscillatory relaxation (blue),
+where the relaxation rate λ has a maximum at critical
+damping [Fig. 3b], similar to a damped harmonic oscil-
+lator. Above the self-propulsion threshold M ∗, droplets
+100
+300
+500
+400
+(a)
+−𝐿
++𝐿
+𝑥𝑑
+0
+50
+25
+25
+5
+0
+0
+0
+(c)
+0
+0.1
+10
+1
+5
+(b)
+0
+0
+0
+100
+300 400
+0.3
+0.6
+0.002
+𝑀𝜖0/𝐷
+𝑘1𝑐+/𝑘2
+𝜆 [𝜏0
+−1]
+𝜆 [𝜏0
+−1]
+𝜔 [𝜏0
+−1]
+𝑡 [𝜏0]
+𝑡 [𝜏0]
+𝑡 [𝜏0]
+𝑡 [𝜏0]
+(𝐿−𝑅)/𝑙−
+FIG. 3.
+Self-centering and oscillations.
+(a) Droplet cen-
+ter trajectories in simulations for different values of M; sym-
+bols indicate the position in the diagram shown in Fig. 2c.
+The droplet center is initially at xd(0) = −l0 in a domain of
+size L = 3 l0.
+(b) Decay rate (black dots) and frequency
+(red dots) as functions of M, obtained by fitting the respec-
+tive droplet trajectories. Red triangle indicates overdamped
+regime.
+(c) Relaxation rate λ as a function of k1 (bottom
+axis), for Mϵ0/D = 10 and xd(0) = −0.3l0.
+Top axis re-
+lates the domain size to the length scale of the concentration
+profiles. The analytical predictions (blue line) in the quasi-
+steady-state approximation [25] match our simulations (black
+dots).
+autonomously accelerate to a terminal velocity v [Fig. 2c].
+Instead of droplet self-centering, one then observes oscil-
+lations (green) with frequency ω ≈ v/(L − l−), where the
+domain boundaries cause droplets to slow down and re-
+verse. Droplets with strong self-propulsion (purple) can
+overcome this repulsion and attach to the boundary.
+To elucidate how droplet self-centering depends on the
+reaction rates, we analyzed the overdamped regime, in
+which droplet motion is much slower than the relaxation
+of the substrate and product concentration profiles. This
+time scale separation allows to solve Eqs. (2) analytically
+using a quasi-stationary approximation, where the sub-
+strate and product concentration profiles are in steady
+state with the droplet center xd(t) considered as slowly
+varying.
+The obtained steady-state profiles are asym-
+metric when the droplet is not centered in the domain,
+if and only if the characteristic length l+ is neither much
+larger nor much smaller than the droplet size R (lest
+both droplet interfaces have equal concentrations). These
+asymmetric concentration profiles induce droplet motion
+towards the domain center, see Eq. (3), with a velocity
+v(xd) that we linearize as a function of the distance to
+the domain center [25], |xd| ≪ l−. The resulting approx-
+imation for the relaxation rate λ agrees well with our
+simulations [Fig. 3c], and demonstrates that droplet self-
+centering is fastest for a finite value of k1c+/k2. More-
+over, our analysis and simulations show that droplet self-
+centering proceeds fastest when the distance between the
+droplet interface and the domain boundary is comparable
+to the range of repulsion, L − R ∼ l− [Fig. 3c].
+
+4
+A state with multiple droplets cannot be stable in a
+thermodynamic system.
+Instead, a coarsening process
+driven by interfacial energy minimization takes place,
+causing smaller droplets to shrink and larger droplets
+to grow until there is complete phase separation [11–
+13]. As our system is out of equilibrium, it can result
+in a stable coexistence of multiple droplets. Specifically
+for the system we are considering, larger droplets have a
+larger enzymatic activity and hence consume more sub-
+strate, leading to a reduced substrate concentration at
+their interfaces. This results in a gradient of substrate
+(and product) in the low-concentration phase between
+droplets of different sizes, thereby transporting enzymes
+from the larger to the smaller droplets and thus counter-
+acting the coarsening process [Eq. (1)].
+For a 1d system, the thermodynamic coarsening pro-
+cess described by the CH model is extraordinarily slow
+with the average droplet radius growing only logarithmi-
+cally with time [13]. Hence, one expects that (even weak)
+enzymatic processes can interrupt coarsening.
+Indeed,
+solving the dynamics of multiple droplets analytically in
+the adiabatic limit [25], we find enzyme fluxes between
+pairs of differently sized droplets, which are proportional
+to the difference in the substrate concentration at their
+closest interfaces.
+These currents stop the coarsening
+process and lead to a steady state where the droplets po-
+sition themselves equidistantly to each other to even out
+concentration imbalances between all interfaces [Video 4].
+For 2d and 3d systems, Ostwald ripening is dominated
+by surface tension effects (Laplace pressure) and the en-
+suing law for droplet growth becomes a power law [13].
+In this case, one intuitively expects that the coarsening
+process can be interrupted only if the mass fluxes of the
+enzymes are sufficiently strongly coupled to the concen-
+tration of the products and substrates [48, 49]. Figure 4a
+shows FEM simulation results for a 3d system with a pair
+of droplets, which confirm this intuitive argument. The
+existence of a coarsening threshold for 3d systems can
+also be understood analytically as a balance between a
+coarsening current due to surface tension and a mass flux
+of enzymes driven by reaction-maintained product and
+substrate concentration gradients. We estimate the for-
+mer using the standard Gibbs-Thomson relation [13, 14]
+and the latter by adapting the above results for the 1d
+system [25]. By comparing the two currents we find the
+following estimate for the critical difference between the
+FH parameters ∆χ = χp − χs above which one expects
+droplet coexistence, i.e., interrupted coarsening:
+∆χ∗ = 2
+3
+rw∆c/∆s⋆ �
+l−1
++ cosh(ξ) + l−1
+− sinh(ξ)
+�2
+l−1
++ [sinh(2ξ) − 2ξ] + 2l−1
+− [sinh2(ξ) − ξ2] ,
+(5)
+where ξ := ¯R/l+ is the ratio of the average droplet radius
+to the typical length scale of the concentration gradients
+inside of a droplet, and ∆s⋆ is the difference between
+the local equilibria of substrate in the two phases [25].
+(b)0
+10
+43
+44
+95
+concentration
+0
+0
+10
+0.25
+Δ𝜒 [𝑟]
+𝑐−
+𝑐+
+(a)
+𝑘1𝑐+/𝑘2
+𝑡 [𝜏0]
+FIG. 4.
+Coexistence and division of 3d droplets. We consider
+Mϵ0/D = 10 and w = 0.05l0. Scale bars indicate unit length
+l0 :=
+�
+D/k2. (a) Simulated pairs of droplets with different
+initial radii R1 = 1.1l0 and R2 = 0.9l0 either stay separated
+(cyan regime) or coalesce (yellow regime) depending on k1 and
+on the difference in FH parameters ∆χ = χp − χs. Solid black
+line corresponds to analytical estimate, Eq. (5), for ∆χ∗(k1).
+(b) Simulation snapshots demonstrating a droplet division in
+3d. We observed droplet divisions only for very strong attrac-
+tion of enzymes towards substrates, here χs/r = −0.5.
+Notwithstanding the partially heuristic nature of the
+derivation, our estimate yields a good approximation for
+the boundary between coexistence and coarsening in pa-
+rameter space [Fig. 4a].
+In our numerical simulations, we also observed that
+initially spherical droplets can undergo a shape instabil-
+ity and elongate in one direction for large enough values
+of ∆χ [25]. Once sufficiently elongated, 3d droplets form
+a neck and divide [Fig. 4, Video 5], which we speculate to
+occur through a pearling instability driven by surface ten-
+sion [50] independent of the preceding shape instability.
+This droplet division process is driven by intermolecular
+interactions that induce conservative enzyme fluxes, as
+opposed to Ref. [21] where the droplet material is cycli-
+cally produced and degraded, leading to non-conservative
+fluxes and droplet growth.
+We have analyzed the nonequilibrium dynamics of
+enzyme-enriched condensates, whose enzymatic activity
+guides
+the
+generation
+of
+inhomogeneous
+substrate
+and product concentration profiles that, in turn drive
+condensate motion.
+Conceptually, this corresponds
+to a feedback mechanism in which, for example, an
+NTPase such as PomZ undergoing a cycle of hydrolysis
+(s → p, catalyzed by a NAP c) and nucleotide exchange
+(p → s) generates concentration gradients of its two
+different chemical states (s and p) that drive droplet
+movement through a process akin to chemophoresis. Our
+results show that such a generic mechanism results in
+equidistant positioning of condensates in closed domains,
+persistent condensate motion, and even shape insta-
+bilities that lead to condensate division. We speculate
+that this mechanism, in its basic form, may be relevant
+for processes like midcell localization of protein clusters
+in some prokaryotic cells [9, 26], directed motion of
+partition complexes [28], equidistant placing of plasmids
+
+5
+along nucleoids [51],
+and maybe even transcription
+regulation [52].
+We thank Dominik Schumacher and Lotte Søgaard-
+Andersen for helpful discussions.
+We acknowledge fi-
+nancial support by the German Research Foundation
+(DFG) through TRR 174 (Project ID No. 269423233)
+and SFB1032 (Project ID No. 201269156) and the Ex-
+cellence Cluster ORIGINS under Germany’s Excellence
+Strategy (EXC-2094-390783311). AG was supported by
+a DFG fellowship through the Graduate School of Quan-
+titative Biosciences Munich (QBM). During his time
+at the Massachusetts Institute of Technology, AG was
+supported by the National Science Foundation (NSF)
+through grant number 2044895. IM has received funding
+from the European Union’s Framework Programme for
+Research and Innovation Horizon 2020 under the Marie
+Sk�lodowska-Curie Grant Agreement No. 754388 (LMU
+Research Fellows) and from LMU excellent, funded by
+the Federal Ministry of Education and Research (BMBF)
+and the Free State of Bavaria under the Excellence Strat-
+egy of the German Federal Government and the L¨ander.
+∗ These authors contributed equally to this work.; Present
+address: Sorbonne Universit´e, CNRS, Institut de Biolo-
+gie Paris-Seine (IBPS), Laboratoire Jean Perrin (LJP),
+F-75005 Paris, France
+† These authors contributed equally to this work.; Present
+address: Institute for Medical Engineering and Science,
+Massachusetts Institute of Technology, Cambridge, MA
+02139, United States; andriy.goychuk@gmail.com
+‡ frey@lmu.de
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+Enzyme-enriched condensates show self-propulsion, positioning,
+and coexistence
+Leonardo Demarchi,1, ∗ Andriy Goychuk,1, † Ivan Maryshev,1 and Erwin Frey1, 2, ‡
+1Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience,
+Department of Physics, Ludwig-Maximilians-Universit¨at M¨unchen,
+Theresienstraße 37, D-80333 M¨unchen, Germany
+2Max Planck School Matter to Life, Hofgartenstraße 8, D-80539 M¨unchen, Germany
+CONTENTS
+I. Nonequilibrium enzyme dynamics coupled to product and substrate
+3
+II. Numerical simulations of single droplet dynamics
+5
+A. Self-propulsion of a droplet
+6
+B. Self-centering of a droplet in a finite domain
+6
+III. Analytic theory of a single droplet in a one-dimensional domain
+8
+A. Substrate and product redistribution by a stationary droplet
+8
+B. Substrate and product redistribution by a moving droplet
+11
+C. Gradients in substrate and product concentration drive droplet motion
+12
+D. A self-consistency relation for the self-propulsion instability
+13
+E. Quasi-steady state approximation for droplet self-centering
+16
+IV. Enzymatic activity of droplets leads to coexistence and arrests the coarsening
+process
+18
+A. Localization of multiple coexisting droplets
+19
+B. Coexistence of droplets in a one-dimensional domain
+19
+∗ These authors contributed equally to this work.; Present address: Sorbonne Universit´e, CNRS, Institut
+de Biologie Paris-Seine (IBPS), Laboratoire Jean Perrin (LJP), F-75005 Paris, France
+† These authors contributed equally to this work.; Present address:
+Institute for Medical Engineer-
+ing and Science, Massachusetts Institute of Technology, Cambridge, MA 02139, United States; an-
+driy.goychuk@gmail.com
+‡ frey@lmu.de
+arXiv:2301.00392v1  [physics.bio-ph]  1 Jan 2023
+
+2
+C. Coexistence of droplets in a three-dimensional domain
+23
+D. Droplet division
+26
+V. Multi-component condensates
+28
+VI. Supplemental Videos
+30
+References
+32
+
+3
+I.
+NONEQUILIBRIUM ENZYME DYNAMICS COUPLED TO PRODUCT AND
+SUBSTRATE
+We consider a scenario where there is an interplay between equilibrium phase separation
+and nonequilibrium chemical reactions. To model the aspect of equilibrium phase separa-
+tion, we construct a free energy functional that describes a regular solution consisting of
+solvent, enzymes, substrates, and products. For simplicity, we choose the concentrations of
+substrates, s(x, t), and products, p(x, t), to be very small, so that they each contribute to
+the entropy independent of other concentrations. Then, gradients in the concentration of
+substrates and products are also small, so that we can neglect the corresponding interfacial
+energy terms. Furthermore, we assume that the interactions between solvent, substrates,
+and products are negligible. Under these assumptions, neither the substrate nor the product
+molecules can show phase separation. In contrast, we expand the free energy functional that
+describes the entropy and interactions between solvent and enzymes1, which are present at
+much larger concentrations than substrates and products and are assumed to exhibit phase
+separation, up to fourth order in the enzyme concentration, c(x, t), and use the Ginzburg-
+Landau functional
+f0(c) = u
+4(c − ˜c)4 − r
+2(c − ˜c)2 + κ
+2|∇c|2 ,
+(S1)
+with a positive stiffness, κ > 0, and parameters u > 0 and r > 0 (corresponding to a
+double-well potential).
+Finally, we parameterize enzyme–substrate and enzyme–product
+interactions with the Flory-Huggins (FH) parameters χs and χp, respectively. In summary,
+the statistics of the system is characterized by the following effective free energy functional,
+F =
+�
+ddx
+�
+kBT
+�
+s log(s ν) + p log(p ν)
+�
++ χs c s + χp c p + f0(c)
+�
+,
+(S2)
+where kBT is the thermal energy and d refers to the number of spatial dimensions. For
+simplicity, we have here assumed that substrates and products have the same molecular
+volume ν.
+Given the above constraints on the model parameters, the enzymes will show spontaneous
+phase separation into droplets or labyrinth-like patterns if the average enzyme concentration
+1 We have chosen to perform simulations with comparable values of the three concentration fields, but one
+can rescale s and p without changing the dynamics, by rescaling χs,p and Λ accordingly. Fig. S1 shows
+the results of simulations with rescaled parameters.
+
+4
+lies in the spinodal regime, or can separate through nucleation and growth in the binodal
+regime. For our simulations, we initialize the system with a single predefined droplet to
+study its dynamics, or with multiple droplets to study their coarsening dynamics.
+An exchange of particles (substrates, products, or enzymes) modifies the free energy of
+the system and is thus associated with the following chemical potentials:
+µs = δF
+δs = kBT
+�
+log(s ν) + ν−1�
++ χs c ,
+(S3a)
+µp = δF
+δp = kBT
+�
+log(p ν) + ν−1�
++ χp c ,
+(S3b)
+µc = δF
+δc = µ0(c) + χs s + χp p ,
+(S3c)
+where we have defined the chemical potential of the Cahn-Hilliard model,
+µ0(c) := u (c − ˜c)3 − r (c − ˜c) − κ ∇2c
+(S4)
+to simplify our notation. For our analysis, we consider a scenario where enzymes interact
+with substrates and products weakly, |χss| + |χpp| ≪ r (c+ − c−), where c± refers to the
+concentrations in the high-density and in the low-density phase, respectively. Then, the
+chemical potential µ0 dominates the phase separation of enzymes and maintains their sharp
+concentration profile. Following the general ideas of nonequilibrium thermodynamics [1, 2],
+gradients in the chemical potentials (S3) drive conservative currents that gradually minimize
+the free energy functional F,
+�
+����
+js
+jp
+jc
+�
+���� = −
+�
+����
+Λ s
+0
+0
+0
+Λ p
+0
+0
+0
+Mc
+�
+���� ·
+�
+����
+∇µs
+∇µp
+∇µc
+�
+���� ,
+(S5)
+where substrates and products are for simplicity assumed to have identical mobility Λ, and
+enzymes have mobility M (Onsager coefficients). Equation (S5) can be interpreted as the
+local concentration of each species multiplied with an average drift velocity induced by
+driving forces ∇µ, so that the diagonal entries in the response matrix are the quotient of
+the local concentration and the viscous friction of each species.
+We consider a system that is driven out of equilibrium by chemical reactions that are
+
+5
+not derived from the free energy functional F and therefore provide an external energy
+influx. In particular, we consider a scenario where enzymes catalyze turnover of substrates
+into products with rate k1c s, while products gradually decay into substrates with rate k2 p.
+Then, the dynamics of substrates, products, and enzymes is given by:
+∂ts + ∇ · js = k2 p − k1 c s ,
+∂tp + ∇ · jp = k1 c s − k2 p ,
+and
+∂tc + ∇ · jc = 0 .
+(S6)
+Taking everything together, one arrives at Eqs. (2) and (3) of the main text:
+∂tc = ∇ ·
+�
+Mc∇
+�
+µ0(c) + χs s + χp p
+��
+,
+(S7a)
+∂ts = D∇2s − k1cs + k2p + Λχs∇ · (s∇c) ,
+(S7b)
+∂tp = D∇2p + k1cs − k2p + Λχp∇ · (p∇c) ,
+(S7c)
+where we have related the mobility of substrates and products to their diffusion coefficient by
+the Einstein relation, D = ΛkBT. In the following, we take the liberty of formally decoupling
+the diffusion coefficient D from the mobility Λ. In doing so, we introduce a further source
+of far from equilibrium dynamics by breaking the fluctuation-dissipation relation valid for
+thermal equilibrium systems.
+II.
+NUMERICAL SIMULATIONS OF SINGLE DROPLET DYNAMICS
+We solved the system of partial differential equations (S7) numerically; the code is avail-
+able on [3]. To that end, we used the implicit Euler method for time discretization and
+the finite element method for spatial discretization; for the latter, we used the FEniCS li-
+braries [4]. We considered a finite-sized domain with no-flux boundary conditions, which in
+one dimension is parameterized by x ∈ [−L, L]. As initial conditions, we chose uniform con-
+centration profiles for the substrates and products, and controlled the average concentration
+of these two species s(x) + p(x) = n. Furthermore, we initialized the enzyme concentration
+profile to resemble a single droplet with a sharp interface:
+csharp(x) =
+�
+�
+�
+�
+�
+c+ ,
+for x ∈ [−R, R] , and
+c− ,
+otherwise.
+(S8)
+
+6
+As indicated in the main text, we use c+ as reference concentration and define the char-
+acteristic time τ0 := k−1
+2 , diffusion length in the absence of enzymes l0 :=
+�
+D/k2, and
+reference energy ϵ0 := rc+.
+The remaining parameters, c− = 0.1c+, w = 0.1l0, R = l0,
+L = 5l0, M = 100D/ϵ0, k1 = k2/c+, χs = −0.05r, χp = −0.01r, and s + p = c+, are fixed
+unless stated otherwise. We then simulated the system for a total time of 100 τ0 to let the
+system reach a steady state. Figure 1 in the main text shows that the enzymes maintain
+a single droplet with a steep interface, enriching products at the expense of substrates in
+regions with high enzyme concentration. Outside of the droplet, where the enzyme concen-
+tration is low, substrates are replenished at the expense of products.
+A.
+Self-propulsion of a droplet
+Our simulations demonstrate that droplets can self-propel and sustain motion for a wide
+range of parameters, in 1d [Fig. 2 in the main text, Supplemental Video 1] as well as 2d and
+3d [Supplemental Video 1]. We initialized droplets at the origin of an interval of half-length
+L = 10 l0 in 1d and at the center of a circular domain of radius Lr = 7 l0 in 2d. To speed
+up simulations in 3d, we chose a rotationally invariant cylindrical coordinate system with
+radius Lr = 4 l0 and height Lz = 7 l0, which reduced the number of degrees of freedom but
+also constrained the space of permitted solutions. In all of our simulations, breaking the
+symmetry of the droplet requires an initial perturbation. We provided such a perturbation
+through spatial inhomogeneities in the concentration profiles of substrates and products,
+which are small compared to the average total concentration n := ⟨s(x) + p(x)⟩x.
+B.
+Self-centering of a droplet in a finite domain
+In a finite domain, our simulations show that a self-propelling droplet can either reach
+and adhere to one of the domain boundaries, or exhibit oscillatory motion by reorienting at
+the domain boundaries [Fig. 3a in the main text]. Reorientation at the domain boundaries
+is mediated by an effective repulsion, which has the following mechanistic origin. In our
+simulations, we observe that the enzymatic depletion of substrate is enhanced in the vicinity
+of a domain boundary, where substrate resupply through diffusive currents is limited. The
+resulting substrate concentration gradients induce net enzyme currents, through attractive
+
+7
+𝑀𝜖0/𝐷
+𝑣 [𝑙0/𝜏0]
+𝑘1𝑐+/𝑘2
+Theory
+1d
+2d
+3d
+5
+10
+1000
+500
+5
+10
+5
+10
+5
+10
+0
+5
+FIG. S1. Theoretical prediction and simulation results for the self-propulsion velocity (color scale)
+with M and k1 as free parameters. The solid black lines indicate the critical mobility M∗. We con-
+sider s + p = 0.01c+, χs = −5r and χp = −r, so that the concentrations of substrate and product
+are comparatively small. These results are analogous to Fig. 2c in the main text.
+substrate–enzyme interactions, away from the domain boundary.
+Next, we study how a droplet that does not meet the parameter criteria for self-propulsion
+[discussed in detail in section III D “A self-consistency relation for the self-propulsion
+instability”] will position itself in the domain.
+To that end, we considered a droplet
+whose center xd(0) = −l0 is initially offset from the domain center.
+We initialized the
+distribution of substrates and products in the steady state that is reached in the absence
+of interactions, χs,p = 0, by performing “pre-simulations” for a duration of 1000 τ0. Then,
+we introduced interactions χs,p ̸= 0 and studied the resulting trajectory of the droplet
+center in a one-dimensional domain as a function of time, xd(t) [Fig. 3a in the main text,
+1d droplet dynamics shown in Supplemental Video 2 while 2d and 3d droplet dynamics
+shown in Supplemental Video 3]. To improve the performance of the simulations, we used
+adaptive time stepping, and confirmed that the total simulation time is sufficiently long for
+the droplets to either attach to the boundary, perform several oscillations, or relax to the
+domain center. For droplets that do not exhibit self-propulsion, instead of sustained oscil-
+lations we observed gradual localization to the domain center akin to a damped harmonic
+oscillator.
+To account for both self-sustained and damped oscillatory motion, we fitted
+each droplet trajectory with an exponentially damped sinusoidal curve using the library
+LMFIT [5]: xd(t) = A e−λt cos(ωt + φ), where A is the amplitude, λ the decay rate, ω the
+frequency and φ the initial value of the phase. Trajectories where the droplet attaches to
+the domain boundary cannot be fitted in such a way, and were therefore excluded. The
+resulting estimates for the decay rate and for the frequency of the oscillations are shown in
+
+8
+−𝐿
++𝐿
+𝑥𝑑
+50
+25
+25
+0
+0
+0
+𝑡 [𝜏0]
+𝑡 [𝜏0]
+𝑡 [𝜏0]
+𝑀𝜖0/𝐷 = 100
+𝑀𝜖0/𝐷 = 300
+𝑀𝜖0/𝐷 = 400
+FIG. S2. Comparison of the droplet trajectories from the simulations (dots) with the best fitting
+curves (solid lines), for the panels shown in Fig. 3a in the main text.
+Fig. 3b in the main text as functions of the droplet mobility M.
+III.
+ANALYTIC THEORY OF A SINGLE DROPLET IN A
+ONE-DIMENSIONAL DOMAIN
+For our theoretical analysis in the present paper, we restrict ourselves to a 1d system. A
+2d and 3d analysis is also possible with a semi-analytical approach, but exceeds the scope of
+the present paper and will be published elsewhere [6]. The idea of our theoretical analysis
+rests on two pillars: (i) In all our simulations, we observed that the droplet maintains a
+steep interface. This suggests that one can well describe our system analytically with a
+sharp-interface approximation.
+(ii) Similar to the analysis of Fisher waves [7], we treat
+moving droplets through a transformation into the co-moving frame. As we discuss next,
+these considerations allow us to derive the concentration profiles of substrates and products,
+as well as a self-consistency relation for the droplet velocity v.
+A.
+Substrate and product redistribution by a stationary droplet
+To perform our theoretical analysis, we first determine the steady-state concentration
+profiles of substrates and products in response to the presence of an enzymatic droplet;
+see Eqs. (S7b) and (S7c). Because the enzyme concentration profile is well described by
+a sharp-interface approximation, Eq. (S8), for now we do not need to explicitly study the
+dynamics of the enzymes, Eq. (S7a).
+In the scenario of nonreciprocal interactions with Λ = 0, by summing Eqs. (S7b) and (S7c)
+
+9
+product 𝑝
+substrate 𝑠
+concentration profiles
+−𝐿
+−𝑅
++𝐿
++𝑅
+left subdomain
+𝑐(𝑥) = 𝑐−
+right subdomain
+𝑐(𝑥) = 𝑐−
+center
+𝑐(𝑥) = 𝑐+
+FIG. S3.
+Illustration of the three sub-domains in the sharp-interface approximation c(x) =
+csharp(x). The total concentration of substrates and products is conserved, s(x, t) + p(x, t) = n. In
+each subdomain, we then only need to solve a Helmholtz equation.
+one finds that the total density of substrates and products is uniform in space, s(x, t) +
+p(x, t) = n; the general scenario with Λ ̸= 0 will be analyzed elsewhere [6]. By substituting
+this conservation law into Eq. (S7b), we find that the steady-state distribution of substrates
+is determined by
+D ∂2
+xs(x) −
+�
+k1csharp(x) + k2
+�
+s(x) + k2n = 0 ,
+(S9)
+where csharp(x) refers to the concentration profile of enzymes in the sharp-interface approx-
+imation [Eq. (S8)]. The resulting steady-state distribution of substrates also defines the
+concentration profile of products p(x, t) = n − s(x, t).
+Because the enzyme concentration is piecewise constant in the sharp-interface approxi-
+mation, Eq. (S9) reduces to a Helmholtz equation defined on three subdomains [Fig. S3].
+The concentration profiles of substrates and products then have different characteristic diffu-
+sion lengths, l+ :=
+�
+D/(k1c+ + k2) inside (center) and l− :=
+�
+D/(k1c− + k2) outside (left
+and right) of the droplet.
+Solving the corresponding Helmholtz equation, Eq. (S9), the
+distribution of substrates for a stationary droplet is given by
+s(x) =
+�
+�
+�
+�
+�
+�
+�
+k2 n
+D l2
++ + 2 Ain cosh
+� x
+l+
+�
+,
+for ∥x∥ ≤ R ,
+k2 n
+D l2
+− + Aout exp
+�
+−|x|
+l−
+�
++ Bout exp
+�|x|
+l−
+�
+,
+otherwise,
+(S10)
+where Ain/out and Bout are integration constants. We determined these integration constants
+
+10
+by imposing smoothness and continuity of the concentration profiles at the droplet interfaces
+x = ±R, which separate different subdomains, as well as no-flux boundary conditions at the
+domain boundaries x = ±L. We provide the full (and rather lengthy) expressions for these
+integration constants in [3]. However, even without having these explicit expressions at hand,
+one can nevertheless analyze generic features of the substrate and product concentration
+profiles dictated by Eq. (S10).
+Because all concentrations must remain finite in the far field x → ±∞, the integration
+constant Bout must vanish (Bout = 0) for an infinitely large domain (L → ∞); see Eq. (S10).
+Therefore, it follows from Eq. (S10) that the substrate and product concentrations far away
+from the droplet (x → ±∞) are given by their local reactive equilibria2, s(±∞) = k2 n
+D l2
+− =
+k2 n
+k1c−+k2, which correspond to the solution of the Helmholtz equation (S9) in the limit D → 0.
+Only the concentrations at the center of the droplet, s(0) = k2 n
+D l2
++ + 2 Ain, are shifted by
+2Ain relative to their local reactive equilibria. The magnitude of this shift depends on the
+relative size of the droplet compared to the characteristic length of the concentration profiles,
+R/l+, and must vanish for very large droplets3 R ≫ l+. The difference between the reactive
+equilibria of substrate at low and at high enzyme concentration is given by:
+∆s⋆ := k2 n
+D
+�
+l2
+− − l2
++
+�
+= k2 n
+D
+k1 ∆c l2
++l2
+−
+D
+.
+(S11)
+Thus, the level of substrate depletion in the droplet is proportional to its enrichment of
+enzymes relative to the surrounding solution, ∆c = c+−c−, and analogously implies different
+characteristic diffusion lengths inside and outside of the droplet, l+ ̸= l−.
+Having discussed these features of the substrate and product concentration profiles, we
+compare our analytic predictions (in the sharp-interface approximation) to our numeric re-
+sults (with a diffuse interface) and find very good agreement [Fig. 1 in the main text].
+Because of this excellent agreement, in the following, we use the sharp-interface approxima-
+tion to also study more intricate scenarios where the droplet becomes mobile and positions
+itself in the domain.
+2 We define the reactive equilibria as the solution of the reaction-diffusion equations for substrates and
+products, Eq. (S7b) and (S7c), in the limit of vanishing transport D → 0 and Λ → 0.
+3 This can be seen by taking the limit of D → 0 in Eq. (S9).
+
+11
+B.
+Substrate and product redistribution by a moving droplet
+We now generalize the results of the former section to an enzymatic droplet that moves
+with constant velocity v, where the enzyme concentration profile can be written in the
+form of a travelling wave c(x, t) = csharp(x − vt); see Eq. (S8). As we have discussed before,
+the total density of substrates and products is uniform in space, s(x, t) + p(x, t) = n, for
+nonreciprocal interactions Λ = 0. Using the substitution z = x − vt to transform Eq. (S7b)
+into the co-moving reference frame of the droplet, the steady-state distribution of substrates
+is then determined by
+D ∂2
+zs(z) −
+�
+k1csharp(z) + k2
+�
+s(z) + k2n = −v ∂zs(z) ,
+(S12)
+which differs from Eq. (S9) only by an advection term.
+To simplify the expressions in
+the following, we introduce the P´eclet number, Pe0 = vR/D, and two modified P´eclet
+numbers that include a correction for the characteristic lengths l± of the concentration
+profiles inside and outside of the droplet, Pe± =
+�
+Pe2
+0 + (2R/l±)2. Furthermore, we note
+that the distribution of substrates must remain finite in the far field z → ±∞. Then, solving
+Eq. (S12) in each of the three subdomains [Fig. S3], imposing smoothness and continuity
+at the subdomain interfaces, and assuming an infinitely large overall domain, yields the
+steady-state distribution of substrates in the co-moving frame of the droplet:
+s(z) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+k2 n
+D l2
++ + Ain exp
+�
+−Pe0 − Pe+
+2
+z
+R
+�
++ Bin exp
+�
+−Pe0 + Pe+
+2
+z
+R
+�
+for ∥z∥ ≤ R ,
+k2 n
+D l2
+− + Aout exp
+�
+−Pe0 − Pe−
+2
+z
+R
+�
+for z < −R ,
+k2 n
+D l2
+− + Bout exp
+�
+−Pe0 + Pe−
+2
+z
+R
+�
+,
+otherwise,
+(S13)
+which also defines the concentration profile of the products p(x, t) = n − s(x, t). As before,
+we determined the integration constants Ain/out and Bin/out by imposing smoothness and con-
+tinuity at the droplet interfaces z = ±R. To that end, we used the Python library Sympy [8]
+for symbolic calculations and confirmed with the computer algebra system Mathematica [9],
+and provide the full expressions in [3].
+The concentration profiles of substrates and products, in the co-moving frame of an
+
+12
+enzymatic droplet with velocity v, are plotted in Fig. 2a of the main text. We observe that
+the droplet enriches the concentration of products at the expense of substrates, which is
+most pronounced at the trailing edge of the droplet. This leads to a difference in substrate
+concentrations between the right and the left edge of the droplet, ∆s(v) := s(R) − s(−R),
+which we quantify after inserting the integration constants [3] in Eq. (S13):
+∆s(v) = ∆s⋆ Pe+Pe0
+�
+cosh(Pe+) − cosh(Pe0)
+�
++ Pe−
+�
+Pe0 sinh(Pe+) − Pe+ sinh(Pe0)
+�
+Pe+Pe− cosh(Pe+) + (Pe2
++ + Pe2
+−) sinh(Pe+)/2
+.
+(S14)
+Here, the substrate concentration difference between the reactive equilibria, ∆s⋆, is given by
+Eq. (S11). Because the total density of substrates and products is uniform in the scenario
+with nonreciprocal interactions (Λ = 0), the difference in product concentrations between
+the right and the left edge of the droplet is given by ∆p(v) = −∆s(v). As we show next, these
+concentration differences can drive a net flux of enzymes from the trailing edge towards the
+leading edge of the moving droplet, thus sustaining its motion. The more generic scenario
+with Λ ̸= 0 exceeds the scope of the present paper and will be published elsewhere [6].
+In short, one finds that Λ ̸= 0 can place further constraints on the parameters to observe
+self-propulsion.
+C.
+Gradients in substrate and product concentration drive droplet motion
+In the discussion so far, we have used the sharp-interface approximation for the enzyme
+concentration profile c(x, t) = csharp(x − vt) of a droplet that moves with constant velocity
+v; see Eq. (S8). Now, we investigate the premises for this sharp-interface approximation to
+be consistent with the enzyme dynamics obeying a continuity equation (S7a). The sharp-
+interface approximation implies that the moving concentration profile of enzymes must be
+in steady state. Using the substitution z = x−vt to transform the continuity equation (S7a)
+into the co-moving reference frame of the droplet, one then has:
+∂z
+�
+Mc(z) ∂z
+�
+µ0(c(z)) + χs s(z) + χp p(z)
+��
+= −v ∂zc(z) ≡ −∂zj(z) .
+(S15)
+
+13
+Indefinite integration of Eq. (S15) yields
+Mc(z) ∂z
+�
+µ0(c(z)) + χs s(z) + χp p(z)
+�
+= −v c(z) + j0 ,
+(S16)
+up to an integration constant j0. To determine this integration constant, we first note that
+far away from the droplet, at z ± ∞, all concentration profiles are homogeneous and the
+enzyme concentration is given by c(±∞) = c−. Comparing these far-field conditions with
+Eq. (S16), one finds that j0 = v c−. To summarize, we have so far
+Mc(z) ∂z
+�
+µ0(c(z)) + χs s(z) + χp p(z)
+�
+= −v
+�
+c(z) − c−
+�
+.
+(S17)
+Now, we use the sharp-interface approximation c(z) = csharp(z); see Eq. (S8). Integrating
+Eq. (S17) over the droplet z ∈ [−R, R] to obtain an expression independent of local gradients,
+we find
+v = − Mc+
+2R ∆c
+�
+χs ∆s(v) + χp ∆p(v)
+�
+,
+(S18)
+where we have defined the enzyme concentration difference ∆c := c+−c−. Note that the bare
+chemical potential of the enzyme, µ0(z), dropped out because it is mirror symmetric in the
+sharp-interface approximation, so that µ0(R) = µ0(−R). Thus, the bare chemical potential
+µ0(z) cannot drive directed net fluxes of enzymes on its own: it instead drives relaxation
+towards an equilibrium state. In contrast, the coupling to substrate and enzymes, which are
+driven out of equilibrium by chemical reactions, drives relaxation towards a non-equilibrium
+steady state. Our result summarized by Eq. (S18) quantifies how asymmetric substrate and
+product concentration profiles (see Eq. (S14)) drive droplet motion.
+D.
+A self-consistency relation for the self-propulsion instability
+In the scenario of nonreciprocal interactions (Λ = 0) where the total density of substrates
+and products is constant, s(z) + p(z) = n, Eq. (S18) simplifies to:
+v = Mc+ ∆χ
+2R ∆c ∆s(v) ,
+(S19)
+
+14
+where ∆χ := χp − χs is a measure for how enzymes are pulled more towards substrates
+than towards products. The droplet velocity is proportional to the overall mobility M of
+enzymes, and for constant ∆s decreases with the overall number of enzymes 2R∆c that are
+translocated.
+The droplet velocity in response to a gradient in the substrate concentration, Eq. (S19),
+and the distribution of substrates in response to a droplet moving with a fixed velocity,
+Eq. (S14), together form a self-consistency relation. This self-consistency relation can be
+solved graphically by identifying a velocity where the left-hand side and the right-hand
+side of Eq. (S19) intersect, as illustrated in Fig. 2b in the main text. We note that the
+substrate concentration difference between the right and the left edge of the droplet, ∆s(v),
+is point symmetric with respect to a reversal of the velocity, ∆s(−v) = −∆s(v), which is
+a feature of the intrinsic symmetry under parity of the system4. Because of this symmetry,
+it is sufficient to discuss a scenario with positive velocities only, v ≥ 0. In the limit of very
+large velocities v → ∞, the substrate and product concentration fields do not have sufficient
+time to respond to the moving enzymatic droplet and therefore remain homogeneous, which
+implies ∆s(∞) → 0.
+Because of the point symmetry of ∆s(v), the self-consistency relation (S19) always per-
+mits a trivial solution with vanishing velocity v = 0. For small velocities, we observe that the
+substrate concentration difference between the leading and the trailing edge of the droplet,
+∆s(v), initially grows with increasing velocity v until it reaches a single maximum, see
+Eq. (S14) and Fig. 2b in the main text. Given these features of the function ∆s(v), two
+additional non-trivial solutions with finite droplet velocity |v| ̸= 0 emerge if the right-hand
+side of Eq. (S19) grows faster than the left-hand side in the limit of small velocities, v → 0:
+1 < Mc+ ∆χ
+2R ∆c ∂v∆s(v)
+����
+v=0
+.
+(S20)
+If a non-trivial solution to the self-consistency relation exists, then an initial inhomogeneity
+of the substrate concentration profile will self-reinforce through a positive feedback loop
+with the motion of the droplet. Because of this feedback loop, the droplet will settle in a
+state that corresponds to the finite velocity v admitted by the self-consistency relation (S19).
+4 One can see this symmetry by making the parity transformations z → −z and v → −v in Eq. (S12), which
+preserves the structure of the equation and corresponds to a mirrored concentration field s(−z). One could
+also argue more heuristically: because there is no global bias for breaking symmetry (polarizing) towards
+the left or to the right, the droplet could go either way with equal velocity.
+
+15
+Substituting Eq. (S14) into Eq. (S20), the criterion to observe self-propelled droplets in our
+simulations can therefore be written as:
+1 < Mc+ ∆χ l−l+
+2RD
+∆s⋆
+∆c
+l+ sinh(2R/l+) + l− cosh(2R/l+) − l− − 2R
+(l2
+− + l2
++) sinh(2R/l+) + 2l+l− cosh(2R/l+) .
+(S21)
+The emergence of two new stable fixed points |v| ̸= 0 as a function of enzyme mobility M
+as control parameter, concomitant with a destabilization of the trivial fixed point v = 0,
+corresponds to a pitchfork bifurcation; see Fig. S4.
+To gain a better understanding on the conditions that are required to observe droplet
+self-propulsion, we study inequality (S21) in the limits of very large or very small droplets:
+1 < Mc+ ∆χ
+2D
+∆s⋆
+∆c
+l−
+R ×
+�
+�
+�
+�
+�
+�
+�
+l+
+l− + l+
+, for R ≫ l± ,
+R2
+l2
++
+, for R ≪ l± .
+(S22)
+As a function of the droplet radius R, the right-hand side of inequality (S22) is clearly non-
+monotonic: it grows with increasing droplet size for small droplets, but then decays with
+increasing droplet size for large droplets. Thus, the self-propulsion instability is suppressed
+both for very large and for very small droplets. Specifically, droplets that are much smaller
+than the characteristic length of the substrate and product concentration profiles, cannot
+build up a sufficient difference in the concentrations of substrates and products between the
+droplet interfaces to sustain self-propulsion. In the opposing limit where droplets are much
+larger than the characteristic length of the substrate and product concentration profiles, the
+diffusion of substrates and products ceases to play a role. Then, substrates and products
+reach their local reactive equilibria, and the droplet also cannot build up a sufficient dif-
+ference in the concentrations of substrates and products between the droplet interfaces to
+sustain self-propulsion. In summary, droplet self-propulsion requires the droplet radius to
+be compatible with the characteristic length of substrate and product concentration profiles,
+and is optimal for R ∼ l+.
+
+16
+𝑀𝜖0/𝐷
+0
+0
+2
+4
+−2
+−4
+200
+400
+600
+800
+1000
+𝑣 [𝑙0/𝜏0]
+FIG. S4. Pitchfork bifurcation for droplet self-propulsion. Tuning enzyme mobility M as control
+parameter, one finds that small enzyme mobilities admit only a single stable solution (solid lines)
+with vanishing droplet velocity. This solution becomes unstable above a critical value of the enzyme
+mobility M⋆ (dashed line), where two new stable solutions with finite velocity appear.
+E.
+Quasi-steady state approximation for droplet self-centering
+From the inequality Eq. (S21), we infer that a droplet can self-propel if certain conditions
+are met, e.g., in the case of large enzyme mobility M. If the droplet exhibits self-propulsion,
+then it will perform oscillatory motion in a closed domain or adhere to one of the domain
+boundaries, as discussed in the main text and in section II B “Self-centering of a droplet in
+a finite domain”. However, even if the conditions for self-propulsion are not met, that is
+when inequality Eq. (S21) is not fulfilled, Eq. (S18) indicates that gradients in the densities
+of substrates and products will still drive droplet motion. Such concentration differences
+between the droplet interfaces can result, for example, from an off-centered position of the
+droplet in its enclosing domain. Then, the droplet will gradually position itself towards
+the domain center. In the following, we study this scenario theoretically, in the overdamped
+limit where the droplet does not overshoot past the domain center. To that end, we consider
+a quasi-steady state approximation where there is a separation of time scales between the
+slow motion of the droplet and the fast relaxation of the substrate and product concentration
+profiles to their pseudo-steady state. We determine the concentration profiles in this pseudo-
+steady state using the Python library Sympy [8] for symbolic calculations and confirmed
+with the computer algebra system Mathematica [9], and provide the full expressions in [3].
+
+17
+We then calculate the substrate and product concentration values at the droplet interfaces,
+and insert them into the self-consistency relation (S18) to determine the resulting droplet
+velocity. This procedure yields the droplet velocity v(xd) as a function of the position of the
+droplet center xd, as plotted in Fig. S5. Figure S5 indicates that the droplet position has a
+single stable fixed point at the center of the domain. Near the domain center, we again use
+the Python library Sympy [8] and the computer algebra system Mathematica [9] to linearize
+the dynamics and find exponential relaxation with the following timescale:
+λ−1 =
+l−R∆c
+Mc+∆χ∆s⋆
+�
+cosh
+�Lfree
+l−
+�
++
+l+ sinh
+� Lfree
+l−
+�
+l− tanh
+� R
+l+
+�
+� �
+sinh
+�Lfree
+l−
+�
++
+l− cosh
+� Lfree
+l−
+�
+l+ tanh
+� R
+l+
+�
+�
+,
+(S23)
+where we have defined Lfree := L−R. We find a very good agreement between these analytic
+results and our finite element simulations [Fig. 3c in the main text].
+To analyze Eq. (S23), we first discuss a scenario where we keep the value of ∆s⋆ fixed.
+Then, Eq. (S23) shows that the characteristic time of self-centering diverges for l− ≪ Lfree
+and for l− ≫ Lfree. Therefore, the typical timescale of finding the domain center is minimal if
+the characteristic length of the concentration profiles outside of the droplet is comparable to
+the typical distance towards the domain boundary, l− ∼ Lfree. Furthermore, Eq. (S23) also
+shows that the characteristic time of self-centering diverges for very large droplets R ≫ l+
+and for very small droplets R ≪ l+. Therefore, droplet self-centering is also optimized if
+R ∼ l+, which allows building up concentration gradients across the droplet. Consistent with
+these arguments for the droplet size, for large domain sizes the relaxation rate Eq. (S23)
+scales as
+λ−1 =
+∆c
+Mc+∆χ∆s⋆ exp
+�2Lfree
+l−
+� R
+4l+
+�
+l− +
+l+
+tanh
+� R
+l+
+�
+� �
+l+ +
+l−
+tanh
+� R
+l+
+�
+�
+,
+(S24)
+which has a minimum as a function of R/l+.
+We next discuss how the features of (S23) relate to Fig. 3c in the main text. Now, the
+value of the concentration difference between the reactive equilibria, ∆s⋆, is not fixed because
+it depends on the catalysis rate k1; see Eq. (S11). When increasing the catalysis rate k1 to
+very high values, the characteristic length l± of the concentration profiles becomes very small
+and one finds that the droplet takes a longer time to find the center of the domain. In the
+opposite scenario, where the catalysis rate k1 is very small, one finds that the droplet also
+
+18
+−2
+−1
+1
+2
+0
+0.00
+−0.02
+0.02
+𝑥𝑑 [𝑙0]
+𝑣 [𝑙0/𝜏0]
+FIG. S5. Droplet velocity as a function of its position in the quasi-steady state approximation, for
+Mϵ0/D = 10. The domain has a half-size of L = 3l0, as in Fig. 3b in the main text.
+takes a longer time to find the domain center because it cannot create significant substrate
+and product concentration gradients (small value of ∆s⋆). Therefore, there is an optimal
+value of the catalysis rate k1 where the droplet finds the center of the domain in the shortest
+amount of time [Fig. 3c in the main text].
+IV.
+ENZYMATIC ACTIVITY OF DROPLETS LEADS TO COEXISTENCE
+AND ARRESTS THE COARSENING PROCESS
+So far we have analyzed the dynamics of a single droplet, and have shown how its en-
+zymatic activity can lead to self-propulsion or self-positioning in a finite domain. Now, we
+extend our analysis to the dynamics of multiple droplets, in the parameter regime where none
+of the droplets self-propel. To that end, we consider small values of the enzyme mobility M
+where inequality Eq. (S21) is not fulfilled. We then analyze under which conditions multiple
+droplets will show arrested coarsening and therefore coexist. Before we proceed with this
+theoretical analysis, we study simulations in the parameter regime where multiple droplets
+coexist without showing self-propulsion, and analyze how the droplets position themselves
+in a finite domain.
+
+19
+A.
+Localization of multiple coexisting droplets
+We have discussed in section II B “Self-centering of a droplet in a finite domain” that enzy-
+matically active droplets exhibit repulsive interactions with domain boundaries, where enzy-
+matic substrate depletion is enhanced. Analogously, two neighboring droplets will strongly
+deplete the substrate in the region between them. The resulting substrate concentration
+gradients will, through attractive enzyme–substrate interactions, drive enzyme fluxes within
+each droplet that lead away from the neighboring droplet. This leads to effective repulsive
+interactions among droplets, which suggest that droplets will position themselves equidis-
+tantly in a finite domain. For a steady state in which N droplets coexist in the same domain
+without showing self-propulsion [Fig. S6], equidistant positioning indicates a replica sym-
+metry in which the domain can be divided into a chain of N identical subdomains, each
+containing only one droplet [Fig. S6]. In such a steady state, there is no net particle ex-
+change across the interfaces between adjacent subdomains, which is equivalent to no-flux
+boundary conditions at each subdomain boundary. Hence, in each individual subdomain,
+the droplet will localize to the center according to sections II B “Self-centering of a droplet
+in a finite domain” and III E “Quasi-steady state approximation for droplet self-centering”.
+Because of this symmetry of replicate subdomains, the distance between one of the domain
+boundaries and the nearest droplet, L/N for a domain of size 2L, will be exactly half of
+the distance between two adjacent droplets, 2L/N. In the next section, we illustrate why
+enzymatic droplets can coexist in the first place.
+B.
+Coexistence of droplets in a one-dimensional domain
+To illustrate the mechanism underlying the coexistence of multiple droplets, we study
+a scenario with only two droplets of differing sizes R1 and R2 ≥ R1 [Fig. S7]. The larger
+droplet depletes more substrate and accumulates more product than the smaller droplet,
+which can be illustrated with the following limiting cases: For droplet sizes much larger
+than the characteristic length of the substrate and product concentration profiles inside of
+the droplet, R2 ≫ l+, the substrate and product concentrations at the droplet boundaries
+are given by their reactive equilibria5. For droplet sizes much smaller than the characteristic
+5 The reactive equilibria correspond to the solution of the reaction-diffusion equations for substrates and
+products, Eq. (S7b) and (S7c), in the limit D → 0 and Λ → 0.
+
+20
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+−15
+−10
+−5
+0
+5
+10
+15
+concentration profiles [𝑐+]
+𝑥 [𝑙0]
+enzyme 𝑐
+substrate 𝑠
+product 𝑝
+FIG. S6.
+Stationary concentration profiles corresponding to 5 droplets coexisting on an inter-
+val, obtained after simulating the system for a total duration of 1000 τ0. Boundaries of replicate
+subdomains are indicated by dotted lines. As initial condition for the enzyme concentration, we
+considered 5 distinct droplets of equal radii R = l0.
+length of the concentration profiles inside of the droplet, R1 ≪ l+, diffusion will homogenize
+the substrate and product concentration profiles. Then, the concentrations at the droplet
+boundaries remain at their ambient concentration values in the far field.
+The resulting
+concentration gradients will, because of strongly attractive enzyme–substrate interactions
+and weaker enzyme–product interactions, drive a net flux of enzymes from the larger to
+the smaller droplet, see Eq. (S7a). Therefore, the size difference between the two droplets,
+∆R = R2 − R1 will shrink over time, indicating coexistence [Supplemental Video 4].
+In the following, we derive the characteristic timescale of droplet size equilibration. For
+droplets that do not exhibit self-propulsion, the dynamics of enzymes is very slow compared
+to the dynamics of substrates and products.
+Therefore, in full analogy to section III E
+“Quasi-steady state approximation for droplet self-centering”, we make a quasi-steady state
+approximation for the fast relaxation of the substrate and product concentration profiles in
+response to the quasi-static enzyme concentration profile:
+c(x) =
+�
+�
+�
+�
+�
+c+
+for ∥x∥ ≤ R1
+and
+∥x − d∥ ≤ R2 ,
+c−
+otherwise,
+(S25)
+where the droplet centers are located at x = 0 and x = d. Given this concentration profile of
+
+21
+concentration profiles
+𝑐(𝑥) = 𝑐−
+𝑐(𝑥) = 𝑐+
+𝑐(𝑥) = 𝑐+
+0 𝑅1
+𝑑
+𝑑−𝑅2
+domain
+FIG. S7. Illustration of the sub-domains in the sharp-interface approximation, for two droplets that
+gradually equilibrate their sizes. Gray areas illustrate subdomains, as discussed in Fig. S3. In each
+subdomain, we only need to solve a Helmholtz equation. To simplify this mathematical problem,
+we construct a scenario where the substrate and product concentration profiles are approximately
+symmetric with respect to the droplet centers.
+Specifically, for our analytical calculations we
+consider a domain that ends at the droplet centers (black vertical lines) and has no-flux boundary
+conditions there. This construction is exact if, for an alternating chain of large and small droplets,
+the substrate and product concentration profiles are symmetric with respect to each droplet center
+(replica symmetry). In our simulations, this would correspond to two droplets that are positioned
+equidistantly in a simulation domain with periodic boundary conditions at the position indicated
+by the dashed lines. For simulation domains with no-flux boundary conditions at the position
+indicated by the dashed lines, our simplification is nevertheless a good approximation.
+enzymes, in the scenario with nonreciprocal interactions (Λ = 0) one can then analytically
+solve Eq. (S9) for the concentration profiles of substrates s(x) and products p(x) = n−s(x).
+This mathematical problem is practically identical to sections III A “Substrate and product
+redistribution by a stationary droplet” and III E “Quasi-steady state approximation for
+droplet self-centering”, as illustrated in Fig. S7, and the full expressions are provided in [3].
+Here, we outline the main assumptions and simplifications of the derivation.
+In principle, for droplets that are close to each other, one observes a net substrate and
+product concentration gradient across each individual droplet. This net concentration gra-
+dient will lead to an effective repulsion between the two droplets, analogous to sections II B
+“Self-centering of a droplet in a finite domain” and IV A “Localization of multiple coex-
+isting droplets”, and prevent coalescence. Here, however, we are interested in the particle
+exchange currents between the two droplets. Therefore, we focus on the substrate and prod-
+uct concentration differences between the nearest interfaces of the two different droplets,
+∆s := s(d−R2)−s(R1) and ∆p := p(d−R2)−p(R1). To simplify our analysis, we construct
+
+22
+a scenario where the substrate and product concentration profiles are approximately sym-
+metric with respect to each individual droplet, so that each droplet will remain immobile as
+discussed in section IV A “Localization of multiple coexisting droplets”. Specifically, we con-
+sider no-flux (reflective) boundary conditions at the droplet centers [Fig. S7], which would
+be exact for a periodic chain of big and small droplets. Then, we solve for the concentration
+profiles of substrates and products in the domain [0, d], and find the substrate concentration
+difference between the closest interfaces of the two droplets [3],
+∆s = −∆s⋆∆R
+l+
+�
+sinh
+� R
+l+
+�
++
+l+ cosh
+� R
+l+
+�
+l− tanh
+� Lfree
+l−
+�
+�−1 �
+cosh
+� R
+l+
+�
++
+l− sinh
+� R
+l+
+�
+l+ tanh
+� Lfree
+l−
+�
+�−1
+, (S26)
+for small differences in the droplet radii ∆R := R2 − R1. Here, we have defined the mean
+droplet radius, R := (R1+R2)/2, and the free half-distance between droplets Lfree := d/2−R.
+These concentration differences drive an exchange current of enzymes between the two
+droplets, which we in analogy to III C “Gradients in substrate and product concentration
+drive droplet motion” determine by integrating the continuity equation Eq. (S7a) over the
+inter-droplet domain [R1, d − R2],
+j× 2Lfree = −Mc−
+�
+χs ∆s + χp ∆p
+�
+,
+(S27)
+where we have omitted the contribution of the Cahn-Hilliard chemical potential µ0(z) as
+explained next.
+In general, the interaction-driven exchange flux (S27) will be superimposed by a coarsen-
+ing current that stems from the Cahn-Hilliard chemical potential µ0(z), which drives slow
+coarsening in one dimension. In the sharp-interface limit and in 1d, however, coarsening
+becomes infinitely slow. Therefore, in 1d the flux given by Eq. (S27) can easily drive a
+dynamics that is opposite to coarsening, by transporting material from the larger droplet
+to the smaller droplet. The size difference between the two droplets then gradually changes
+with time, and is in a finite 1d domain with no-flux boundary conditions determined by6
+j× = ∆c
+�
+∂t(R2 − R1)
+�
+= ∆c ∂t∆R. Taken together, the size difference between the two
+6 In this case, the flux is given by ∆c ∂t(2R2) = −∆c ∂t(2R1) = j×. For periodic domains, an additional
+factor of 2 would enter because one has particle exchange with twice as many droplets.
+
+23
+droplets will evolve as follows:
+∂t∆R = M c− ∆χ
+2 Lfree ∆c ∆s ,
+(S28)
+where we have used ∆p = −∆s and defined ∆χ := χp − χs. After inserting Eq. (S26) into
+Eq. (S28), one finds that the size difference decays exponentially with a characteristic
+timescale
+λ−1 = 2 Lfree l+ ∆c
+M c− ∆χ ∆s⋆
+�
+sinh
+� R
+l+
+�
++
+l+ cosh
+� R
+l+
+�
+l− tanh
+� Lfree
+l−
+�
+� �
+cosh
+� R
+l+
+�
++
+l− sinh
+� R
+l+
+�
+l+ tanh
+� Lfree
+l−
+�
+�
+. (S29)
+This result is equivalent to Eq. (S23) under the replacements c+ ↔ c−, l+ ↔ l−, and Lfree ↔
+R, because it corresponds to the self-centering of an inverted droplet7 Thus, our discussion
+for the self-centering of droplets, see section III E “Quasi-steady state approximation for
+droplet self-centering”, also applies here. In particular, droplet size equilibration is fastest
+when the length scales of the droplets, concentration profiles of substrates and products,
+and the distance between droplets are compatible.
+Comparing our analytical results to
+simulations, we find good agreement [Fig. S8].
+C.
+Coexistence of droplets in a three-dimensional domain
+In the previous section, we have seen that enzyme–substrate and enzyme–product inter-
+actions can easily oppose and reverse the coarsening process in 1d, which becomes infinitely
+slow in the sharp-interface limit in 1d. For 2d and 3d droplets, an additional effect comes
+into play that arises from the curvature of the droplet interfaces: the Laplace pressure leads
+to an evaporation of small droplets and stabilization of large droplets, thus greatly acceler-
+ating the coarsening process [10–12]. This means that in 2d and 3d the interaction-driven
+exchange currents between different droplets are superimposed by much stronger coarsening
+currents than in 1d, which narrows down the parameter regime in which one can observe
+arrested phase separation. In the following, we derive a criterion to still observe arrested
+coarsening in 2d and 3d.
+7 In comparison to Eq. (S23), there is an additional factor of 2. This factor would drop out in a periodic
+domain, where droplet size equilibration is twice as fast because each droplet can exchange particles with
+two neighbors.
+
+24
+×10−3
+0.1
+1
+0
+2
+4
+6
+10
+𝜆 [𝜏0
+−1]
+𝑘1𝑐+/𝑘2
+FIG. S8. Droplet size equilibration rate as a function of the reaction rate k1. The solid line indicates
+the expression obtained in the quasistatic limit (S29), and agrees well with the results of numerical
+simulations (dots). The droplets are initially positioned at x1(0) = −2.5l0 and x2(0) = 2.5l0 in
+the domain [−5l0, 5l0], have radii R1(0) = 0.9l0 and R2(0) = 1.1l0, and have an interface width of
+w = 0.05l0.
+We consider a droplet with radius R and small interface width w =
+�
+2κ/r ≪ R, which
+has the homogeneous free energy density f(c) = u
+4(c−˜c)4− r
+2(c−˜c)2; see Eq. (S1). The surface
+tension of this droplet is given by γ = 2r2w/(3u) and induces a Laplace pressure, which
+leads to an increase in the enzyme concentration just outside of the droplet as described by
+the Gibbs-Thomson relation [12, 13]:
+δc(R) = 2γ
+R
+1
+∆c f ′′(c−) = 1
+6
+w
+R ∆c ,
+(S30)
+where f ′′(c−) = 2r. The increase in the enzyme concentration affects the Cahn-Hilliard
+chemical potential outside of the droplet, µ0 ≃ 2r δc, and is more pronounced for small
+droplets, see Eq. (S30). This leads to a difference in the Cahn-Hilliard chemical potential
+between the two droplets, ∆µ0, which drives diffusive transport of enzymes from small
+
+25
+towards large droplets:
+j0 ≈ −Mc−
+∆µ0
+d
+≈ −2 r Mc−
+δc(R2) − δc(R1)
+d
+= r Mc−w ∆c
+3 R1 R2 d ∆R ,
+(S31)
+where we have assumed that the droplets are far apart, d ≫ R1,2, and defined ∆R := R2−R1.
+As discussed in section IV B “Coexistence of droplets in a one-dimensional domain”, the
+enzymatic activity of the droplets locally depletes substrate and accumulates product, which
+results in substrate and product concentration gradients that can contribute to the exchange
+flux between different droplets, see Eq. (S27). For droplets that are far apart, d ≫ R1,2, one
+then has
+j× ≈ M c− ∆χ
+d
+∆s ,
+(S32)
+where we have again used ∆p = −∆s, and defined ∆χ := χp − χs. If the droplets are far
+apart, then we can determine the local concentration profiles of substrates and products
+independently for each droplet, by solving a Helmholtz equation in spherical coordinates8:
+D
+r2 ∂r
+�
+r2∂rs(r)
+�
+−
+�
+k1c(r) + k2
+�
+s(r) + k2n = 0 ,
+(S33)
+where we make a sharp-interface approximation for the concentration of enzymes, c(r) =
+csharp(r), see Eq. (S8). We determine the substrate and product concentration profiles as
+described in section III A “Substrate and product redistribution by a stationary droplet”;
+the full expressions are provided in [3]. The difference between the substrate concentration
+values at the interfaces of two droplets with a small difference in size, ∆R ≪ l+, are then
+given by:
+∆s ≈ −∆s⋆ l−1
++ [sinh(2R/l+) − 2R/l+] + 2l−1
+− [sinh2(R/l+) − (R/l+)2]
+2R2 �
+l−1
++ cosh(R/l+) + l−1
+− sinh(R/l+)
+�2
+∆R .
+(S34)
+As discussed in section IV B “Coexistence of droplets in a one-dimensional domain”, the
+larger droplet typically depletes more substrate, leading to a substrate concentration dif-
+ference proportional to the size difference: ∆s ∝ −∆R.
+This concentration difference
+drives a flux of enzymes from the larger to the smaller droplet [Fig. S9]. We now substitute
+8 A general analytic solution for droplets that are close to each other would require solving the Helmholtz
+equation in bispherical coordinates.
+However, the Helmholtz equation is not separable in bispherical
+coordinates.
+
+26
+4
+2
+0
+2
+4
+𝑡 = 75 𝜏0
+enzyme concentration 𝑐 [𝑐+]
+radius 𝑟 [𝑙0]
+𝑧 [𝑙0]
+2
+2
+0
+1
+1
+0.2
+0.4
+0.6
+0.8
+1.0
+FIG. S9. Snapshot of a simulation of two coexisting droplets in 3d. The color map indicates the
+local enzyme concentration, and the white arrows are proportional to the local mean velocity of the
+enzymes jc(x)/c(x). The simulation was performed in a rotationally invariant cylindrical domain
+of radius Lr = 2.5l0 and half-height Lz = 5l0. Parameters: Mϵ0/D = 10, w = 0.05l0 as in Fig. 4a
+in the main text, with k1c+/k2 = 1 and χs/r = −0.21.
+Eq. (S34) into the expression for the interaction-driven exchange fluxes that oppose coarsen-
+ing, Eq. (S32), and ask under which conditions these fluxes may dominate (j× +j0 < 0) over
+the coarsening currents given by Eq. (S31). This comparison yields a criterion to observe
+arrested coarsening, which is fulfilled if the interactions are sufficiently strong:
+∆χ ≳
+2rw∆c
+�
+l−1
++ cosh( ¯R/l+) + l−1
+− sinh( ¯R/l+)
+�2
+3∆s⋆ �
+l−1
++ (sinh(2 ¯R/l+) − 2 ¯R/l+) + 2l−1
+− (sinh2( ¯R/l+) − ( ¯R/l+)2)
+� .
+(S35)
+Comparing this analytical criterion with numerical simulations, we find good agreement
+[Fig. 4a in the main text, dynamics shown in Supplemental Video 4].
+D.
+Droplet division
+We have seen that the enzymatic activity of the droplet, coupled with enzyme–substrate
+and enzyme–product interactions, can oppose and stop coarsening. Then, could this mech-
+anism also lead to a shape instability and to divisions of droplets? In fact, in 2d and 3d,
+substrate depletion is enhanced for smaller curvatures of the droplet interface and reduced for
+
+27
+larger curvatures of the droplet interface, by having a smaller interface over which substrate
+can be resupplied from the environment relative to the enclosed volume where substrate is
+depleted by enzymes. This effect is analogous to the stronger substrate depletion by larger
+spherical droplets when compared to smaller droplets, and can drive a net flux of enzymes
+from regions with a small curvature of the droplet interface towards regions with a large
+curvature of the droplet interface. Regions with a large curvature will then move outwards
+while regions with a small curvature will move inwards, elongating the droplet and further
+increasing the differences in curvature (positive feedback mechanism), while the droplet vol-
+ume remains conserved. We observed such droplet elongation in simulations for sufficiently
+large values of the interaction parameter ∆χ [Fig. S10].
+Δ𝜒 [𝑟]
+0.00
+0
+5
+10
+0.25
+0.50
+𝑘1𝑐+/𝑘2
+FIG. S10. Results of numerical simulations starting with a single droplet. The droplet can either
+remain spherical (yellow regime) or start elongating (cyan regime) depending on the values of k1
+and ∆χ. The simulation was performed in a rotationally invariant cylindrical domain of radius
+Lr = 2.5l0 and half-height Lz = 3.5l0. Parameters: Mϵ0/D = 10, and w = 0.05l0 as in Fig. 4b in
+the main text.
+Supplemental Video 5 shows a simulation of a controlled division process in 3d.
+We
+take the catalysis rate k1 of converting products into substrates as a control parameter.
+We start with a large value of k1, for which a single droplet is stable at the center of the
+domain. Then, upon lowering k1, we observe the following dynamics. The droplet first
+elongates (initial shape instability) and then forms a dumbbell shape with a neck. The tube
+
+28
+connecting the two dumbbells gradually becomes thinner until it pinches off, leaving two
+separated droplets (division). These two droplets assume a spherical shape and position
+themselves equidistantly in the domain as expected from the discussion of section IV A. If
+we increase k1 to its initial value, then the two droplets remain stable, showing that the
+process is irreversible. Figure 4b in the main text shows some snapshots of the enzyme
+concentration, with t = 0 being the time when k1 is switched to a smaller value.
+Note that we have only observed droplet divisions in 3d, but never in 2d or 1d. This
+suggests an effect that destabilizes an elongated cylindrical shape in favor of spherical shapes.
+Such an effect is characteristic for classical pearling instabilities driven by surface tension [14],
+where spherical shapes have smaller surface area than cylindrical shapes with the same
+volume. Therefore, cell division should be controlled by the interface width w, which not
+only sets the value of the surface tension, but also defines a length scale over which the two
+sides of a thin cylinder that connects the two dumbbells can interact.
+V.
+MULTI-COMPONENT CONDENSATES
+So far, we have studied a scenario where the enzymes undergo spontaneous phase sepa-
+ration on their own. However, one can also envision a much more general scenario where
+the enzymes do not phase separate spontaneously, but are only enriched in a droplet that
+consists of a scaffold protein with concentration q(x). In this scenario, the scaffold proteins
+spontaneously phase separate driven by the chemical potential µ0(q) of the Cahn-Hilliard
+model, and only interact with the enzymes through a Flory-Huggins coupling χq:
+∂tq = ∇ ·
+�
+Mqq∇
+�
+µ0(q) + χq c
+��
+.
+(S36)
+We assume that the enzymes are present at relatively small concentrations so that they show
+currents driven by diffusion, and effective Flory-Huggins couplings to the scaffold proteins,
+substrates, and products:
+∂tc = ∇ ·
+�
+Dc∇c + Mcc∇
+�
+χqq + χss + χpp
+��
+.
+(S37)
+Attractive effective interactions between the scaffold proteins and the enzymes, χq < 0, lead
+to an enrichment of enzymes in the droplet. In the sharp-interface limit, this enrichment is
+
+29
+quantified by
+c+
+c−
+= exp
+�
+−Mc
+Dc
+χq (q+ − q−)
+�
+,
+(S38)
+where the indices ± indicate the concentrations on the inner and the outer side of each
+droplet interface, respectively. Analogous to droplets that consist mainly of enzymes, which
+we have discussed so far, our simulations show self-propulsion if the mobilities of the scaffold
+proteins (Mq) and enzymes (Mc) are sufficiently large, and self-centering otherwise (Sup-
+plemental Video 6). The enzymes then act as a link, by mediating the spatial organization
+of substrate and product through the droplet, as well as transmitting forces to the droplet
+that arise due to these inhomogeneous substrate and product concentration profiles.
+
+30
+VI.
+SUPPLEMENTAL VIDEOS
+In the following, we describe the videos available as supplemental material.
+1. video_1_self_propulsion_instability.mp4.
+For all simulations, we set M =
+1000D/ϵ0. 1d droplet: Evolution of the concentration profiles of enzymes, substrates
+and products resulting from a numerical simulation of Eqs. (S7) in a one-dimensional
+interval with no-flux boundary conditions and half-length L = 10l0. As initial condi-
+tion we consider a single droplet of enzymes at the center of the interval, the starting
+concentrations of substrates and products are the equilibrium values of the reaction
+terms plus small random perturbations in the droplet region. We observe the self-
+propulsion instability, the droplet starts moving in a random direction determined by
+the initial conditions. 2d droplet: Evolution of the concentration profile of enzymes
+resulting from a numerical simulation performed in a two-dimensional circular domain
+of radius Lr = 7l0. 3d droplet: Evolution of the concentration profile of enzymes
+resulting from a numerical simulation performed in a three-dimensional cylindrical
+domain of radius Lr = 4l0 and half-height Lz = 7l0.
+2. video_2_positioning_1d.mp4. Evolution of the concentration profiles of enzymes,
+substrates and products resulting from numerical simulations analogous to the one
+for the 1d droplet in Supplemental Video 1. As initial condition we consider a single
+droplet of enzymes positioned at xd(0) = −l0, the starting concentrations of sub-
+strates and products are the stationary profiles that they would reach in the absence
+of interactions. The parameter values are the same as for Fig. 3a in the main text.
+3. video_3_positioning_2d_and_3d.mp4.
+2d droplet:
+Evolution of the concentra-
+tion profiles of enzymes resulting from a numerical simulation performed in a two-
+dimensional square domain of half-side length L = 3l0. As initial condition we consider
+a single droplet of enzymes positioned at (x, y) = (−l0, 0), the starting concentrations
+of substrates and products are the equilibrium values of the reaction terms.
+The
+droplet moves to the center of the domain and localizes there. The other parameter
+values are the same as for Supplemental Video 2. 3d droplet: Evolution of the con-
+centration profiles of enzymes resulting from a numerical simulation performed in a
+
+31
+three-dimensional cylindrical domain of radius Lr = 3l0 and half-height Lz = 3l0. The
+droplet is initially positioned at z = −l0.
+4. video_4_coexistence.mp4. 1d droplets: Evolution of the concentration profiles of
+enzymes, substrates and products resulting from a numerical simulation analogous to
+the one for the 1d droplet in Supplemental Video 1. As initial condition we consider
+two distinct droplets of enzymes of radii R1(0) = 0.5l0 and R2(0) = 1.5l0 positioned at
+x = ∓2.5l0. The starting concentrations of substrates and products are the equilibrium
+values of the reaction terms. Enzymes are transported from the larger droplet to the
+smaller one until the radii of the two droplets become equal. 3d droplets: Evolution of
+the concentration profiles of enzymes resulting from a numerical simulation analogous
+to the one for the 3d droplet in Supplemental Video 1. The initial conditions are
+analogous to the one for the 1d droplets.
+Lr = 2.5l0, Lz = 5l0, χs/r = −0.21,
+w = 0.05l0.
+5. video_5_droplet_division.mp4. Evolution of the concentration profiles of enzymes
+resulting from a numerical simulation performed in a three-dimensional cylindrical do-
+main of radius Lr = 1.5l0 and half-height Lz = 4l0. As initial condition we considered
+a droplet of enzymes positioned at the center of the domain. The starting concentra-
+tions of substrates and products are the equilibrium values of the reaction terms. The
+simulation starts with a catalysis rate k1 = 100 k2/c+ for which the droplet is stable.
+Then the catalysis rate is switched to k1 = 1 k2/c+ and the droplet divides into two.
+Finally, the catalysis rate is switched again to k1 = 100 k2/c+ and the two droplets
+maintain their stability. Parameters: M = 10D/ϵ0, χs = −0.5r, w = 0.05l0.
+6. video_6_multicomponent_droplets.mp4. Evolution of the concentration profiles of
+scaffold proteins, enzymes, substrates and products resulting from numerical simula-
+tions of Eqs. (S7b, S7c, S36, S37). We consider as initial condition a droplet of scaffold
+proteins and a uniform concentration of enzymes c(t = 0) = 0.25q+. Self-propulsion:
+Simulation analogous to the 1d droplet shown in Supplemental Video 1. Parameters:
+χq/r = −0.2, Mq = Mc = 1000D/ϵ0, Dc/D = 100. Self-centering: Simulation anal-
+ogous to Supplemental Video 2. Parameters: χq/r = −0.2, Mq = Mc = 100D/ϵ0,
+Dc/D = 10.
+
+32
+[1] S. R. De Groot and P. Mazur, Non-Equilibrium Thermodynamics, Dover Books on Physics
+(Dover Publications, 2013).
+[2] R. Balian, From Microphysics to Macrophysics (Springer Berlin Heidelberg, 2006).
+[3] L. Demarchi, A. Goychuk, I. Maryshev, and E. Frey, Source code - Enzyme-enriched conden-
+sates show self-propulsion, positioning, and coexistence (2022).
+[4] A. Logg, K.-A. Mardal, G. N. Wells, et al., Automated Solution of Differential Equations by
+the Finite Element Method, edited by A. Logg, K.-A. Mardal, and G. N. Wells (Springer,
+2012).
+[5] M. Newville, T. Stensitzki, D. B. Allen, and A. Ingargiola, LMFIT: Non-linear least-square
+minimization and curve-fitting for Python (2014).
+[6] A. Goychuk, L. Demarchi, I. Maryshev, and E. Frey, The role of non-reciprocity in the fluid-like
+dynamics of enzyme-enriched droplets (to be published).
+[7] R. A. Fisher, The wave of advance of advantageous genes, Ann. Eugen. 7, 355 (1937).
+[8] A. Meurer, C. P. Smith, M. Paprocki, O. ˇCert´ık, S. B. Kirpichev, M. Rocklin, A. Kumar,
+S. Ivanov, J. K. Moore, S. Singh, T. Rathnayake, S. Vig, B. E. Granger, R. P. Muller,
+F. Bonazzi, H. Gupta, S. Vats, F. Johansson, F. Pedregosa, M. J. Curry, A. R. Terrel,
+ˇS. Rouˇcka, A. Saboo, I. Fernando, S. Kulal, R. Cimrman, and A. Scopatz, SymPy: sym-
+bolic computing in Python, PeerJ Comput. Sci. 3, e103 (2017).
+[9] W. R. Inc., Mathematica, Version 12.2, champaign, IL, 2020.
+[10] I. M. Lifshitz and V. V. Slyozov, The kinetics of precipitation from supersaturated solid
+solutions, J. Phys. Chem. Solids 19, 35 (1961).
+[11] C. Wagner, Theorie der Alterung von Niederschl¨agen durch Uml¨osen (Ostwald-Reifung), Z.
+Elektrochem. 65, 581 (1961).
+[12] A. J. Bray, Theory of phase-ordering kinetics, Adv. Phys. 43, 357 (1994).
+[13] C. A. Weber, D. Zwicker, F. J¨ulicher, and C. F. Lee, Physics of active emulsions, Rep. Prog.
+Phys. 82, 064601 (2019).
+[14] L. Rayleigh, On the instability of jets, Proc. London Math. Soc. s1-10, 4 (1878).
+
diff --git a/YdAyT4oBgHgl3EQfiPiV/content/tmp_files/load_file.txt b/YdAyT4oBgHgl3EQfiPiV/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..ced8e9b9d02b5569edb2ecb4b5bd51ed549594c7
--- /dev/null
+++ b/YdAyT4oBgHgl3EQfiPiV/content/tmp_files/load_file.txt
@@ -0,0 +1,1283 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf,len=1282
+page_content='Enzyme-enriched condensates show self-propulsion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' positioning,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' and coexistence  Leonardo Demarchi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' ∗ Andriy Goychuk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' † Ivan Maryshev,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='1 and Erwin Frey1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' ‡  1Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Ludwig-Maximilians-Universit¨at M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Theresienstraße 37,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' D-80333 M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Germany  2Max Planck School Matter to Life,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Hofgartenstraße 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' D-80539 M¨unchen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Germany  Enzyme-enriched condensates can organize the spatial distribution of their substrates by catalyz-  ing non-equilibrium reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Conversely, an inhomogeneous substrate distribution induces enzyme  fluxes through substrate-enzyme interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We find that condensates move towards the center  of a confining domain when this feedback is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Above a feedback threshold, they exhibit self-  propulsion, leading to oscillatory dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Moreover, catalysis-driven enzyme fluxes can lead to  interrupted coarsening, resulting in equidistant condensate positioning, and to condensate division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Liquid-liquid phase separation in living cells can lead  to the formation of biomolecular condensates that aid  intracellular organization [1–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' These condensates have  different functions such as compartmentalization of re-  actions [4], buffering of molecules [8], and midcell local-  ization during cell division [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' However, in a thermal  equilibrium system, the liquids will completely segregate  through a coarsening process (Ostwald ripening) [10–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  To arrest this process, the system must be brought out  of equilibrium by supplying energy, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=', via fuel-driven  chemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' This has been shown to lead to ‘ac-  tive droplet’ systems that exhibit a wealth of novel phe-  nomena not encountered in thermal equilibrium [3, 6, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Previous studies have considered systems with a con-  tinuous turnover of condensate (droplet) material by  chemical reactions [14–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The resulting material fluxes  lead to multi-droplet coexistence [18, 19, 22–24] and  droplet division [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Here, we study a different class of  systems where conserved enzymes spontaneously phase  separate, or localize to an existing condensate [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' These  enzymes then regulate reactions among other molecules,  by transiently binding substrate and catalyzing its con-  version into product via a lower activation barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  For example, in the bacterium Myxococcus xanthus, a  PomXY cluster (moving on the nucleoid) regulates the  cycling of PomZ between two conformations [9, 26–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  We show that such substrate turnover and the result-  ing enzyme fluxes lead to condensate self-propulsion, po-  sitioning, interrupted coarsening, and condensate divi-  sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Interestingly, previous studies have shown that liq-  uid droplets can self-propel on a surface through active  stresses [29–34], altering their wetting properties [35, 36],  or in viscous fluids through Marangoni flows [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In con-  trast, in our case, condensate motion is driven by the  bulk interactions between the various chemical species  and does not require surfaces or hydrodynamic coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  While condensates might consist of several compo-  nents, here we focus on the enzyme concentration c(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  To describe the dynamics of liquid-liquid phase sepa-  ration, we take the Cahn-Hilliard (CH) equation as a  starting point with the following chemical potential [38]:  µ0(c) = −r(c − ˜c) + u(c − ˜c)3 − κ∇2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  This chemical  potential µ0(c) = δF[c]/δc corresponds to the Ginzburg-  Landau free energy functional F[c] for a symmetric bi-  nary mixture with the critical density ˜c and phenomeno-  logical parameters r, u, and κ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' in particular, the con-  trol parameter r measures the distance from the critical  point [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  The enzymes interact with substrates and  products, which are present at concentrations s(x, t) and  p(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' These couplings, quantified by the Flory-Huggins  (FH) parameters χs and χp, modify the local chemical  potential of enzymes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Assuming that the particle cur-  rents are proportional to gradients in the chemical po-  tential [39, 40], the enzyme dynamics is given by  ∂tc(x, t) = ∇ ·  �  M c ∇  �  µ0(c) + χss + χpp  ��  ,  (1)  where M denotes a mobility and the term in the square  brackets is the enzyme flux j(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' This gradient dynam-  ics leads to a gradual minimization of the free energy  functional from which it is derived [25], a hallmark of  systems close to thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Analogously, the  thermodynamic fluxes of substrates and products can be  derived from the same free energy functional [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Active systems, however, exhibit processes that break  detailed balance in protein reaction networks [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' For  example, in the conversion of nucleoside triphosphatases  (NTPases) between an NDP-bound (“product”) and a  less stable NTP-bound (“substrate”) state, abundance  of NTP (“fuel”) in solution may shift the equilibrium  toward the latter [43] and replenish substrate with the  net rate k2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Transient binding of an enzyme (NTPase-  activating protein, NAP) to substrate can, by lowering  the activation barrier of hydrolysis, kinetically select the  fuel-independent reaction pathway and replenish product  with the net rate k1c s, which follows from the law of  mass action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  We assume that these separate reaction  pathways are far from their respective equilibria, so that  we can disregard thermodynamic constraints [39, 44] and  treat the rate constants k1,2 as independent parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Hence, we write for the dynamics of the substrate and  the product:  ∂ts = ∇ · (D ∇s + Λ s χs∇c) − k1 c s + k2 p ,  (2a)  ∂tp = ∇ · (D ∇p + Λ p χp∇c) + k1 c s − k2 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (2b)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='00392v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='bio-ph]  1 Jan 2023  2  Catalytic substrate turnover is preceded by enzyme  binding, suggesting effective pairwise attraction, χs < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Converting substrate into product reduces its affinity for  the enzymes, χs < χp, leading to unbinding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Note that in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (2) we have taken the liberty of formally decoupling  the diffusion coefficient D from the mobility Λ, thus intro-  ducing a further source of far from equilibrium dynamics  by breaking the fluctuation-dissipation relation valid for  thermal equilibrium systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In the present context, this  is the Einstein-Smoluchowsky relation D = Λ kBT [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  With the aim of simplifying the analysis, in the present  work we consider Λ = 0, an approximation that is valid  for weak FH parameters χs,p and will be addressed else-  where [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  For our initial exploration of the dynamics, we con-  sider a finite-sized domain [−L, L] in a one-dimensional  (1d) geometry with no-flux boundary conditions at  x = ±L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  A droplet then corresponds to a plateau  with a high enzyme concentration, surrounded by an  enzyme-poor phase [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  If the width of the inter-  face between these phases, w =  �  2κ/r, is much smaller  than all other length scales, one can use a sharp inter-  face approximation with piecewise constant concentra-  tion c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' For our analysis, we consider weak interactions,  |χss| + |χpp| ≪ r (c+ − c−), and therefore approximate  the enzyme concentrations in the two phases by their  equilibrium values, c± = ˜c ±  �  r/u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  First, consider a stationary droplet, where a closed an-  alytic solution of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (1) and (2) can be obtained [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  The stationary state is maintained by a balance of re-  active and diffusive fluxes [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 1]: In the droplet, the  enzymes catalyze the conversion of substrate to prod-  uct, consuming the former and accumulating the lat-  enzyme 𝑐  substrate 𝑠  product 𝑝  −𝐷𝜕𝑥𝑠  −𝐷𝜕𝑥𝑝  𝑘1𝑐𝑠  𝑘2𝑝  concentration profiles  −𝐿  −𝑅  +𝐿  +𝑅  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Steady state concentration profiles for a 1d  droplet with no-flux boundary conditions at x = ±L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Ar-  rows indicate reactive (vertical) and diffusive (horizontal)  fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  The analytical solutions in the sharp interface ap-  proximation (lines) match our simulations (dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  We use  c+ as reference concentration and define the characteris-  tic time τ0 := k−1  2 , diffusion length in the absence of en-  zymes l0 :=  �  D/k2, and reference energy ϵ0 := rc+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The re-  maining parameters, c− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='1c+, w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='1l0, R = l0, L = 5l0,  M = 100D/ϵ0, k1 = k2/c+, χs = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='05r, χp = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='01r, and  s + p = c+, are fixed for all figures unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  −5  concentration  5  10  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='2  −10  𝑣  𝑐  𝑠  𝑝  𝑐+  𝑐−  𝑣 [𝑙0/𝜏0]  space 𝑥 [𝑙0]  Δ𝑠 [𝑐+]  Theory  1d  2d  3d  5  10  1000  500  5  10  5  10  5  10  0  5  𝑣 [𝑙0/𝜏0]  𝑀𝜖0/𝐷  𝑘1𝑐+/𝑘2  (b)  (a)  (c)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Self-propulsion instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (a) Analytical profiles  for a droplet moving with velocity v = 2 l0/τ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Lighter col-  ors indicate earlier times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (b) Graphical analysis of the self-  consistency relation (3) for Mϵ0/D = 1000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The solid curve  indicates the substrate imbalance ∆s(v), while the slope of  the dashed line corresponds to the right-hand side of inequal-  ity (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Stationary droplets correspond to unstable solutions  (empty circle), while self-propelling droplets are stable (filled  circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (c) Theoretical prediction and simulation results for  the self-propulsion velocity (color scale) with M and k1 as  free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  The solid black lines indicate the critical  mobility M ∗ (for the explicit closed expression see [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Diffusive fluxes in turn replenish substrate in the  droplet while expelling product, resulting in concentra-  tion gradients over the characteristic diffusion lengths  l± =  �  D/(k1c± + k2) inside and outside the droplet, re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  This leads to cyclic diffusive and reactive  fluxes such that time-reversal symmetry is broken and  one has a reaction-driven non-equilibrium steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  If there is an appreciable difference in substrate or  product concentration between the two droplet inter-  faces (henceforth referred to as “imbalance”), this gen-  erally results in a chemical potential gradient that can  drive droplet motion through a net flux of enzymes  [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Indeed, using finite element (FEM) simula-  tions of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (1) and (2), we find a broad parameter  regime with ballistic droplet motion [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 2c, Video 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  To analytically determine the conditions for the onset of  this self-propulsion instability, we next study the sharp-  interface limit of a single 1d droplet in an infinite domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Similar to the analysis of Fisher waves [47], we solve  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (2) in the reference frame of a moving droplet to  obtain the concentration profiles of substrates and prod-  ucts, and then use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (1) to derive a self-consistency  relation for the droplet velocity v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Specifically, the conti-  nuity equation (1) implies that the moving steady-state  enzyme profile, c(z) with z := x − vt and constant ve-  locity v, can only be maintained if ∂zj(z) = v ∂zc(z)  holds at all times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The local flux of enzymes is there-  fore given by j(z) = v [c(z) − c−], and vanishes in the far  3  field where all concentrations become homogeneous [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  The concentration of enzymes in the droplet is en-  riched by ∆c = c+ − c− with respect to the far-field value  c(±∞) = c−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' While the enzyme flux is driven by the local  chemical potential and concentration gradients [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (1)],  integrating the flux over the droplet domain [−R, R]  yields an expression that depends only on the values  at the droplet boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In the sharp-interface limit,  the µ0-term becomes mirror-symmetric relative to the  droplet center and hence doesn’t contribute to the self-  consistency relation for the droplet velocity:  2R ∆c v = −Mc+  �  χs∆s(v) + χp∆p(v)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (3)  Here, ∆s(v) = s(R) − s(−R) is the substrate concentra-  tion imbalance between the two opposite sides of the  droplet, with an analogous expression ∆p(v) for the prod-  uct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' This result quantifies how asymmetric substrate and  product concentration profiles drive droplet motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Using the closed analytic expressions for the substrate  concentration profiles [25], shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 2a, one can  graphically solve the self-consistency relation (3) for the  droplet velocity v [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 2b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Specifically, for Λ = 0, where  the total concentration of substrates and products is con-  stant [Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (2)], the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (3) simplifies  to Mc+(χp − χs)∆s(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Based on the graphical form of  ∆s(v), a non-vanishing solution to the self-consistency  relation for the velocity v exists if [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 2b]:  ∂  ∂v ∆s(v)  ���  v=0 >  2R∆c  Mc+(χp − χs) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (4)  Thus, traveling wave solutions emerge if, for example, the  mobility M or the difference between the FH parameters  χp − χs are sufficiently large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Then, enzymes are pulled  more towards substrates than products, so that enzy-  matic substrate depletion can induce a chemophoretic  effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' This theoretical analysis quantitatively explains  the onset of the self-propulsion instability that we ob-  served in our simulations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 2c for a comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Droplet movement is driven by asymmetries in the con-  centration profiles of substrates and products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Droplets  induce such asymmetries autonomously during self-  propulsion, but also near impermeable domain bound-  aries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Figure 3a and Video 2 show the results of FEM  simulations in a closed domain for four characteristic val-  ues of the mobility M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Below the self-propulsion thresh-  old M ∗, the droplet exhibits an overdamped relaxation  toward the domain center (red), where the concentration  profiles become symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Thus, the impermeable do-  main boundaries effectively repel the droplet, due to sub-  strate depletion and product enrichment within a range  l−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' When increasing M ∗, there is a transition from over-  damped to underdamped oscillatory relaxation (blue),  where the relaxation rate λ has a maximum at critical  damping [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 3b], similar to a damped harmonic oscil-  lator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Above the self-propulsion threshold M ∗, droplets  100  300  500  400  (a)  −𝐿  +𝐿  𝑥𝑑  0  50  25  25  5  0  0  0  (c)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='1  10  1  5  (b)  0  0  0  100  300 400  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='002  𝑀𝜖0/𝐷  𝑘1𝑐+/𝑘2  𝜆 [𝜏0  −1]  𝜆 [𝜏0  −1]  𝜔 [𝜏0  −1]  𝑡 [𝜏0]  𝑡 [𝜏0]  𝑡 [𝜏0]  𝑡 [𝜏0]  (𝐿−𝑅)/𝑙−  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Self-centering and oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (a) Droplet cen-  ter trajectories in simulations for different values of M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' sym-  bols indicate the position in the diagram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 2c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  The droplet center is initially at xd(0) = −l0 in a domain of  size L = 3 l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (b) Decay rate (black dots) and frequency  (red dots) as functions of M, obtained by fitting the respec-  tive droplet trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Red triangle indicates overdamped  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (c) Relaxation rate λ as a function of k1 (bottom  axis), for Mϵ0/D = 10 and xd(0) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='3l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Top axis re-  lates the domain size to the length scale of the concentration  profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The analytical predictions (blue line) in the quasi-  steady-state approximation [25] match our simulations (black  dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  autonomously accelerate to a terminal velocity v [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 2c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Instead of droplet self-centering, one then observes oscil-  lations (green) with frequency ω ≈ v/(L − l−), where the  domain boundaries cause droplets to slow down and re-  verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Droplets with strong self-propulsion (purple) can  overcome this repulsion and attach to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  To elucidate how droplet self-centering depends on the  reaction rates, we analyzed the overdamped regime, in  which droplet motion is much slower than the relaxation  of the substrate and product concentration profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' This  time scale separation allows to solve Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (2) analytically  using a quasi-stationary approximation, where the sub-  strate and product concentration profiles are in steady  state with the droplet center xd(t) considered as slowly  varying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  The obtained steady-state profiles are asym-  metric when the droplet is not centered in the domain,  if and only if the characteristic length l+ is neither much  larger nor much smaller than the droplet size R (lest  both droplet interfaces have equal concentrations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' These  asymmetric concentration profiles induce droplet motion  towards the domain center, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (3), with a velocity  v(xd) that we linearize as a function of the distance to  the domain center [25], |xd| ≪ l−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The resulting approx-  imation for the relaxation rate λ agrees well with our  simulations [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 3c], and demonstrates that droplet self-  centering is fastest for a finite value of k1c+/k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' More-  over, our analysis and simulations show that droplet self-  centering proceeds fastest when the distance between the  droplet interface and the domain boundary is comparable  to the range of repulsion, L − R ∼ l− [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 3c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  4  A state with multiple droplets cannot be stable in a  thermodynamic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Instead, a coarsening process  driven by interfacial energy minimization takes place,  causing smaller droplets to shrink and larger droplets  to grow until there is complete phase separation [11–  13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' As our system is out of equilibrium, it can result  in a stable coexistence of multiple droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Specifically  for the system we are considering, larger droplets have a  larger enzymatic activity and hence consume more sub-  strate, leading to a reduced substrate concentration at  their interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' This results in a gradient of substrate  (and product) in the low-concentration phase between  droplets of different sizes, thereby transporting enzymes  from the larger to the smaller droplets and thus counter-  acting the coarsening process [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  For a 1d system, the thermodynamic coarsening pro-  cess described by the CH model is extraordinarily slow  with the average droplet radius growing only logarithmi-  cally with time [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Hence, one expects that (even weak)  enzymatic processes can interrupt coarsening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Indeed,  solving the dynamics of multiple droplets analytically in  the adiabatic limit [25], we find enzyme fluxes between  pairs of differently sized droplets, which are proportional  to the difference in the substrate concentration at their  closest interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  These currents stop the coarsening  process and lead to a steady state where the droplets po-  sition themselves equidistantly to each other to even out  concentration imbalances between all interfaces [Video 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  For 2d and 3d systems, Ostwald ripening is dominated  by surface tension effects (Laplace pressure) and the en-  suing law for droplet growth becomes a power law [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  In this case, one intuitively expects that the coarsening  process can be interrupted only if the mass fluxes of the  enzymes are sufficiently strongly coupled to the concen-  tration of the products and substrates [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Figure 4a  shows FEM simulation results for a 3d system with a pair  of droplets, which confirm this intuitive argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The  existence of a coarsening threshold for 3d systems can  also be understood analytically as a balance between a  coarsening current due to surface tension and a mass flux  of enzymes driven by reaction-maintained product and  substrate concentration gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We estimate the for-  mer using the standard Gibbs-Thomson relation [13, 14]  and the latter by adapting the above results for the 1d  system [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' By comparing the two currents we find the  following estimate for the critical difference between the  FH parameters ∆χ = χp − χs above which one expects  droplet coexistence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=', interrupted coarsening:  ∆χ∗ = 2  3  rw∆c/∆s⋆ �  l−1  + cosh(ξ) + l−1  − sinh(ξ)  �2  l−1  + [sinh(2ξ) − 2ξ] + 2l−1  − [sinh2(ξ) − ξ2] ,  (5)  where ξ := ¯R/l+ is the ratio of the average droplet radius  to the typical length scale of the concentration gradients  inside of a droplet, and ∆s⋆ is the difference between  the local equilibria of substrate in the two phases [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (b)0  10  43  44  95  concentration  0  0  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='25  Δ𝜒 [𝑟]  𝑐−  𝑐+  (a)  𝑘1𝑐+/𝑘2  𝑡 [𝜏0]  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Coexistence and division of 3d droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We consider  Mϵ0/D = 10 and w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='05l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Scale bars indicate unit length  l0 :=  �  D/k2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (a) Simulated pairs of droplets with different  initial radii R1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='1l0 and R2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='9l0 either stay separated  (cyan regime) or coalesce (yellow regime) depending on k1 and  on the difference in FH parameters ∆χ = χp − χs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Solid black  line corresponds to analytical estimate, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (5), for ∆χ∗(k1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (b) Simulation snapshots demonstrating a droplet division in  3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We observed droplet divisions only for very strong attrac-  tion of enzymes towards substrates, here χs/r = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Notwithstanding the partially heuristic nature of the  derivation, our estimate yields a good approximation for  the boundary between coexistence and coarsening in pa-  rameter space [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 4a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  In our numerical simulations, we also observed that  initially spherical droplets can undergo a shape instabil-  ity and elongate in one direction for large enough values  of ∆χ [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Once sufficiently elongated, 3d droplets form  a neck and divide [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 4, Video 5], which we speculate to  occur through a pearling instability driven by surface ten-  sion [50] independent of the preceding shape instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  This droplet division process is driven by intermolecular  interactions that induce conservative enzyme fluxes, as  opposed to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' [21] where the droplet material is cycli-  cally produced and degraded, leading to non-conservative  fluxes and droplet growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  We have analyzed the nonequilibrium dynamics of  enzyme-enriched condensates, whose enzymatic activity  guides  the  generation  of  inhomogeneous  substrate  and product concentration profiles that, in turn drive  condensate motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Conceptually, this corresponds  to a feedback mechanism in which, for example, an  NTPase such as PomZ undergoing a cycle of hydrolysis  (s → p, catalyzed by a NAP c) and nucleotide exchange  (p → s) generates concentration gradients of its two  different chemical states (s and p) that drive droplet  movement through a process akin to chemophoresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Our  results show that such a generic mechanism results in  equidistant positioning of condensates in closed domains,  persistent condensate motion, and even shape insta-  bilities that lead to condensate division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We speculate  that this mechanism, in its basic form, may be relevant  for processes like midcell localization of protein clusters  in some prokaryotic cells [9, 26], directed motion of  partition complexes [28], equidistant placing of plasmids  5  along nucleoids [51],  and maybe even transcription  regulation [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  We thank Dominik Schumacher and Lotte Søgaard-  Andersen for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  We acknowledge fi-  nancial support by the German Research Foundation  (DFG) through TRR 174 (Project ID No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 269423233)  and SFB1032 (Project ID No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 201269156) and the Ex-  cellence Cluster ORIGINS under Germany’s Excellence  Strategy (EXC-2094-390783311).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' AG was supported by  a DFG fellowship through the Graduate School of Quan-  titative Biosciences Munich (QBM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' During his time  at the Massachusetts Institute of Technology, AG was  supported by the National Science Foundation (NSF)  through grant number 2044895.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' IM has received funding  from the European Union’s Framework Programme for  Research and Innovation Horizon 2020 under the Marie  Sk�lodowska-Curie Grant Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 754388 (LMU  Research Fellows) and from LMU excellent, funded by  the Federal Ministry of Education and Research (BMBF)  and the Free State of Bavaria under the Excellence Strat-  egy of the German Federal Government and the L¨ander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  ∗ These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Present  address: Sorbonne Universit´e, CNRS, Institut de Biolo-  gie Paris-Seine (IBPS), Laboratoire Jean Perrin (LJP),  F-75005 Paris, France  † These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Present  address: Institute for Medical Engineering and Science,  Massachusetts Institute of Technology, Cambridge, MA  02139, United States;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' andriy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='goychuk@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='com  ‡ frey@lmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='de  [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Brangwynne, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
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+page_content=' Chakraborty, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Young, RNA-  mediated feedback control of transcriptional condensates,  Cell 184, 207 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Enzyme-enriched condensates show self-propulsion, positioning,  and coexistence  Leonardo Demarchi,1, ∗ Andriy Goychuk,1, † Ivan Maryshev,1 and Erwin Frey1, 2, ‡  1Arnold Sommerfeld Center for Theoretical Physics and Center for NanoScience,  Department of Physics, Ludwig-Maximilians-Universit¨at M¨unchen,  Theresienstraße 37, D-80333 M¨unchen, Germany  2Max Planck School Matter to Life, Hofgartenstraße 8, D-80539 M¨unchen, Germany  CONTENTS  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Nonequilibrium enzyme dynamics coupled to product and substrate  3  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Numerical simulations of single droplet dynamics  5  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Self-propulsion of a droplet  6  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Self-centering of a droplet in a finite domain  6  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Analytic theory of a single droplet in a one-dimensional domain  8  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Substrate and product redistribution by a stationary droplet  8  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Substrate and product redistribution by a moving droplet  11  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Gradients in substrate and product concentration drive droplet motion  12  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' A self-consistency relation for the self-propulsion instability  13  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Quasi-steady state approximation for droplet self-centering  16  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Enzymatic activity of droplets leads to coexistence and arrests the coarsening  process  18  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Localization of multiple coexisting droplets  19  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Coexistence of droplets in a one-dimensional domain  19  ∗ These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Present address: Sorbonne Universit´e, CNRS, Institut  de Biologie Paris-Seine (IBPS), Laboratoire Jean Perrin (LJP), F-75005 Paris, France  † These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Present address:  Institute for Medical Engineer-  ing and Science, Massachusetts Institute of Technology, Cambridge, MA 02139, United States;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' an-  driy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='goychuk@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='com  ‡ frey@lmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='de  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='00392v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='bio-ph]  1 Jan 2023  2  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Coexistence of droplets in a three-dimensional domain  23  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Droplet division  26  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Multi-component condensates  28  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Supplemental Videos  30  References  32  3  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  NONEQUILIBRIUM ENZYME DYNAMICS COUPLED TO PRODUCT AND  SUBSTRATE  We consider a scenario where there is an interplay between equilibrium phase separation  and nonequilibrium chemical reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' To model the aspect of equilibrium phase separa-  tion, we construct a free energy functional that describes a regular solution consisting of  solvent, enzymes, substrates, and products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' For simplicity, we choose the concentrations of  substrates, s(x, t), and products, p(x, t), to be very small, so that they each contribute to  the entropy independent of other concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Then, gradients in the concentration of  substrates and products are also small, so that we can neglect the corresponding interfacial  energy terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Furthermore, we assume that the interactions between solvent, substrates,  and products are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Under these assumptions, neither the substrate nor the product  molecules can show phase separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In contrast,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' we expand the free energy functional that  describes the entropy and interactions between solvent and enzymes1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' which are present at  much larger concentrations than substrates and products and are assumed to exhibit phase  separation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' up to fourth order in the enzyme concentration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' c(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' and use the Ginzburg-  Landau functional  f0(c) = u  4(c − ˜c)4 − r  2(c − ˜c)2 + κ  2|∇c|2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (S1)  with a positive stiffness,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' κ > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' and parameters u > 0 and r > 0 (corresponding to a  double-well potential).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Finally, we parameterize enzyme–substrate and enzyme–product  interactions with the Flory-Huggins (FH) parameters χs and χp, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In summary,  the statistics of the system is characterized by the following effective free energy functional,  F =  �  ddx  �  kBT  �  s log(s ν) + p log(p ν)  �  + χs c s + χp c p + f0(c)  �  ,  (S2)  where kBT is the thermal energy and d refers to the number of spatial dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' For  simplicity, we have here assumed that substrates and products have the same molecular  volume ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Given the above constraints on the model parameters, the enzymes will show spontaneous  phase separation into droplets or labyrinth-like patterns if the average enzyme concentration  1 We have chosen to perform simulations with comparable values of the three concentration fields, but one  can rescale s and p without changing the dynamics, by rescaling χs,p and Λ accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S1 shows  the results of simulations with rescaled parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  4  lies in the spinodal regime, or can separate through nucleation and growth in the binodal  regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' For our simulations, we initialize the system with a single predefined droplet to  study its dynamics, or with multiple droplets to study their coarsening dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  An exchange of particles (substrates, products, or enzymes) modifies the free energy of  the system and is thus associated with the following chemical potentials:  µs = δF  δs = kBT  �  log(s ν) + ν−1�  + χs c ,  (S3a)  µp = δF  δp = kBT  �  log(p ν) + ν−1�  + χp c ,  (S3b)  µc = δF  δc = µ0(c) + χs s + χp p ,  (S3c)  where we have defined the chemical potential of the Cahn-Hilliard model,  µ0(c) := u (c − ˜c)3 − r (c − ˜c) − κ ∇2c  (S4)  to simplify our notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' For our analysis, we consider a scenario where enzymes interact  with substrates and products weakly, |χss| + |χpp| ≪ r (c+ − c−), where c± refers to the  concentrations in the high-density and in the low-density phase, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Then, the  chemical potential µ0 dominates the phase separation of enzymes and maintains their sharp  concentration profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Following the general ideas of nonequilibrium thermodynamics [1, 2],  gradients in the chemical potentials (S3) drive conservative currents that gradually minimize  the free energy functional F,  �  ����  js  jp  jc  �  ���� = −  �  ����  Λ s  0  0  0  Λ p  0  0  0  Mc  �  ���� ·  �  ����  ∇µs  ∇µp  ∇µc  �  ���� ,  (S5)  where substrates and products are for simplicity assumed to have identical mobility Λ, and  enzymes have mobility M (Onsager coefficients).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Equation (S5) can be interpreted as the  local concentration of each species multiplied with an average drift velocity induced by  driving forces ∇µ, so that the diagonal entries in the response matrix are the quotient of  the local concentration and the viscous friction of each species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  We consider a system that is driven out of equilibrium by chemical reactions that are  5  not derived from the free energy functional F and therefore provide an external energy  influx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In particular, we consider a scenario where enzymes catalyze turnover of substrates  into products with rate k1c s, while products gradually decay into substrates with rate k2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Then, the dynamics of substrates, products, and enzymes is given by:  ∂ts + ∇ · js = k2 p − k1 c s ,  ∂tp + ∇ · jp = k1 c s − k2 p ,  and  ∂tc + ∇ · jc = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (S6)  Taking everything together, one arrives at Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (2) and (3) of the main text:  ∂tc = ∇ ·  �  Mc∇  �  µ0(c) + χs s + χp p  ��  ,  (S7a)  ∂ts = D∇2s − k1cs + k2p + Λχs∇ · (s∇c) ,  (S7b)  ∂tp = D∇2p + k1cs − k2p + Λχp∇ · (p∇c) ,  (S7c)  where we have related the mobility of substrates and products to their diffusion coefficient by  the Einstein relation, D = ΛkBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In the following, we take the liberty of formally decoupling  the diffusion coefficient D from the mobility Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In doing so, we introduce a further source  of far from equilibrium dynamics by breaking the fluctuation-dissipation relation valid for  thermal equilibrium systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  NUMERICAL SIMULATIONS OF SINGLE DROPLET DYNAMICS  We solved the system of partial differential equations (S7) numerically;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' the code is avail-  able on [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' To that end, we used the implicit Euler method for time discretization and  the finite element method for spatial discretization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' for the latter, we used the FEniCS li-  braries [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We considered a finite-sized domain with no-flux boundary conditions, which in  one dimension is parameterized by x ∈ [−L, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' As initial conditions, we chose uniform con-  centration profiles for the substrates and products, and controlled the average concentration  of these two species s(x) + p(x) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Furthermore, we initialized the enzyme concentration  profile to resemble a single droplet with a sharp interface:  csharp(x) =  �  �  �  �  �  c+ ,  for x ∈ [−R, R] , and  c− ,  otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (S8)  6  As indicated in the main text, we use c+ as reference concentration and define the char-  acteristic time τ0 := k−1  2 , diffusion length in the absence of enzymes l0 :=  �  D/k2, and  reference energy ϵ0 := rc+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  The remaining parameters, c− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='1c+, w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='1l0, R = l0,  L = 5l0, M = 100D/ϵ0, k1 = k2/c+, χs = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='05r, χp = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='01r, and s + p = c+, are fixed  unless stated otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We then simulated the system for a total time of 100 τ0 to let the  system reach a steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Figure 1 in the main text shows that the enzymes maintain  a single droplet with a steep interface, enriching products at the expense of substrates in  regions with high enzyme concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Outside of the droplet, where the enzyme concen-  tration is low, substrates are replenished at the expense of products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Self-propulsion of a droplet  Our simulations demonstrate that droplets can self-propel and sustain motion for a wide  range of parameters, in 1d [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 2 in the main text, Supplemental Video 1] as well as 2d and  3d [Supplemental Video 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We initialized droplets at the origin of an interval of half-length  L = 10 l0 in 1d and at the center of a circular domain of radius Lr = 7 l0 in 2d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' To speed  up simulations in 3d, we chose a rotationally invariant cylindrical coordinate system with  radius Lr = 4 l0 and height Lz = 7 l0, which reduced the number of degrees of freedom but  also constrained the space of permitted solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In all of our simulations, breaking the  symmetry of the droplet requires an initial perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We provided such a perturbation  through spatial inhomogeneities in the concentration profiles of substrates and products,  which are small compared to the average total concentration n := ⟨s(x) + p(x)⟩x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Self-centering of a droplet in a finite domain  In a finite domain, our simulations show that a self-propelling droplet can either reach  and adhere to one of the domain boundaries, or exhibit oscillatory motion by reorienting at  the domain boundaries [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 3a in the main text].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Reorientation at the domain boundaries  is mediated by an effective repulsion, which has the following mechanistic origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In our  simulations, we observe that the enzymatic depletion of substrate is enhanced in the vicinity  of a domain boundary, where substrate resupply through diffusive currents is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The  resulting substrate concentration gradients induce net enzyme currents, through attractive  7  𝑀𝜖0/𝐷  𝑣 [𝑙0/𝜏0]  𝑘1𝑐+/𝑘2  Theory  1d  2d  3d  5  10  1000  500  5  10  5  10  5  10  0  5  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Theoretical prediction and simulation results for the self-propulsion velocity (color scale)  with M and k1 as free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The solid black lines indicate the critical mobility M∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We con-  sider s + p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='01c+, χs = −5r and χp = −r, so that the concentrations of substrate and product  are comparatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' These results are analogous to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 2c in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  substrate–enzyme interactions, away from the domain boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Next, we study how a droplet that does not meet the parameter criteria for self-propulsion  [discussed in detail in section III D “A self-consistency relation for the self-propulsion  instability”] will position itself in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  To that end, we considered a droplet  whose center xd(0) = −l0 is initially offset from the domain center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  We initialized the  distribution of substrates and products in the steady state that is reached in the absence  of interactions, χs,p = 0, by performing “pre-simulations” for a duration of 1000 τ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Then,  we introduced interactions χs,p ̸= 0 and studied the resulting trajectory of the droplet  center in a one-dimensional domain as a function of time, xd(t) [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 3a in the main text,  1d droplet dynamics shown in Supplemental Video 2 while 2d and 3d droplet dynamics  shown in Supplemental Video 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' To improve the performance of the simulations, we used  adaptive time stepping, and confirmed that the total simulation time is sufficiently long for  the droplets to either attach to the boundary, perform several oscillations, or relax to the  domain center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' For droplets that do not exhibit self-propulsion, instead of sustained oscil-  lations we observed gradual localization to the domain center akin to a damped harmonic  oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  To account for both self-sustained and damped oscillatory motion, we fitted  each droplet trajectory with an exponentially damped sinusoidal curve using the library  LMFIT [5]: xd(t) = A e−λt cos(ωt + φ), where A is the amplitude, λ the decay rate, ω the  frequency and φ the initial value of the phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Trajectories where the droplet attaches to  the domain boundary cannot be fitted in such a way, and were therefore excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The  resulting estimates for the decay rate and for the frequency of the oscillations are shown in  8  −𝐿  +𝐿  𝑥𝑑  50  25  25  0  0  0  𝑡 [𝜏0]  𝑡 [𝜏0]  𝑡 [𝜏0]  𝑀𝜖0/𝐷 = 100  𝑀𝜖0/𝐷 = 300  𝑀𝜖0/𝐷 = 400  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Comparison of the droplet trajectories from the simulations (dots) with the best fitting  curves (solid lines), for the panels shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 3a in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 3b in the main text as functions of the droplet mobility M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  ANALYTIC THEORY OF A SINGLE DROPLET IN A  ONE-DIMENSIONAL DOMAIN  For our theoretical analysis in the present paper, we restrict ourselves to a 1d system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' A  2d and 3d analysis is also possible with a semi-analytical approach, but exceeds the scope of  the present paper and will be published elsewhere [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The idea of our theoretical analysis  rests on two pillars: (i) In all our simulations, we observed that the droplet maintains a  steep interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' This suggests that one can well describe our system analytically with a  sharp-interface approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (ii) Similar to the analysis of Fisher waves [7], we treat  moving droplets through a transformation into the co-moving frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' As we discuss next,  these considerations allow us to derive the concentration profiles of substrates and products,  as well as a self-consistency relation for the droplet velocity v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Substrate and product redistribution by a stationary droplet  To perform our theoretical analysis, we first determine the steady-state concentration  profiles of substrates and products in response to the presence of an enzymatic droplet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S7b) and (S7c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Because the enzyme concentration profile is well described by  a sharp-interface approximation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S8), for now we do not need to explicitly study the  dynamics of the enzymes, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S7a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  In the scenario of nonreciprocal interactions with Λ = 0, by summing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S7b) and (S7c)  9  product 𝑝  substrate 𝑠  concentration profiles  −𝐿  −𝑅  +𝐿  +𝑅  left subdomain  𝑐(𝑥) = 𝑐−  right subdomain  𝑐(𝑥) = 𝑐−  center  𝑐(𝑥) = 𝑐+  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Illustration of the three sub-domains in the sharp-interface approximation c(x) =  csharp(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The total concentration of substrates and products is conserved, s(x, t) + p(x, t) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In  each subdomain, we then only need to solve a Helmholtz equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  one finds that the total density of substrates and products is uniform in space, s(x, t) +  p(x, t) = n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' the general scenario with Λ ̸= 0 will be analyzed elsewhere [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' By substituting  this conservation law into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S7b), we find that the steady-state distribution of substrates  is determined by  D ∂2  xs(x) −  �  k1csharp(x) + k2  �  s(x) + k2n = 0 ,  (S9)  where csharp(x) refers to the concentration profile of enzymes in the sharp-interface approx-  imation [Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S8)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The resulting steady-state distribution of substrates also defines the  concentration profile of products p(x, t) = n − s(x, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Because the enzyme concentration is piecewise constant in the sharp-interface approxi-  mation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S9) reduces to a Helmholtz equation defined on three subdomains [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  The concentration profiles of substrates and products then have different characteristic diffu-  sion lengths, l+ :=  �  D/(k1c+ + k2) inside (center) and l− :=  �  D/(k1c− + k2) outside (left  and right) of the droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Solving the corresponding Helmholtz equation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S9), the  distribution of substrates for a stationary droplet is given by  s(x) =  �  �  �  �  �  �  �  k2 n  D l2  + + 2 Ain cosh  � x  l+  �  ,  for ∥x∥ ≤ R ,  k2 n  D l2  − + Aout exp  �  −|x|  l−  �  + Bout exp  �|x|  l−  �  ,  otherwise,  (S10)  where Ain/out and Bout are integration constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We determined these integration constants  10  by imposing smoothness and continuity of the concentration profiles at the droplet interfaces  x = ±R, which separate different subdomains, as well as no-flux boundary conditions at the  domain boundaries x = ±L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We provide the full (and rather lengthy) expressions for these  integration constants in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' However, even without having these explicit expressions at hand,  one can nevertheless analyze generic features of the substrate and product concentration  profiles dictated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Because all concentrations must remain finite in the far field x → ±∞, the integration  constant Bout must vanish (Bout = 0) for an infinitely large domain (L → ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Therefore, it follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S10) that the substrate and product concentrations far away  from the droplet (x → ±∞) are given by their local reactive equilibria2, s(±∞) = k2 n  D l2  − =  k2 n  k1c−+k2, which correspond to the solution of the Helmholtz equation (S9) in the limit D → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Only the concentrations at the center of the droplet, s(0) = k2 n  D l2  + + 2 Ain, are shifted by  2Ain relative to their local reactive equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The magnitude of this shift depends on the  relative size of the droplet compared to the characteristic length of the concentration profiles,  R/l+, and must vanish for very large droplets3 R ≫ l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The difference between the reactive  equilibria of substrate at low and at high enzyme concentration is given by:  ∆s⋆ := k2 n  D  �  l2  − − l2  +  �  = k2 n  D  k1 ∆c l2  +l2  −  D  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (S11)  Thus, the level of substrate depletion in the droplet is proportional to its enrichment of  enzymes relative to the surrounding solution, ∆c = c+−c−, and analogously implies different  characteristic diffusion lengths inside and outside of the droplet, l+ ̸= l−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Having discussed these features of the substrate and product concentration profiles, we  compare our analytic predictions (in the sharp-interface approximation) to our numeric re-  sults (with a diffuse interface) and find very good agreement [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 1 in the main text].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Because of this excellent agreement, in the following, we use the sharp-interface approxima-  tion to also study more intricate scenarios where the droplet becomes mobile and positions  itself in the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  2 We define the reactive equilibria as the solution of the reaction-diffusion equations for substrates and  products, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S7b) and (S7c), in the limit of vanishing transport D → 0 and Λ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  3 This can be seen by taking the limit of D → 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  11  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Substrate and product redistribution by a moving droplet  We now generalize the results of the former section to an enzymatic droplet that moves  with constant velocity v, where the enzyme concentration profile can be written in the  form of a travelling wave c(x, t) = csharp(x − vt);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' As we have discussed before,  the total density of substrates and products is uniform in space, s(x, t) + p(x, t) = n, for  nonreciprocal interactions Λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Using the substitution z = x − vt to transform Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S7b)  into the co-moving reference frame of the droplet, the steady-state distribution of substrates  is then determined by  D ∂2  zs(z) −  �  k1csharp(z) + k2  �  s(z) + k2n = −v ∂zs(z) ,  (S12)  which differs from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S9) only by an advection term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  To simplify the expressions in  the following, we introduce the P´eclet number, Pe0 = vR/D, and two modified P´eclet  numbers that include a correction for the characteristic lengths l± of the concentration  profiles inside and outside of the droplet, Pe± =  �  Pe2  0 + (2R/l±)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Furthermore, we note  that the distribution of substrates must remain finite in the far field z → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Then, solving  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S12) in each of the three subdomains [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S3],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' imposing smoothness and continuity  at the subdomain interfaces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' and assuming an infinitely large overall domain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' yields the  steady-state distribution of substrates in the co-moving frame of the droplet:  s(z) =  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  k2 n  D l2  + + Ain exp  �  −Pe0 − Pe+  2  z  R  �  + Bin exp  �  −Pe0 + Pe+  2  z  R  �  for ∥z∥ ≤ R ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  k2 n  D l2  − + Aout exp  �  −Pe0 − Pe−  2  z  R  �  for z < −R ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  k2 n  D l2  − + Bout exp  �  −Pe0 + Pe−  2  z  R  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  otherwise,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (S13)  which also defines the concentration profile of the products p(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' t) = n − s(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' As before,  we determined the integration constants Ain/out and Bin/out by imposing smoothness and con-  tinuity at the droplet interfaces z = ±R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' To that end, we used the Python library Sympy [8]  for symbolic calculations and confirmed with the computer algebra system Mathematica [9],  and provide the full expressions in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  The concentration profiles of substrates and products, in the co-moving frame of an  12  enzymatic droplet with velocity v, are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 2a of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We observe that  the droplet enriches the concentration of products at the expense of substrates, which is  most pronounced at the trailing edge of the droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' This leads to a difference in substrate  concentrations between the right and the left edge of the droplet, ∆s(v) := s(R) − s(−R),  which we quantify after inserting the integration constants [3] in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S13):  ∆s(v) = ∆s⋆ Pe+Pe0  �  cosh(Pe+) − cosh(Pe0)  �  + Pe−  �  Pe0 sinh(Pe+) − Pe+ sinh(Pe0)  �  Pe+Pe− cosh(Pe+) + (Pe2  + + Pe2  −) sinh(Pe+)/2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (S14)  Here, the substrate concentration difference between the reactive equilibria, ∆s⋆, is given by  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Because the total density of substrates and products is uniform in the scenario  with nonreciprocal interactions (Λ = 0), the difference in product concentrations between  the right and the left edge of the droplet is given by ∆p(v) = −∆s(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' As we show next, these  concentration differences can drive a net flux of enzymes from the trailing edge towards the  leading edge of the moving droplet, thus sustaining its motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The more generic scenario  with Λ ̸= 0 exceeds the scope of the present paper and will be published elsewhere [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  In short, one finds that Λ ̸= 0 can place further constraints on the parameters to observe  self-propulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Gradients in substrate and product concentration drive droplet motion  In the discussion so far, we have used the sharp-interface approximation for the enzyme  concentration profile c(x, t) = csharp(x − vt) of a droplet that moves with constant velocity  v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Now, we investigate the premises for this sharp-interface approximation to  be consistent with the enzyme dynamics obeying a continuity equation (S7a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The sharp-  interface approximation implies that the moving concentration profile of enzymes must be  in steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Using the substitution z = x−vt to transform the continuity equation (S7a)  into the co-moving reference frame of the droplet, one then has:  ∂z  �  Mc(z) ∂z  �  µ0(c(z)) + χs s(z) + χp p(z)  ��  = −v ∂zc(z) ≡ −∂zj(z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (S15)  13  Indefinite integration of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S15) yields  Mc(z) ∂z  �  µ0(c(z)) + χs s(z) + χp p(z)  �  = −v c(z) + j0 ,  (S16)  up to an integration constant j0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' To determine this integration constant, we first note that  far away from the droplet, at z ± ∞, all concentration profiles are homogeneous and the  enzyme concentration is given by c(±∞) = c−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Comparing these far-field conditions with  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S16), one finds that j0 = v c−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' To summarize, we have so far  Mc(z) ∂z  �  µ0(c(z)) + χs s(z) + χp p(z)  �  = −v  �  c(z) − c−  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (S17)  Now, we use the sharp-interface approximation c(z) = csharp(z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Integrating  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S17) over the droplet z ∈ [−R, R] to obtain an expression independent of local gradients,  we find  v = − Mc+  2R ∆c  �  χs ∆s(v) + χp ∆p(v)  �  ,  (S18)  where we have defined the enzyme concentration difference ∆c := c+−c−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Note that the bare  chemical potential of the enzyme, µ0(z), dropped out because it is mirror symmetric in the  sharp-interface approximation, so that µ0(R) = µ0(−R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Thus, the bare chemical potential  µ0(z) cannot drive directed net fluxes of enzymes on its own: it instead drives relaxation  towards an equilibrium state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In contrast, the coupling to substrate and enzymes, which are  driven out of equilibrium by chemical reactions, drives relaxation towards a non-equilibrium  steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Our result summarized by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S18) quantifies how asymmetric substrate and  product concentration profiles (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S14)) drive droplet motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  A self-consistency relation for the self-propulsion instability  In the scenario of nonreciprocal interactions (Λ = 0) where the total density of substrates  and products is constant, s(z) + p(z) = n, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S18) simplifies to:  v = Mc+ ∆χ  2R ∆c ∆s(v) ,  (S19)  14  where ∆χ := χp − χs is a measure for how enzymes are pulled more towards substrates  than towards products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The droplet velocity is proportional to the overall mobility M of  enzymes, and for constant ∆s decreases with the overall number of enzymes 2R∆c that are  translocated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  The droplet velocity in response to a gradient in the substrate concentration, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S19),  and the distribution of substrates in response to a droplet moving with a fixed velocity,  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S14), together form a self-consistency relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' This self-consistency relation can be  solved graphically by identifying a velocity where the left-hand side and the right-hand  side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S19) intersect, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 2b in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We note that the  substrate concentration difference between the right and the left edge of the droplet, ∆s(v),  is point symmetric with respect to a reversal of the velocity, ∆s(−v) = −∆s(v), which is  a feature of the intrinsic symmetry under parity of the system4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Because of this symmetry,  it is sufficient to discuss a scenario with positive velocities only, v ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In the limit of very  large velocities v → ∞, the substrate and product concentration fields do not have sufficient  time to respond to the moving enzymatic droplet and therefore remain homogeneous, which  implies ∆s(∞) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Because of the point symmetry of ∆s(v), the self-consistency relation (S19) always per-  mits a trivial solution with vanishing velocity v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' For small velocities, we observe that the  substrate concentration difference between the leading and the trailing edge of the droplet,  ∆s(v), initially grows with increasing velocity v until it reaches a single maximum, see  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S14) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 2b in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Given these features of the function ∆s(v), two  additional non-trivial solutions with finite droplet velocity |v| ̸= 0 emerge if the right-hand  side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S19) grows faster than the left-hand side in the limit of small velocities, v → 0:  1 < Mc+ ∆χ  2R ∆c ∂v∆s(v)  ����  v=0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (S20)  If a non-trivial solution to the self-consistency relation exists, then an initial inhomogeneity  of the substrate concentration profile will self-reinforce through a positive feedback loop  with the motion of the droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Because of this feedback loop, the droplet will settle in a  state that corresponds to the finite velocity v admitted by the self-consistency relation (S19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  4 One can see this symmetry by making the parity transformations z → −z and v → −v in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S12), which  preserves the structure of the equation and corresponds to a mirrored concentration field s(−z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' One could  also argue more heuristically: because there is no global bias for breaking symmetry (polarizing) towards  the left or to the right, the droplet could go either way with equal velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  15  Substituting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S14) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S20), the criterion to observe self-propelled droplets in our  simulations can therefore be written as:  1 < Mc+ ∆χ l−l+  2RD  ∆s⋆  ∆c  l+ sinh(2R/l+) + l− cosh(2R/l+) − l− − 2R  (l2  − + l2  +) sinh(2R/l+) + 2l+l− cosh(2R/l+) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (S21)  The emergence of two new stable fixed points |v| ̸= 0 as a function of enzyme mobility M  as control parameter, concomitant with a destabilization of the trivial fixed point v = 0,  corresponds to a pitchfork bifurcation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  To gain a better understanding on the conditions that are required to observe droplet  self-propulsion, we study inequality (S21) in the limits of very large or very small droplets:  1 < Mc+ ∆χ  2D  ∆s⋆  ∆c  l−  R ×  �  �  �  �  �  �  �  l+  l− + l+  , for R ≫ l± ,  R2  l2  +  , for R ≪ l± .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (S22)  As a function of the droplet radius R, the right-hand side of inequality (S22) is clearly non-  monotonic: it grows with increasing droplet size for small droplets, but then decays with  increasing droplet size for large droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Thus, the self-propulsion instability is suppressed  both for very large and for very small droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Specifically, droplets that are much smaller  than the characteristic length of the substrate and product concentration profiles, cannot  build up a sufficient difference in the concentrations of substrates and products between the  droplet interfaces to sustain self-propulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In the opposing limit where droplets are much  larger than the characteristic length of the substrate and product concentration profiles, the  diffusion of substrates and products ceases to play a role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Then, substrates and products  reach their local reactive equilibria, and the droplet also cannot build up a sufficient dif-  ference in the concentrations of substrates and products between the droplet interfaces to  sustain self-propulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In summary, droplet self-propulsion requires the droplet radius to  be compatible with the characteristic length of substrate and product concentration profiles,  and is optimal for R ∼ l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  16  𝑀𝜖0/𝐷  0  0  2  4  −2  −4  200  400  600  800  1000  𝑣 [𝑙0/𝜏0]  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Pitchfork bifurcation for droplet self-propulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Tuning enzyme mobility M as control  parameter, one finds that small enzyme mobilities admit only a single stable solution (solid lines)  with vanishing droplet velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' This solution becomes unstable above a critical value of the enzyme  mobility M⋆ (dashed line), where two new stable solutions with finite velocity appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Quasi-steady state approximation for droplet self-centering  From the inequality Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S21), we infer that a droplet can self-propel if certain conditions  are met, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=', in the case of large enzyme mobility M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' If the droplet exhibits self-propulsion,  then it will perform oscillatory motion in a closed domain or adhere to one of the domain  boundaries, as discussed in the main text and in section II B “Self-centering of a droplet in  a finite domain”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' However, even if the conditions for self-propulsion are not met, that is  when inequality Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S21) is not fulfilled, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S18) indicates that gradients in the densities  of substrates and products will still drive droplet motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Such concentration differences  between the droplet interfaces can result, for example, from an off-centered position of the  droplet in its enclosing domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Then, the droplet will gradually position itself towards  the domain center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In the following, we study this scenario theoretically, in the overdamped  limit where the droplet does not overshoot past the domain center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' To that end, we consider  a quasi-steady state approximation where there is a separation of time scales between the  slow motion of the droplet and the fast relaxation of the substrate and product concentration  profiles to their pseudo-steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We determine the concentration profiles in this pseudo-  steady state using the Python library Sympy [8] for symbolic calculations and confirmed  with the computer algebra system Mathematica [9], and provide the full expressions in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  17  We then calculate the substrate and product concentration values at the droplet interfaces,  and insert them into the self-consistency relation (S18) to determine the resulting droplet  velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' This procedure yields the droplet velocity v(xd) as a function of the position of the  droplet center xd, as plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Figure S5 indicates that the droplet position has a  single stable fixed point at the center of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Near the domain center, we again use  the Python library Sympy [8] and the computer algebra system Mathematica [9] to linearize  the dynamics and find exponential relaxation with the following timescale:  λ−1 =  l−R∆c  Mc+∆χ∆s⋆  �  cosh  �Lfree  l−  �  +  l+ sinh  � Lfree  l−  �  l− tanh  � R  l+  �  � �  sinh  �Lfree  l−  �  +  l− cosh  � Lfree  l−  �  l+ tanh  � R  l+  �  �  ,  (S23)  where we have defined Lfree := L−R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We find a very good agreement between these analytic  results and our finite element simulations [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 3c in the main text].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  To analyze Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S23), we first discuss a scenario where we keep the value of ∆s⋆ fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Then, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S23) shows that the characteristic time of self-centering diverges for l− ≪ Lfree  and for l− ≫ Lfree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Therefore, the typical timescale of finding the domain center is minimal if  the characteristic length of the concentration profiles outside of the droplet is comparable to  the typical distance towards the domain boundary, l− ∼ Lfree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Furthermore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S23) also  shows that the characteristic time of self-centering diverges for very large droplets R ≫ l+  and for very small droplets R ≪ l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Therefore, droplet self-centering is also optimized if  R ∼ l+, which allows building up concentration gradients across the droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Consistent with  these arguments for the droplet size, for large domain sizes the relaxation rate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S23)  scales as  λ−1 =  ∆c  Mc+∆χ∆s⋆ exp  �2Lfree  l−  � R  4l+  �  l− +  l+  tanh  � R  l+  �  � �  l+ +  l−  tanh  � R  l+  �  �  ,  (S24)  which has a minimum as a function of R/l+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  We next discuss how the features of (S23) relate to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 3c in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Now, the  value of the concentration difference between the reactive equilibria, ∆s⋆, is not fixed because  it depends on the catalysis rate k1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' When increasing the catalysis rate k1 to  very high values, the characteristic length l± of the concentration profiles becomes very small  and one finds that the droplet takes a longer time to find the center of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In the  opposite scenario, where the catalysis rate k1 is very small, one finds that the droplet also  18  −2  −1  1  2  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='00  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='02  𝑥𝑑 [𝑙0]  𝑣 [𝑙0/𝜏0]  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Droplet velocity as a function of its position in the quasi-steady state approximation, for  Mϵ0/D = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The domain has a half-size of L = 3l0, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 3b in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  takes a longer time to find the domain center because it cannot create significant substrate  and product concentration gradients (small value of ∆s⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Therefore, there is an optimal  value of the catalysis rate k1 where the droplet finds the center of the domain in the shortest  amount of time [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 3c in the main text].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  ENZYMATIC ACTIVITY OF DROPLETS LEADS TO COEXISTENCE  AND ARRESTS THE COARSENING PROCESS  So far we have analyzed the dynamics of a single droplet, and have shown how its en-  zymatic activity can lead to self-propulsion or self-positioning in a finite domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Now, we  extend our analysis to the dynamics of multiple droplets, in the parameter regime where none  of the droplets self-propel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' To that end, we consider small values of the enzyme mobility M  where inequality Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S21) is not fulfilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We then analyze under which conditions multiple  droplets will show arrested coarsening and therefore coexist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Before we proceed with this  theoretical analysis, we study simulations in the parameter regime where multiple droplets  coexist without showing self-propulsion, and analyze how the droplets position themselves  in a finite domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  19  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Localization of multiple coexisting droplets  We have discussed in section II B “Self-centering of a droplet in a finite domain” that enzy-  matically active droplets exhibit repulsive interactions with domain boundaries, where enzy-  matic substrate depletion is enhanced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Analogously, two neighboring droplets will strongly  deplete the substrate in the region between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The resulting substrate concentration  gradients will, through attractive enzyme–substrate interactions, drive enzyme fluxes within  each droplet that lead away from the neighboring droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' This leads to effective repulsive  interactions among droplets, which suggest that droplets will position themselves equidis-  tantly in a finite domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' For a steady state in which N droplets coexist in the same domain  without showing self-propulsion [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S6], equidistant positioning indicates a replica sym-  metry in which the domain can be divided into a chain of N identical subdomains, each  containing only one droplet [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In such a steady state, there is no net particle ex-  change across the interfaces between adjacent subdomains, which is equivalent to no-flux  boundary conditions at each subdomain boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Hence, in each individual subdomain,  the droplet will localize to the center according to sections II B “Self-centering of a droplet  in a finite domain” and III E “Quasi-steady state approximation for droplet self-centering”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Because of this symmetry of replicate subdomains, the distance between one of the domain  boundaries and the nearest droplet, L/N for a domain of size 2L, will be exactly half of  the distance between two adjacent droplets, 2L/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In the next section, we illustrate why  enzymatic droplets can coexist in the first place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Coexistence of droplets in a one-dimensional domain  To illustrate the mechanism underlying the coexistence of multiple droplets, we study  a scenario with only two droplets of differing sizes R1 and R2 ≥ R1 [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The larger  droplet depletes more substrate and accumulates more product than the smaller droplet,  which can be illustrated with the following limiting cases: For droplet sizes much larger  than the characteristic length of the substrate and product concentration profiles inside of  the droplet, R2 ≫ l+, the substrate and product concentrations at the droplet boundaries  are given by their reactive equilibria5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' For droplet sizes much smaller than the characteristic  5 The reactive equilibria correspond to the solution of the reaction-diffusion equations for substrates and  products, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S7b) and (S7c), in the limit D → 0 and Λ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='0  −15  −10  −5  0  5  10  15  concentration profiles [𝑐+]  𝑥 [𝑙0]  enzyme 𝑐  substrate 𝑠  product 𝑝  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Stationary concentration profiles corresponding to 5 droplets coexisting on an inter-  val, obtained after simulating the system for a total duration of 1000 τ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Boundaries of replicate  subdomains are indicated by dotted lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' As initial condition for the enzyme concentration, we  considered 5 distinct droplets of equal radii R = l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  length of the concentration profiles inside of the droplet, R1 ≪ l+, diffusion will homogenize  the substrate and product concentration profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Then, the concentrations at the droplet  boundaries remain at their ambient concentration values in the far field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  The resulting  concentration gradients will, because of strongly attractive enzyme–substrate interactions  and weaker enzyme–product interactions, drive a net flux of enzymes from the larger to  the smaller droplet, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S7a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Therefore, the size difference between the two droplets,  ∆R = R2 − R1 will shrink over time, indicating coexistence [Supplemental Video 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  In the following, we derive the characteristic timescale of droplet size equilibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' For  droplets that do not exhibit self-propulsion, the dynamics of enzymes is very slow compared  to the dynamics of substrates and products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Therefore, in full analogy to section III E  “Quasi-steady state approximation for droplet self-centering”, we make a quasi-steady state  approximation for the fast relaxation of the substrate and product concentration profiles in  response to the quasi-static enzyme concentration profile:  c(x) =  �  �  �  �  �  c+  for ∥x∥ ≤ R1  and  ∥x − d∥ ≤ R2 ,  c−  otherwise,  (S25)  where the droplet centers are located at x = 0 and x = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Given this concentration profile of  21  concentration profiles  𝑐(𝑥) = 𝑐−  𝑐(𝑥) = 𝑐+  𝑐(𝑥) = 𝑐+  0 𝑅1  𝑑  𝑑−𝑅2  domain  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Illustration of the sub-domains in the sharp-interface approximation, for two droplets that  gradually equilibrate their sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Gray areas illustrate subdomains, as discussed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In each  subdomain, we only need to solve a Helmholtz equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' To simplify this mathematical problem,  we construct a scenario where the substrate and product concentration profiles are approximately  symmetric with respect to the droplet centers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Specifically, for our analytical calculations we  consider a domain that ends at the droplet centers (black vertical lines) and has no-flux boundary  conditions there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' This construction is exact if, for an alternating chain of large and small droplets,  the substrate and product concentration profiles are symmetric with respect to each droplet center  (replica symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In our simulations, this would correspond to two droplets that are positioned  equidistantly in a simulation domain with periodic boundary conditions at the position indicated  by the dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' For simulation domains with no-flux boundary conditions at the position  indicated by the dashed lines, our simplification is nevertheless a good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  enzymes, in the scenario with nonreciprocal interactions (Λ = 0) one can then analytically  solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S9) for the concentration profiles of substrates s(x) and products p(x) = n−s(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  This mathematical problem is practically identical to sections III A “Substrate and product  redistribution by a stationary droplet” and III E “Quasi-steady state approximation for  droplet self-centering”, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S7, and the full expressions are provided in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Here, we outline the main assumptions and simplifications of the derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  In principle, for droplets that are close to each other, one observes a net substrate and  product concentration gradient across each individual droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' This net concentration gra-  dient will lead to an effective repulsion between the two droplets, analogous to sections II B  “Self-centering of a droplet in a finite domain” and IV A “Localization of multiple coex-  isting droplets”, and prevent coalescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Here, however, we are interested in the particle  exchange currents between the two droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Therefore, we focus on the substrate and prod-  uct concentration differences between the nearest interfaces of the two different droplets,  ∆s := s(d−R2)−s(R1) and ∆p := p(d−R2)−p(R1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' To simplify our analysis, we construct  22  a scenario where the substrate and product concentration profiles are approximately sym-  metric with respect to each individual droplet, so that each droplet will remain immobile as  discussed in section IV A “Localization of multiple coexisting droplets”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Specifically, we con-  sider no-flux (reflective) boundary conditions at the droplet centers [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S7], which would  be exact for a periodic chain of big and small droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Then, we solve for the concentration  profiles of substrates and products in the domain [0, d], and find the substrate concentration  difference between the closest interfaces of the two droplets [3],  ∆s = −∆s⋆∆R  l+  �  sinh  � R  l+  �  +  l+ cosh  � R  l+  �  l− tanh  � Lfree  l−  �  �−1 �  cosh  � R  l+  �  +  l− sinh  � R  l+  �  l+ tanh  � Lfree  l−  �  �−1  , (S26)  for small differences in the droplet radii ∆R := R2 − R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Here, we have defined the mean  droplet radius, R := (R1+R2)/2, and the free half-distance between droplets Lfree := d/2−R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  These concentration differences drive an exchange current of enzymes between the two  droplets, which we in analogy to III C “Gradients in substrate and product concentration  drive droplet motion” determine by integrating the continuity equation Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S7a) over the  inter-droplet domain [R1, d − R2],  j× 2Lfree = −Mc−  �  χs ∆s + χp ∆p  �  ,  (S27)  where we have omitted the contribution of the Cahn-Hilliard chemical potential µ0(z) as  explained next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  In general, the interaction-driven exchange flux (S27) will be superimposed by a coarsen-  ing current that stems from the Cahn-Hilliard chemical potential µ0(z), which drives slow  coarsening in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In the sharp-interface limit and in 1d, however, coarsening  becomes infinitely slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Therefore, in 1d the flux given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S27) can easily drive a  dynamics that is opposite to coarsening, by transporting material from the larger droplet  to the smaller droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The size difference between the two droplets then gradually changes  with time, and is in a finite 1d domain with no-flux boundary conditions determined by6  j× = ∆c  �  ∂t(R2 − R1)  �  = ∆c ∂t∆R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Taken together, the size difference between the two  6 In this case, the flux is given by ∆c ∂t(2R2) = −∆c ∂t(2R1) = j×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' For periodic domains, an additional  factor of 2 would enter because one has particle exchange with twice as many droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  23  droplets will evolve as follows:  ∂t∆R = M c− ∆χ  2 Lfree ∆c ∆s ,  (S28)  where we have used ∆p = −∆s and defined ∆χ := χp − χs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' After inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S26) into  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S28), one finds that the size difference decays exponentially with a characteristic  timescale  λ−1 = 2 Lfree l+ ∆c  M c− ∆χ ∆s⋆  �  sinh  � R  l+  �  +  l+ cosh  � R  l+  �  l− tanh  � Lfree  l−  �  � �  cosh  � R  l+  �  +  l− sinh  � R  l+  �  l+ tanh  � Lfree  l−  �  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S29)  This result is equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S23) under the replacements c+ ↔ c−, l+ ↔ l−, and Lfree ↔  R, because it corresponds to the self-centering of an inverted droplet7 Thus, our discussion  for the self-centering of droplets, see section III E “Quasi-steady state approximation for  droplet self-centering”, also applies here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In particular, droplet size equilibration is fastest  when the length scales of the droplets, concentration profiles of substrates and products,  and the distance between droplets are compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Comparing our analytical results to  simulations, we find good agreement [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Coexistence of droplets in a three-dimensional domain  In the previous section, we have seen that enzyme–substrate and enzyme–product inter-  actions can easily oppose and reverse the coarsening process in 1d, which becomes infinitely  slow in the sharp-interface limit in 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' For 2d and 3d droplets, an additional effect comes  into play that arises from the curvature of the droplet interfaces: the Laplace pressure leads  to an evaporation of small droplets and stabilization of large droplets, thus greatly acceler-  ating the coarsening process [10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' This means that in 2d and 3d the interaction-driven  exchange currents between different droplets are superimposed by much stronger coarsening  currents than in 1d, which narrows down the parameter regime in which one can observe  arrested phase separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In the following, we derive a criterion to still observe arrested  coarsening in 2d and 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  7 In comparison to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S23), there is an additional factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' This factor would drop out in a periodic  domain, where droplet size equilibration is twice as fast because each droplet can exchange particles with  two neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  24  ×10−3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='1  1  0  2  4  6  10  𝜆 [𝜏0  −1]  𝑘1𝑐+/𝑘2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Droplet size equilibration rate as a function of the reaction rate k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The solid line indicates  the expression obtained in the quasistatic limit (S29), and agrees well with the results of numerical  simulations (dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The droplets are initially positioned at x1(0) = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='5l0 and x2(0) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='5l0 in  the domain [−5l0, 5l0], have radii R1(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='9l0 and R2(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='1l0, and have an interface width of  w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='05l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  We consider a droplet with radius R and small interface width w =  �  2κ/r ≪ R, which  has the homogeneous free energy density f(c) = u  4(c−˜c)4− r  2(c−˜c)2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The surface  tension of this droplet is given by γ = 2r2w/(3u) and induces a Laplace pressure, which  leads to an increase in the enzyme concentration just outside of the droplet as described by  the Gibbs-Thomson relation [12, 13]:  δc(R) = 2γ  R  1  ∆c f ′′(c−) = 1  6  w  R ∆c ,  (S30)  where f ′′(c−) = 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The increase in the enzyme concentration affects the Cahn-Hilliard  chemical potential outside of the droplet, µ0 ≃ 2r δc, and is more pronounced for small  droplets, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' This leads to a difference in the Cahn-Hilliard chemical potential  between the two droplets, ∆µ0, which drives diffusive transport of enzymes from small  25  towards large droplets:  j0 ≈ −Mc−  ∆µ0  d  ≈ −2 r Mc−  δc(R2) − δc(R1)  d  = r Mc−w ∆c  3 R1 R2 d ∆R ,  (S31)  where we have assumed that the droplets are far apart, d ≫ R1,2, and defined ∆R := R2−R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  As discussed in section IV B “Coexistence of droplets in a one-dimensional domain”, the  enzymatic activity of the droplets locally depletes substrate and accumulates product, which  results in substrate and product concentration gradients that can contribute to the exchange  flux between different droplets, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' For droplets that are far apart, d ≫ R1,2, one  then has  j× ≈ M c− ∆χ  d  ∆s ,  (S32)  where we have again used ∆p = −∆s, and defined ∆χ := χp − χs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' If the droplets are far  apart, then we can determine the local concentration profiles of substrates and products  independently for each droplet, by solving a Helmholtz equation in spherical coordinates8:  D  r2 ∂r  �  r2∂rs(r)  �  −  �  k1c(r) + k2  �  s(r) + k2n = 0 ,  (S33)  where we make a sharp-interface approximation for the concentration of enzymes, c(r) =  csharp(r), see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We determine the substrate and product concentration profiles as  described in section III A “Substrate and product redistribution by a stationary droplet”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  the full expressions are provided in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The difference between the substrate concentration  values at the interfaces of two droplets with a small difference in size, ∆R ≪ l+, are then  given by:  ∆s ≈ −∆s⋆ l−1  + [sinh(2R/l+) − 2R/l+] + 2l−1  − [sinh2(R/l+) − (R/l+)2]  2R2 �  l−1  + cosh(R/l+) + l−1  − sinh(R/l+)  �2  ∆R .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (S34)  As discussed in section IV B “Coexistence of droplets in a one-dimensional domain”, the  larger droplet typically depletes more substrate, leading to a substrate concentration dif-  ference proportional to the size difference: ∆s ∝ −∆R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  This concentration difference  drives a flux of enzymes from the larger to the smaller droplet [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We now substitute  8 A general analytic solution for droplets that are close to each other would require solving the Helmholtz  equation in bispherical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  However, the Helmholtz equation is not separable in bispherical  coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  26  4  2  0  2  4  𝑡 = 75 𝜏0  enzyme concentration 𝑐 [𝑐+]  radius 𝑟 [𝑙0]  𝑧 [𝑙0]  2  2  0  1  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='0  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Snapshot of a simulation of two coexisting droplets in 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The color map indicates the  local enzyme concentration, and the white arrows are proportional to the local mean velocity of the  enzymes jc(x)/c(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The simulation was performed in a rotationally invariant cylindrical domain  of radius Lr = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='5l0 and half-height Lz = 5l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Parameters: Mϵ0/D = 10, w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='05l0 as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 4a  in the main text, with k1c+/k2 = 1 and χs/r = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S34) into the expression for the interaction-driven exchange fluxes that oppose coarsen-  ing, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S32), and ask under which conditions these fluxes may dominate (j× +j0 < 0) over  the coarsening currents given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' This comparison yields a criterion to observe  arrested coarsening, which is fulfilled if the interactions are sufficiently strong:  ∆χ ≳  2rw∆c  �  l−1  + cosh( ¯R/l+) + l−1  − sinh( ¯R/l+)  �2  3∆s⋆ �  l−1  + (sinh(2 ¯R/l+) − 2 ¯R/l+) + 2l−1  − (sinh2( ¯R/l+) − ( ¯R/l+)2)  � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (S35)  Comparing this analytical criterion with numerical simulations, we find good agreement  [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 4a in the main text, dynamics shown in Supplemental Video 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Droplet division  We have seen that the enzymatic activity of the droplet, coupled with enzyme–substrate  and enzyme–product interactions, can oppose and stop coarsening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Then, could this mech-  anism also lead to a shape instability and to divisions of droplets?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In fact, in 2d and 3d,  substrate depletion is enhanced for smaller curvatures of the droplet interface and reduced for  27  larger curvatures of the droplet interface, by having a smaller interface over which substrate  can be resupplied from the environment relative to the enclosed volume where substrate is  depleted by enzymes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' This effect is analogous to the stronger substrate depletion by larger  spherical droplets when compared to smaller droplets, and can drive a net flux of enzymes  from regions with a small curvature of the droplet interface towards regions with a large  curvature of the droplet interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Regions with a large curvature will then move outwards  while regions with a small curvature will move inwards, elongating the droplet and further  increasing the differences in curvature (positive feedback mechanism), while the droplet vol-  ume remains conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We observed such droplet elongation in simulations for sufficiently  large values of the interaction parameter ∆χ [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Δ𝜒 [𝑟]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='00  0  5  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='50  𝑘1𝑐+/𝑘2  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Results of numerical simulations starting with a single droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The droplet can either  remain spherical (yellow regime) or start elongating (cyan regime) depending on the values of k1  and ∆χ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The simulation was performed in a rotationally invariant cylindrical domain of radius  Lr = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='5l0 and half-height Lz = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='5l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Parameters: Mϵ0/D = 10, and w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='05l0 as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 4b in  the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Supplemental Video 5 shows a simulation of a controlled division process in 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  We  take the catalysis rate k1 of converting products into substrates as a control parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  We start with a large value of k1, for which a single droplet is stable at the center of the  domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Then, upon lowering k1, we observe the following dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The droplet first  elongates (initial shape instability) and then forms a dumbbell shape with a neck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The tube  28  connecting the two dumbbells gradually becomes thinner until it pinches off, leaving two  separated droplets (division).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' These two droplets assume a spherical shape and position  themselves equidistantly in the domain as expected from the discussion of section IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' If  we increase k1 to its initial value, then the two droplets remain stable, showing that the  process is irreversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Figure 4b in the main text shows some snapshots of the enzyme  concentration, with t = 0 being the time when k1 is switched to a smaller value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Note that we have only observed droplet divisions in 3d, but never in 2d or 1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' This  suggests an effect that destabilizes an elongated cylindrical shape in favor of spherical shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Such an effect is characteristic for classical pearling instabilities driven by surface tension [14],  where spherical shapes have smaller surface area than cylindrical shapes with the same  volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Therefore, cell division should be controlled by the interface width w, which not  only sets the value of the surface tension, but also defines a length scale over which the two  sides of a thin cylinder that connects the two dumbbells can interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  MULTI-COMPONENT CONDENSATES  So far, we have studied a scenario where the enzymes undergo spontaneous phase sepa-  ration on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' However, one can also envision a much more general scenario where  the enzymes do not phase separate spontaneously, but are only enriched in a droplet that  consists of a scaffold protein with concentration q(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In this scenario, the scaffold proteins  spontaneously phase separate driven by the chemical potential µ0(q) of the Cahn-Hilliard  model, and only interact with the enzymes through a Flory-Huggins coupling χq:  ∂tq = ∇ ·  �  Mqq∇  �  µ0(q) + χq c  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (S36)  We assume that the enzymes are present at relatively small concentrations so that they show  currents driven by diffusion, and effective Flory-Huggins couplings to the scaffold proteins,  substrates, and products:  ∂tc = ∇ ·  �  Dc∇c + Mcc∇  �  χqq + χss + χpp  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  (S37)  Attractive effective interactions between the scaffold proteins and the enzymes, χq < 0, lead  to an enrichment of enzymes in the droplet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' In the sharp-interface limit, this enrichment is  29  quantified by  c+  c−  = exp  �  −Mc  Dc  χq (q+ − q−)  �  ,  (S38)  where the indices ± indicate the concentrations on the inner and the outer side of each  droplet interface, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Analogous to droplets that consist mainly of enzymes, which  we have discussed so far, our simulations show self-propulsion if the mobilities of the scaffold  proteins (Mq) and enzymes (Mc) are sufficiently large, and self-centering otherwise (Sup-  plemental Video 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The enzymes then act as a link, by mediating the spatial organization  of substrate and product through the droplet, as well as transmitting forces to the droplet  that arise due to these inhomogeneous substrate and product concentration profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  30  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  SUPPLEMENTAL VIDEOS  In the following, we describe the videos available as supplemental material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' video_1_self_propulsion_instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='mp4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  For all simulations, we set M =  1000D/ϵ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 1d droplet: Evolution of the concentration profiles of enzymes, substrates  and products resulting from a numerical simulation of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S7) in a one-dimensional  interval with no-flux boundary conditions and half-length L = 10l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' As initial condi-  tion we consider a single droplet of enzymes at the center of the interval, the starting  concentrations of substrates and products are the equilibrium values of the reaction  terms plus small random perturbations in the droplet region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We observe the self-  propulsion instability, the droplet starts moving in a random direction determined by  the initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 2d droplet: Evolution of the concentration profile of enzymes  resulting from a numerical simulation performed in a two-dimensional circular domain  of radius Lr = 7l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 3d droplet: Evolution of the concentration profile of enzymes  resulting from a numerical simulation performed in a three-dimensional cylindrical  domain of radius Lr = 4l0 and half-height Lz = 7l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' video_2_positioning_1d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='mp4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Evolution of the concentration profiles of enzymes,  substrates and products resulting from numerical simulations analogous to the one  for the 1d droplet in Supplemental Video 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' As initial condition we consider a single  droplet of enzymes positioned at xd(0) = −l0, the starting concentrations of sub-  strates and products are the stationary profiles that they would reach in the absence  of interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The parameter values are the same as for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 3a in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' video_3_positioning_2d_and_3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='mp4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  2d droplet:  Evolution of the concentra-  tion profiles of enzymes resulting from a numerical simulation performed in a two-  dimensional square domain of half-side length L = 3l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' As initial condition we consider  a single droplet of enzymes positioned at (x, y) = (−l0, 0), the starting concentrations  of substrates and products are the equilibrium values of the reaction terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  The  droplet moves to the center of the domain and localizes there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The other parameter  values are the same as for Supplemental Video 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 3d droplet: Evolution of the con-  centration profiles of enzymes resulting from a numerical simulation performed in a  31  three-dimensional cylindrical domain of radius Lr = 3l0 and half-height Lz = 3l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The  droplet is initially positioned at z = −l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' video_4_coexistence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='mp4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 1d droplets: Evolution of the concentration profiles of  enzymes, substrates and products resulting from a numerical simulation analogous to  the one for the 1d droplet in Supplemental Video 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' As initial condition we consider  two distinct droplets of enzymes of radii R1(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='5l0 and R2(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='5l0 positioned at  x = ∓2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='5l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The starting concentrations of substrates and products are the equilibrium  values of the reaction terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Enzymes are transported from the larger droplet to the  smaller one until the radii of the two droplets become equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 3d droplets: Evolution of  the concentration profiles of enzymes resulting from a numerical simulation analogous  to the one for the 3d droplet in Supplemental Video 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The initial conditions are  analogous to the one for the 1d droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Lr = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='5l0, Lz = 5l0, χs/r = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='21,  w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='05l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' video_5_droplet_division.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='mp4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Evolution of the concentration profiles of enzymes  resulting from a numerical simulation performed in a three-dimensional cylindrical do-  main of radius Lr = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='5l0 and half-height Lz = 4l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' As initial condition we considered  a droplet of enzymes positioned at the center of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The starting concentra-  tions of substrates and products are the equilibrium values of the reaction terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' The  simulation starts with a catalysis rate k1 = 100 k2/c+ for which the droplet is stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Then the catalysis rate is switched to k1 = 1 k2/c+ and the droplet divides into two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  Finally, the catalysis rate is switched again to k1 = 100 k2/c+ and the two droplets  maintain their stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Parameters: M = 10D/ϵ0, χs = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='5r, w = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='05l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' video_6_multicomponent_droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='mp4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Evolution of the concentration profiles of  scaffold proteins, enzymes, substrates and products resulting from numerical simula-  tions of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' (S7b, S7c, S36, S37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' We consider as initial condition a droplet of scaffold  proteins and a uniform concentration of enzymes c(t = 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='25q+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Self-propulsion:  Simulation analogous to the 1d droplet shown in Supplemental Video 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Parameters:  χq/r = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='2, Mq = Mc = 1000D/ϵ0, Dc/D = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Self-centering: Simulation anal-  ogous to Supplemental Video 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Parameters: χq/r = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='2, Mq = Mc = 100D/ϵ0,  Dc/D = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  32  [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' De Groot and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Mazur, Non-Equilibrium Thermodynamics, Dover Books on Physics  (Dover Publications, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Balian, From Microphysics to Macrophysics (Springer Berlin Heidelberg, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  [3] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Demarchi, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Goychuk, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Maryshev, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Frey, Source code - Enzyme-enriched conden-  sates show self-propulsion, positioning, and coexistence (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Logg, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Mardal, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Wells, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=', Automated Solution of Differential Equations by  the Finite Element Method, edited by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Logg, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Mardal, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Wells (Springer,  2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Newville, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Stensitzki, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Allen, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Ingargiola, LMFIT: Non-linear least-square  minimization and curve-fitting for Python (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Goychuk, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Demarchi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Maryshev, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Frey, The role of non-reciprocity in the fluid-like  dynamics of enzyme-enriched droplets (to be published).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  [7] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Fisher, The wave of advance of advantageous genes, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Eugen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' 7, 355 (1937).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content='  [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Meurer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Smith, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Paprocki, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' ˇCert´ık, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Kirpichev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Rocklin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Kumar,  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Ivanov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Moore, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Singh, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Rathnayake, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Vig, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Granger, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Muller,  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Bonazzi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Gupta, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Vats, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Johansson, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Pedregosa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Curry, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Terrel,  ˇS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
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+page_content=' Zwicker, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
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+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
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+page_content=' Rayleigh, On the instability of jets, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
+page_content=' s1-10, 4 (1878).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YdAyT4oBgHgl3EQfiPiV/content/2301.00392v1.pdf'}
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+GROWTH RATES OF THE NUMBER OF INDECOMPOSABLE SUMMANDS IN
+TENSOR POWERS
+KEVIN COULEMBIER, VICTOR OSTRIK AND DANIEL TUBBENHAUER
+Abstract. In this paper we study the asymptotic behavior of the number of summands in tensor products
+of finite dimensional representations of affine (semi)group (super)schemes and related objects.
+Contents
+1.
+Introduction and main results
+1
+2.
+The general linear group and consequences
+4
+3.
+The general linear super group and consequences
+11
+4.
+The general linear quantum group and consequences
+14
+5.
+Counterexamples
+17
+6.
+Questions
+18
+References
+20
+1. Introduction and main results
+A central, yet hard, problem in representation theory is the decomposition of tensor products of repre-
+sentations into indecomposable summands. Computations of these decomposition numbers are often major
+unsolved problems in representation theory. In this paper we take a different perspective and we are interested
+in asymptotic properties of the number of indecomposables in tensor products of representation rather than
+explicit decompositions. In contrast to the question of explicit decompositions, we obtain results in extensive
+generality.
+To get started, let Γ be a finite group with a finite dimensional representation V over some field k.
+Definition 1.1. We define
+bΓ,V
+n
+:= #indecomposable summands in V ⊗n counted with multiplicities
+(sometimes simply denoted by bn when Γ and V are clear from the context), where ‘indecomposable’ means
+as Γ-representation. Let further
+βΓ,V := lim
+n→∞
+n�
+bΓ,V
+n
+.
+Note that bΓ,V
+n
+bΓ,V
+m
+≤ bΓ,V
+n+m, so that βΓ,V is well-defined by (a version of) Fekete’s Subadditive Lemma.
+There is no chance to compute bΓ,V
+n
+explicitly in this generality, but it turns out that the ‘limit’ βΓ,V can
+be understood. As a first step we note that we have the following lemma whose the proof is immediate.
+Lemma 1.2. We have bΓ,V
+n
+≤ (dim V )n, and consequently
+βΓ,V ≤ dim V.
+□
+A classical result of Bryant–Kovács [BK72, Theorem 1] shows that there exists some n for which V ⊗n
+contains a projective direct summand (projective over Γ if V is a faithful and projective over the appropriate
+factor group of Γ otherwise). As observed in [BS20, Theorem 6.3], a consequence of this is that the bound
+for βΓ,V in Lemma 1.2 is actually an equality, that is:
+Theorem 1.3. We have
+βΓ,V = dim V.
+□
+Mathematics Subject Classification 2020. Primary: 17B10, 18M05; Secondary: 16T05, 17B37, 20C25.
+Keywords. Tensor products, asymptotic behavior, affine group schemes, affine semigroup schemes, semigroups, supergroups,
+Hopf algebras, (symmetric) monoidal categories.
+1
+arXiv:2301.00885v1  [math.RT]  2 Jan 2023
+
+2
+K. COULEMBIER, V. OSTRIK AND D. TUBBENHAUER
+It is a natural question to which extent this result generalizes. Concretely, one can let Γ be an infinite group,
+an infinite semigroup, an affine group scheme, super versions, a finite dimensional Hopf algebra or any other
+algebraic structure for which we have a notion of tensor products of representations, and V a finite dimensional
+Γ-representation. Definition 1.1 and Lemma 1.2 work verbatim, including that βΓ,V is well-defined, although
+clearly the above proof of Theorem 1.3 does not extend.
+We show that the theorem remains valid in great generality, but has limitations:
+Theorem 1.4.
+(a) Theorem 1.3 holds true for Γ any affine semigroup superscheme as defined in Definition 2.1. Theo-
+rem 1.3 holds also true for the quantum groups Uq(slM) and Uq(glM), where we allow any q ∈ k∗ for
+k∗ = k \ {0, −1} if char(k) ̸= 2 and k∗ = k \ {0} otherwise.
+(b) Theorem 1.3 does not hold true in general for Hopf algebras.
+Remark 1.5.
+(a) Note that Theorem 1.4.(a) includes the cases where Γ is any, possibly infinite, abstract semigroup (for
+instance a monoid or group – as observed above, Theorem 1.4.(a) is classical for finite groups, but
+already for infinite abstract groups it appears to be new). Indeed, using Tannakian reconstruction we
+can associate an affine semigroup scheme to an abstract semigroup which has equivalent representation
+theory. The result can of course also be obtained directly without this observation and we elaborate
+on special cases like these in the main body of the paper.
+(b) We have not sought completeness in Theorem 1.4(a). We can, for example, include algebraic objects
+like transitive groupoids in schemes, see Theorem 2.6. We omit such cases here for clarity and because
+they are included in Theorem 1.9(a) below.
+(c) The proof of Theorem 1.4.(a), excluding the quantum group case, can be reduced to the specific cases
+of the general linear group GLM and the general linear supergroup GLM|N.
+(d) While proving Theorem 1.4.(a) we will also give asymptotic formulas in some cases, which improves
+Theorem 1.4.(a).
+Let us also comment on some variations of Theorem 1.4.(a) which have appeared in the literature.
+Remark 1.6.
+(a) There are known variants of βΓ,V for which the analog of Theorem 1.4.(a) is not valid. For instance if,
+in characteristic p > 0, one only counts direct summands of dimension not divisible by p (categorical
+dimension zero), it is shown in [CEO21, Theorem 8.1] for affine group superschemes that the corre-
+sponding limit yields a ring homomorphism from the Grothendieck ring which takes values in the ring
+of integers of a particular cyclotomic extension of Q. For example, for p = 5, Γ = Z/5Z and V its
+indecomposable of dimension three the limit is the golden ratio. In the case of a finite group where one
+only counts non-projective summands one obtains the variant studied for instance in [BS20]. (Note
+that these results do not cover the monoid and semigroup case and it would be interesting to know
+whether there are similar statements for monoids and semigroups.) Our result that βΓ,V = dim V
+for affine semigroup superschemes might be useful to compute the various variants of βΓ,V in the
+literature.
+(b) Let k = k and consider TLn = EndSL2
+�
+(k2)⊗n�
+which is known as the Temperley–Lieb algebra. By
+Schur–Weyl duality, the dimensions of simple TLn-representations correspond to the decomposition
+multiplicities of indecomposables in (k2)⊗n, and thus, Theorem 1.4.(a) and its proof given below imply
+that some of the simple TLn-representations are very large for n ≫ 1. In the spirit of this example,
+the paper [KST22] proposes to study large simple representations in the setting of monoids that arise
+in a ‘Schur–Weyl dual way’ from Theorem 1.4.(a) such as the Temperley–Lieb monoid, the Brauer
+monoid etc. We hope that Theorem 1.4.(a) can be used to generalize [KST22] beyond the case of the
+monoids discussed in that work.
+Note that Theorem 1.3 is a weaker statement than an asymptotic formula for the growth of bΓ,V
+n
+. We show
+this in the following example illustrating Theorem 1.3, which is in some sense the key example.
+Example 1.7. Take k = C, Γ2 = SL2 and V2 = C2, its vector representation. Then the first numbers bΓ2,V2
+n
+are:
+{1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252},
+bΓ2,V2
+n
+for n = 0, . . . , 10.
+Here, as throughout, we view bn = bΓ2,V2
+n
+as a function in n.
+
+GROWTH RATES OF THE NUMBER OF INDECOMPOSABLE SUMMANDS IN TENSOR POWERS
+3
+A Mathematica loglog plot of
+n√bn (y-axis) for n ∈ {1, . . . , 1000} (x-axis) gives
+bn
+1/n
+2
+1
+5
+10
+50
+100
+500 1000
+1.00
+1.25
+1.50
+1.75
+2.00
+,
+and indeed the limit is two, as predicted by Theorem 1.3. Precisely,
+1000√b1000 ≈ 1.99265.
+However, the asymptotic growth rate of bn is different than 2n. As we will see in Example 2B.4, we get
+bΓ2,V2
+n
+∼
+�
+2/π · 2n/√n.
+(Here and throughout, we use f ∼ g for f is equal to g asymptotically, meaning the ratio of f and g converges
+to one.) We have
+�
+2/π ≈ 0.798 and Mathematica’s log plot gives:
+bn
+2n
+0.798·2n
+n
+5
+10
+15
+20
+1
+10
+100
+1000
+104
+105
+106
+,
+bn
+2n
+0.798·2n
+n
+20
+40
+60
+80
+100
+10
+1011
+1021
+1031
+.
+For a precise statement see Example 2B.4 below.
+3
+A broader formulation of the ideas in Definition 1.1 and Theorem 1.3 is the following setup.
+Notation 1.8. Let D be a k-linear Karoubian monoidal category that is Krull–Schmidt, with a k-linear
+faithful monoidal functor F : D → VectK to the category VectK of finite dimensional vector spaces over a field
+extension K of k.
+Note that, if K is a finite extension of k, then the existence of F implies that morphism spaces in D are
+finite dimensional, so D is Krull–Schmidt automatically Krull–Schmidt.
+For any object X ∈ D, we can define bX
+n similarly as in Definition 1.1 as the number of indecomposable
+direct summands in X⊗n and we have
+βD,X := lim
+n→∞
+n�
+bX
+n ≤ dim F(X).
+Again, βD,X is well-defined by (a version of) Fekete’s Subadditive Lemma. We prove the following result,
+which also generalizes most of the examples in Theorem 1.4.
+Theorem 1.9.
+(a) If D and F are symmetric monoidal, then Theorem 1.3 holds, i.e.:
+βD,X = dim F(X).
+(b) Assume that char(k) = 0. If D is symmetric and F can be lifted to a symmetric monoidal functor
+F ′ : D → SV ectK to the category of super vector spaces SV ectK, then Theorem 1.3 holds, i.e.:
+βD,X = dim F(X).
+(c) Let k = C and D = RepCSL2. For every m ∈ Z≥2, there exists a faithful monoidal functor Fm : D →
+VectC, which sends the vector SL2-representation X to Cm. Hence, for m ≥ 3 we have
+βD,X = 2 < m = dim Fm(X).
+Here and throughout, RepkΓ denotes the category of finite dimensional (rational) Γ-representations over k.
+Remark 1.10.
+(a) Note that the functors Fm in Theorem 1.9.(c) are not symmetric for m ≥ 3, so Theorem 1.9.(a) does
+not apply.
+
+4
+K. COULEMBIER, V. OSTRIK AND D. TUBBENHAUER
+(b) Faithfulness of F in Theorem 1.9.(a) is required to ensure the estimate bX
+n ≤
+�
+dim F(X)
+�n and cannot
+be dropped, since it is easy to construct counterexamples with Deligne’s categories of [Del07]. Of
+course, even without faithfulness, the bound dim F(X) ≤ βD,X remains valid.
+Acknowledgments. Want to thank Pavel Etingof for comments on a draft of this paper, in particular for
+Remark 6.2 which is Pavel’s observation, Andrew Mathas for help with the literature on symmetric groups
+and Schur algebras, Volodymyr Mazorchuk for email exchanges about monoids, and Jonathan Gruber and
+Arun Ram for discussions about growth rates. We also thank the MFO workshop 2235 “Character Theory and
+Categorification” for bringing us together in Oberwolfach in August/September 2022 – this project started
+during this fantastic workshop.
+K.C. was partly supported by ARC grant DP200100712. D.T. was supported, in part, by the Australian
+Research Council, and V.O. and D.T. were supported by their depressions.
+2. The general linear group and consequences
+Let us start by defining some of our main players:
+Definition 2.1. We define an affine semigroup scheme (over k) to be a semigroup object in the category of
+affine k-schemes (the opposite of the category of commutative k-algebras). Equivalently, we can think of it as
+a representable functor from the category of k-algebras to the category of semigroups. Concretely, an affine
+semigroup scheme corresponds to a commutative bialgebra, potentially without counit. The special case of
+an affine monoid scheme corresponds precisely to a bialgebra with a counit, and the further special case of an
+affine group scheme corresponds to a Hopf algebra.
+We have the same notions in the ‘super’ version, by replacing the category of commutative k-algebras with
+the category of graded commutative Z/2Z-graded k-algebras.
+Let M ∈ Z>0, consider GLM and let VM be the tautological representation of GLM with dim VM = M.
+Set bM
+n = bGLM,VM
+n
+and βM = βGLM,VM = limn→∞
+n�
+bM
+n .
+Proposition 2.2. For any M ∈ Z>0 we have βM = M.
+Proof. The case M = 1 is immediate. In Section 2C and Section 2D we prove the statement for M > 1.
+□
+We also fix some notation:
+Notation 2.3.
+(a) Recall that k denotes an arbitrary field, and we thus have that its characteristic char(k) ∈ N. We
+always let p ∈ Z>1 ∪ {∞} denote the additive order of 1 ∈ k×. Thus, char(k) = p except that for
+char(k) = 0 we set p = ∞. When we state char(k) = p > 0 it thus unambiguously means positive
+characteristic.
+(b) Throughout this section, we will set ΓM = SLM. Since the decomposition of V ⊗n
+M
+into indecomposable
+summands is identical for SLM and GLM, we can focus just on ΓM Proposition 2.2.
+Remark 2.4. The proof of Proposition 2.2 will crucially exploit the theory of tilting representations, see
+for example [Jan03, Part II.E] (note that the relevant section in [Jan03] works over an arbitrary field of
+characteristic char(k) = p > 0) or [AST18] and the appendix for its arXiv version for background. The point
+is that VM is tilting, and by abstract theory direct summands of tensor products of tilting representations are
+tilting. Thus, the summands of V ⊗n
+M
+are tilting.
+Before proving Proposition 2.2, we extract some of its consequences.
+Proof of Theorem 1.9.(a). Let D and F be as in Theorem 1.9.(a), and let X be an object of D. We have
+βX ≤ dim F(X) by the analog of Lemma 1.2. We will show that bdim F (X)
+n
+≤ bX
+n , which implies the claim by
+Proposition 2.2.
+To this end, recall that p = char(k) ∈ N ∪ {∞}.
+Recall that simple representations of the symmetric
+group Sn are labeled by p-regular partitions λ of n, see e.g. [Mat99, Theorem 3.43] for an even more general
+statement. Let Dλ be the simple representation labeled by λ. Then Schur–Weyl duality, see e.g. [Jan03, E.17],
+gives
+bM
+n =
+�
+λ
+dim Dλ,
+where the sum runs over all p-regular partitions of n with ≤ M rows.
+By assumption, we have algebra
+morphisms
+kSn → EndD(X⊗n) → Endk
+�
+F(X)⊗n�
+,
+
+GROWTH RATES OF THE NUMBER OF INDECOMPOSABLE SUMMANDS IN TENSOR POWERS
+5
+where the composite is the usual permutation action of Sn. For any p-regular partition λ let eλ ∈ k[Sn] be
+a primitive idempotent such that k[Sn]eλ is the projective cover of Dλ and let ΠλX be the direct summand
+eλ(X⊗n) of X⊗n.
+Clearly ΠλX is not zero whenever ΠλF(X) := eλ
+�
+F(X)⊗n�
+is not zero.
+Hence X⊗n
+decomposes as a direct sum of ΠλX (where the latter need not be indecomposable) with multiplicities dim Dλ.
+By the above, this shows that bX
+n is indeed bounded below by bdim F (X)
+n
+.
+□
+Proof of Theorem 1.4.(a) – affine group schemes. For any affine group scheme (or any abstract group) Γ we
+have
+bdim V
+n
+≤ bΓ,V
+n
+,
+(2.5)
+since V ⊗n, considered as a Γ-representation, is the restriction of the tensor power of the tautological GL(V )-
+representation under the usual homomorphism Γ → GL(V ). Thus, in order to prove Theorem 1.3 we just
+need to combine the estimate bΓ,V
+n
+≤ (dim V )n from Lemma 1.2 with Proposition 2.2 and (2.5).
+□
+Proof of Theorem 1.4.(a) – affine semigroup schemes. For any affine semigroup scheme (or any semigroup) Γ
+let D = RepkΓ. We have a k-linear faithful monoidal functor F : D → Vectk sending X to its underlying
+k-vector space. Since this functor is symmetric, Theorem 1.9.(a) applies and we are done.
+□
+There is also a proof for semigroups which does not rely on Schur–Weyl duality. Details will be given in
+Section 3 because Schur–Weyl duality fails in the super case in positive characteristic.
+Finally, we conclude the list of consequences of Proposition 2.2 with the case of groupoids. Consider a
+groupoid (S : G) in the category of k-schemes, see [Del90, §1.6], with source and target morphisms G ⇒ S.
+If G → S × S is faithfully flat, then we say (S : G) is transitive. A representation of a groupoid (G : S) is
+a quasi-coherent sheaf on S with an action of G. Tensor products of representations are taken over OS. We
+denote by b(S:G),V
+n
+the analog of Definition 1.1.
+Theorem 2.6. Consider a transitive groupoid (S : G) in k-schemes with a representation V on a locally free
+OS-module of finite rank. Then
+β(S:G),V = lim
+n→∞
+n�
+b(S:G),V
+n
+= rank(V ).
+□
+Proof. As observed in [Del90, §1], the monoidal category Repk(S : G) of representations on locally free
+modules is actually abelian, and exact monoidal functors out of it are automatically faithful. For any field
+extension K of k for which S(K) ̸= 0, taking stalks at the corresponding point of S therefore yields a faithful
+symmetric monoidal functor
+Repk(S : G) → VectK,
+which sends V to a vector space of dimension rank(V ). We can thus apply Theorem 1.9(a).
+□
+2A. Finite groups.
+Notation 2A.1. Recall the Landau–Bachmann notation, which we adjust as follows.
+A function f satisfies f ∈ Θ′(g) if there exists a constant A ∈ R>0 such that A · g(n) ≤ f(n) ≤ g(n) for
+all n0 < n for some fixed n0 ∈ N. Without the prime this is the classical Landau–Bachmann notation where
+A · g(n) ≤ f(n) ≤ B · g(n) for A, B ∈ R>0. Similarly, but only for either the lower or the upper bound, we
+write f ∈ Ω(g) and f ∈ O′(g), respectively. Finally, we write f ∈ Ω′(g) if g ∈ O′(f).
+We also use f ∼ g, f is asymptotic to g, meaning limn→∞ f(n)/g(n) = 1 (with g(n) ̸= 0 for all n ≫ 1).
+Proposition 2A.2. If Γ is a finite group, then
+bΓ,V
+n
+∈ Θ′�
+(dim V )n�
+.
+Proof. Without loss of generality, we assume that Γ acts faithfully on V . By [BK72, Theorem 1], there exists
+r ∈ N for which V ⊗r contains a projective direct summand. Using that projective representations form a
+tensor ideal, and the fact that projective indecomposables P satisfy dim P ≤ |Γ|, it then follows that
+(dim V )n
+(dim V )r|Γ| ≤ bΓ,V
+n
+,
+which, by Lemma 1.2, concludes the proof.
+□
+This implies Theorem 1.4 for finite groups.
+Example 2A.3. Let Γ2 = SL2(k) and let V2 be its vector representation.
+For k = C we have seen in
+Example 1.7 that bΓ2,V2
+n
+∼ A · 2n/√n for A ∈ R>0.
+Let Fpr be the finite Galois field with pr elements.
+For k = Fpr, since the number of indecomposable summands is bounded from above by the number of
+indecomposable summands for k = C, Example 1.7 implies that the growth rate of bΓ2,V2
+n
+is bounded by
+A · 2n/√n for A ∈ R>0. With contrast, in the case k = Fpr Proposition 2A.2 shows that bΓ2,V2
+n
+∈ Θ′(2n) and
+
+6
+K. COULEMBIER, V. OSTRIK AND D. TUBBENHAUER
+hence, we do not have an upper bound by A · 2n/√n for A ∈ R>0. This in turn implies that Schur–Weyl
+duality fails over finite fields. The latter was observed in [BD09, Section 5].
+3
+Remark 2A.4. The argument in the proof of Proposition 2A.2 is that, for n ≫ 0, most indecomposable
+summands are projective, their number of appearance can be bound from below and grows already fast
+enough. As we will see in Section 2D below, for the special and general linear groups the role of projective
+representations will be played by certain tilting representations.
+Remark 2A.5. Let Γ be a monoid or semigroup. Note that kΓ is not a Hopf algebra, but only a bialgebra
+(potentially without unit). In particular, we can tensor representations, but the projective representation do
+not form a tensor ideal, see [Ste16, Exercise 17.15] for an explicit counterexample. Thus, the arguments in
+the proof of Proposition 2A.2 do not apply, not even for the case of finite monoids or semigroups. From this
+point of view it is surprising that Theorem 1.3 remains valid.
+2B. Proof of Proposition 2.2 – semisimple case. Let char(k) = 0, which we call the semisimple case.
+Notation 2B.1. For m1, . . . , mM−1 ∈ NM−1 we denote by ∆(m1, . . . , mM−1) the Weyl representations of
+ΓM = SLM of highest weight (m1, . . . , mM−1). This is in terms of the fundamental weights meaning that the
+highest weight is �M−1
+i=1 miωi where the ωi the fundamental weights.
+These are ΓM-representations defined integrally and these are simple for char(k) = 0, and we also have
+∆(1, 0, . . . , 0) = VM. See [Jan03, Part II] for some background.
+For char(k) = 0 the tensor product V ⊗n
+M
+decomposes into the simple summands for m1, . . . , mM−1 ∈ NM.
+Recall Weyl’s character formula, see e.g. [FH91, Section 24], which shows that
+dim ∆(m1, . . . , mM−1) ∈ N‘[m1, . . . , mM−1],
+i.e. dim ∆(m1, . . . , mM−1) is a polynomial in m1, . . . , mM−1.
+Example 2B.2. For example, for M = 3 one has dim ∆(m1, m2) = 1
+2(m1 + 1)(m2 + 1)(m1 + m2 + 2).
+3
+The following implies Proposition 2.2 for char(k) = 0:
+Proposition 2B.3. We have
+Ω
+�
+M n/nM(M−1)/2�
+∋ bM
+n ∈ O′(M n).
+Proof. All the weights of the representation V ⊗n
+M
+are bounded by n in the sense that any coefficient of the
+expansion with respect to fundamental weights is less than n in absolute value, meaning that m1+· · ·+mM−1 ≤
+n. Thus, all simple summands of V ⊗n
+M
+have dimensions bounded by a polynomial in n, and a closer look at
+Weyl’s character formula then implies that the polynomial is of degree M(M−1)
+2
+. See Example 2B.2 for an
+example. Thus, from this and Lemma 1.2 we get
+M n
+f(n) ≤ bM
+n ≤ M n,
+f(x) ∈ N[x], deg f = M(M−1)
+2
+.
+The claim follows.
+□
+Example 2B.4. We now strengthen Proposition 2B.3 for Γ2.
+For a Γ2-representation W, with weight
+spaces {Wi ⊂ W|i ∈ Z}, its character is the Laurent polynomial with non-negative coefficients ch W =
+�
+i(dim Wi)vi ∈ N[v, v−1].
+Let V be a representation of Γ2 over k with char(k) = 0. Then, by the classification of simple SL2(k)-
+representations, bG,V
+n
+equals the sum of dimensions of zero weight space and one weight space of V ⊗n. For
+example if V = V2 is the vector representation, then ch V2 = v + v−1 and ch V ⊗n
+2
+= (v + v−1)n. This implies
+that bΓ2,V2
+n
+is the constant term or the coefficient of v of (v + v−1)n, depending on parity of n. Using the
+binomial theorem we see that
+bΓ2,V2
+n
+=
+�� n
+n/2
+�
+n even,
+�
+n
+(n−1)/2
+�
+n odd.
+By applying Stirling’s formula we get that asymptotically
+bΓ2,V2
+n
+∼
+�
+2/π · 2n
+√n,
+which implies the claim in Example 1.7. Note that this is better than what we get from Proposition 2B.3 for
+M = 2. That is, the lower bound 2n/n is what we get from Proposition 2B.3 and Mathematica’s log plot
+
+GROWTH RATES OF THE NUMBER OF INDECOMPOSABLE SUMMANDS IN TENSOR POWERS
+7
+gives:
+bn
+2n
+2n
+n
+2n
+n
+5
+10
+15
+20
+1
+10
+100
+1000
+104
+105
+106
+,
+bn
+2n
+2n
+n
+2n
+n
+20
+40
+60
+80
+100
+10
+1011
+1021
+1031
+.
+More generally, let V be any nontrivial representation of Γ2. Then bΓ2,V
+n
+equals the sum of the constant
+term and the coefficient of v in (ch V )n. The asymptotic of this number can be computed by using the central
+limit theorem. We get that
+bΓ2,V
+n
+∼ (dim V )n/(2πnA2)1/2,
+where A = A(V ) ∈ R>0 is an easily computable constant depending on V . In the case when V is simple
+multinomials appear and one can use e.g. the results from [Ege14]. The same approach applies in general. 3
+Remark 2B.5. In the semisimple case many related results are known, in particular for Lie algebras and Lie
+groups, see e.g. [PR20] for a recent publication. That paper studies the problem of finding the asymptotic of
+multiplicities of fixed simple representations instead of all simple representations.
+A growth rate bΓ,V
+n
+∈ Θ′�
+(dim V )n�
+as in Proposition 2A.2 is very rarely the case, as the following result
+indicates:
+Proposition 2B.6. Recall that char(k) = 0. For an abstract group Γ with a finite dimensional representation
+V , the following are equivalent:
+(a) We have bΓ,V
+n
+∈ Θ′�
+(dim V )n�
+.
+(b) The connected component of the Zariski closure of the image of Γ in GL(V ) is a torus.
+Proof. We start by proving that (a) implies (b). Note that, with W = ¯k ⊗k V equipped with the canonical
+structure of a Γ-representation over ¯k, we have bΓ,V
+n
+≤ bΓ,W
+n
+. By definition, an algebraic group is a torus over
+k if and only if its extension of scalars to ¯k is a torus (a finite product of copies of the multiplicative group).
+Consequently, for this implication, we might as well assume that k is algebraically closed.
+Replacing Γ by the Zariski closure of its image in GL(V ) does not change the numbers bΓ,V
+n
+, so we can
+assume that Γ is an algebraic group and V is a faithful Γ-representation.
+Now we argue by contradiction. We assume that (a) is satisfied. If (b) is not satisfied, then by [Mil17,
+Corollary 17.25] and the fact that every connected one-dimensional unipotent algebraic group is isomorphic to
+the additive group, it follows that Γ contains a copy of the additive group Ga. We can restrict V to Ga and
+get
+A · (dim V )n ≤ bΓ,V
+n
+≤ bGa,V
+n
+,
+for some A ∈ R>0. We can apply the Jacobson–Morozov theorem and find Γ2 = SL2 ⊂ GL(V ) containing Ga
+and such that bGa,V
+n
+= bΓ2,V
+n
+, and thus,
+A · (dim V )n ≤ bΓ2,V
+n
+,
+for the same A ∈ R>0. This gives a contradiction with Example 2B.4, concluding the proof of this direction.
+Now we prove that (b) implies (a).
+For a finite field extension K of k, we can again consider a Γ-
+representation on U = K⊗k V . Restricting the K-vector space U ⊗Kn to k yields a direct sum of [K : k] copies
+of V ⊗n. Since this isomorphism respects the Γ-action, it follows that
+bΓ,U
+n
+≤ [K : k] · bΓ,V
+n
+.
+We can again replace Γ by the Zariski closure of its image. Since any torus splits after a finite field extension,
+we can thus assume that
+Γ = (G×d
+m ) ⋊ H
+for d ∈ N and a finite group H. It is well-known that for such groups the (rational) representation theory is
+semisimple and the dimension of the simple representations are bounded by |H|. Conclusion (a) from this in
+the same way as in Proposition 2A.2.
+□
+
+8
+K. COULEMBIER, V. OSTRIK AND D. TUBBENHAUER
+2C. Proof of Proposition 2.2 – for M = 2. The case M = 2 is special since we have full access to the
+characters of tilting representations and these are the direct summands of V ⊗n
+2
+. As before, Γ2 = SL2.
+It is crucial that V ⊗n
+2
+is a tilting Γ2-representation, see Remark 2.4.
+Thus, its direct summands are
+indecomposable tilting representations T(m) parameterized by dominant weights m ∈ N.
+Moreover, the
+Γ2-representation V ⊗n
+2
+decomposes into direct summands T(m) with m ≤ n.
+Example 2C.1. Donkin’s tensor product theorem [Don93, Proposition 2.1] allows us to describe tilting
+characters explicitly for Γ2. As observed in e.g. [TW21] or [STWZ21, Section 2], we can reformulate Donkin’s
+result for Γ2 as follows. Let m+1 = adpd +· · ·+a1p+a0 = (ad, . . . , a0) be the p-adic expansion of m+1 where
+ai ∈ {0, . . . , p−1} and ad ̸= 0. (The a0 digit is the one for p0, where we use the convention that ∞0 = 1.) We use
+the convention for characters from Example 2B.4. For b ∈ N write [b]x = x−(b−1) +x−(b−3) +· · ·+xb−3 +xb−1.
+We then have
+ch T(m) = [adpd]v ·
+�
+ai̸=0,i̸=d
+[2]vaipi .
+For example, for m = 52 and p = 2 we have m + 1 = (1, 1, 0, 1, 0, 1) so
+ch T(52) = [p5]v · [2]v24 [2]v22 [2]v20 .
+In particular, for v = 1 we get dim T(52) = 256.
+3
+This example implies:
+Lemma 2C.2. Let α = 1 + (log2 p)−1. Then we have
+dim T(m) ≤ (m + 1)α.
+Proof. It follows from Example 2C.1 that
+dim T(m) = 2kadpd,
+where k is the number of non-zero digits among the ad−1, . . . , a1, a0 in the p-adic extension of m + 1. This
+implies dim T(m) ≤ (m+1)α: First, we have adpd ≤ m+1 so it remains to argue that 2k ≤ (m+1)log2 2/ log2 p.
+Note secondly that 2k ≤ 2d−1 and 2d−1 = (m + 1)b for b = (d − 1) log2 2/ log2(m + 1). However, m + 1 =
+adpd + · · · + a1p + a0 for ad ̸= 0 which gives log2(m + 1) ≥ log2(pd−1) = (d − 1) log2 p so that b ≤ log2 2/ log2 p.
+The result follows.
+□
+Remark 2C.3. The number α = 1 + (log2 p)−1 in Lemma 2C.2 converges to 1 for p → ∞, and p = ∞ is the
+semisimple case where dim T(m) = m + 1.
+Example 2C.4. For p = 2 (left) and p = 3 (right) we get
+dim T(m)
+(m + 1)α
+20
+40
+60
+80
+100
+10
+100
+1000
+104
+,
+dim T(m)
+(m + 1)α
+20
+40
+60
+80
+100
+5
+10
+50
+100
+500
+1000
+which are again Mathematica log plots.
+3
+The following is a finer result than Proposition 2.2 itself, and thus, also implies Proposition 2.2.
+Proposition 2C.5. We have
+Ω′�
+2n/n2�
+∋ b2
+n ∈ O′(2n).
+Proof. Since 1 ≤ α ≤ 2, Lemma 1.2 and Lemma 2C.2 give us
+2n/(n + 1)2 ≤ 2n/(n + 1)α ≤ b2
+n ≤ 2n,
+and the statement follows.
+□
+
+GROWTH RATES OF THE NUMBER OF INDECOMPOSABLE SUMMANDS IN TENSOR POWERS
+9
+2D. Proof of Proposition 2.2 – for M ≥ 2. We assume M ≥ 2 and char(k) = p > 0 as the case char(k) = 0
+is dealt with in Section 2B. In particular, the results from [Jan03, Part II.E] apply. Let ΓM = SLM.
+Remark 2D.1. The bound as in Lemma 2C.2 is unavailable for M ≥ 3; the billiards conjecture in [LW18]
+and [Jen21] suggests that dimensions of tilting representations along the boundary grow exponentially already
+for Γ3 = SL3(k).
+Our proof below “ignores” these tilting representations: we argue that we already have enough summands
+in the part where Donkin’s tensor product formula applies up to a certain degree.
+We will use the following to only consider the case when M is odd since this case has slightly nicer combi-
+natorics:
+Lemma 2D.2. If Proposition 2.2 holds for M + 1, then it holds for M as well.
+Proof. Recall that the ΓM+1-representation VM+1 restricts to the ΓM-representation VM ⊕ k under the usual
+embedding ΓM �→ ΓM+1. It follows that
+bM+1
+n
+≤
+n
+�
+i=0
+�n
+i
+�
+bM
+i .
+Using that Proposition 2.2 holds for M + 1 we get:
+M + 1 ≤ lim
+n→∞ an,
+an =
+n
+�
+�
+�
+�
+n
+�
+i=0
+�n
+i
+�
+bM
+i .
+We claim that thus limn→∞
+n�
+bM
+n = M, as required. This can be seen as follows. Assume that for some fixed
+ϵ ∈ R>0 and all δN ∈ R>0 there exists N ∈ N such that |(M − ϵ) − (bM
+i )1/i| < δN for i ≥ N.
+Since �n
+i=0
+�n
+i
+�
+(M − ϵ)i = (M − ϵ + 1)n and bM
+i
+≈ (M − ϵ)i for i ≫ 0, it follows that an = M − ϵ + 1 − δ′
+n
+for some δ′
+n ∈ R>0 with limn→∞ δ′
+n = 0.
+Hence, limn→∞ an = M − ϵ + 1 < M + 1, which contradicts
+M + 1 ≤ limn→∞ an and the proof completes.
+□
+Recall that the category of finite dimensional ΓM = SLM-representations, considered as an abelian category,
+has a direct summand STr(ΓM) = ST p
+r (ΓM) consisting of representations which are linked with the Steinberg
+representation Str = Stp
+r = T
+�
+(pr − 1)ρ
+�
+(note that these depend on p), see [And18, Section 3.5]. These
+Steinberg representations are tilting and Weyl representations at the same time, and we will use this below.
+We plan to choose r = r(n) in such a way that the number of summands of V ⊗n
+M
+from STr(ΓM) is still
+about M n. Let us first estimate the number of occurrences of Str as a subquotient of a good filtration of
+V ⊗n
+M . This number depends only on the character of V ⊗n
+M
+and hence, is independent of the characteristic in
+the sense that the characters of both, V ⊗n
+M
+and Str ∼= ∆
+�
+(pr − 1)ρ
+�
+, are as in characteristic zero.
+In fundamental weight coordinates and SLM notation, we let ρSLM = (1, . . . , 1) and there are choices
+involved how to lift this to GLM notation in standard coordinates. We will use
+ρ = (ρ1, . . . , ρM) =
+�M − 1
+2
+, M − 3
+2
+, . . . , −M + 3
+2
+, −M + 1
+2
+�
+.
+Now the number of times that Str appears in V ⊗r
+M
+over SLM is at least the number of times it appears when
+we work over GLM, so we estimate the latter number.
+In characteristic zero we can compute the involved characters via Schur–Weyl duality by applying the hook
+length formula to the partition
+λ = λ(p, r) = n
+M (1, . . . , 1) + (pr − 1)ρ,
+where we from now on assume that n is divisible by M (which is sufficient to calculate the limit of
+n�
+bM
+n ),
+that M is odd (which is justified by Lemma 2D.2) and that (pr−1)(M−1)
+2
+≤
+n
+M .
+Example 2D.3. Let n = 12, M = 3, p = 2 and r = 2. Note that 3 = (pr−1)(M−1)
+2
+≤
+n
+M = 4 is satisfied. Then
+λ is
+λ =
+.
+This is the partition (7, 4, 1) so that the row differences are pr − 1 = 3.
+3
+Now let us make the following concrete choice for r:
+r(n) = ⌊logp(√n)⌋.
+
+10
+K. COULEMBIER, V. OSTRIK AND D. TUBBENHAUER
+Remark 2D.4. In fact, we could use r(n) =
+�
+logp
+�
+f(n)
+��
+for every function f which grows slower than n.
+The choice r(n) = ⌊logp(√n)⌋ is mostly for convenience as the formulas come out nicely.
+The hook formula implies that, up to factors which will not contribute to the limit of the nth root, the
+number of times that Str(n) = ∆(λ) for λ as above appears in V ⊗n over GLM is approximately
+a(n) =
+n!
+� n
+M + (pr(n) − 1)ρ1
+�
+! . . .
+� n
+M + (pr(n) − 1)ρM
+�
+!.
+Let us write xi = xi(n) =
+n
+M + (pr(n) − 1)ρi (so �
+i xi = n). To approximate the above formula recall that,
+for all a ∈ Z≥1, we have
+√
+2πa
+�a
+e
+�a
+e
+1
+12a+1 < a! <
+√
+2πa
+�a
+e
+�a
+e
+1
+12a .
+Hence, we get that
+√
+2πn(n)n
+�
+i
+√2πxi(xi)xi · e
+1
+12n+1 −�
+i
+1
+12xi < a(n) <
+√
+2πn(n)n
+�
+i
+√2πxi(xi)xi · e
+1
+12n −�
+i
+1
+12xi+1 .
+We claim that the greenish and reddish colored parts (the two right-hand sides for the reader with a black-
+and-white version) then converge to one if n → ∞. Indeed, because of our choice for r(n), |(pr(n) − 1)ρi| is
+bounded from above by B · √n for some B ∈ R>0 and thus, limn→∞ xi = ∞. We then can also see that the
+exponents of the marked terms converge to zero, and the claim follows. Consequently, using also xi(n) ∼
+n
+M ,
+we find
+a(n) ∼
+1
+(2π)
+M−1
+2
+·
+nn+1/2
+�
+i xxi+1/2
+i
+∼
+M M/2
+(2π)
+M−1
+2
+· nn+ 1−M
+2
+�
+i xxi
+i
+.
+Let f(n) denote a function with f(n) ∈ Θ(n−1/2) in Landau–Bachmann notation, see Notation 2A.1. We get
+a(n) ∼ A · ef(n)n(1−M)/2 M n ,
+for A ∈ R>0. This can be seen by using that
+�
+i
+1/xxi
+i
+∼ B · eg(n)n−nM n
+for some B ∈ R>0 and g(n) ∈ Θ(n−1/2).
+Thus, since the limit n → ∞ of the nth root of the (marked in a blueish color) left-hand side is one, we see
+that nth root of this sequence converges to M and we get:
+lim
+n→∞
+n�
+a(n) = lim
+n→∞
+n�
+A · ef(n)n(1−M)/2M n = M.
+Now let tn be the total dimension of summands of V ⊗n
+M
+which are in STr(n). Clearly, we have a(n) ≤ tn ≤
+(dim V )n and thus in conclusion
+lim
+n→∞
+n√tn = M.
+Next, we estimate the dimensions of the indecomposable summands of V ⊗n
+M
+which are from STr(ΓM). We
+start with a general and well-known lemma.
+Lemma 2D.5. We have dim T(a1, . . . , aM−1) ≤ �M−1
+i=1
+�M
+i
+�ai.
+Proof. Recall that �i VM is a tilting ΓM-representation for all i ∈ {1, . . . , M − 1} (this follows since �i VM
+is the Weyl representation ∆(0, . . . , 0, 1, 0, . . . , 0) for the ith fundamental weight ωi and this weight is mini-
+mal in the set of dominant integral weights). Now, essentially by their construction, the ΓM-representation
+T(a1, . . . , aM−1) is a direct summand of the ΓM-representation (�1 VM)⊗a1 ⊗ · · · ⊗ (�M−1 VM)⊗aM−1, and
+dim �i VM =
+�M
+i
+�
+.
+□
+Lemma 2D.6. Let Dn denote the maximum of the dimensions of the indecomposable summands of V ⊗n
+M
+from
+STr(n)(ΓM). Then
+lim
+n→∞
+n�
+Dn = 1.
+Proof. Every such summand is of the form Str ⊗ T (r) where T is an indecomposable tilting representation
+and (−)(r) is the rth Frobenius twist, see [And18, Remark 2(1)]. Hence, the highest weight of T should be
+bounded by n/pr(n) in the sense that sum of coefficients of the fundamental weights is bounded by this number
+(more restrictively even, if λ is the highest weight of T, then the weight (pr −1)ρ+prλ should appear in V ⊗n
+M ).
+
+GROWTH RATES OF THE NUMBER OF INDECOMPOSABLE SUMMANDS IN TENSOR POWERS
+11
+If we let A denote the maximum A = maxi{
+�M
+i
+�
+}, so that A =
+�
+M
+(M−1)/2
+�
+, then by Lemma 2D.5, we know
+that if the relevant tilting module in STr(n)(ΓM) is to appear in V ⊗n
+M , then
+dim Str ⊗ T (r) ≤ prA
+�
+i ai ≤ prAn/pr.
+Now we can calculate
+lim
+n→∞
+n�
+pr(n)An/pr(n) ≤ lim
+n→∞
+n�√nA
+√n = 1,
+which concludes the proof.
+□
+Now the total number of summands of V ⊗n
+M
+coming from STr(ΓM) is at least
+tn
+Dn . Hence,
+tn/Dn ≤ bM
+n ≤ M n,
+lim
+n→∞
+n�
+tn/Dn = M.
+Hence, Proposition 2.2 follows for M odd. Then Lemma 2D.2 implies Proposition 2.2 for M even.
+3. The general linear super group and consequences
+In this section we will work in the category of super vector spaces over k (although we sometimes omit the
+word ‘super’‘ to avoid too cumbersome phrasings). Since the latter reduces to the ordinary category of vector
+spaces in characteristic 2, we assume char(k) ̸= 2.
+Recall from Definition 2.1 that an affine group superscheme (in short: a supergroup) G over k is a repre-
+sentable functor from the category of commutative superalgebras (associative Z/2Z = {¯0, ¯1}-graded algebras
+which are graded commutative) over k to the category of groups. For general background on the theory of
+supergroups we refer to, for example, [BK03a], [BK03b], [Mas12] and [Mus12].
+We refer to (M, N) ∈ N×2 as the ‘super dimension’ of the Z/2Z-graded vector space kM|N, and M + N
+as the ‘dimension’. This should not lead to confusion as we will have no need for the ‘categorical dimension’
+M − N, also something referred to as the (super) dimension.
+Proposition 3.1. For a representation V of a supergroup G on a super vector space V of super dimension
+(M, N), we have
+βG,V = lim
+n→∞
+n�
+bG,V
+n
+= M + N.
+Here the numbers βG,V and bG,V
+n
+have the same meaning as before, i.e.
+they refer to the number of
+indecomposable summands in the G-representation V ⊗n. If we denote the representing commutative Hopf
+superalgebra for G by O(G), then a representation of G can either be interpreted as a Z/2Z-graded comodule
+for O(G), or equivalently as a homomorphism G → GLM|N of supergroups. By the latter interpretation, it is
+clearly sufficient to prove Proposition 3.1 for G = GLM|N and V = VM|N its vector representation on kM|N.
+Before getting to the proof of Proposition 3.1, we derive some consequences.
+Proof of Theorem 1.9.(b). If char(k) = 0, then
+kSn → EndGLM|N (V ⊗n
+M|N)
+is surjective, see [BR87, Section 4]. We can therefore repeat the proof of Theorem 1.9.(a) from Section 2 to
+reduce to Proposition 3.1 for G = GLM|N. If one wants to write things out explicitly, the set of partitions is
+now those for which λM+1 ≤ N.
+□
+Remark 3.2. The proof of Theorem 1.9.(b) does not extend to positive characteristic. Indeed, in this case
+kSn → EndGLM|N (V ⊗n
+M|N) need not be surjective, see [CEKO22, Theorem C]. More concretely, it is observed
+in [CEKO22, §4], that for p = 3, M = 2 and N = 1, the number of indecomposable summands in V ⊗5 is 17,
+while the number of primitive idempotents in a decomposition of unity in kS5 is only 16. Hence the action of
+the symmetric group on tensor powers of the vector representation of GLM|N is not sufficient to account for
+all indecomposable summands.
+Because of this remark, we need an alternative proof for semigroups compared to the non-super case:
+Proof of Theorem 1.4.(a) – affine semigroup superschemes. A representation of an affine semigroup super-
+scheme Γ corresponds to a semigroup homomorphism Γ → MatM|N, with MatM|N denoting the monoid
+superscheme of square (M + N)-matrices. In particular, the number of summands in V ⊗n over Γ is bounded
+from below by the number of summands over MatM|N. By considering O(MatM|N) as a subcoalgebra of
+O(GLM|N), we can identify the category of MatM|N-representations with the category of polynomial GLM|N-
+representations, so the number of‘ direct summands in V ⊗n over MatM|N is the same as over GLM|N.
+In conclusion, the number of direct summands over Γ is bounded from below by the number of summands
+over GLM|N. The result thus follows from Proposition 3.1 for G = GLM|N.
+□
+
+12
+K. COULEMBIER, V. OSTRIK AND D. TUBBENHAUER
+3A. Proof of Proposition 3.1 – semisimple case. Assume that char(k) = 0. We get a stronger statement:
+Lemma 3A.1. The GLM|N-representation V ⊗n
+M|N is semisimple and the dimension of the simple representa-
+tions occurring in V ⊗n
+M|N is bounded by a polynomial in n of degree M(M−1)+N(N−1)
+2
+.
+Proof. That the tensor powers are semisimple is proved in [BR87, Theorem 5.14]. The dimension of these
+simple representations is bounded by that of the Kac modules with same highest weight, see for instance
+[Mus12, §8.2]. As induced modules, the dimension of the latter is given by a constant (depending on MN,
+see for instance Lemma 3B.1(b)) times the dimension of the simple (GLM × GLN)-representation with same
+highest weight, which is a weight appearing in V ⊗n
+M|N. The latter can be bounded by a polynomial in n of
+degree M(M−1)+N(N−1)
+2
+, as explained in Section 2B.
+□
+Let bM,N
+n
+be the analog of bM
+n for GLM|N.
+Proposition 3A.2. We have
+Ω
+�
+(M + N)n/n(M(M−1)+N(N−1))/2�
+∋ bM,N
+n
+∈ O′�
+(M + N)n�
+.
+Proof. As before, this follows from Lemma 3A.1 and the super analog of Lemma 1.2.
+□
+3B. Preparation for the proof: distributions and induction. By a ‘subgroup’ H < G of a supergroup
+we refer to a representable subgroup functor, or equivalently a closed subsuperscheme which is also closed
+under the group operation.
+For the general linear supergroup GLM|N we consider the subgroups P + < GLM|N > P −. Here, for any
+superalgebra A
+P +(A) < GLM|N(A) = AutA(AM|N)
+consists of all automorphisms which are expressed as (M + N)-block matrices in a way that the left down
+block of size N × M is zero. The subgroup P − corresponds similarly to a zero (M × N)-block.
+For a supergroup G, we have the underlying affine group scheme G0, which can be defined as the restriction
+of the functor G to k-algebras (viewed as superalgebras contained in degree ¯0) or via the quotient of O(G) by
+the ideal generated by all odd elements. For G = GLM|N we have G0 = P + ∩ P − = GLM × GLN.
+For an affine group scheme G, one defines the distribution superalgebra as a subalgebra Dist G ⊂ O(G)∗
+which is a cocommutative Hopf superalgebra, similarly to the classical case, see [BK03b, §3]. Explicit de-
+scriptions of these algebras for GLM|N and subgroups as P ± are also given loc. cit.
+We also have the Lie superalgebra Lie G as a subspace of Dist G. For GLM|N this is the general linear Lie
+superalgebra glM|N of square (M +N)-matrices with supercommutator. We have a vector space decomposition
+glM|N = g− ⊕ g¯0 ⊕ g+,
+where g¯0 ⊕ g± is the subalgebra corresponding to P ± < G.
+For a supergroup G we denote by RepkG the rigid monoidal category of finite dimensional (super) repre-
+sentations. In particular, for G an ordinary affine group scheme interpreted as a supergroup, this category
+is equivalent (as a k-linear additive category) to a direct sum of two copies of the classical representation
+category.
+Lemma 3B.1. Set G = GLM|N.
+(a) The forgetful functors resG
+P : RepkG → RepkP and resP −
+G0 : RepkP − → RepkG0 have left adjoint
+functors indG
+P and indP −
+G0 .
+(b) As G0-representations and g−-representations, we have
+indG
+P M ≃ Λg− ⊗ M.
+(c) We have a natural isomorphism
+indP −
+G0 resP
+G0 ⇒ resG
+P −indG
+P .
+Proof. This can be proved by relying either on the theory of Harish-Chandra pairs from [Mas12] or the
+distribution algebras from [BK03b]. We choose the latter approach.
+In [BK03b] it is proved that, for H denoting any of the supergroups in the lemma, RepkH is equivalent
+to the category of integrable finite dimensional modules of Dist H. Here, ‘integrable’ essentially means weight
+module.
+Clearly, on the level of (Dist H)-representations, restriction has a left adjoint functor given by
+induction, for instance Dist G ⊗Dist P −. By the explicit realization in [BK03b, §4], it follows that
+Dist G ≃ Λg− ⊗ Dist P,
+respectively
+Dist P − ≃ Λg− ⊗ Dist G0
+
+GROWTH RATES OF THE NUMBER OF INDECOMPOSABLE SUMMANDS IN TENSOR POWERS
+13
+as right (Dist P)-representations respectively right (Dist G0)-representations. It follows easily that induction
+sends integrable modules to integrable modules, providing the desired left adjoints in (a). Statements (b) and
+(c) then follow again from the above and the explicit forms of the distribution algebras in [BK03b, §4].
+□
+Lemma 3B.2. Consider a supergroup G with subgroup H < G and a representation V of G on kM|N. Assume
+that:
+(i) βH,V = M + N.
+(ii) resG
+H : RepkG → RepkH has a left adjoint indG
+H and there exist j ∈ N and U ∈ RepkH for which
+indG
+HU is a G-summand of V ⊗j.
+Then it follows that βG,V = M + N.
+Proof. As a direct consequence of adjunction and the definition of indG
+H, we find for any W ∈ RepkG
+indG
+H(U) ⊗ W ≃ indG
+H
+�
+U ⊗ resG
+H(W)
+�
+.
+By assumption, bG,V
+n+j is at least the number of G-summands in
+indG
+H(U) ⊗ V ⊗n ≃ indG
+H(U ⊗ V ⊗n),
+which in particular shows that bH,V
+n
+≤ bG,V
+n+j. Thus, assumption (i) and the super analog of Lemma 1.2 imply
+the claim.
+□
+3C. Proof of Proposition 3.1 – positive characteristic. Now we fix M, N > 0, G = GLM|N, P ± < G
+from Section 3B and thus G0 = P0 = GLM × GLN. Let V = kM|N be the vector representation of G. Since
+we will use comparison with characteristic zero, we do not yet make assumptions on char(k).
+For partitions λ = (λi)1≤i≤M, µ = (µj)1≤j≤N of length at most M, N, we denote by L0(λ|µ) the corre-
+sponding simple polynomial G0-representation, which we also interpret as a P-representation in the usual way
+(for instance with trivial action of g+).
+Lemma 3C.1. The G-representation indG
+P L0(λ|µ) is simple if the integer
+cij(λ|µ) := λi − i + µj − j + M + 1
+is not zero in k, for all 1 ≤ i ≤ M and 1 ≤ j ≤ N.
+Proof. By Lemma 3B.1(b), indG
+P L0(λ|µ) is free as a Λg−-representation, so every P −-submodule contains
+u ⊗ L0(λ|µ) for u a non-zero element (unique up to constant) in the top degree of Λg−. Let v+ be a highest
+weight vector of L0(λ|µ). It suffices to prove that the g+-submodule of indG
+P L0(λ|µ) generated by u ⊗ v+
+contains 1 ⊗ v+, where we use indG
+P L0(λ|µ) ≃ Λg− ⊗ L0(λ|µ). By choosing a conveniently ordered product of
+all root vectors in g− for u and a mirrored product of root vectors for g+ for v, it follows that
+vu ⊗ v+ =
+�
+ij
+cij(λ|µ) (1 ⊗ v+),
+which concludes the proof.
+□
+Now we fix a prime p and consider the partitions α, ν of lengths M, N − 1 given by
+αi = N + (p − 1)(M − i), 1 ≤ i ≤ M,
+νj = (p − 1)(N − j), 1 ≤ j ≤ N.
+Since αM = N is greater than the length of ν, the partition κ := ανt of length M +ν1 makes sense. Concretely
+κi =
+�
+αi
+if i ≤ M
+(νt)i−M
+if i > M.
+It follows that κ is a p-core.
+Before coming to a crucial proposition, we need the following (well-known) lemma.
+Lemma 3C.2. Let λ be a p-core partition, and T be a standard λ-tableaux. The primitive idempotent eT in
+QSr associated to T via Young symmetrizers (so that in particular (QSr)eT is isomorphic to the simple Specht
+module Sλ) belongs to Z(p)Sr ⊂ QSr.
+Proof. According to the theory of Young symmetrizers, see for example [Ful97, Section 7.2], we can clear
+denominators in eT and get a pseudo-idempotent ˜eT ∈ ZSr. This pseudo-idempotent satisfies ˜eT ˜eT = nλ · ˜eT
+(in other words eT = ˜eT /nλ), where nλ ∈ Z is the product of the hook length in λ by e.g. [Ful97, Section 7.4,
+Exercises 18 and 19]. Moreover, the hook length of p-cores are never divisible by p, see e.g. [JK81, Statement
+2.7.40]. Hence, for a p-core we have nλ /∈ pZ, and we are done.
+□
+
+14
+K. COULEMBIER, V. OSTRIK AND D. TUBBENHAUER
+For the remainder of the section, we assume that char(k) = p > 2. By the above lemma, we can take an
+idempotent in e0 ∈ Z(p)Sr, with r = |κ|, which is a primitive idempotent corresponding to κ, when considered
+in QSr, and we denote by e ∈ FpSr ⊂ kSr its image modulo p. We denote by SκV the corresponding direct
+summand e(V ⊗r) in V ⊗r and we use the same notation SκV for the summand e0(V ⊗r) when working over Q.
+Proposition 3C.3. Working either over k′ = Q or k′ = k and with α, ν, κ as just introduced, we have
+SκV ≃ indG
+P L0(α|ν)
+in
+Repk′GLM|N.
+Proof. By construction (and realizing V as the extension of scalars over a free Z(p)-module), the character of
+the left-hand side is identical for k and Q. The GLM × GLN-representation L0(α|ν) has the Weyl character,
+because ν is a p-core and α is just a ‘shifted’ p-core. Indeed, this follows from [JM97, Theorem 4.5], or from
+a direct application of tilting theory and Lemma 3C.2. It therefore follows from Lemma 3B.1(2) that also the
+character of indG
+P L0(α|ν) is identical over k and Q.
+By Lemma 3C.1, the right-hand side is a simple representation. Indeed, we can calculate
+cij(α|ν) = (M + N − i − j)p + 1,
+which is never zero in k or in Q. In particular, over Q, the right-hand side is the simple representation with
+highest weight α|ν. The isomorphism over Q now follows from [BR87, Section 4].
+Now working over k, by the combination of the previous two paragraphs, the two representations have the
+same character and the right-hand side is simple. They must thus be isomorphic.
+□
+Now we can prove the main result.
+Proof of Proposition 3.1. By the combination of Proposition 3C.3 and Lemma 3B.2 it suffices to prove Propo-
+sition 3.1 for the supergroup P acting on V . We prove the equivalent formulation in terms of P −.
+Proposition 3C.3 and Lemma 3B.1(c) imply that
+SκV ≃ indP −
+G0 L0(α|ν)
+as P −-representations. In particular, the result for (P −, V ) follows, via Lemma 3B.2 from the purely even
+case (G0, V ) in Section 2.
+□
+4. The general linear quantum group and consequences
+Let k still denote an arbitrary field. Further, fix q ∈ k∗. We consider either of the following objects.
+To (k, q) we associate the pair, often called the mixed characteristic of (k, q), by
+(p, ℓ) := (|1|, |q2|) ∈ (N ∪ {∞})×2
+of the orders |−| of 1 and q2 in the additive group underlying k. For example, the case p = ℓ ∈ N corresponds
+to a field k of positive characteristic and q = 1 and the case p = ℓ = ∞ corresponds to a field of characteristic
+zero with generic q. We also use the quantum numbers for a ∈ N and x ∈ k:
+[a]x = x−(a−1) + x−(a−3) + · · · + xa−3 + xa−1 = xa − x−a
+x − x−1 ∈ k,
+where the second equality is only applicable for q2 ̸= 1. It then follows that ℓ is also the minimal value of n
+for which [n] = 0, or ∞ when no such value exists.
+We call q ∈ k∗ generic if q is not a root of unity in k, e.g. q could be the formal variable in the field C(q)
+of rational complex functions.
+Example 4.1. Let us give a few examples:
+(a) For k = C(q) and generic q we have (p, ℓ) = (∞, ∞). For k = F7(q) and generic q we have (p, ℓ) =
+(7, ∞). These cases are both ‘semisimple’, in the terminology of Remark 4.2.
+(b) For k = C and q = exp(πi/3) we have (p, ℓ) = (∞, 3).
+(c) For k = F7 and q = 2 we have (p, ℓ) = (7, 3).
+(d) For k = F7 and q = 1 we have (p, ℓ) = (7, 7).
+In general, the representation category of ΓM is semisimple if and only if ℓ = ∞.
+3
+We consider Lusztig’s divided power quantum group over k as in [Lus90] associated to a type A Cartan
+datum. We use ΓM = ΓM = U k
+q (slM) or ΓM = U k
+q (glM) as the notation.
+Remark 4.2. Following Example 4.1, all of our discussions regarding ΓM split into four distinct cases:
+(a) For (p, ∞) the quantum group representations are semisimple for any p, and their combinatorics is the
+same as for G(C), for G = SLM or G = GLM. We call this the semisimple case. We discuss this case
+in Section 4A.
+
+GROWTH RATES OF THE NUMBER OF INDECOMPOSABLE SUMMANDS IN TENSOR POWERS
+15
+(b) The case (∞, ℓ) for ℓ < ∞ can be combinatorially identified with its special case k = C. We call this
+the complex quantum group case. We discuss this case in Section 4D.
+(c) The strictly mixed case is p, ℓ < ∞ and p ̸= ℓ. We discuss this case in Section 4C.
+(d) The situation p = ℓ < ∞ prime is characteristic p. This reduces to the case in Section 2.
+This list is ordered in increasing order of difficulty, in the sense that the dimensions of the indecomposable
+summands are increasing, reading from top to bottom (and also more difficult to compute). Hence, using the
+same definitions as before, the convergence rate of
+n�
+bΓM,V
+n
+is slower for the bottom cases compared to the
+top ones.
+Let VM denote the quantum vector representation of ΓM.
+With the same notation as in the previous
+sections we have the analog of Proposition 2.2:
+Proposition 4.3. For any M ∈ Z>0 we have
+lim
+n→∞
+n�
+bM
+n = M.
+Proof. The case M = 1 is again immediate, and M > 1 is proven in Section 4B and Section 4C.
+□
+Remark 4.4. As before, we make use of the fact that V ⊗n
+M
+is tilting, see e.g. [AST17, Proposition 2.3]. In
+particular, V ⊗n
+M
+is a direct sum of indecomposable tilting representations. For (m1, . . . , mM−1) ∈ NM−1 we use
+the Weyl representations ∆(m1, . . . , mM−1) and the indecomposable tilting representations T(m1, . . . , mM−1)
+of highest weight (m1, . . . , mM−1).
+We now consider any k-subalgebra Γ ⊂ U k
+q (glM). In this case V ⊗n is a Γ-representation by restriction,
+although the tensor product of representations does not need to exist in general.
+Proof of Theorem 1.4 – quantum groups of type A and k-subalgebras. The only difference to the proof in Sec-
+tion 2 is that we use Proposition 4.3 instead of Proposition 2.2.
+□
+Remark 4.5. As observed in the 1990s or even earlier, embeddings of Lie subalgebras g �→ glM do not quantize
+properly. Consequently, contrary to Section 2 and Section 3, proving Theorem 1.4 for the general linear case
+does not imply it for other quantum groups. For this reason, quantum groups that are not of type A are
+outside of the scope of this paper. On the other hand, Theorem 1.4 does include coideal subalgebras, with
+the most prominent example being quantum symmetric pairs (also called ıquantum groups), which have been
+studied many people, see e.g. [NS95], [Let99] or [Kol14].
+4A. Proof of Proposition 4.3 – semisimple case. In this case, the indecomposable tilting representation
+Tq(m1, . . . , mM−1) is isomorphic to the Weyl representation ∆q(m1, . . . , mM−1), and the latter has the quan-
+tum Weyl character, see e.g. [Saw06, Equation 2]. We therefore get the same bound as for the group schemes,
+namely
+M n
+f(n) ≤ bM
+n for f ∈ N[x] with deg f = M(M−1)
+2
+. As before, an analog of Proposition 2B.3 follows from
+the above. This in turn implies Proposition 4.3 and Theorem 1.4.
+Example 4A.1. Let Γ2 = U k
+q (sl2) for (p, ∞).
+As in Example 2B.4, we see that bG,V2
+n
+∼ 2n/(πn/2)1/2
+where V2 is the quantum vector representation. Moreover, for any nontrivial representation of Γ2 we get that
+bG,V
+n
+∼ (dim V )n/(2πnA2)1/2 for some A = A(V ) ∈ R>0.
+3
+4B. Proof of Proposition 4.3 – for M = 2. First, in the case M = 2 we can use the known character
+formulas for Γ2:
+Example 4B.1. The quantum version of Example 2C.1 is as follows, see e.g. [STWZ21, Section 2] for details.
+We will use the above quantum numbers for x = v a formal variable for the characters.
+Let p(i) = pi−1ℓ for i > 0 and p(0) = 1. Let m + 1 = anp(n) + · · · + a1p(1) + a0p(0) = (an, . . . , a0) the
+(p, ℓ)-adic expansion of m + 1 where a0 ∈ {0, 1, . . . , ℓ − 1} and ai ∈ {0, 1, . . . , p − 1} for i > 0. (Note that the
+cases p = ∞ or ℓ = ∞ are covered by the notation as well.) We then have
+ch T(m) = [anp(n)]v ·
+�
+ai̸=0,i̸=n
+[2]vaip(i) .
+For example, for m = 52 and p = 2 and ℓ = 3 we have m + 1 = (1, 0, 0, 0, 1, 2) so
+ch T(52) = [p(5)]v · [2]vp(1)[2]v2p(0).
+For v = 1 we get dim T(52) = 192.
+3
+This proves:
+
+16
+K. COULEMBIER, V. OSTRIK AND D. TUBBENHAUER
+Lemma 4B.2. Let α = 1 for ℓ = ∞, and otherwise let α = 1 + (log2 p′)−1 where p′ = min{p, ℓ}. Then for
+any m ≤ n we have
+dim T(m) ≤ (n + 1)α.
+Proof. By Example 4B.1, as in the proof of Lemma 2C.2.
+□
+As before, we get the analog of Proposition 2C.5 from Lemma 4B.2. This in turn implies Proposition 4.3.
+Example 4B.3. Let m = 52, so m+1 = 53. Here are some (p, ℓ)-adic expansions and the dimension of T(m):
+(p, ∞): 53 = (53),
+dim T(52) = 53,
+(∞, 3): 53 = (17, 2),
+dim T(52) = 102 = 21 · 17 · 3,
+(2, 3): 53 = (1, 0, 0, 0, 1, 2),
+dim T(52) = 192 = 22 · (24 · 3),
+(2, 2): 53 = (1, 1, 0, 1, 0, 1),
+dim T(52) = 256 = 23 · 25.
+Note that the smaller (p, ℓ) the bigger dim T(m).
+3
+With α as in Lemma 4B.2 we then again get
+2n
+(n+1)2 ≤
+2n
+(n+1)α ≤ b2
+n ≤ 2n, which again proves Theorem 1.4
+and gives a bit finer information.
+4C. Proof of Proposition 4.3 – for M ≥ 2. We now show that Proposition 4.3 holds for (p, ℓ) with p, ℓ < ∞
+and p ̸= ℓ, and arbitrary M ≥ 2.
+We can use the same arguments as in the reductive group case above, with the following adaptations (the
+reader should compare Example 2C.1 and Example 4B.1 while reading the below):
+(a) Recall that p(i) = pi−1ℓ for i > 0 and p(0) = 1. All appearances of pr should be replaced by p(r).
+(b) The results we need from [And18] hold, mutatis mutandis, for the quantum group as well, see [And18,
+Remark 2.2]. That is, instead of the Frobenius twist one uses the Frobenius–Lusztig twist which,
+roughly speaking, acts as the Frobenius twist on digits ai for i > 0 and as its quantum analog on the
+zeroth digit, and the rest is the same.
+Taking all of the above together, Proposition 4.3 follows from the same arguments as for SLM, which proves
+Theorem 1.4.
+4D. Some extra observations for the complex quantum group case. For p = ∞ the combinatorics
+(in particular, the multiplicities of the decompositions) are the same as for the complex root of unity case.
+Here, we consider (p = ∞, ℓ < ∞), since the case (∞, ∞) is semisimple and has the same combinatorics as the
+complex group case. This case is therefore already addressed in Section 2B and Section 4A.
+Proposition 4D.1. For (m1, . . . , mM−1) ∈ NM−1. Then there exists A ∈ R>0 such that
+dim T(m1, . . . , mM−1) ≤ A · dim ∆(m1, . . . , mM−1)
+which implies
+M n
+f(n) ≤ bM
+n ,
+f ∈ N[x], deg f = M(M−1)
+2
+and
+Ω
+�
+M n/nM(M−1)�
+∋ bM
+n ∈ O′(M n).
+Proof. In this case the numbers (tilting:Weyl) are known to be given by parabolic Kazhdan–Lusztig polyno-
+mials, see [Soe97] and [Soe98]. Even better, for ΓM = SLM the results of [Str97] imply that the parabolic
+Kazhdan–Lusztig polynomials are bounded. This in turn implies, again using the quantum Weyl character
+formula as in Section 4A, that the dimension of the tilting representation T(m1, . . . , mM−1) is a polynomial
+in m1, . . . , mM−1 of degree M(M−1)
+2
+, as before.
+□
+Thus, Proposition 4D.1 implies that Proposition 4.3 holds for (∞, ℓ) and arbitrary M, but the result is even
+a bit stronger. We can even say a little more for M ∈ {2, 3}:
+Proposition 4D.2.
+(a) Let M = 2. For m ∈ N we have
+dim T(m) ≤ 2(m + 1),
+M n
+2(n + 1) ≤ bM
+n .
+(b) Let M = 3. For (m1, m2) ∈ N2 we have
+dim T(m1, m2) ≤ 12(m1 + 1)(m2 + 1)(m1 + m2 + 2),
+M n
+2(n3 + 3n2 + 3n + 1) ≤ bM
+n .
+
+GROWTH RATES OF THE NUMBER OF INDECOMPOSABLE SUMMANDS IN TENSOR POWERS
+17
+Proof. (a). By Example 4B.1.
+(b). We will crucially use that the tilting characters (the Weyl multiplicities with the indecomposable tiling
+representations) are known explicitly by, for example, [Soe97], [Soe98] and [Str97]. This explicit description
+of the characters is known as periodic patterns.
+The quantum version of Weyl character formula gives dim ∆(m1, m2) = (m1 + 1)(m2 + 1)(m1 + m2 + 2).
+We will use this as follows. The periodic pattern for U k
+q (sl3) tilting representations are given by
+These patterns mean that e.g. the tilting representation with highest weight in a star pattern at the north east
+(the position of the highest weight is indicated by zero) has the twelve Weyl representations indicated by the
+circles in its Weyl filtration. All appearing highest weights of the Weyl representation are in the action orbit
+of the affine Weyl group on this alcove picture. Thus, dim ∆(m1, m2) = (m1 +1)(m2 +1)(m1 +m2 +2) implies
+that dim T(m1, m2) ≤ 12(m1 + 1)(m2 + 1)(m1 + m2 + 2) for the star pattern. All other patterns have fewer
+Weyl factors and the claim follows. The same bound, as one easily checks, works for the periodic patterns
+along the boundary of the Weyl alcove as well.
+□
+Similar to Section 4C we have certain summands that appear often enough to imply Theorem 1.3:
+Proposition 4D.3. Let M = 2 and ℓ = 3. Then there exist a family of summands of V ⊗n
+2
+whose number of
+appearance tn in V ⊗n
+2
+satisfies
+2n/n5/2 ≤ tn.
+Proof. Via quantum Schur–Weyl duality, this is [KST22, Theorem 4E.2].
+□
+Remark 4D.4. The restriction to ℓ = 3 in Proposition 4D.3 is used as [KST22] study the monoid version of
+the Temperley–Lieb calculus. Similar (and sharper) results can be obtained for any ℓ ∈ N by using Schur–Weyl
+duality and [Spe20, Propositions 9.4 and 9.5].
+5. Counterexamples
+In this section we will show that Theorem 1.3 does not extend arbitrarily, even over k = C. We will use
+analog notion as before.
+Theorem 5.1. Let k = C.
+For every A ∈ R≥0 there exists a cotriangular Hopf algebra Γ with a finite
+dimensional Γ-corepresentation V such that
+βΓ,V + A < dim V.
+Proof. The main player in this proof is the Temperley–Lieb category TL(−2) of Rumer–Teller–Weyl [RTW32]
+with circle parameter −2 ∈ C. It is the diagrammatic incarnation of D = RepCSL2(C).
+Let m ≥ 2 and let X = C2 be the vector representation of SL2(C). It follows from [Bic03, Theorem 1.1]
+that every matrix E ∈ GLm(C) with
+tr(ET E−1) = −2 ∈ C
+gives a nonsymmetric fiber functor
+FE : D → VectC,
+X �→ Cm.
+Here we use the notion fiber functor as in e.g. [EGNO15, Definition 5.1.1].
+Since the number of indecomposable summands bD,X
+n
+does not depend on FE but only on RepCSL2(C),
+the above implies that
+βD,X = 2 < dim X = m.
+Finally, [Bic03, Theorem 1.1] and reconstruction theory as in e.g. [EGNO15, Theorem 5.4.1] provide a
+cotriangular Hopf algebra ΓE = H(FE) for E ∈ GLm(C) as above such that, as monoidal categories,
+coRepCΓE ∼= D
+and such that FE becomes the forgetful functor.
+
+s,0
+S.118
+K. COULEMBIER, V. OSTRIK AND D. TUBBENHAUER
+It remains to argue that tr(ET E−1) = −2 admits a solution for every m ∈ N≥2. Indeed, we can take
+E =
+�
+�
+idm−2
+0
+0
+0
+x
+−1
+0
+�
+� ,
+for x a solution to the equation x2 − x(m − 1) + 2 = 0 if m ̸= 2 (which always has two solutions for m ∈ N≥3),
+and x = 1 for m = 2.
+□
+Theorem 5.1 implies Theorem 1.4.(b) and Theorem 1.9.(c).
+Remark 5.2. There is also a quantum version of Theorem 5.1 where one replaces the trace condition by
+qtr(ET E−1) = −1 − q2 ∈ k.
+6. Questions
+We list a few open questions regarding bΓ,V
+n
+and βΓ,V .
+Question 6.1. Let Γ2 = SL2 and let V2 denote its vector representation. It is easy to observe that the number
+of summands in V ⊗n
+2
+in positive characteristic is bounded by the corresponding number in characteristic zero.
+In particular, Example 2B.4 and Lemma 2C.2 imply that for arbitrary p we have
+A · 2n/nα ≤ bΓ2,V2
+n
+≤ B · 2n/n1/2
+for A, B ∈ R>0 and α = 1 + (log2 p)−1.
+So we ask: For fixed p, is there some δ = δ(p) ∈ R>0 for which
+bΓ2,V2
+n
+∈ Θ(2nn−δ)?
+Note that in characteristic zero we have δ = 1/2 by Example 2B.4.
+Remark 6.2. The following observation was communicated to us by Pavel Etingof. For p = 2 the value δ ∈ R>0
+in Question 6.1 appears to be δ = 1
+2 log2(8/3), which is approximately 0.708. For example,
+,
+.
+are Mathematica log plots for p = 2 with α as in Lemma 2C.2.
+The motivation for δ = 1
+2 log2(8/3) is as follows. Consider the random variable
+�
+dim T(m)
+�s, where T(m)
+as before denotes an indecomposable tilting Γ2-representation with highest weight m in [n/2, n − 1], and
+consider the uniform distribution of m. Then the expectation value E
+��
+dim T(m)
+�s�
+of this is, for n ≫ 0,
+proportional to nf(s), where f(s) = s − 1 + log2(1 + 2s). This can be proven using the character formula
+for T(m) in Example 2C.1. In particular, the average of 1/ dim T(m) is nf(−1) = n−2+log2(3/2) = n− log2(8/3).
+Now, in characteristic zero the same type of argument would give that n−1 is proportional to E
+�
+1/ dim T(m)
+�
+.
+By Example 1.7 this suggests to take the square root, which gives the correct result up to a factor, i.e. in
+characteristic zero we have
+�
+E
+�
+1/ dim T(m)
+�
+=
+√
+n−1 = n−δ.
+For char(k) = 2 this then suggests to take δ = 1
+2 log2(8/3) ≈ 0.708 which, empirically speaking, seems to be
+correct, see the Mathematica output above.
+
+2
+1028
+n0.708
+bn
+1018
+108
+20
+40
+60
+80
+1002h
+105
+n0.708
+bn
+104
+2" n-α
+1000
+100
+10
+5
+10
+15
+20GROWTH RATES OF THE NUMBER OF INDECOMPOSABLE SUMMANDS IN TENSOR POWERS
+19
+A similar calculation could work for any prime.
+For example, for p = 3 the above strategy gives δ =
+1
+2 log2(9/2) ≈ 0.6845 and
+bn
+2n n-α
+2n
+n0.6845
+5
+10
+15
+20
+1
+10
+100
+1000
+104
+105
+,
+is the associated Mathematica log plot.
+More general than Question 6.1, namely for all algebraic objects where a version of Theorem 1.3 holds, we
+could ask:
+Question 6.3. For fixed p, is it true that
+bΓ,V
+n
+∈ Θ
+�
+(dim V )nn−δ�
+for some δ = δ(p) ∈ R>0?
+For example even for Γ3 = SL3 and V3 its vector representation we do not know whether bΓ3,V3
+n
+is bounded
+from below by A · 3n/nδ for some A ∈ R>0.
+Question 6.4. Assume that there exists A ∈ R>0 such that
+A · (dim V )n ≤ bΓ,V
+n
+?
+What can we say about Γ and V where Γ is assumed to be a group scheme?
+Recall that in the case char(k) = 0, Question 6.4 is answered in Proposition 2B.6. Note that the same
+answer cannot hold verbatim in positive characteristic (even after generalizing ‘torus’ to group of multiplicative
+type; as we could also do in characteristic zero at no additional cost): there are finite group schemes which
+have a connected component which is not of multiplicative type, for instance infinitesimal group schemes.
+Question 6.5. Let Verp be the universal Verlinde category, see for instance [Ost20, Section 3], and G an
+affine group scheme in Verp with a representation X on an object X0 ∈ Verp. Does Theorem 1.4(a) extend to
+the property that
+βG,X = FPdim(X0),
+with FPdim the Frobenius–Perron (sometimes called Perron–Frobenius) dimension. This formulation, with
+Verp replaced by its subcategories of (super) vector spaces corresponds precisely to Theorem 1.4(a) for (su-
+per)groups.
+Let D be a Karoubian symmetric monoidal category with the property that the tensor product of two non-
+zero objects is not zero. We say that D is of moderate decomposition growth if for any object X the sequence
+bΓ,X
+n
+is bounded by a geometric progression. In this case the limit βD,X exists, as before, by (a version of)
+Fekete’s Subadditive lemma.
+Question 6.6. Does βD,X have nice properties? Is it additive? Is it multiplicative? Does it take values in
+algebraic numbers?
+With respect to Remark 3.2 we can ask:
+Question 6.7. Does Theorem 1.9(c) extend to positive characteristic? In other words, is the number b′
+n ≤
+b
+GLM|N,VM|N
+n
+of direct summands in V ⊗n
+M|N which are forced to exist by considering idempotents in kSn (and
+the action of kSn on V ⊗n
+M|N) sufficient to ensure limn→∞
+n�
+b′n = M + N.
+Regarding the quantum case in Section 4 one could ask:
+Question 6.8. Let Γ = U k
+q (g) with g not of type A, e.g. g = soM. Do we still have βΓ,V = dim V ?
+
+20
+K. COULEMBIER, V. OSTRIK AND D. TUBBENHAUER
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+K.C.: The University of Sydney, School of Mathematics and Statistics F07, Office Carslaw 717, NSW 2006,
+Australia
+Email address: kevin.coulembier@sydney.edu.au
+V.O.: University of Oregon, Department of Mathematics, Eugene, OR 97403, USA
+Email address: vostrik@math.uoregon.edu
+D.T.: The University of Sydney, School of Mathematics and Statistics F07, Office Carslaw 827, NSW 2006,
+Australia, www.dtubbenhauer.com
+Email address: daniel.tubbenhauer@sydney.edu.au
+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf,len=1662
+page_content='GROWTH RATES OF THE NUMBER OF INDECOMPOSABLE SUMMANDS IN  TENSOR POWERS  KEVIN COULEMBIER, VICTOR OSTRIK AND DANIEL TUBBENHAUER  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In this paper we study the asymptotic behavior of the number of summands in tensor products  of finite dimensional representations of affine (semi)group (super)schemes and related objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Introduction and main results  1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  The general linear group and consequences  4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  The general linear super group and consequences  11  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  The general linear quantum group and consequences  14  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Counterexamples  17  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Questions  18  References  20  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Introduction and main results  A central, yet hard, problem in representation theory is the decomposition of tensor products of repre-  sentations into indecomposable summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Computations of these decomposition numbers are often major  unsolved problems in representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In this paper we take a different perspective and we are interested  in asymptotic properties of the number of indecomposables in tensor products of representation rather than  explicit decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In contrast to the question of explicit decompositions, we obtain results in extensive  generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  To get started, let Γ be a finite group with a finite dimensional representation V over some field k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We define  bΓ,V  n  := #indecomposable summands in V ⊗n counted with multiplicities  (sometimes simply denoted by bn when Γ and V are clear from the context), where ‘indecomposable’ means  as Γ-representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let further  βΓ,V := lim  n→∞  n�  bΓ,V  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Note that bΓ,V  n  bΓ,V  m  ≤ bΓ,V  n+m, so that βΓ,V is well-defined by (a version of) Fekete’s Subadditive Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  There is no chance to compute bΓ,V  n  explicitly in this generality, but it turns out that the ‘limit’ βΓ,V can  be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' As a first step we note that we have the following lemma whose the proof is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We have bΓ,V  n  ≤ (dim V )n, and consequently  βΓ,V ≤ dim V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  A classical result of Bryant–Kovács [BK72, Theorem 1] shows that there exists some n for which V ⊗n  contains a projective direct summand (projective over Γ if V is a faithful and projective over the appropriate  factor group of Γ otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' As observed in [BS20, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3], a consequence of this is that the bound  for βΓ,V in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 is actually an equality, that is:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We have  βΓ,V = dim V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  Mathematics Subject Classification 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Primary: 17B10, 18M05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Secondary: 16T05, 17B37, 20C25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Tensor products, asymptotic behavior, affine group schemes, affine semigroup schemes, semigroups, supergroups,  Hopf algebras, (symmetric) monoidal categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='00885v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='RT]  2 Jan 2023  2  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' COULEMBIER, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' OSTRIK AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' TUBBENHAUER  It is a natural question to which extent this result generalizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Concretely, one can let Γ be an infinite group,  an infinite semigroup, an affine group scheme, super versions, a finite dimensional Hopf algebra or any other  algebraic structure for which we have a notion of tensor products of representations, and V a finite dimensional  Γ-representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1 and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 work verbatim, including that βΓ,V is well-defined, although  clearly the above proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 does not extend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  We show that the theorem remains valid in great generality, but has limitations:  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (a) Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 holds true for Γ any affine semigroup superscheme as defined in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Theo-  rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 holds also true for the quantum groups Uq(slM) and Uq(glM), where we allow any q ∈ k∗ for  k∗ = k \\ {0, −1} if char(k) ̸= 2 and k∗ = k \\ {0} otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (b) Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 does not hold true in general for Hopf algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (a) Note that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (a) includes the cases where Γ is any, possibly infinite, abstract semigroup (for  instance a monoid or group – as observed above, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (a) is classical for finite groups, but  already for infinite abstract groups it appears to be new).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Indeed, using Tannakian reconstruction we  can associate an affine semigroup scheme to an abstract semigroup which has equivalent representation  theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The result can of course also be obtained directly without this observation and we elaborate  on special cases like these in the main body of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (b) We have not sought completeness in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We can, for example, include algebraic objects  like transitive groupoids in schemes, see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We omit such cases here for clarity and because  they are included in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='9(a) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (c) The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (a), excluding the quantum group case, can be reduced to the specific cases  of the general linear group GLM and the general linear supergroup GLM|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (d) While proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (a) we will also give asymptotic formulas in some cases, which improves  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Let us also comment on some variations of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (a) which have appeared in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (a) There are known variants of βΓ,V for which the analog of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (a) is not valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For instance if,  in characteristic p > 0, one only counts direct summands of dimension not divisible by p (categorical  dimension zero), it is shown in [CEO21, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1] for affine group superschemes that the corre-  sponding limit yields a ring homomorphism from the Grothendieck ring which takes values in the ring  of integers of a particular cyclotomic extension of Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For example, for p = 5, Γ = Z/5Z and V its  indecomposable of dimension three the limit is the golden ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In the case of a finite group where one  only counts non-projective summands one obtains the variant studied for instance in [BS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (Note  that these results do not cover the monoid and semigroup case and it would be interesting to know  whether there are similar statements for monoids and semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=') Our result that βΓ,V = dim V  for affine semigroup superschemes might be useful to compute the various variants of βΓ,V in the  literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (b) Let k = k and consider TLn = EndSL2  �  (k2)⊗n�  which is known as the Temperley–Lieb algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' By  Schur–Weyl duality, the dimensions of simple TLn-representations correspond to the decomposition  multiplicities of indecomposables in (k2)⊗n, and thus, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (a) and its proof given below imply  that some of the simple TLn-representations are very large for n ≫ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In the spirit of this example,  the paper [KST22] proposes to study large simple representations in the setting of monoids that arise  in a ‘Schur–Weyl dual way’ from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (a) such as the Temperley–Lieb monoid, the Brauer  monoid etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We hope that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (a) can be used to generalize [KST22] beyond the case of the  monoids discussed in that work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Note that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 is a weaker statement than an asymptotic formula for the growth of bΓ,V  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We show  this in the following example illustrating Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3, which is in some sense the key example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Take k = C, Γ2 = SL2 and V2 = C2, its vector representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Then the first numbers bΓ2,V2  n  are:  {1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 252},  bΓ2,V2  n  for n = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Here, as throughout, we view bn = bΓ2,V2  n  as a function in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  GROWTH RATES OF THE NUMBER OF INDECOMPOSABLE SUMMANDS IN TENSOR POWERS  3  A Mathematica loglog plot of  n√bn (y-axis) for n ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , 1000} (x-axis) gives  bn  1/n  2  1  5  10  50  100  500 1000  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='00  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='25  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='00  ,  and indeed the limit is two, as predicted by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Precisely,  1000√b1000 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='99265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  However, the asymptotic growth rate of bn is different than 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' As we will see in Example 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4, we get  bΓ2,V2  n  ∼  �  2/π · 2n/√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (Here and throughout, we use f ∼ g for f is equal to g asymptotically, meaning the ratio of f and g converges  to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=') We have  �  2/π ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='798 and Mathematica’s log plot gives:  bn  2n  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='798·2n  n  5  10  15  20  1  10  100  1000  104  105  106  ,  bn  2n  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='798·2n  n  20  40  60  80  100  10  1011  1021  1031  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  For a precise statement see Example 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  3  A broader formulation of the ideas in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 is the following setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Notation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let D be a k-linear Karoubian monoidal category that is Krull–Schmidt, with a k-linear  faithful monoidal functor F : D → VectK to the category VectK of finite dimensional vector spaces over a field  extension K of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Note that, if K is a finite extension of k, then the existence of F implies that morphism spaces in D are  finite dimensional, so D is Krull–Schmidt automatically Krull–Schmidt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  For any object X ∈ D, we can define bX  n similarly as in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1 as the number of indecomposable  direct summands in X⊗n and we have  βD,X := lim  n→∞  n�  bX  n ≤ dim F(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Again, βD,X is well-defined by (a version of) Fekete’s Subadditive Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We prove the following result,  which also generalizes most of the examples in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (a) If D and F are symmetric monoidal, then Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' :  βD,X = dim F(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (b) Assume that char(k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' If D is symmetric and F can be lifted to a symmetric monoidal functor  F ′ : D → SV ectK to the category of super vector spaces SV ectK, then Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 holds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' :  βD,X = dim F(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (c) Let k = C and D = RepCSL2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For every m ∈ Z≥2, there exists a faithful monoidal functor Fm : D →  VectC, which sends the vector SL2-representation X to Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Hence, for m ≥ 3 we have  βD,X = 2 < m = dim Fm(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Here and throughout, RepkΓ denotes the category of finite dimensional (rational) Γ-representations over k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (a) Note that the functors Fm in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (c) are not symmetric for m ≥ 3, so Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (a) does  not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  4  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' COULEMBIER, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' OSTRIK AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' TUBBENHAUER  (b) Faithfulness of F in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (a) is required to ensure the estimate bX  n ≤  �  dim F(X)  �n and cannot  be dropped, since it is easy to construct counterexamples with Deligne’s categories of [Del07].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Of  course, even without faithfulness, the bound dim F(X) ≤ βD,X remains valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Want to thank Pavel Etingof for comments on a draft of this paper, in particular for  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 which is Pavel’s observation, Andrew Mathas for help with the literature on symmetric groups  and Schur algebras, Volodymyr Mazorchuk for email exchanges about monoids, and Jonathan Gruber and  Arun Ram for discussions about growth rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We also thank the MFO workshop 2235 “Character Theory and  Categorification” for bringing us together in Oberwolfach in August/September 2022 – this project started  during this fantastic workshop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' was partly supported by ARC grant DP200100712.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' was supported, in part, by the Australian  Research Council, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' were supported by their depressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The general linear group and consequences  Let us start by defining some of our main players:  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We define an affine semigroup scheme (over k) to be a semigroup object in the category of  affine k-schemes (the opposite of the category of commutative k-algebras).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Equivalently, we can think of it as  a representable functor from the category of k-algebras to the category of semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Concretely, an affine  semigroup scheme corresponds to a commutative bialgebra, potentially without counit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The special case of  an affine monoid scheme corresponds precisely to a bialgebra with a counit, and the further special case of an  affine group scheme corresponds to a Hopf algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  We have the same notions in the ‘super’ version, by replacing the category of commutative k-algebras with  the category of graded commutative Z/2Z-graded k-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Let M ∈ Z>0, consider GLM and let VM be the tautological representation of GLM with dim VM = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Set bM  n = bGLM,VM  n  and βM = βGLM,VM = limn→∞  n�  bM  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For any M ∈ Z>0 we have βM = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The case M = 1 is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In Section 2C and Section 2D we prove the statement for M > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  We also fix some notation:  Notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (a) Recall that k denotes an arbitrary field, and we thus have that its characteristic char(k) ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We  always let p ∈ Z>1 ∪ {∞} denote the additive order of 1 ∈ k×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Thus, char(k) = p except that for  char(k) = 0 we set p = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' When we state char(k) = p > 0 it thus unambiguously means positive  characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (b) Throughout this section, we will set ΓM = SLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Since the decomposition of V ⊗n  M  into indecomposable  summands is identical for SLM and GLM, we can focus just on ΓM Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 will crucially exploit the theory of tilting representations, see  for example [Jan03, Part II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='E] (note that the relevant section in [Jan03] works over an arbitrary field of  characteristic char(k) = p > 0) or [AST18] and the appendix for its arXiv version for background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The point  is that VM is tilting, and by abstract theory direct summands of tensor products of tilting representations are  tilting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Thus, the summands of V ⊗n  M  are tilting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Before proving Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2, we extract some of its consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let D and F be as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (a), and let X be an object of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We have  βX ≤ dim F(X) by the analog of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We will show that bdim F (X)  n  ≤ bX  n , which implies the claim by  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  To this end, recall that p = char(k) ∈ N ∪ {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Recall that simple representations of the symmetric  group Sn are labeled by p-regular partitions λ of n, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' [Mat99, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='43] for an even more general  statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let Dλ be the simple representation labeled by λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Then Schur–Weyl duality, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' [Jan03, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='17],  gives  bM  n =  �  λ  dim Dλ,  where the sum runs over all p-regular partitions of n with ≤ M rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  By assumption, we have algebra  morphisms  kSn → EndD(X⊗n) → Endk  �  F(X)⊗n�  ,  GROWTH RATES OF THE NUMBER OF INDECOMPOSABLE SUMMANDS IN TENSOR POWERS  5  where the composite is the usual permutation action of Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For any p-regular partition λ let eλ ∈ k[Sn] be  a primitive idempotent such that k[Sn]eλ is the projective cover of Dλ and let ΠλX be the direct summand  eλ(X⊗n) of X⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Clearly ΠλX is not zero whenever ΠλF(X) := eλ  �  F(X)⊗n�  is not zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Hence X⊗n  decomposes as a direct sum of ΠλX (where the latter need not be indecomposable) with multiplicities dim Dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  By the above, this shows that bX  n is indeed bounded below by bdim F (X)  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (a) – affine group schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For any affine group scheme (or any abstract group) Γ we  have  bdim V  n  ≤ bΓ,V  n  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='5)  since V ⊗n, considered as a Γ-representation, is the restriction of the tensor power of the tautological GL(V )-  representation under the usual homomorphism Γ → GL(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Thus, in order to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 we just  need to combine the estimate bΓ,V  n  ≤ (dim V )n from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (a) – affine semigroup schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For any affine semigroup scheme (or any semigroup) Γ  let D = RepkΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We have a k-linear faithful monoidal functor F : D → Vectk sending X to its underlying  k-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Since this functor is symmetric, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (a) applies and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  There is also a proof for semigroups which does not rely on Schur–Weyl duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Details will be given in  Section 3 because Schur–Weyl duality fails in the super case in positive characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Finally, we conclude the list of consequences of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 with the case of groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Consider a  groupoid (S : G) in the category of k-schemes, see [Del90, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='6], with source and target morphisms G ⇒ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  If G → S × S is faithfully flat, then we say (S : G) is transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' A representation of a groupoid (G : S) is  a quasi-coherent sheaf on S with an action of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Tensor products of representations are taken over OS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We  denote by b(S:G),V  n  the analog of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Consider a transitive groupoid (S : G) in k-schemes with a representation V on a locally free  OS-module of finite rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Then  β(S:G),V = lim  n→∞  n�  b(S:G),V  n  = rank(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' As observed in [Del90, §1], the monoidal category Repk(S : G) of representations on locally free  modules is actually abelian, and exact monoidal functors out of it are automatically faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For any field  extension K of k for which S(K) ̸= 0, taking stalks at the corresponding point of S therefore yields a faithful  symmetric monoidal functor  Repk(S : G) → VectK,  which sends V to a vector space of dimension rank(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We can thus apply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='9(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Notation 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Recall the Landau–Bachmann notation, which we adjust as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  A function f satisfies f ∈ Θ′(g) if there exists a constant A ∈ R>0 such that A · g(n) ≤ f(n) ≤ g(n) for  all n0 < n for some fixed n0 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Without the prime this is the classical Landau–Bachmann notation where  A · g(n) ≤ f(n) ≤ B · g(n) for A, B ∈ R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Similarly, but only for either the lower or the upper bound, we  write f ∈ Ω(g) and f ∈ O′(g), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Finally, we write f ∈ Ω′(g) if g ∈ O′(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  We also use f ∼ g, f is asymptotic to g, meaning limn→∞ f(n)/g(n) = 1 (with g(n) ̸= 0 for all n ≫ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proposition 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' If Γ is a finite group, then  bΓ,V  n  ∈ Θ′�  (dim V )n�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Without loss of generality, we assume that Γ acts faithfully on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' By [BK72, Theorem 1], there exists  r ∈ N for which V ⊗r contains a projective direct summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Using that projective representations form a  tensor ideal, and the fact that projective indecomposables P satisfy dim P ≤ |Γ|, it then follows that  (dim V )n  (dim V )r|Γ| ≤ bΓ,V  n  ,  which, by Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2, concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  This implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4 for finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Example 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let Γ2 = SL2(k) and let V2 be its vector representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  For k = C we have seen in  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='7 that bΓ2,V2  n  ∼ A · 2n/√n for A ∈ R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Let Fpr be the finite Galois field with pr elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  For k = Fpr, since the number of indecomposable summands is bounded from above by the number of  indecomposable summands for k = C, Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='7 implies that the growth rate of bΓ2,V2  n  is bounded by  A · 2n/√n for A ∈ R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' With contrast, in the case k = Fpr Proposition 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 shows that bΓ2,V2  n  ∈ Θ′(2n) and  6  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' COULEMBIER, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' OSTRIK AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' TUBBENHAUER  hence, we do not have an upper bound by A · 2n/√n for A ∈ R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' This in turn implies that Schur–Weyl  duality fails over finite fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The latter was observed in [BD09, Section 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  3  Remark 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The argument in the proof of Proposition 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 is that, for n ≫ 0, most indecomposable  summands are projective, their number of appearance can be bound from below and grows already fast  enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' As we will see in Section 2D below, for the special and general linear groups the role of projective  representations will be played by certain tilting representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Remark 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let Γ be a monoid or semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Note that kΓ is not a Hopf algebra, but only a bialgebra  (potentially without unit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In particular, we can tensor representations, but the projective representation do  not form a tensor ideal, see [Ste16, Exercise 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='15] for an explicit counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Thus, the arguments in  the proof of Proposition 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 do not apply, not even for the case of finite monoids or semigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' From this  point of view it is surprising that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 remains valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 – semisimple case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let char(k) = 0, which we call the semisimple case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Notation 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , mM−1 ∈ NM−1 we denote by ∆(m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , mM−1) the Weyl representations of  ΓM = SLM of highest weight (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , mM−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' This is in terms of the fundamental weights meaning that the  highest weight is �M−1  i=1 miωi where the ωi the fundamental weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  These are ΓM-representations defined integrally and these are simple for char(k) = 0, and we also have  ∆(1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , 0) = VM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' See [Jan03, Part II] for some background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  For char(k) = 0 the tensor product V ⊗n  M  decomposes into the simple summands for m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , mM−1 ∈ NM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Recall Weyl’s character formula, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' [FH91, Section 24], which shows that  dim ∆(m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , mM−1) ∈ N‘[m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , mM−1],  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' dim ∆(m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , mM−1) is a polynomial in m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , mM−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Example 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For example, for M = 3 one has dim ∆(m1, m2) = 1  2(m1 + 1)(m2 + 1)(m1 + m2 + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  3  The following implies Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 for char(k) = 0:  Proposition 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We have  Ω  �  M n/nM(M−1)/2�  ∋ bM  n ∈ O′(M n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' All the weights of the representation V ⊗n  M  are bounded by n in the sense that any coefficient of the  expansion with respect to fundamental weights is less than n in absolute value, meaning that m1+· · ·+mM−1 ≤  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Thus, all simple summands of V ⊗n  M  have dimensions bounded by a polynomial in n, and a closer look at  Weyl’s character formula then implies that the polynomial is of degree M(M−1)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' See Example 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 for an  example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Thus, from this and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 we get  M n  f(n) ≤ bM  n ≤ M n,  f(x) ∈ N[x], deg f = M(M−1)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  The claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  Example 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We now strengthen Proposition 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 for Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  For a Γ2-representation W, with weight  spaces {Wi ⊂ W|i ∈ Z}, its character is the Laurent polynomial with non-negative coefficients ch W =  �  i(dim Wi)vi ∈ N[v, v−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Let V be a representation of Γ2 over k with char(k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Then, by the classification of simple SL2(k)-  representations, bG,V  n  equals the sum of dimensions of zero weight space and one weight space of V ⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For  example if V = V2 is the vector representation, then ch V2 = v + v−1 and ch V ⊗n  2  = (v + v−1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' This implies  that bΓ2,V2  n  is the constant term or the coefficient of v of (v + v−1)n, depending on parity of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Using the  binomial theorem we see that  bΓ2,V2  n  =  �� n  n/2  �  n even,  �  n  (n−1)/2  �  n odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  By applying Stirling’s formula we get that asymptotically  bΓ2,V2  n  ∼  �  2/π · 2n  √n,  which implies the claim in Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Note that this is better than what we get from Proposition 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 for  M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' That is, the lower bound 2n/n is what we get from Proposition 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 and Mathematica’s log plot  GROWTH RATES OF THE NUMBER OF INDECOMPOSABLE SUMMANDS IN TENSOR POWERS  7  gives:  bn  2n  2n  n  2n  n  5  10  15  20  1  10  100  1000  104  105  106  ,  bn  2n  2n  n  2n  n  20  40  60  80  100  10  1011  1021  1031  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  More generally, let V be any nontrivial representation of Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Then bΓ2,V  n  equals the sum of the constant  term and the coefficient of v in (ch V )n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The asymptotic of this number can be computed by using the central  limit theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We get that  bΓ2,V  n  ∼ (dim V )n/(2πnA2)1/2,  where A = A(V ) ∈ R>0 is an easily computable constant depending on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In the case when V is simple  multinomials appear and one can use e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' the results from [Ege14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The same approach applies in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' 3  Remark 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In the semisimple case many related results are known, in particular for Lie algebras and Lie  groups, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' [PR20] for a recent publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' That paper studies the problem of finding the asymptotic of  multiplicities of fixed simple representations instead of all simple representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  A growth rate bΓ,V  n  ∈ Θ′�  (dim V )n�  as in Proposition 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 is very rarely the case, as the following result  indicates:  Proposition 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Recall that char(k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For an abstract group Γ with a finite dimensional representation  V , the following are equivalent:  (a) We have bΓ,V  n  ∈ Θ′�  (dim V )n�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (b) The connected component of the Zariski closure of the image of Γ in GL(V ) is a torus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We start by proving that (a) implies (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Note that, with W = ¯k ⊗k V equipped with the canonical  structure of a Γ-representation over ¯k, we have bΓ,V  n  ≤ bΓ,W  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' By definition, an algebraic group is a torus over  k if and only if its extension of scalars to ¯k is a torus (a finite product of copies of the multiplicative group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Consequently, for this implication, we might as well assume that k is algebraically closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Replacing Γ by the Zariski closure of its image in GL(V ) does not change the numbers bΓ,V  n  , so we can  assume that Γ is an algebraic group and V is a faithful Γ-representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Now we argue by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We assume that (a) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' If (b) is not satisfied, then by [Mil17,  Corollary 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='25] and the fact that every connected one-dimensional unipotent algebraic group is isomorphic to  the additive group, it follows that Γ contains a copy of the additive group Ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We can restrict V to Ga and  get  A · (dim V )n ≤ bΓ,V  n  ≤ bGa,V  n  ,  for some A ∈ R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We can apply the Jacobson–Morozov theorem and find Γ2 = SL2 ⊂ GL(V ) containing Ga  and such that bGa,V  n  = bΓ2,V  n  , and thus,  A · (dim V )n ≤ bΓ2,V  n  ,  for the same A ∈ R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' This gives a contradiction with Example 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4, concluding the proof of this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Now we prove that (b) implies (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  For a finite field extension K of k, we can again consider a Γ-  representation on U = K⊗k V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Restricting the K-vector space U ⊗Kn to k yields a direct sum of [K : k] copies  of V ⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Since this isomorphism respects the Γ-action, it follows that  bΓ,U  n  ≤ [K : k] · bΓ,V  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  We can again replace Γ by the Zariski closure of its image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Since any torus splits after a finite field extension,  we can thus assume that  Γ = (G×d  m ) ⋊ H  for d ∈ N and a finite group H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' It is well-known that for such groups the (rational) representation theory is  semisimple and the dimension of the simple representations are bounded by |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Conclusion (a) from this in  the same way as in Proposition 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  8  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' COULEMBIER, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' OSTRIK AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' TUBBENHAUER  2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 – for M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The case M = 2 is special since we have full access to the  characters of tilting representations and these are the direct summands of V ⊗n  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' As before, Γ2 = SL2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  It is crucial that V ⊗n  2  is a tilting Γ2-representation, see Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Thus, its direct summands are  indecomposable tilting representations T(m) parameterized by dominant weights m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Moreover, the  Γ2-representation V ⊗n  2  decomposes into direct summands T(m) with m ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Example 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Donkin’s tensor product theorem [Don93, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1] allows us to describe tilting  characters explicitly for Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' As observed in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' [TW21] or [STWZ21, Section 2], we can reformulate Donkin’s  result for Γ2 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let m+1 = adpd +· · ·+a1p+a0 = (ad, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , a0) be the p-adic expansion of m+1 where  ai ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , p−1} and ad ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (The a0 digit is the one for p0, where we use the convention that ∞0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=') We use  the convention for characters from Example 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For b ∈ N write [b]x = x−(b−1) +x−(b−3) +· · ·+xb−3 +xb−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  We then have  ch T(m) = [adpd]v ·  �  ai̸=0,i̸=d  [2]vaipi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  For example, for m = 52 and p = 2 we have m + 1 = (1, 1, 0, 1, 0, 1) so  ch T(52) = [p5]v · [2]v24 [2]v22 [2]v20 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  In particular, for v = 1 we get dim T(52) = 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  3  This example implies:  Lemma 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let α = 1 + (log2 p)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Then we have  dim T(m) ≤ (m + 1)α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' It follows from Example 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1 that  dim T(m) = 2kadpd,  where k is the number of non-zero digits among the ad−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , a1, a0 in the p-adic extension of m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' This  implies dim T(m) ≤ (m+1)α: First, we have adpd ≤ m+1 so it remains to argue that 2k ≤ (m+1)log2 2/ log2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Note secondly that 2k ≤ 2d−1 and 2d−1 = (m + 1)b for b = (d − 1) log2 2/ log2(m + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' However, m + 1 =  adpd + · · · + a1p + a0 for ad ̸= 0 which gives log2(m + 1) ≥ log2(pd−1) = (d − 1) log2 p so that b ≤ log2 2/ log2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  The result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  Remark 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The number α = 1 + (log2 p)−1 in Lemma 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 converges to 1 for p → ∞, and p = ∞ is the  semisimple case where dim T(m) = m + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Example 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For p = 2 (left) and p = 3 (right) we get  dim T(m)  (m + 1)α  20  40  60  80  100  10  100  1000  104  ,  dim T(m)  (m + 1)α  20  40  60  80  100  5  10  50  100  500  1000  which are again Mathematica log plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  3  The following is a finer result than Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 itself, and thus, also implies Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proposition 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We have  Ω′�  2n/n2�  ∋ b2  n ∈ O′(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Since 1 ≤ α ≤ 2, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 and Lemma 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 give us  2n/(n + 1)2 ≤ 2n/(n + 1)α ≤ b2  n ≤ 2n,  and the statement follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  GROWTH RATES OF THE NUMBER OF INDECOMPOSABLE SUMMANDS IN TENSOR POWERS  9  2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 – for M ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We assume M ≥ 2 and char(k) = p > 0 as the case char(k) = 0  is dealt with in Section 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In particular, the results from [Jan03, Part II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='E] apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let ΓM = SLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Remark 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The bound as in Lemma 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 is unavailable for M ≥ 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' the billiards conjecture in [LW18]  and [Jen21] suggests that dimensions of tilting representations along the boundary grow exponentially already  for Γ3 = SL3(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Our proof below “ignores” these tilting representations: we argue that we already have enough summands  in the part where Donkin’s tensor product formula applies up to a certain degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  We will use the following to only consider the case when M is odd since this case has slightly nicer combi-  natorics:  Lemma 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' If Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 holds for M + 1, then it holds for M as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Recall that the ΓM+1-representation VM+1 restricts to the ΓM-representation VM ⊕ k under the usual  embedding ΓM �→ ΓM+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' It follows that  bM+1  n  ≤  n  �  i=0  �n  i  �  bM  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Using that Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 holds for M + 1 we get:  M + 1 ≤ lim  n→∞ an,  an =  n  �  �  �  �  n  �  i=0  �n  i  �  bM  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  We claim that thus limn→∞  n�  bM  n = M, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' This can be seen as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Assume that for some fixed  ϵ ∈ R>0 and all δN ∈ R>0 there exists N ∈ N such that |(M − ϵ) − (bM  i )1/i| < δN for i ≥ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Since �n  i=0  �n  i  �  (M − ϵ)i = (M − ϵ + 1)n and bM  i  ≈ (M − ϵ)i for i ≫ 0, it follows that an = M − ϵ + 1 − δ′  n  for some δ′  n ∈ R>0 with limn→∞ δ′  n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Hence, limn→∞ an = M − ϵ + 1 < M + 1, which contradicts  M + 1 ≤ limn→∞ an and the proof completes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  Recall that the category of finite dimensional ΓM = SLM-representations, considered as an abelian category,  has a direct summand STr(ΓM) = ST p  r (ΓM) consisting of representations which are linked with the Steinberg  representation Str = Stp  r = T  �  (pr − 1)ρ  �  (note that these depend on p), see [And18, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' These  Steinberg representations are tilting and Weyl representations at the same time, and we will use this below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  We plan to choose r = r(n) in such a way that the number of summands of V ⊗n  M  from STr(ΓM) is still  about M n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let us first estimate the number of occurrences of Str as a subquotient of a good filtration of  V ⊗n  M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' This number depends only on the character of V ⊗n  M  and hence, is independent of the characteristic in  the sense that the characters of both, V ⊗n  M  and Str ∼= ∆  �  (pr − 1)ρ  �  , are as in characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  In fundamental weight coordinates and SLM notation, we let ρSLM = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , 1) and there are choices  involved how to lift this to GLM notation in standard coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We will use  ρ = (ρ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , ρM) =  �M − 1  2  , M − 3  2  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , −M + 3  2  , −M + 1  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Now the number of times that Str appears in V ⊗r  M  over SLM is at least the number of times it appears when  we work over GLM, so we estimate the latter number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  In characteristic zero we can compute the involved characters via Schur–Weyl duality by applying the hook  length formula to the partition  λ = λ(p, r) = n  M (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , 1) + (pr − 1)ρ,  where we from now on assume that n is divisible by M (which is sufficient to calculate the limit of  n�  bM  n ),  that M is odd (which is justified by Lemma 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2) and that (pr−1)(M−1)  2  ≤  n  M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Example 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let n = 12, M = 3, p = 2 and r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Note that 3 = (pr−1)(M−1)  2  ≤  n  M = 4 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Then  λ is  λ =  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  This is the partition (7, 4, 1) so that the row differences are pr − 1 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  3  Now let us make the following concrete choice for r:  r(n) = ⌊logp(√n)⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  10  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' COULEMBIER, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' OSTRIK AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' TUBBENHAUER  Remark 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In fact, we could use r(n) =  �  logp  �  f(n)  ��  for every function f which grows slower than n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  The choice r(n) = ⌊logp(√n)⌋ is mostly for convenience as the formulas come out nicely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  The hook formula implies that, up to factors which will not contribute to the limit of the nth root, the  number of times that Str(n) = ∆(λ) for λ as above appears in V ⊗n over GLM is approximately  a(n) =  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  � n  M + (pr(n) − 1)ρ1  �  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  � n  M + (pr(n) − 1)ρM  �  !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='.  Let us write xi = xi(n) =  n  M + (pr(n) − 1)ρi (so �  i xi = n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' To approximate the above formula recall that,  for all a ∈ Z≥1, we have  √  2πa  �a  e  �a  e  1  12a+1 < a!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' <  √  2πa  �a  e  �a  e  1  12a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Hence, we get that  √  2πn(n)n  �  i  √2πxi(xi)xi · e  1  12n+1 −�  i  1  12xi < a(n) <  √  2πn(n)n  �  i  √2πxi(xi)xi · e  1  12n −�  i  1  12xi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  We claim that the greenish and reddish colored parts (the two right-hand sides for the reader with a black-  and-white version) then converge to one if n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Indeed, because of our choice for r(n), |(pr(n) − 1)ρi| is  bounded from above by B · √n for some B ∈ R>0 and thus, limn→∞ xi = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We then can also see that the  exponents of the marked terms converge to zero, and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Consequently, using also xi(n) ∼  n  M ,  we find  a(n) ∼  1  (2π)  M−1  2 nn+1/2  �  i xxi+1/2  i  ∼  M M/2  (2π)  M−1  2  nn+ 1−M  2  �  i xxi  i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Let f(n) denote a function with f(n) ∈ Θ(n−1/2) in Landau–Bachmann notation, see Notation 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We get  a(n) ∼ A · ef(n)n(1−M)/2 M n ,  for A ∈ R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' This can be seen by using that  �  i  1/xxi  i  ∼ B · eg(n)n−nM n  for some B ∈ R>0 and g(n) ∈ Θ(n−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Thus, since the limit n → ∞ of the nth root of the (marked in a blueish color) left-hand side is one, we see  that nth root of this sequence converges to M and we get:  lim  n→∞  n�  a(n) = lim  n→∞  n�  A · ef(n)n(1−M)/2M n = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Now let tn be the total dimension of summands of V ⊗n  M  which are in STr(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Clearly, we have a(n) ≤ tn ≤  (dim V )n and thus in conclusion  lim  n→∞  n√tn = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Next, we estimate the dimensions of the indecomposable summands of V ⊗n  M  which are from STr(ΓM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We  start with a general and well-known lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Lemma 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We have dim T(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , aM−1) ≤ �M−1  i=1  �M  i  �ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Recall that �i VM is a tilting ΓM-representation for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , M − 1} (this follows since �i VM  is the Weyl representation ∆(0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , 0, 1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , 0) for the ith fundamental weight ωi and this weight is mini-  mal in the set of dominant integral weights).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Now, essentially by their construction, the ΓM-representation  T(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , aM−1) is a direct summand of the ΓM-representation (�1 VM)⊗a1 ⊗ · · · ⊗ (�M−1 VM)⊗aM−1, and  dim �i VM =  �M  i  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  Lemma 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let Dn denote the maximum of the dimensions of the indecomposable summands of V ⊗n  M  from  STr(n)(ΓM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Then  lim  n→∞  n�  Dn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Every such summand is of the form Str ⊗ T (r) where T is an indecomposable tilting representation  and (−)(r) is the rth Frobenius twist, see [And18, Remark 2(1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Hence, the highest weight of T should be  bounded by n/pr(n) in the sense that sum of coefficients of the fundamental weights is bounded by this number  (more restrictively even, if λ is the highest weight of T, then the weight (pr −1)ρ+prλ should appear in V ⊗n  M ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  GROWTH RATES OF THE NUMBER OF INDECOMPOSABLE SUMMANDS IN TENSOR POWERS  11  If we let A denote the maximum A = maxi{  �M  i  �  }, so that A =  �  M  (M−1)/2  �  , then by Lemma 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='5, we know  that if the relevant tilting module in STr(n)(ΓM) is to appear in V ⊗n  M , then  dim Str ⊗ T (r) ≤ prA  �  i ai ≤ prAn/pr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Now we can calculate  lim  n→∞  n�  pr(n)An/pr(n) ≤ lim  n→∞  n�√nA  √n = 1,  which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  Now the total number of summands of V ⊗n  M  coming from STr(ΓM) is at least  tn  Dn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Hence,  tn/Dn ≤ bM  n ≤ M n,  lim  n→∞  n�  tn/Dn = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Hence, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 follows for M odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Then Lemma 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 implies Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 for M even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The general linear super group and consequences  In this section we will work in the category of super vector spaces over k (although we sometimes omit the  word ‘super’‘ to avoid too cumbersome phrasings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Since the latter reduces to the ordinary category of vector  spaces in characteristic 2, we assume char(k) ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Recall from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1 that an affine group superscheme (in short: a supergroup) G over k is a repre-  sentable functor from the category of commutative superalgebras (associative Z/2Z = {¯0, ¯1}-graded algebras  which are graded commutative) over k to the category of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For general background on the theory of  supergroups we refer to, for example, [BK03a], [BK03b], [Mas12] and [Mus12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  We refer to (M, N) ∈ N×2 as the ‘super dimension’ of the Z/2Z-graded vector space kM|N, and M + N  as the ‘dimension’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' This should not lead to confusion as we will have no need for the ‘categorical dimension’  M − N, also something referred to as the (super) dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For a representation V of a supergroup G on a super vector space V of super dimension  (M, N), we have  βG,V = lim  n→∞  n�  bG,V  n  = M + N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Here the numbers βG,V and bG,V  n  have the same meaning as before, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  they refer to the number of  indecomposable summands in the G-representation V ⊗n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' If we denote the representing commutative Hopf  superalgebra for G by O(G), then a representation of G can either be interpreted as a Z/2Z-graded comodule  for O(G), or equivalently as a homomorphism G → GLM|N of supergroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' By the latter interpretation, it is  clearly sufficient to prove Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1 for G = GLM|N and V = VM|N its vector representation on kM|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Before getting to the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1, we derive some consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' If char(k) = 0, then  kSn → EndGLM|N (V ⊗n  M|N)  is surjective, see [BR87, Section 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We can therefore repeat the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (a) from Section 2 to  reduce to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1 for G = GLM|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' If one wants to write things out explicitly, the set of partitions is  now those for which λM+1 ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (b) does not extend to positive characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Indeed, in this case  kSn → EndGLM|N (V ⊗n  M|N) need not be surjective, see [CEKO22, Theorem C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' More concretely, it is observed  in [CEKO22, §4], that for p = 3, M = 2 and N = 1, the number of indecomposable summands in V ⊗5 is 17,  while the number of primitive idempotents in a decomposition of unity in kS5 is only 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Hence the action of  the symmetric group on tensor powers of the vector representation of GLM|N is not sufficient to account for  all indecomposable summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Because of this remark, we need an alternative proof for semigroups compared to the non-super case:  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (a) – affine semigroup superschemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' A representation of an affine semigroup super-  scheme Γ corresponds to a semigroup homomorphism Γ → MatM|N, with MatM|N denoting the monoid  superscheme of square (M + N)-matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In particular, the number of summands in V ⊗n over Γ is bounded  from below by the number of summands over MatM|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' By considering O(MatM|N) as a subcoalgebra of  O(GLM|N), we can identify the category of MatM|N-representations with the category of polynomial GLM|N-  representations, so the number of‘ direct summands in V ⊗n over MatM|N is the same as over GLM|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  In conclusion, the number of direct summands over Γ is bounded from below by the number of summands  over GLM|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The result thus follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1 for G = GLM|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  12  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' COULEMBIER, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' OSTRIK AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' TUBBENHAUER  3A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1 – semisimple case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Assume that char(k) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We get a stronger statement:  Lemma 3A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The GLM|N-representation V ⊗n  M|N is semisimple and the dimension of the simple representa-  tions occurring in V ⊗n  M|N is bounded by a polynomial in n of degree M(M−1)+N(N−1)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' That the tensor powers are semisimple is proved in [BR87, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The dimension of these  simple representations is bounded by that of the Kac modules with same highest weight, see for instance  [Mus12, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' As induced modules, the dimension of the latter is given by a constant (depending on MN,  see for instance Lemma 3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1(b)) times the dimension of the simple (GLM × GLN)-representation with same  highest weight, which is a weight appearing in V ⊗n  M|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The latter can be bounded by a polynomial in n of  degree M(M−1)+N(N−1)  2  , as explained in Section 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  Let bM,N  n  be the analog of bM  n for GLM|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proposition 3A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We have  Ω  �  (M + N)n/n(M(M−1)+N(N−1))/2�  ∋ bM,N  n  ∈ O′�  (M + N)n�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' As before, this follows from Lemma 3A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1 and the super analog of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Preparation for the proof: distributions and induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' By a ‘subgroup’ H < G of a supergroup  we refer to a representable subgroup functor, or equivalently a closed subsuperscheme which is also closed  under the group operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  For the general linear supergroup GLM|N we consider the subgroups P + < GLM|N > P −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Here, for any  superalgebra A  P +(A) < GLM|N(A) = AutA(AM|N)  consists of all automorphisms which are expressed as (M + N)-block matrices in a way that the left down  block of size N × M is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The subgroup P − corresponds similarly to a zero (M × N)-block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  For a supergroup G, we have the underlying affine group scheme G0, which can be defined as the restriction  of the functor G to k-algebras (viewed as superalgebras contained in degree ¯0) or via the quotient of O(G) by  the ideal generated by all odd elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For G = GLM|N we have G0 = P + ∩ P − = GLM × GLN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  For an affine group scheme G, one defines the distribution superalgebra as a subalgebra Dist G ⊂ O(G)∗  which is a cocommutative Hopf superalgebra, similarly to the classical case, see [BK03b, §3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Explicit de-  scriptions of these algebras for GLM|N and subgroups as P ± are also given loc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  We also have the Lie superalgebra Lie G as a subspace of Dist G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For GLM|N this is the general linear Lie  superalgebra glM|N of square (M +N)-matrices with supercommutator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We have a vector space decomposition  glM|N = g− ⊕ g¯0 ⊕ g+,  where g¯0 ⊕ g± is the subalgebra corresponding to P ± < G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  For a supergroup G we denote by RepkG the rigid monoidal category of finite dimensional (super) repre-  sentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In particular, for G an ordinary affine group scheme interpreted as a supergroup, this category  is equivalent (as a k-linear additive category) to a direct sum of two copies of the classical representation  category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Lemma 3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Set G = GLM|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (a) The forgetful functors resG  P : RepkG → RepkP and resP −  G0 : RepkP − → RepkG0 have left adjoint  functors indG  P and indP −  G0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (b) As G0-representations and g−-representations, we have  indG  P M ≃ Λg− ⊗ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (c) We have a natural isomorphism  indP −  G0 resP  G0 ⇒ resG  P −indG  P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' This can be proved by relying either on the theory of Harish-Chandra pairs from [Mas12] or the  distribution algebras from [BK03b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We choose the latter approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  In [BK03b] it is proved that, for H denoting any of the supergroups in the lemma, RepkH is equivalent  to the category of integrable finite dimensional modules of Dist H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Here, ‘integrable’ essentially means weight  module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Clearly, on the level of (Dist H)-representations, restriction has a left adjoint functor given by  induction, for instance Dist G ⊗Dist P −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' By the explicit realization in [BK03b, §4], it follows that  Dist G ≃ Λg− ⊗ Dist P,  respectively  Dist P − ≃ Λg− ⊗ Dist G0  GROWTH RATES OF THE NUMBER OF INDECOMPOSABLE SUMMANDS IN TENSOR POWERS  13  as right (Dist P)-representations respectively right (Dist G0)-representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' It follows easily that induction  sends integrable modules to integrable modules, providing the desired left adjoints in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Statements (b) and  (c) then follow again from the above and the explicit forms of the distribution algebras in [BK03b, §4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  Lemma 3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Consider a supergroup G with subgroup H < G and a representation V of G on kM|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Assume  that:  (i) βH,V = M + N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (ii) resG  H : RepkG → RepkH has a left adjoint indG  H and there exist j ∈ N and U ∈ RepkH for which  indG  HU is a G-summand of V ⊗j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Then it follows that βG,V = M + N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' As a direct consequence of adjunction and the definition of indG  H, we find for any W ∈ RepkG  indG  H(U) ⊗ W ≃ indG  H  �  U ⊗ resG  H(W)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  By assumption, bG,V  n+j is at least the number of G-summands in  indG  H(U) ⊗ V ⊗n ≃ indG  H(U ⊗ V ⊗n),  which in particular shows that bH,V  n  ≤ bG,V  n+j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Thus, assumption (i) and the super analog of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 imply  the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  3C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1 – positive characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Now we fix M, N > 0, G = GLM|N, P ± < G  from Section 3B and thus G0 = P0 = GLM × GLN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let V = kM|N be the vector representation of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Since  we will use comparison with characteristic zero, we do not yet make assumptions on char(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  For partitions λ = (λi)1≤i≤M, µ = (µj)1≤j≤N of length at most M, N, we denote by L0(λ|µ) the corre-  sponding simple polynomial G0-representation, which we also interpret as a P-representation in the usual way  (for instance with trivial action of g+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Lemma 3C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The G-representation indG  P L0(λ|µ) is simple if the integer  cij(λ|µ) := λi − i + µj − j + M + 1  is not zero in k, for all 1 ≤ i ≤ M and 1 ≤ j ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' By Lemma 3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1(b), indG  P L0(λ|µ) is free as a Λg−-representation, so every P −-submodule contains  u ⊗ L0(λ|µ) for u a non-zero element (unique up to constant) in the top degree of Λg−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let v+ be a highest  weight vector of L0(λ|µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' It suffices to prove that the g+-submodule of indG  P L0(λ|µ) generated by u ⊗ v+  contains 1 ⊗ v+, where we use indG  P L0(λ|µ) ≃ Λg− ⊗ L0(λ|µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' By choosing a conveniently ordered product of  all root vectors in g− for u and a mirrored product of root vectors for g+ for v, it follows that  vu ⊗ v+ =  �  ij  cij(λ|µ) (1 ⊗ v+),  which concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  Now we fix a prime p and consider the partitions α, ν of lengths M, N − 1 given by  αi = N + (p − 1)(M − i), 1 ≤ i ≤ M,  νj = (p − 1)(N − j), 1 ≤ j ≤ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Since αM = N is greater than the length of ν, the partition κ := ανt of length M +ν1 makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Concretely  κi =  �  αi  if i ≤ M  (νt)i−M  if i > M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  It follows that κ is a p-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Before coming to a crucial proposition, we need the following (well-known) lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Lemma 3C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let λ be a p-core partition, and T be a standard λ-tableaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The primitive idempotent eT in  QSr associated to T via Young symmetrizers (so that in particular (QSr)eT is isomorphic to the simple Specht  module Sλ) belongs to Z(p)Sr ⊂ QSr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' According to the theory of Young symmetrizers, see for example [Ful97, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2], we can clear  denominators in eT and get a pseudo-idempotent ˜eT ∈ ZSr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' This pseudo-idempotent satisfies ˜eT ˜eT = nλ · ˜eT  (in other words eT = ˜eT /nλ), where nλ ∈ Z is the product of the hook length in λ by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' [Ful97, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4,  Exercises 18 and 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Moreover, the hook length of p-cores are never divisible by p, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' [JK81, Statement  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Hence, for a p-core we have nλ /∈ pZ, and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  14  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' COULEMBIER, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' OSTRIK AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' TUBBENHAUER  For the remainder of the section, we assume that char(k) = p > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' By the above lemma, we can take an  idempotent in e0 ∈ Z(p)Sr, with r = |κ|, which is a primitive idempotent corresponding to κ, when considered  in QSr, and we denote by e ∈ FpSr ⊂ kSr its image modulo p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We denote by SκV the corresponding direct  summand e(V ⊗r) in V ⊗r and we use the same notation SκV for the summand e0(V ⊗r) when working over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proposition 3C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Working either over k′ = Q or k′ = k and with α, ν, κ as just introduced, we have  SκV ≃ indG  P L0(α|ν)  in  Repk′GLM|N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' By construction (and realizing V as the extension of scalars over a free Z(p)-module), the character of  the left-hand side is identical for k and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The GLM × GLN-representation L0(α|ν) has the Weyl character,  because ν is a p-core and α is just a ‘shifted’ p-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Indeed, this follows from [JM97, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='5], or from  a direct application of tilting theory and Lemma 3C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' It therefore follows from Lemma 3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1(2) that also the  character of indG  P L0(α|ν) is identical over k and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  By Lemma 3C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1, the right-hand side is a simple representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Indeed, we can calculate  cij(α|ν) = (M + N − i − j)p + 1,  which is never zero in k or in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In particular, over Q, the right-hand side is the simple representation with  highest weight α|ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The isomorphism over Q now follows from [BR87, Section 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Now working over k, by the combination of the previous two paragraphs, the two representations have the  same character and the right-hand side is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' They must thus be isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  Now we can prove the main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' By the combination of Proposition 3C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 and Lemma 3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 it suffices to prove Propo-  sition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1 for the supergroup P acting on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We prove the equivalent formulation in terms of P −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proposition 3C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 and Lemma 3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1(c) imply that  SκV ≃ indP −  G0 L0(α|ν)  as P −-representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In particular, the result for (P −, V ) follows, via Lemma 3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 from the purely even  case (G0, V ) in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The general linear quantum group and consequences  Let k still denote an arbitrary field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Further, fix q ∈ k∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We consider either of the following objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  To (k, q) we associate the pair, often called the mixed characteristic of (k, q), by  (p, ℓ) := (|1|, |q2|) ∈ (N ∪ {∞})×2  of the orders |−| of 1 and q2 in the additive group underlying k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For example, the case p = ℓ ∈ N corresponds  to a field k of positive characteristic and q = 1 and the case p = ℓ = ∞ corresponds to a field of characteristic  zero with generic q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We also use the quantum numbers for a ∈ N and x ∈ k:  [a]x = x−(a−1) + x−(a−3) + · · · + xa−3 + xa−1 = xa − x−a  x − x−1 ∈ k,  where the second equality is only applicable for q2 ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' It then follows that ℓ is also the minimal value of n  for which [n] = 0, or ∞ when no such value exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  We call q ∈ k∗ generic if q is not a root of unity in k, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' q could be the formal variable in the field C(q)  of rational complex functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let us give a few examples:  (a) For k = C(q) and generic q we have (p, ℓ) = (∞, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For k = F7(q) and generic q we have (p, ℓ) =  (7, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' These cases are both ‘semisimple’, in the terminology of Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (b) For k = C and q = exp(πi/3) we have (p, ℓ) = (∞, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (c) For k = F7 and q = 2 we have (p, ℓ) = (7, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (d) For k = F7 and q = 1 we have (p, ℓ) = (7, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  In general, the representation category of ΓM is semisimple if and only if ℓ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  3  We consider Lusztig’s divided power quantum group over k as in [Lus90] associated to a type A Cartan  datum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We use ΓM = ΓM = U k  q (slM) or ΓM = U k  q (glM) as the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Following Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1, all of our discussions regarding ΓM split into four distinct cases:  (a) For (p, ∞) the quantum group representations are semisimple for any p, and their combinatorics is the  same as for G(C), for G = SLM or G = GLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We call this the semisimple case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We discuss this case  in Section 4A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  GROWTH RATES OF THE NUMBER OF INDECOMPOSABLE SUMMANDS IN TENSOR POWERS  15  (b) The case (∞, ℓ) for ℓ < ∞ can be combinatorially identified with its special case k = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We call this  the complex quantum group case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We discuss this case in Section 4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (c) The strictly mixed case is p, ℓ < ∞ and p ̸= ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We discuss this case in Section 4C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (d) The situation p = ℓ < ∞ prime is characteristic p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' This reduces to the case in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  This list is ordered in increasing order of difficulty, in the sense that the dimensions of the indecomposable  summands are increasing, reading from top to bottom (and also more difficult to compute).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Hence, using the  same definitions as before, the convergence rate of  n�  bΓM,V  n  is slower for the bottom cases compared to the  top ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Let VM denote the quantum vector representation of ΓM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  With the same notation as in the previous  sections we have the analog of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2:  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For any M ∈ Z>0 we have  lim  n→∞  n�  bM  n = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The case M = 1 is again immediate, and M > 1 is proven in Section 4B and Section 4C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' As before, we make use of the fact that V ⊗n  M  is tilting, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' [AST17, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In  particular, V ⊗n  M  is a direct sum of indecomposable tilting representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , mM−1) ∈ NM−1 we use  the Weyl representations ∆(m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , mM−1) and the indecomposable tilting representations T(m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , mM−1)  of highest weight (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , mM−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  We now consider any k-subalgebra Γ ⊂ U k  q (glM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In this case V ⊗n is a Γ-representation by restriction,  although the tensor product of representations does not need to exist in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4 – quantum groups of type A and k-subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The only difference to the proof in Sec-  tion 2 is that we use Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 instead of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' As observed in the 1990s or even earlier, embeddings of Lie subalgebras g �→ glM do not quantize  properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Consequently, contrary to Section 2 and Section 3, proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4 for the general linear case  does not imply it for other quantum groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For this reason, quantum groups that are not of type A are  outside of the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' On the other hand, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4 does include coideal subalgebras, with  the most prominent example being quantum symmetric pairs (also called ıquantum groups), which have been  studied many people, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' [NS95], [Let99] or [Kol14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  4A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 – semisimple case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In this case, the indecomposable tilting representation  Tq(m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , mM−1) is isomorphic to the Weyl representation ∆q(m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , mM−1), and the latter has the quan-  tum Weyl character, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' [Saw06, Equation 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We therefore get the same bound as for the group schemes,  namely  M n  f(n) ≤ bM  n for f ∈ N[x] with deg f = M(M−1)  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' As before, an analog of Proposition 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 follows from  the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' This in turn implies Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Example 4A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let Γ2 = U k  q (sl2) for (p, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  As in Example 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4, we see that bG,V2  n  ∼ 2n/(πn/2)1/2  where V2 is the quantum vector representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Moreover, for any nontrivial representation of Γ2 we get that  bG,V  n  ∼ (dim V )n/(2πnA2)1/2 for some A = A(V ) ∈ R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  3  4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 – for M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' First, in the case M = 2 we can use the known character  formulas for Γ2:  Example 4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The quantum version of Example 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1 is as follows, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' [STWZ21, Section 2] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  We will use the above quantum numbers for x = v a formal variable for the characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Let p(i) = pi−1ℓ for i > 0 and p(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let m + 1 = anp(n) + · · · + a1p(1) + a0p(0) = (an, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , a0) the  (p, ℓ)-adic expansion of m + 1 where a0 ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , ℓ − 1} and ai ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , p − 1} for i > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (Note that the  cases p = ∞ or ℓ = ∞ are covered by the notation as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=') We then have  ch T(m) = [anp(n)]v ·  �  ai̸=0,i̸=n  [2]vaip(i) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  For example, for m = 52 and p = 2 and ℓ = 3 we have m + 1 = (1, 0, 0, 0, 1, 2) so  ch T(52) = [p(5)]v · [2]vp(1)[2]v2p(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  For v = 1 we get dim T(52) = 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  3  This proves:  16  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' COULEMBIER, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' OSTRIK AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' TUBBENHAUER  Lemma 4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let α = 1 for ℓ = ∞, and otherwise let α = 1 + (log2 p′)−1 where p′ = min{p, ℓ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Then for  any m ≤ n we have  dim T(m) ≤ (n + 1)α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' By Example 4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1, as in the proof of Lemma 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  As before, we get the analog of Proposition 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='5 from Lemma 4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' This in turn implies Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Example 4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let m = 52, so m+1 = 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Here are some (p, ℓ)-adic expansions and the dimension of T(m):  (p, ∞): 53 = (53),  dim T(52) = 53,  (∞, 3): 53 = (17, 2),  dim T(52) = 102 = 21 · 17 · 3,  (2, 3): 53 = (1, 0, 0, 0, 1, 2),  dim T(52) = 192 = 22 · (24 · 3),  (2, 2): 53 = (1, 1, 0, 1, 0, 1),  dim T(52) = 256 = 23 · 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Note that the smaller (p, ℓ) the bigger dim T(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  3  With α as in Lemma 4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 we then again get  2n  (n+1)2 ≤  2n  (n+1)α ≤ b2  n ≤ 2n, which again proves Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4  and gives a bit finer information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  4C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 – for M ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We now show that Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 holds for (p, ℓ) with p, ℓ < ∞  and p ̸= ℓ, and arbitrary M ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  We can use the same arguments as in the reductive group case above, with the following adaptations (the  reader should compare Example 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1 and Example 4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1 while reading the below):  (a) Recall that p(i) = pi−1ℓ for i > 0 and p(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' All appearances of pr should be replaced by p(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (b) The results we need from [And18] hold, mutatis mutandis, for the quantum group as well, see [And18,  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' That is, instead of the Frobenius twist one uses the Frobenius–Lusztig twist which,  roughly speaking, acts as the Frobenius twist on digits ai for i > 0 and as its quantum analog on the  zeroth digit, and the rest is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Taking all of the above together, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 follows from the same arguments as for SLM, which proves  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Some extra observations for the complex quantum group case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For p = ∞ the combinatorics  (in particular, the multiplicities of the decompositions) are the same as for the complex root of unity case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Here, we consider (p = ∞, ℓ < ∞), since the case (∞, ∞) is semisimple and has the same combinatorics as the  complex group case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' This case is therefore already addressed in Section 2B and Section 4A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proposition 4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For (m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , mM−1) ∈ NM−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Then there exists A ∈ R>0 such that  dim T(m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , mM−1) ≤ A · dim ∆(m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , mM−1)  which implies  M n  f(n) ≤ bM  n ,  f ∈ N[x], deg f = M(M−1)  2  and  Ω  �  M n/nM(M−1)�  ∋ bM  n ∈ O′(M n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In this case the numbers (tilting:Weyl) are known to be given by parabolic Kazhdan–Lusztig polyno-  mials, see [Soe97] and [Soe98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Even better, for ΓM = SLM the results of [Str97] imply that the parabolic  Kazhdan–Lusztig polynomials are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' This in turn implies, again using the quantum Weyl character  formula as in Section 4A, that the dimension of the tilting representation T(m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , mM−1) is a polynomial  in m1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' , mM−1 of degree M(M−1)  2  , as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  Thus, Proposition 4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1 implies that Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 holds for (∞, ℓ) and arbitrary M, but the result is even  a bit stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We can even say a little more for M ∈ {2, 3}:  Proposition 4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (a) Let M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For m ∈ N we have  dim T(m) ≤ 2(m + 1),  M n  2(n + 1) ≤ bM  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (b) Let M = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For (m1, m2) ∈ N2 we have  dim T(m1, m2) ≤ 12(m1 + 1)(m2 + 1)(m1 + m2 + 2),  M n  2(n3 + 3n2 + 3n + 1) ≤ bM  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  GROWTH RATES OF THE NUMBER OF INDECOMPOSABLE SUMMANDS IN TENSOR POWERS  17  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' By Example 4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We will crucially use that the tilting characters (the Weyl multiplicities with the indecomposable tiling  representations) are known explicitly by, for example, [Soe97], [Soe98] and [Str97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' This explicit description  of the characters is known as periodic patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  The quantum version of Weyl character formula gives dim ∆(m1, m2) = (m1 + 1)(m2 + 1)(m1 + m2 + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  We will use this as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The periodic pattern for U k  q (sl3) tilting representations are given by  These patterns mean that e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' the tilting representation with highest weight in a star pattern at the north east  (the position of the highest weight is indicated by zero) has the twelve Weyl representations indicated by the  circles in its Weyl filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' All appearing highest weights of the Weyl representation are in the action orbit  of the affine Weyl group on this alcove picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Thus, dim ∆(m1, m2) = (m1 +1)(m2 +1)(m1 +m2 +2) implies  that dim T(m1, m2) ≤ 12(m1 + 1)(m2 + 1)(m1 + m2 + 2) for the star pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' All other patterns have fewer  Weyl factors and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The same bound, as one easily checks, works for the periodic patterns  along the boundary of the Weyl alcove as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  Similar to Section 4C we have certain summands that appear often enough to imply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3:  Proposition 4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let M = 2 and ℓ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Then there exist a family of summands of V ⊗n  2  whose number of  appearance tn in V ⊗n  2  satisfies  2n/n5/2 ≤ tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Via quantum Schur–Weyl duality, this is [KST22, Theorem 4E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  Remark 4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The restriction to ℓ = 3 in Proposition 4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 is used as [KST22] study the monoid version of  the Temperley–Lieb calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Similar (and sharper) results can be obtained for any ℓ ∈ N by using Schur–Weyl  duality and [Spe20, Propositions 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4 and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Counterexamples  In this section we will show that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 does not extend arbitrarily, even over k = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We will use  analog notion as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let k = C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  For every A ∈ R≥0 there exists a cotriangular Hopf algebra Γ with a finite  dimensional Γ-corepresentation V such that  βΓ,V + A < dim V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The main player in this proof is the Temperley–Lieb category TL(−2) of Rumer–Teller–Weyl [RTW32]  with circle parameter −2 ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' It is the diagrammatic incarnation of D = RepCSL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Let m ≥ 2 and let X = C2 be the vector representation of SL2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' It follows from [Bic03, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1]  that every matrix E ∈ GLm(C) with  tr(ET E−1) = −2 ∈ C  gives a nonsymmetric fiber functor  FE : D → VectC,  X �→ Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Here we use the notion fiber functor as in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' [EGNO15, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Since the number of indecomposable summands bD,X  n  does not depend on FE but only on RepCSL2(C),  the above implies that  βD,X = 2 < dim X = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Finally, [Bic03, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1] and reconstruction theory as in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' [EGNO15, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1] provide a  cotriangular Hopf algebra ΓE = H(FE) for E ∈ GLm(C) as above such that, as monoidal categories,  coRepCΓE ∼= D  and such that FE becomes the forgetful functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  s,0  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='118  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' COULEMBIER, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' OSTRIK AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' TUBBENHAUER  It remains to argue that tr(ET E−1) = −2 admits a solution for every m ∈ N≥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Indeed, we can take  E =  �  �  idm−2  0  0  0  x  −1  0  �  � ,  for x a solution to the equation x2 − x(m − 1) + 2 = 0 if m ̸= 2 (which always has two solutions for m ∈ N≥3),  and x = 1 for m = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  □  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1 implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (b) and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' There is also a quantum version of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1 where one replaces the trace condition by  qtr(ET E−1) = −1 − q2 ∈ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Questions  We list a few open questions regarding bΓ,V  n  and βΓ,V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let Γ2 = SL2 and let V2 denote its vector representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' It is easy to observe that the number  of summands in V ⊗n  2  in positive characteristic is bounded by the corresponding number in characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  In particular, Example 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4 and Lemma 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 imply that for arbitrary p we have  A · 2n/nα ≤ bΓ2,V2  n  ≤ B · 2n/n1/2  for A, B ∈ R>0 and α = 1 + (log2 p)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  So we ask: For fixed p, is there some δ = δ(p) ∈ R>0 for which  bΓ2,V2  n  ∈ Θ(2nn−δ)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Note that in characteristic zero we have δ = 1/2 by Example 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The following observation was communicated to us by Pavel Etingof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For p = 2 the value δ ∈ R>0  in Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1 appears to be δ = 1  2 log2(8/3), which is approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For example,  ,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  are Mathematica log plots for p = 2 with α as in Lemma 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  The motivation for δ = 1  2 log2(8/3) is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Consider the random variable  �  dim T(m)  �s, where T(m)  as before denotes an indecomposable tilting Γ2-representation with highest weight m in [n/2, n − 1], and  consider the uniform distribution of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Then the expectation value E  ��  dim T(m)  �s�  of this is, for n ≫ 0,  proportional to nf(s), where f(s) = s − 1 + log2(1 + 2s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' This can be proven using the character formula  for T(m) in Example 2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In particular, the average of 1/ dim T(m) is nf(−1) = n−2+log2(3/2) = n− log2(8/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Now, in characteristic zero the same type of argument would give that n−1 is proportional to E  �  1/ dim T(m)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  By Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='7 this suggests to take the square root, which gives the correct result up to a factor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' in  characteristic zero we have  �  E  �  1/ dim T(m)  �  =  √  n−1 = n−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  For char(k) = 2 this then suggests to take δ = 1  2 log2(8/3) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='708 which, empirically speaking, seems to be  correct, see the Mathematica output above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  2  1028  n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='708  bn  1018  108  20  40  60  80  1002h  105  n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='708  bn  104  2" n-α  1000  100  10  5  10  15  20GROWTH RATES OF THE NUMBER OF INDECOMPOSABLE SUMMANDS IN TENSOR POWERS  19  A similar calculation could work for any prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  For example, for p = 3 the above strategy gives δ =  1  2 log2(9/2) ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='6845 and  bn  2n n-α  2n  n0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='6845  5  10  15  20  1  10  100  1000  104  105  ,  is the associated Mathematica log plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  More general than Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='1, namely for all algebraic objects where a version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3 holds, we  could ask:  Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' For fixed p, is it true that  bΓ,V  n  ∈ Θ  �  (dim V )nn−δ�  for some δ = δ(p) ∈ R>0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  For example even for Γ3 = SL3 and V3 its vector representation we do not know whether bΓ3,V3  n  is bounded  from below by A · 3n/nδ for some A ∈ R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Assume that there exists A ∈ R>0 such that  A · (dim V )n ≤ bΓ,V  n  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  What can we say about Γ and V where Γ is assumed to be a group scheme?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Recall that in the case char(k) = 0, Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4 is answered in Proposition 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Note that the same  answer cannot hold verbatim in positive characteristic (even after generalizing ‘torus’ to group of multiplicative  type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' as we could also do in characteristic zero at no additional cost): there are finite group schemes which  have a connected component which is not of multiplicative type, for instance infinitesimal group schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let Verp be the universal Verlinde category, see for instance [Ost20, Section 3], and G an  affine group scheme in Verp with a representation X on an object X0 ∈ Verp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Does Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4(a) extend to  the property that  βG,X = FPdim(X0),  with FPdim the Frobenius–Perron (sometimes called Perron–Frobenius) dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' This formulation, with  Verp replaced by its subcategories of (super) vector spaces corresponds precisely to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='4(a) for (su-  per)groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Let D be a Karoubian symmetric monoidal category with the property that the tensor product of two non-  zero objects is not zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' We say that D is of moderate decomposition growth if for any object X the sequence  bΓ,X  n  is bounded by a geometric progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In this case the limit βD,X exists, as before, by (a version of)  Fekete’s Subadditive lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Does βD,X have nice properties?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Is it additive?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Is it multiplicative?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Does it take values in  algebraic numbers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  With respect to Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='2 we can ask:  Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Does Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='9(c) extend to positive characteristic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' In other words, is the number b′  n ≤  b  GLM|N,VM|N  n  of direct summands in V ⊗n  M|N which are forced to exist by considering idempotents in kSn (and  the action of kSn on V ⊗n  M|N) sufficient to ensure limn→∞  n�  b′n = M + N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  Regarding the quantum case in Section 4 one could ask:  Question 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Let Γ = U k  q (g) with g not of type A, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' g = soM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Do we still have βΓ,V = dim V ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='  20  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' COULEMBIER, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' OSTRIK AND D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' TUBBENHAUER  References  [And18]  H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' Andersen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' The Steinberg linkage class for a reductive algebraic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
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+page_content=' Birkhäuser Boston, Boston, MA, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
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+page_content=' With applications to representation theory and geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
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+page_content=' Springer-Verlag, New  York, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
+page_content=' A first course, Readings in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
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+page_content=' Robinson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
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+page_content=' Jantzen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
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+page_content=' Theory, 2:432–448, 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/a9AyT4oBgHgl3EQf9_qg/content/2301.00885v1.pdf'}
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+ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION MODEL
+FOR HUMAN BEHAVIORS DURING A CATASTROPHIC EVENT
+K. KHALIL1, V. LANZA1, D. MANCEAU1, M.A. AZIZ-ALAOUI1, D. PROVITOLO2
+Abstract. In this work, using the theory of first-order macroscopic crowd models, we intro-
+duce a compartmental advection-diffusion type model, describing the spatio-temporal dynamics
+of a population in different behaviors (alert, panic and control behaviors) during a catastrophic
+event. For this model, we prove the local existence, regularity and uniqueness of a solution,
+as well as the positivity and boundedness of this solution that allows the global existence.
+Then, in order to study the spatio-temporal behavioural dynamics of a population during a
+catastrophic event, we present several numerical simulations for different evacuation scenarios.
+Contents
+1.
+Introduction
+2
+2.
+A spatio-temporal advection-diffusion model for human behaviors during a
+catastrophic event
+3
+2.1.
+The temporal model of the human behavior of a population during a catastrophic
+event
+3
+2.2.
+First-order macroscopic pedestrian models
+6
+2.3.
+The spatio-temporal model corresponding to (2.1)
+7
+3.
+Well-posedness, positivity and positively invariant sets of solutions for the
+spatio-temporal model spatio-temporal model (2.14), (2.15) and (2.16)
+10
+3.1.
+The abstract formulation and the associated boundary value Cauchy problem
+10
+3.2.
+Preliminary results
+12
+3.3.
+Local existence and regularity
+14
+3.4.
+Positivity
+16
+3.5.
+L1-boundedness of the total population density
+17
+3.6.
+Uniform boundedness and global existence
+18
+4.
+Numerical Simulations
+21
+5.
+Conclusion
+27
+Appendices
+28
+A.
+Proofs of the preliminary results of Section 3
+28
+B.
+Tables of the functions and the parameters of the APC model
+31
+ACKNOWLEDGMENT
+32
+References
+32
+1LMAH, University of Le Havre Normandie, FR-CNRS-3335, ISCN, Le Havre 76600, France.
+2000 Mathematics Subject Classification. 34G20, 47D06.
+Key words and phrases. First-order macroscopic models; pedestrians crowd models; human behaviors; mathe-
+matical modeling; panic; semigroup theory.
+∗Corresponding author: K. Khalil; kamal.khalil@univ-lehavre.fr.
+1
+arXiv:2301.02520v1  [math.AP]  6 Jan 2023
+
+2
+K. KHALIL ET AL.
+1. Introduction
+In the last decades the world has known some radical changes at almost all levels such as
+technological developments, climatic changes and human evolution. Due to these factors, pop-
+ulations (in both, developed or undeveloped countries) are aggressively facing many natural
+disasters (tsunamis, earthquakes), technological events and terrorist attacks. In particular sit-
+uations of sudden, unexpected and without alert disasters require high security measures and
+strategies in order to predict and manage the movement and the behavior of a crowd. During a
+catastrophe, people may experience many different behaviors, but there is still few information
+about the dynamics and the succession of such behaviors during the event, see [15, 36, 37]. Thus,
+for the development of an efficient disaster management strategy, it becomes necessary not only
+to take into account the disaster features but also the different psychological human behaviors
+during the disaster event.
+Recently, several pedestrians crowd models have been developed. Their main objective is to
+predict the movements of a crowd in different environmental situations. Mathematical models of
+human crowds are mainly divided into two categories, namely, microscopic models and macro-
+scopic ones. In the microscopic approach, individuals are treated separately as particles and the
+evolution is determined using Newton’s second law and by considering physical and social forces
+that describe the interaction among individuals as well as their interactions with the physical sur-
+rounding (for more details we refer to the works of Helbing described in [33]). The macroscopic
+approach, that we adopt in this paper, considers a crowd as a whole quantity, without recogniz-
+ing individual differences, and it is therefore more suitable to the study of the movement of an
+extremely large number of pedestrians. In particular, first-order macroscopic models introduced
+by Hughes [25] (see also [14]) are based on a mass conservation equation and a density-velocity
+closure equation with suitable boundary conditions. Furthermore, several models are devoted
+to study the dynamics of multiple pedestrian species in the context of macroscopic first-order
+systems (see [8, 9, 13, 18, 19, 24, 25, 38, 40] and references therein). In [25] Hughes studied
+crowds with large density of multiple pedestrian classes with different walking characteristics and
+destinations (identified by the index i). The system reads as
+�
+�
+�
+�
+�
+∂tρi + ∇ · qi(ρ) = 0
+in [0, T) × Ω,
+qi · n = q0
+i · n
+in [0, T) × ∂Ω,
+ρi(0) = ρ0
+i
+in Ω,
+(1.1)
+for i = 1, . . . , N (N ≥ 2), where Ω ⊂ R2 is a bounded domain with smooth boundary ∂Ω and
+qi(ρ) = ρiv(ρ)νi. The velocity is defined as v(ρ) = A − B˜ρ, thus it is linear with respect to the
+total population ˜ρ and is the same for all populations. On the contrary, each population can
+have a different direction of the movement νi. Finally, for each population i, q0
+i is the outflow
+from the boundary in the direction of the normal vector n and ρ0
+i is the initial data. Moreover,
+in [8] authors studied a nonlinear drift-diffusion model with in-outflow boundary conditions for
+the transport of particles. Notice that these systems are consisting of conservative equations (i.e.
+with no reactions terms). A non-conservative system is proposed in [27] but the authors consider
+
+ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL
+3
+only one population species and Neumann homogeneous boundary conditions, namely
+�
+�
+�
+�
+�
+∂tρ + ∇ · q(ρ) = α(t, x)f(ρ) − β(t, x)ρ
+in [0, T) × Ω,
+∇ρ · n = 0
+in [0, T) × ∂Ω,
+ρ(0) = ρ0
+in Ω,
+(1.2)
+where q(ρ) = −∇ρ + f(ρ)∇V (ρ) where f(ρ) = ρ(1 − ρ) and V : Rn −→ R is a potential. In
+all these papers, either there is no mention about the behaviors of the pedestrians or all the
+pedestrians have the same emotional state (mainly panic).
+In the recent years the RCP (Reflex-Panic-Control) [10] and the APC (Alert-Panic-Control)
+[30] models have been proposed in order to describe the evolution in time of human behaviors
+during a catastrophe. They both consist in systems of nonlinear ODEs and have been devised
+following the structure of the compartmental models in mathematical epidemiology. In [29] the
+spatial dynamics has been integrated in the APC model, by considering the space as a discrete
+variable. In [11] the first system of reaction-diffusion equations describing a population with
+several behaviors has been proposed.
+The aim of this present paper is to introduce a spatio-temporal macroscopic first-order non-
+conservative pedestrians model describing the evolution of a population in a sudden, unexpected
+and without warning signs disaster. For this purpose, starting from the nonlinear ODE APC
+model proposed in [29, 30], we introduce a non-conservative first-order macroscopic model to
+describe the spatio-temporal dynamics of a population exhibiting different behavioral states.
+Our model reads as �
+�
+�
+�
+�
+∂tρi + ∇ · qi(ρ) = fi(ρ1, . . . , ρ5)
+in [0, T) × Ω,
+qi · n = q0
+i · n
+in [0, T) × ∂Ω,
+ρi(0) = ρ0
+i
+in Ω,
+(1.3)
+with qi := −di∇ρi + ρi⃗vi(ρ), for i = 1, . . . , 5.
+We prove the well-posedness of this system and we present different numerical simulations for
+several scenarios of evacuation of populations in emergency.
+The organization of this paper is as follows. In Section 3 we briefly present the APC model
+equations, we recall the structure of a first-order macroscopic pedestrian model and we introduce
+our advection-diffusion pedestrian APC model. Section 3 is devoted to the mathematical analysis
+of our model. Using the semigroup approach, we prove a local existence and uniqueness result and
+the positivity of the solution. Moreover, we prove the boundedness of this solution, under some
+assumptions on the parameters, which yields the global existence. Finally, Section 4 presents
+numerical results about different scenarios of evacuation.
+2. A spatio-temporal advection-diffusion model for human behaviors during a
+catastrophic event
+2.1. The temporal model of the human behavior of a population during a catastrophic
+event. In this section, we briefly present a model describing the evolution of a population during
+a sudden, rapid and unpredictable catastrophic event.
+The model describes the evolution of
+different human behaviors during a disaster, see [29, 30]. Depending on the emotional charges
+and their regulation, the different human reactions of a population in an emergency situation are
+
+4
+K. KHALIL ET AL.
+Daily 
+population 
+density
+Alert
+population 
+density
+Panic 
+population 
+density
+Control 
+population 
+density
+Back to Daily 
+population 
+density
+Daily 
+population
+density
+b4
+b3
+c2
+γ(t)
+ϕ(t)
+G
+c1
+b2
+F
+H
+b1
+Figure 1. The transfer diagram of the APC model. The arrows indicate the
+transitions among the compartments.
+here subdivided into three main categories, namely, alert, panic and control behaviors. The APC
+model considers the time evolution of the following five variables:
+• the density of individuals in an alert state ρ1(t),
+• the density of individuals that exhibit panic behaviors ρ2(t),
+• the density of individuals in a state of control ρ3(t),
+• the density of individuals in a daily behavior before the catastrophe ρ4(t),
+• the density of individuals in a behavior of everyday life after the disaster ρ5(t),
+• the density of individuals who die during the disaster ρ6(t).
+The corresponding model is given by the following nonlinear ODE system that matches with the
+classical compartmental SIR models (t ≥ 0):
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+ρ′
+1 =
+−(b1 + b2 + δ1)ρ1 + γ(t)q + b3ρ3 + b4ρ2 − F(ρ1, ρ3) − G(ρ1, ρ2),
+ρ′
+2 =
+−(b4 + c1 + δ2)ρ2 + b2ρ1 + c2ρ3 + G(ρ1, ρ2) − H(ρ2, ρ3),
+ρ′
+3 =
+−(b3 + c2 + δ3)ρ3 + b1ρ1 + c1ρ5 − φ(t)ρ3 + F(ρ1, ρ3) + H(ρ2, ρ3),
+ρ′
+4 =
+−γ(t)q,
+ρ′
+5 =
+φ(t)ρ3,
+ρ′
+6 =
+δ1ρ1 + δ2ρ2 + δ3ρ3,
+(2.1)
+with the initial condition (ρ1, ρ2, ρ3, ρ4, ρ5, ρ6)(0) = (0, 0, 0, 1, 0, 0), since the population is sup-
+posed to be in a daily behavior before the onset of the disaster. The detailed description of all
+the parameters is given in Table 3 in the Appendix. In particular, the transitions among the
+compartments are of two types since they model two fundamental phenomena (see Figure 1):
+• The intrinsic transitions: They represent the behavioral transitions that depend on
+the individual properties (past experiences, level of risk culture etc.) They are modeled
+by linear terms in system (2.1).
+The parameters of these transitions are bi > 0 for
+i = 1, . . . , 4 and cj > 0 for j = 1, 2.
+
+ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL
+5
+2
+4
+6
+8
+0.2
+0.4
+0.6
+0.8
+1.0
+ξ(w)
+w
+Figure 2. The function ξ involved in the imitation terms: the imitation starts
+very slowly, then it accelerates before slowing down and saturating.
+• The imitation phenomenon: Individuals have a tendency to imitate the behaviors of
+people around. Here we follow the dominant behavior principle, i.e. in the case of two
+populations in interaction, the most adopted behavior is the most imitated one. Thus,
+imitation between two behaviors depend on the ratio of the two populations. Only alert
+behaviors are not imitable. In system (2.1) the behavioral transitions due to imitation
+are represented by nonlinear terms defined as:
+F(ρ1, ρ3) := α13ξ
+�
+ρ3
+ρ1 + ε
+�
+ρ1ρ3,
+(2.2)
+G(ρ1, ρ2) := α12ξ
+�
+ρ2
+ρ1 + ε
+�
+ρ1ρ2,
+(2.3)
+H(ρ2, ρ3) :=
+�
+α23ξ
+�
+ρ3
+ρ2 + ε
+�
+− α32ξ
+�
+ρ2
+ρ3 + ε
+��
+ρ2ρ3.
+(2.4)
+The parameter 0 < ε ≪ 1 is considered here to avoid singularities, and the following
+function
+ξ(w) :=
+w2
+1 + w2 , w ∈ R.
+(2.5)
+takes into account the dominant behavior principle, that is the fact that the rate of
+imitation depends on the ratio of the corresponding populations (see Figure 2).
+For
+example, if we consider the imitation phenomenon from alert to panic, we remark that
+if
+ρ2
+ρ1+ε < 1 is small, then ξ
+�
+ρ2
+ρ1+ε
+�
+is almost equal to zero, so the imitation is weak.
+Conversely, if
+ρ2
+ρ1+ε ≫ 1 is large, it means that we have a majority of individuals in panic.
+In this case, ξ
+�
+ρ2
+ρ1+ε
+�
+goes to 1 and alerted individuals would imitate the panic ones.
+The same situation holds for the other imitation transitions.
+Finally, function γ describes the transition from the daily to the alert behavior at the beginning
+of the catastrophic event, while function φ represents the transition from a control behavior to
+an everyday life behavior at the end of the event. It is assumed that they are time-dependent
+functions that depend on the nature of the disaster. In [29] the authors consider the functions
+
+6
+K. KHALIL ET AL.
+2
+4
+6
+8
+0.2
+0.4
+0.6
+0.8
+1.0
+t
+γ(t)
+20
+40
+60
+80
+100
+120
+0.2
+0.4
+0.6
+0.8
+1.0
+t
+(t)
+ϕ
+Figure 3. Example of the functions γ and φ, which describe the transition from
+the daily to the alert behaviors, and from the control to everyday life behaviors:
+γ(t) = ζ(t, 1, 3) and φ(t) = ζ(t, 20, 70) respectively.
+φ, γ : [0, ∞) −→ [0, 1] defined as
+φ(t) := ζ(t, τ0, τ1)
+for
+τ0 < τ1,
+(2.6)
+and
+γ(t) := ζ(t, σ0, σ1)
+for
+σ0 < σ1,
+(2.7)
+where
+ζ(t, z0, z1) :=
+�
+�
+�
+�
+�
+�
+�
+�
+�
+0
+if t < z0,
+1
+2 − 1
+2 cos
+� t − z0
+z1 − z0
+π
+�
+if z0 ≤ t ≤ z1,
+1
+elswhere.
+(2.8)
+Here τ0 is the time at which the daily population starts to be impacted by the event, and τ1 is
+the time at which the total daily population becomes alert. Additionally, σ0 represents the time
+at which the first individuals in a control state can go back to a pseudo-daily behavior, while σ1
+is the time where this transition is highest. See Figure 3 for an example of these two functions.
+Summing up the equations of (2.1), we notice that
+6
+�
+i=1
+ρi(t) = 1 (the total population density), ∀t ≥ 0.
+(2.9)
+For this reason we only need to solve the system (2.1) without considering the victims density ρ6,
+since the last equation is a linear combination of the others.
+2.2. First-order macroscopic pedestrian models. In this section we give the mathematical
+framework about first-order macroscopic pedestrian models [14, 25]. We recall that the continuity
+principle indicates that the variation of the density ρ of a certain quantity, in Ω ⊂ R2, is given
+by the balance of the flow q of this quantity across the boundary ∂Ω and the amount of quantity
+produced or removed inside the domain. Mathematically, this can be expressed as follows:
+∂tρ + ∇ · q = S(t, x, ρ),
+t ≥ 0, x ∈ Ω,
+(2.10)
+
+ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL
+7
+where S is the source term. To be more precise, the flux q can be advective (qadv), proportional
+to a velocity ⃗v(ρ) where ρ is the transported scalar quantity, i.e. qadv = ρ⃗v(ρ); but it may also
+be diffusive (qdiff), that is, it consists of a diffusion term qdiff = −d∇ρ corresponding to the
+transportation of the scalar quantity according to its gradient, where d is the constant diffusion
+rate. Thus, we have
+q = qadv + qdiff := −d∇ρ + ρ⃗v(ρ).
+Moreover, the source term S can be divided into a pure and a reaction source terms:
+S = Sp + Sr.
+The pure source term denoted by Sp represents the self-creation/destruction rate inside the
+domain (using population dynamics terminology, it corresponds to the birth and death terms for
+example). This term will be denoted by Sp(t, x, ρ) = g(t, x, ρ) where g is a given linear function
+with respect to ρ. The reaction term Sr describes the creation/destruction processes as a reaction
+to this quantity (corresponding to the reaction and interaction terms). This term will be denoted
+by Sr(t, x, ρ) = f(t, x, ρ), where f is a given nonlinear function with respect to ρ. Moreover, the
+associated boundary conditions are expressed as follows
+q · n = q0 · n,
+where n is the outward unit normal vector to ∂Ω.
+These boundary conditions mean that the flux crossing the boundary part ∂Ω in the direction
+of the normal vector n is given by an observed flux q0 in the same direction n. According to that,
+the complete first-order equation (2.10) in its non-conservative form is given by:
+�
+�
+�
+∂tρ
+= d∆ρ − ∇ · (ρ⃗v(ρ)) + g(t, x, ρ) + f(t, x, ρ),
+t ≥ 0, in Ω,
+q · n
+= q0 · n,
+t ≥ 0, on ∂Ω,
+ρ(0)
+= ρ0,
+in Ω,
+(2.11)
+where ρ0 is the initial condition. For more details we refer to [3] and references therein.
+2.3. The spatio-temporal model corresponding to (2.1). In this section, we present our
+advection-diffusion APC (Alert-Panic-Control) model using the first-order continuity equations
+(2.11) presented in Section 2.2. Notice that system (2.11) can be generalized to the case where
+several populations are in interaction (as is the case for the system (1.1), where each population
+can for example have different directions of movement). Moreover, our new model takes into
+account the transitions and the imitations characteristics which are not considered in the conser-
+vative system (1.1).
+Let Ω ⊂ R2 be a non-empty bounded domain with Lipschitz boundary. Consider the local
+population densities ρi : [0, +∞) × Ω −→ R for i = 1, . . . , 5 where
+• ρ1(t, x) is the local density of individuals in the alert situation,
+• ρ2(t, x) is the local density of individuals in the panic situation,
+• ρ3(t, x) is the local density of individuals in the control situation,
+• ρ4(t, x) is the local density of individuals in the daily behaviour before the disaster,
+• ρ5(t, x) is the local density of individuals corresponding to the daily situation after the
+disaster,
+
+8
+K. KHALIL ET AL.
+and let ρ be given by
+ρ := (ρ1, ρ2, ρ3, ρ4, ρ5)∗.
+As for the model (2.11) we will consider an advective flux and a diffusive flux. The advection
+phenomenon models the movement of a population in a chosen direction, typically to escape the
+Ω domain. It is therefore natural to incorporate advection terms in our case.
+We set
+qi,adv := ρi⃗vi(ρ),
+the advective flux, where ρ⃗vi(ρ), i = 1, . . . , 5 is the corresponding velocity for each population
+of density ρi, i = 1, . . . , 5. The alert population corresponds to the set of behaviors such as
+information seeking and hazard identification, and therefore here it is assumed that it cannot
+undergo the advection phenomenon. Thus, it is assumed that only the populations in a situation
+of panic or control are concerned by the advection phenomenon, hence
+⃗vi(ρ) = 0,
+for i = 1, 4, 5.
+For i = 2, 3, we assume that the velocities ⃗vi satisfy
+⃗vi(ρ) = Vi(ρ)⃗ν,
+for i = 2, 3,
+where the vector ⃗ν : Ω → R2 that represents the direction of the movement, and satisfies certain
+conditions that will specified later.
+Moreover, V2, V3 are the scalar speed-density functions.
+Several different type of speed-density functions are used in the literature (see a.e. [7, 14, 41]).
+Here, we choose a linear dependence:
+V2(ρ) = V2,max (1 − ˜ρ)
+and
+V3(ρ) = V3,max (1 − ˜ρ) ,
+where V2,max, V3,max are two positive constants and
+˜ρ :=
+5
+�
+i=1
+ρi.
+Similar assumptions on the panic and the control maximum speeds are used in [31].
+Moreover, each population diffuses in the spatial domain Ω according to the density gradient
+qi,diff := −di∇ρi,
+with constant diffusion coefficients di for i = 1, . . . , 5. It is assumed that the whole crowd diffuses
+with different diffusion coefficients depending on the type of behavior.
+With these assumptions and notations, the associated fluxes are given by
+qi := −di∇ρi + ρi⃗vi(ρ),
+for i = 1, . . . , 5.
+The source (pure and reaction) terms correspond to the intrinsic transitions and the imitation
+terms described in Section 2.1.
+Moreover, we divide the boundary ∂Ω of Ω in two parts, each of them corresponding to different
+boundary conditions:
+∂Ω := Γ1 ∪ Γ2
+with
+Γ1 ∩ Γ2 = ∅.
+
+ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL
+9
+Here Γ1 corresponds to the part of boundary that could not be crossed by the population, while Γ2
+corresponds to an escape. We define the observed fluxes q0
+i on the boundary ∂Ω by
+q0
+i (ρi) :=
+�
+0
+on Γ1
+−ρivi,out ⃗ν
+on Γ2
+where vi,out ≥ 0 is the constant speed at the boundary Γ2 and ⃗ν is the direction of the movement.
+This means that each population cannot cross along Γ1 and cross the escape Γ2 with speed vi,out.
+We assume that the function ⃗ν satisfies:
+⃗ν(x) =
+�
+�
+�
+�
+�
+(0, 0)∗,
+x ∈ Γ1
+(νx1(x), νx2(x))∗,
+x ∈ Ω
+n(x),
+x ∈ Γ2
+(2.12)
+where n is the unit normal vector at the boundary part Γ2. This choice of vector ⃗ν means that,
+at the part of the boundary where pedestrians cannot cross, i.e. Γ1, the direction of movement
+vanishes, but at the target exit, i.e., Γ2, the pedestrians cross this part of the boundary in a
+direction parallel to the normal vector n, while the desired direction of motion inside the domain
+Ω is given in a suitable way that depends on the regularity of Ω, and it satisfies the following
+assumption:
+⃗ν|Ω ∈ W 1,∞(Ω, R2),
+such that ∇(ν(x)) ≤ 0 for all x ∈ Ω. For example, we can take ⃗ν|Ω to be normalized vectors
+between any point x ∈ Ω and a centered (fixed) target point that lies outside Ω, see (4.1). Thus,
+the observed fluxes on the boundary ∂Ω are given by
+q0
+i (ρi) = ρivi,out ⃗ν.
+We assume that at the beginning t = 0, the whole population is in a daily behavior, so we consider
+the following initial conditions: ρ(t = 0, 0) = ρ0 where ρ0 is given by
+ρ0 := (0, 0, 0, θ, 0)∗
+on Ω,
+(2.13)
+and θ : Ω → [0, ∞) is such that, the integral exists, and that
+�
+Ω
+θ(x)dx = 1.
+Thus, from (2.1) and (2.11), we obtain the following system:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+∂tρ1 =
+d1∆ρ1 − (b1 + b2 + δ1)ρ1 + γ(t)ρ4 + b3ρ3 + b4ρ2
+−F(ρ1, ρ3) − G(ρ1, ρ2)
+in Ω, t ≥ 0,
+∂tρ2 =
+d2∆ρ2 − (b4 + c1 + δ2)ρ2 + b2ρ1 + c2ρ3
+−∇ · (ρ2⃗v2(ρ)) + G(ρ1, ρ2) − H(ρ2, ρ3)
+in Ω, t ≥ 0,
+∂tρ3 =
+d3∆ρ3 − (b3 + c2 + δ3)ρ3 + b1ρ1 + c1ρ2
+−φ(t)ρ3 − ∇ · (ρ3⃗v3(ρ)) + F(ρ1, ρ3) + H(ρ2, ρ3)
+in Ω, t ≥ 0,
+∂tρ4 =
+d4∆ρ4 − γ(t)ρ4
+in Ω, t ≥ 0
+∂tρ5 =
+d5∆ρ5 + φ(t)ρ3
+in Ω, t ≥ 0.
+(2.14)
+with the boundary conditions
+di∇ρi · n = (ρi⃗vi(ρ) − ρivi,out⃗ν) · n
+on ∂Ω, t ≥ 0,
+i = 1, . . . 5,
+(2.15)
+
+10
+K. KHALIL ET AL.
+or more explicitly, using the definition of ⃗ν:
+�
+di∇ρi · n = 0
+on Γ1 for i = 1, · · · , 5,
+di∇ρi · n = −viρi + ρiVi(ρ)
+on Γ2 for i = 1, · · · , 5,
+and the initial condition
+ρ(t = 0, ·) = ρ0
+in Ω.
+(2.16)
+3. Well-posedness, positivity and positively invariant sets of solutions for the
+spatio-temporal model spatio-temporal model (2.14), (2.15) and (2.16)
+In this section, we prove the well-posedness of our spatio-temporal APC model (2.14), (2.15)
+and (2.16) introduced in Section . Then we establish the positivity of the solutions, the L1-
+boundedness of the sum of the population densities and the boundedness of the solution which
+gives the global existence.
+3.1. The abstract formulation and the associated boundary value Cauchy problem.
+To study the existence and uniqueness of solutions to the system spatio-temporal APC model
+(2.14)-(2.16) we use the abstract formulation and semigroup theory [20, 35]. In order to do that,
+for p > 2, we define the Banach space X := Lp(Ω)5, the product of the Lebesgue spaces of order
+p, equipped with the following norm
+∥ϕ := (ϕ1, · · · , ϕ5)∗∥ :=
+5
+�
+i=1
+∥ϕi∥,
+where ∥ · ∥ is the usual norm in Lp(Ω). It is clear that X is a Banach lattice, i.e.,
+|ϕi(x)| ≤ |ψi(x)| for a.e. x ∈ Ω for all i = 1, · · · , 5 implies that ∥ϕ∥ ≤ ∥ψ∥.
+Moreover, we define the linear closed operator (A, D(A)) on X by
+�
+A = diag(d1∆, · · · , d5∆)
+D(A) = W 2,p(Ω)5.
+(3.1)
+The nonlinear function K : [0, ∞) × Xα −→ X is defined by
+K(t, ϕ) =
+�
+�
+�
+�
+�
+�
+−(b1 + b2 + δ1)ϕ1 + γ(t)ϕ4 + b3ϕ3 + b4ϕ2 − F(ϕ1, ϕ3) − G(ϕ1, ϕ2)
+−(b4 + c1 + δ2)ϕ2 + b2ϕ1 + c2ϕ3 − ∇ · (ϕ2vp(ϕ)ν) + G(ϕ1, ϕ2) − H(ϕ2, ϕ3)
+−(b3 + c2 + δ3)ϕ3 + b1ϕ1 + c1ϕ2 − φ(t)ϕ3 − ∇ · (ϕ3v3(ϕ)ν) + F(ϕ1, ϕ3) + H(ϕ2, ϕ3)
+−γ(t)ϕ4
+φ(t)ϕ3
+�
+�
+�
+�
+�
+�
+,
+(3.2)
+where we take
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+K1(t, ϕ1, ∇ϕ) = −(b1 + b2 + δ1)ϕ1 + γ(t)ϕ4 + b3ϕ3 + b4ϕ2 − F(ϕ1, ϕ3) − G(ϕ1, ϕ2),
+K2(t, ϕ2, ∇ϕ) = −(b4 + c1 + δ2)ϕ2 + b2ϕ1 + c2ϕ3 − ∇ · (ϕ2vp(ϕ)ν) + G(ϕ1, ϕ2) − H(ϕ2, ϕ3),
+K3(t, ϕ3, ∇ϕ) = −(b3 + c2 + δ3)ϕ3 + b1ϕ1 + c1ϕ2 − φ(t)ϕ3 − ∇ · (ϕ3v3(ϕ)ν) + F(ϕ1, ϕ3) + H(ϕ2, ϕ3),
+K4(t, ϕ4, ∇ϕ) = −γ(t)ϕ4,
+K5(t, ϕ5, ∇ϕ) = φ(t)ϕ3,
+(3.3)
+
+ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL
+11
+and Xα := {ϕ ∈ W 2α,p(Ω)5 : di∂nϕi|∂Ω = 0} for some (fixed) α ∈ (1/p + 1/2, 1) equipped with
+the norm ∥ · ∥0,α := ∥ · ∥ + ∥∇ · ∥ + [·]ζ where
+[ϕ]ζ :=
+��
+Ω×Ω
+|ϕ(x) − ϕ(y)|p
+|x − y|2+pζ
+dxdy
+�1/p
+,
+ζ = 2α − 1.
+Hence, ∥ϕ∥α := �5
+i=1 ∥ϕi∥0,α defines a norm on Xα which make it a Banach space. Then from
+the Sobolev embedding, we have
+Xα �→ C1(Ω)5.
+Define the boundary space ∂X := W 1−1/p,p(∂Ω)5 which is equipped with the norm
+∥ϕ∥∂X :=
+5
+�
+i=1
+|ϕi|p
+where
+|ϕ|p =
+��
+∂Ω
+|ϕ(x)|pdx +
+�
+∂Ω×∂Ω
+|ϕ(x) − ϕ(y)|p
+|x − y|p
+dxdy
+�1/p
+.
+Since 1 − 2/p > 0, we obtain the continuous embedding ∂X �→ C(∂Ω)5. Moreover, we define the
+boundary operator L : Z −→ ∂X by
+Lϕ = (d1∂nϕ1, · · · , d5∂nϕ5)∗
+on ∂Ω.
+(3.4)
+The nonlinear boundary term M : Xα −→ ∂X is given by
+Mϕ =
+�
+(0, 0, 0, 0, 0)∗
+on Γ1
+(−v1ρ1, −v2ρ2 + ϕ2V2(ϕ), −v3ρ3 + ϕ3V3(ϕ), −v4ρ4, −v5ρ5)∗
+on Γ2.
+(3.5)
+Let the Banach space Z := D(A) equipped with its usual norm, so the continuous embedding
+Z �→ Xα holds. Hence the linear operator A : Z −→ X is bounded and L : Z −→ ∂X is bounded
+and surjective (see [1, 2]). The initial conditions are given by the following vector
+ρ0 = (0, 0, 0, θ, 0)∗.
+(3.6)
+Now we can write our boundary evolution system as
+�
+�
+�
+�
+�
+ut(t) =Au(t) + K(t, u(t)),
+t ≥ 0,
+Lu(t) =M(u(t)),
+t ≥ 0,
+u(0) =u0,
+(3.7)
+where
+u(t) := (ρ1(t, ·), ρ2(t, ·), ρ3(t, ·), ρ4(t, ·), ρ5(t, ·))∗,
+and
+u0 = ρ0.
+
+12
+K. KHALIL ET AL.
+3.2. Preliminary results. In this section, we give our preliminary results, the proofs are given
+for the sake of completeness for a curious reader, and will be available in Appendix A. In the
+following, we define A0 := A| ker(L).
+Definition 3.1. We recall that X is a Banach lattice.
+(i) A vector ϕ = (ϕ1, · · · , ϕ5)∗ ∈ X is said to be positive, i.e., ϕ(x) ≥ 0, if and only if, ϕi(x) ≥ 0
+for a.e. x ∈ Ω for all i = 1, · · · , 5. So that, X+ denotes the positive cone of X.
+(ii) A bounded operator T , in the Banach lattice X, is said to be positive if and only if, for every
+ϕ ∈ X, ϕ(x) ≥ 0 implies T ϕ(x) ≥ 0 for a.e. x ∈ Ω.
+(iii) A semigroup (T (t))t≥0, in the Banach lattice X, is said to be positive if and only if, for
+every ϕ ∈ X, ϕ(x) ≥ 0 implies T (t)ϕ(x) ≥ 0 for all t ≥ 0 for a.e. x ∈ Ω.
+Proposition 3.2. The following assertions hold:
+(i) The closed operator A0 generates a contraction holomorphic C0-semigroup (T (t))t≥0 on X.
+(ii) The semigroup (T (t))t≥0 generated by A0 := A| ker(L) is compact and positive (i.e., T (t)X+ ⊂
+X+).
+Moreover, the semigroup (T (t))t≥0 is given by the following matrix-valued operators
+T (t) = diag(T1(t), · · · , T5(t))∗,
+t ≥ 0.
+Now, we present the inter- and extrapolation spaces associated to the generator A0. We define
+on X the norm ∥x∥−1 = ∥R(λ, A0)x∥, for x ∈ X. Then the completion of (X, ∥ · ∥−1) is called
+the extrapolation space of X associated to A0 and will be denoted by X−1. This means that A0
+has a unique extension A−1 : D(A−1) = X −→ X−1. Since for every t ≥ 0, (T (t))t≥0 commutes
+with the operator resolvent R(λ, A0), the extensions of (T (t))t≥0 to X−1 exists and defines an
+analytic semigroup (T−1(t))t≥0 which is generated by A−1. Let α ∈ (0, 1), we define the following
+interpolated extrapolation spaces by:
+Xα−1 = X
+∥·∥α−1,
+where
+∥x∥α−1 := sup
+ω>0
+∥(ωαR(ω, A−1 − λ)x∥.
+Then, we have the following continuous embeddings:
+D(A0) �→ Xα �→ Xβ �→ X
+X �→ Xα−1 �→ Xβ−1 �→ X−1,
+for all 0 < β < α < 1, where D(A0) is equipped with the graph norm that makes it a Banach
+space.
+Remark 3.3. (i) Note that the extrapolated spaces introduced here do not depend on any choice
+of λ ∈ ρ(A0), which means that, any other choice of λ gives the same extrapolated space with an
+equivalent norm, this holds by virtue of the resolvent equation, see [4, 20]. We recall that, the
+spectrum of A0 satisfies σ(A0) ⊂ (−∞, 0]. So that C \ (−∞, 0] ⊂ ϱ(A0), see [16].
+Remark 3.4. It follows from [42, Sections 4.3.3 and 4.6.1], that the spaces Xα for 0 < α <
+1 introduced here coincide with real interpolation spaces (of order α) between D(A0) and X.
+Moreover, the embedding Z �→ Xα also holds.
+Proposition 3.5. For each 0 ≤ δ ≤ 1, (Tδ−1(t))t≥0 is the unique extension semigroup of
+(T (t))t≥0 with the associated generator Aδ−1 satisfying D(Aδ−1) = Xδ. Moreover, the semigroup
+(Tδ−1(t))t≥0 inherits all the properties of (T (t))t≥0. That is, (Tδ−1(t))t≥0 is strongly continuous,
+analytic, compact and positive.
+
+ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL
+13
+Definition 3.6. We define the positive cone of Xα by
+X+
+α = X+ ∩ Xα,
+Similarly, the positive cone of Xβ−1 is defined by
+X+
+β−1 = X+ ∩ Xβ−1.
+Let 0 < ai < +∞ for i = 1, · · · , 5 and Λa ⊂ R5 be such that
+Λa := Π5
+i=1[0, ai],
+A function ϕ ∈ X belongs to Λa if and only if 0 ≤ ϕi(x) ≤ ai for a.e. x ∈ Ω, i = 1, · · · , 5.
+Moreover,
+Λ+∞ := Π4
+i=1[0, +∞).
+In particular, if only a5 = +∞, we give
+Λa,+∞ := Π4
+i=1[0, ai] × [0, +∞).
+Using Definition 3.6, we may observe that
+Definition 3.7. The bounded positive cones of X, Xβ−1 and Xα are defined respectively by:
+XΛa = {ϕ ∈ X : ϕ(x) ∈ Λa, a.e. x ∈ Ω},
+XΛa
+β−1 = {ϕ ∈ Xβ−1 : ϕ(x) ∈ Λa, a.e. x ∈ Ω},
+and
+XΛa
+α
+= {ϕ ∈ Xα : ϕ(x) ∈ Λa, a.e. x ∈ Ω}.
+Remark 3.8. (i) Note that, since Xα �→ C1(Ω)5, the "a.e. x ∈ Ω" in the definition of X+
+α (resp.
+of XΛa
+α ) becomes "for all x ∈ Ω".
+(ii) By definition, we have X+ = XΛ+∞, X+
+α = XΛ+∞
+α
+and X+
+β−1 = XΛ+∞
+β−1 .
+In order to study the boundary evolution equation (2.14), we proceed as in [2] and later [17]
+(see [2, Section 12] and also [17, Section 4]), namely, the nonlinear boundary evolution problem
+(2.14) admits a solution, if and only if, the following semilinear Cauchy problem admits a solution
+�
+ut(t) =Aβ−1u(t) + ˜K(t, u(t)),
+t ≥ 0,
+u(0) =u0,
+(3.8)
+where ˜K := K + (λ − Aβ−1)DM with D is the Dirichlet map associated to the operator (λ − A)
+i.e. v = Dw is the unique solution of the abstract boundary value problem
+�(λ − A)v = 0
+Lv = w
+(3.9)
+for each w ∈ ∂X for some λ ∈ ϱ(A). In fact, let u ∈ X and w ∈ ∂X. Then, the equation
+�(λ − A)v = u
+Lv = w
+(3.10)
+admits the solution v = R(λ, A)u + Dw. This solution is unique in Z since λ − A is injective on
+D(A0) := ker(L).
+
+14
+K. KHALIL ET AL.
+This approach of studying the boundary evolution equation (3.7) by equivalently studying the
+Cauchy problem (3.8) was first introduced separately in [2, 21], and was later perfected in [17]
+and others (see the references therein). The conditions under which this approach is used are
+cited in all the references [2, 17, 21]. However, for the sake of completeness, we will cite these
+conditions as follows:
+(C1) There exists a new norm | · |m which is finer than the norm of X, such that the space
+Z := (D(A), | · |m) is complete, i.e, Z is continuously embedded in X and A ∈ L(Z, X).
+(C2) The restriction operator A0 = A|ker(L) generates a strongly continuous analytic semigroup.
+(C3) The operator L : Z −→ ∂X is bounded and surjective.
+(C4) Z is continuously embedded in Xα, i.e., Z �→ Xα for some 0 < α < 1.
+(C5) The functions K : [0, +∞)×Xα −→ X and M : [0, +∞)×Xα −→ ∂X are locally integrable
+in the first variable and continuous with respect to the second one.
+We mention that all the conditions (C1)-(C5) are satisfied.
+Lemma 3.9. The operator (λ − A−1)D is bounded from ∂X to X−1 with its norm denoted by
+∥(λ − A−1)D∥∂X→X−1 ≤ c.
+Proposition 3.10. The function ˜K : [0, +∞) × Xα −→ Xβ−1 is Lipschitzian in bounded sets
+i.e., for all R > 0 there exists LR ≥ 0 such that
+∥ ˜K(t, ρ) − ˜K(s, υ)∥β−1 ≤ LR(| t − s | +∥ρ − υ∥α)
+for all ρ, υ ∈ B(0, R) for all t, s ≥ 0. (3.11)
+Remark 3.11. Notice that if 1/p + 1/2 < β < α < 1, then Xβ �→ C1(Ω)5. So, (3.11) holds also
+in Xβ.
+The following Lemma is needed to show that, even equation in the following.
+Lemma 3.12. Let 0 < β < 1 and B : [0, T] −→ Xβ−1 such that there exist 0 < η ≤ 1 and l ≥ 0
+satisfying
+∥B(t) − B(s)∥β−1 ≤ l|t − s|η,
+t, s ∈ [0, T].
+(3.12)
+Then,
+v(t) =
+� t
+0
+Tβ−1(t − s)B(s)ds ∈ D(Aβ−1) = Xβ
+for 0 ≤ t ≤ T.
+Moreover, v ∈ C1((0, T], Xβ).
+Definition 3.13. [2, 17] Let u0 ∈ Xα and T > 0. By a solution to equation (3.8), we mean
+a function u ∈ C([0, T], Xα) ∩ C1([0, T], Xβ−1), such that u(t) ∈ Xβ for 0 ≤ t ≤ T and such
+that (3.8) is pointwisely satisfied. In particular, this solution must satisfy the following integral
+formula:
+u(t) = T (t)u0 +
+� t
+0
+Tβ−1(t − s) (K(s, u(s)) + (ω − Aβ−1)DM(u(s))) ds,
+t ∈ [0, T].
+(3.13)
+3.3. Local existence and regularity. In this Section, using the our we prove the local existence,
+uniqueness and regularity of solutions to equation (3.8) which yields the local well-posedness for
+the model (2.14)-(2.16).
+
+ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL
+15
+Theorem 3.14 (Local existence and regularity). For each u0 ∈ Xα there exist a maximal time
+T(u0) > 0 and a unique maximal solution u(·) := u(·, u0) ∈ C([0, T(u0)), Xα)∩C1([0, T(u0)), Xβ)
+of equation (3.8) such that
+u(t) = T (t)u0 +
+� t
+0
+Tβ−1(t − s) (K(s, u(s)) + (ω − Aβ−1)DM(u(s)))
+�
+��
+�
+= ˜K(s,u(s))
+ds,
+t ∈ [0, T(u0)). (3.14)
+Moreover the solution u satisfies the following blow-up property:
+T(u0) = +∞
+or
+lim sup
+t→T (u0)− ∥u(t)∥ = +∞.
+(3.15)
+Proof. Let u0 ∈ Xα. So, using Proposition 3.10, it yields from [32, Section 7.1] (by taking Xβ−1
+instead of X and Tβ−1 instead of T ), that there exist T > 0 (small enough) and a unique solution
+u ∈ C([0, T], Xα) ∩ C1([0, T], Xβ−1) of equation (3.8) satisfying (3.14). Note that, in our case,
+A0 is densely defined in X, so that the continuity at t = 0 holds. Now, to conclude, we use the
+integral formula of our solution (3.14) and Lemma 3.12 to prove that u ∈ C1([0, T], Xβ).
+First, note that u0 ∈ Xα �→ Xβ implies that T (t)u0 ∈ C1([0, T], Xβ). Then, it suffices to prove
+that
+t �→ v(t) =
+� t
+0
+Tβ−1(t − s) ˜K(s, u(s))ds ∈ C1([0, T], Xβ).
+Remark that, u is Hölder continuous in Xβ−1 (since it is C1), i.e, there exist ˜l ≥ 0 and 0 < ϑ ≤ 1,
+such that
+∥u(t) − u(s)∥β−1 ≤ ˜l|t − s|ϑ,
+t, s ∈ [0, T].
+(3.16)
+Moreover, since u0 ∈ Xα �→ Xβ, it yields, using Remark 3.11, that
+u ∈ C([0, T], Xβ) ∩ C1([0, T], Xβ−1).
+Hence, u is bounded in Xβ, since it is continuous. Furthermore, using the reiteration theorem,
+we obtain that Xβ = (Xα, Xβ−1)˜θ, with 0 < ˜θ < 1. That is,
+∥u(t) − u(s)∥β ≤ c(α, β)∥u(t) − u(s)∥1−˜θ
+α
+∥u(t) − u(s)∥
+˜θ
+1−β,
+t, s ∈ [0, T].
+Therefore, we have
+∥u(t) − u(s)∥β ≤ ˜c(α, β)|t − s|
+˜θϑ,
+t, s ∈ [0, T].
+Note that u is bounded in Xβ. Hence, by (3.16) and Remark 3.11 (using Proposition 3.10 for Xβ
+instead of Xα), we obtain that
+∥ ˜K(t, u(t)) − ˜K(s, u(s))∥β−1 ≤ LR(| t − s | +∥u(t) − u(s)∥β)
+≤ ˜
+LR(| t − s | +|t − s|
+˜θϑ),
+t, s ∈ [0, T].
+This proves that ˜K(·, u(·)) is Hölder continuous in Xβ−1. Then, we conclude using Lemma 3.12,
+by taking B(·) = ˜K(·, u(·)).
+Henceforth, we can argue similarly as in [32, Proposition 7.1.8] to prove that the solution u can
+be extended continuously to a maximal interval [0, T(u0)), where T(u0) > 0 is the maximal time,
+such that the property (3.15) is also satisfied.
+□
+
+16
+K. KHALIL ET AL.
+Remark 3.15. We mention that, in [32, Theorem 7.1.2], the result of existence of a solution
+(without regularity) of equation (3.8) uses the fractional power space D(Aα
+0 ) as an intermediate
+space Xα. However, this fact does not affect our existence result since the proof can be given in
+a similar way for any intermediate Banach space, see the proof of [32, Theorem 7.1.2].
+3.4. Positivity. This section aims to show the positivity of the solution of our model (2.14)-
+(2.16) obtained in Section (3.3). Results and the proofs of the present section are inspired from
+those in [28, Section 2], see also [12, Section 6.3] in the case of homogeneous boundary conditions.
+Let ϕ ∈ Xα, we define
+[(λ − A−1)DM]ϕ(x) = ([(λ − A−1)DM]ϕ1(x), · · · , [(λ − A−1)DM]ϕ5(x))∗,
+and, then
+˜K(t, ϕ)(x) := ˜K(t, ϕ(x))
+=
+�
+K1(t, ϕ1(x), ∇ϕ(x)) + [(λ − A−1)DM]1ϕ1(x), · · · , K5(t, ϕ5(x), ∇ϕ(x)) + [(λ − A−1)DM]5ϕ5(x)
+�∗
+,
+for a.e. x ∈ Ω. Then, we have the following positivity result.
+Theorem 3.16 (Positivity). For each u0 ∈ X+
+α equation (3.8) has a unique maximal solution
+u(·, u0) ∈ C([0, T(u0)), Xα) ∩ C1([0, T(u0)), Xβ) such that u(t) ∈ X+
+α for all t ∈ [0, T(u0)).
+Proof. From Proposition 3.2, it is clear that T (t)X+
+α ⊂ X+
+α for all t ≥ 0. Let ϕ ∈ X+
+α . So from
+[28, Corollary 4] it suffices to show that
+lim
+h→0 h−1d(ϕ + h ˜K(t, ϕ); X+
+β−1) = 0
+for each t ≥ 0.
+(3.17)
+First, we prove (pointwisely) that
+lim
+h→0 h−1d(sup
+ω>0
+ωβ(R(ω, A−1 − λ)
+�
+ϕ(x) + h[ ˜K(t, ϕ)](x)
+�
+; Λ+∞) = 0
+for each t ≥ 0, a.e. x ∈ Ω.
+(3.18)
+Then, the formula (3.18) holds since the transformation supω>0 ωβR(ω, A−1 − λ) preserves the
+positivity, and due to [28, Remark 1.2] by the fact that Ki(t, 0) ≥ 0 and [(ω − A−1)DM]i0 = 0
+for all t ≥ 0 which gives that ˜Ki(t, 0) ≥ 0. Note that the operator R(ω, A−1 − λ) is positive (see
+[4]) which yields the positivity of supω>0 ωβR(ω, A−1 − λ). Hence, we aim to prove that (3.17)
+holds. Let | · |p be the p-norm in R5 defined as |(x1, · · · , x5)|p = (�5
+k=1 |xi|p)
+1
+p . Then, for ϕ ∈ X,
+the norm
+∥ϕ∥p := (
+�
+x∈Ω
+| ϕ(x) |p dx)
+1
+p
+is equivalent to the norm on X, this is due the fact that all the norms in R5 are equivalent.
+Similarly, using this new norm ∥ · ∥p, we can define an associated equivalent norm for Xα, and
+the new equivalent norm on Xβ−1 which is given by
+∥ · ∥β−1,p = sup
+ω>0
+∥ωβR(ω, λ − A−1) · ∥p.
+Let us define the Euclidean projection onto Λ+∞, πΛ : R5 −→ Λ0,+∞, by
+| x − πΛx |= d(x, Λ+∞).
+
+ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL
+17
+Notice that the mapping πΛ is well-defined and continuous on Rn (eventually it is 1-Lipschitzian).
+Let ε > 0 such that there exists δ > 0 and define
+ϕh(x) := πΛ(ϕ(x) + h[ ˜K(t, ϕ)](x))
+for t ≥ 0, x ∈ Ω, h > 0.
+So, ϕh ∈ XΛ+∞
+β−1 and
+d(ϕ + h ˜K(t, ϕ); XΛ+∞
+β−1 )p ≤ ∥ϕ + h ˜K(t, ϕ) − ϕh∥p
+β−1
+≤ sup
+ω>0
+ωβ
+�
+x∈Ω
+| R(ω, λ − A−1)
+�
+ϕ(x) + h[ ˜K(t, ϕ)](x) − ϕh(x)
+�
+|p
+p dx
+=
+�
+x∈Ω
+d(sup
+ω>0
+ωβR(ω, λ − A−1)
+�
+ϕ(x) + h[ ˜K(t, ϕ)](x)
+�
+; Λ0,+∞)pdx
+(3.19)
+Moreover, for 0 < h ≤ δ it follows in view of the convexity of the operator distance, by the
+continuity of ˜K and using (3.18), that
+�
+Ω
+d(sup
+ω>0
+ωβ(R(ω, A−1 − λ)
+�
+ϕ(x) + h[ ˜K(t, ϕ)](x)
+�
+; Λ0,+∞)pdx ≤ |Ω|(hε)p.
+Hence, by (3.19) we have
+d(ϕ + h ˜K(t, ϕ); X+
+β−1) ≤ |Ω|
+1
+p hε
+for all 0 < h ≤ δ.
+Which proves the result.
+□
+3.5. L1-boundedness of the total population density. In this Section, we show the L1-
+boundedness of the total population density of our model (2.14)-(2.16). Let u0 ∈ X+
+α and let
+u(t, u0) = (ρ1(t, ·), · · · , ρ5(t, ·))∗ for all t ∈ [0, T(u0)) be the corresponding maximal solution. It
+is clear, from Theorem 3.14, that u(·, u0) ∈ Xα �→ C1(Ω)5 �→ L1(Ω)5. This means that each ρi
+is bounded with respect to Ω and then it is L1(Ω). Hence, the following map
+t ∈ [0, T(u0)) �−→ U(t) :=
+�
+Ω
+[ρ1(t, x) + · · · + ρ5(t, x)] dx ∈ R,
+is well-defined. Furthermore, we have
+Proposition 3.17 (L1-boundedness).
+0 ≤ U(t) ≤ 1
+for all t ∈ [0, T(u0)).
+Proof. By assumption, we have
+U(0) =
+�
+Ω
+θ(x)dx = 1.
+In otherwise, the positivity result in Theorem 3.16 gives
+U(t) ≥ 0
+for all t ∈ [0, T(u0)).
+Moreover, the mapping U is well-defined and it is continuously differentiable on [0, T(u0)). Hence,
+we show that
+d
+dtU(t) ≤ 0,
+t ∈ [0, T(u0)).
+
+18
+K. KHALIL ET AL.
+Indeed, using the Green-Ostrogradski formula, we obtain that
+d
+dtU(t) =
+�
+Ω
+∂t [ρ1 + ρ2 + ρ3 + ρ4 + ρ5] dx
+=
+5
+�
+i=1
+di
+�
+Ω
+∆ρidx −
+�
+Ω
+∇(ρ2v(ρ))dx −
+�
+Ω
+∇(ρ3v(ρ))dx −
+3
+�
+i=1
+δi
+�
+Ω
+ρi
+=
+5
+�
+i=1
+di
+�
+∂Ω
+∇ρi · ndx −
+�
+∂Ω
+ρ2v(ρ) · ndx −
+�
+∂Ω
+ρ3v(ρ) · ndx −
+3
+�
+i=1
+δi
+�
+Ω
+ρidx
+= −
+5
+�
+i=1
+vi
+�
+Γ2
+ρidx −
+3
+�
+i=1
+δi
+�
+Ω
+ρidx ≤ 0,
+t ∈ [0, T(u0)).
+The last estimate is a consequence of the positivity of the terms ρi, i = 1, · · · , 5. That is, U is
+decreasing and then, we have
+U(t) ≤ U(0) = 1
+for all t ∈ [0, T(u0)).
+□
+3.6. Uniform boundedness and global existence. In this Section, using the method of pos-
+itively invariant regions, we prove that the maximal solution of equation (3.8) have a bounded
+positive invariant region. This fact guarantees the uniform boundedness of the solution (see The-
+orem 3.18) that yields the global existence (see Corollary 3.21). We recall also that the result
+obtained in this section is new and generalize those in [28, Section 2] and [12, Section 6.3] in the
+case of inhomogeneous boundary type conditions. Moreover, we assume that
+5
+�
+i=1
+ai > 1.
+Theorem 3.18 (Uniform boundedness). For each u0 ∈ XΛa equation (3.8) has a unique max-
+imal solution u(·, u0) ∈ C([0, T(u0)), Xα) ∩ C1([0, T(u0)), Xβ) such that u(t) ∈ XΛa for all
+t ∈ [0, T(u0)) provided that,
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+a4 ≤ (b1 + b2 + δ1)a1;
+b3 ≤ αa→cξ(
+a3
+a1 + ε)a1;
+b4 ≤ αa→pξ(
+a2
+a1 + ε)a1,
+b2a1 + αc→pξ(
+a2
+a3 + ε)a2a3 ≤ (b4 + c1 + δ2)a2;
+c2a3 + αa→pξ(
+a2
+a1 + ε)a1a2 ≤ αp→cξ(
+a3
+a2 + ε)a2a3,
+b1a1 + αp→cξ(
+a3
+a2 + ε)a2a3 ≤ (b3 + c2 + δ3)a3;
+c1a2 + αa→cξ(
+a3
+a1 + ε)a1a3 ≤ αc→pξ(
+a2
+a3 + ε)a2a3.
+(3.20)
+Proof. Let ϕ ∈ XΛa. First, we prove that T (t)XΛa ⊂ XΛa. That is, from the invariance result
+[35, Theorem 5.1], it suffices to show that R(λ, A0)XΛa ⊂ XΛa for some λ ∈ ρ(A0) large enough.
+This last follows immediately since we have
+(λ − A0)a = λa ≥ a,
+for large λ ∈ ρ(A0). Remark that R(λ, A0) is a positive operator.
+Furthermore, we show the invariance for the solution under the set
+Λa,∞ := Π4
+i=1[0, ai] × [0, ∞).
+
+ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL
+19
+which leads to the invariance under the bounded region Π4
+i=1[0, ai] for the vector components
+(ρ1, · · · , ρ4)∗. For the case ρ5 ≤ a5 we treat it separately. So it suffices to prove that
+lim
+h→0 h−1d(ϕ + h ˜K(t, ϕ); XΛa,∞
+−1
+) = 0,
+(3.21)
+which amounts, in view of the proof of Theorem 3.16, to proving that
+lim
+h→0 h−1d(sup
+ω>0
+ωβ(R(ω, A−1 − λ)
+�
+ϕ(x) + h[ ˜K(t, ϕ)](x)
+�
+; Λa,∞) = 0
+for all t ≥ 0, a.e. x ∈ Ω.
+(3.22)
+So, in view of [23, Proposition 12], formula (3.22) holds if we check that for ρ = a, we have
+˜Ki(t, ai, ∇a) ≤ 0.
+Thus, it suffices to prove that Ki(t, ai, ∇a) + [(λ − A−1)DM]iai ≤ 0 for i = 1, · · · , 4. Note that,
+by construction, ˜K4(t, a4, ∇β) ≤ 0. Moreover, since ∇ai · n = 0, for i = 1, · · · , 5, we obtain that
+La = (d1∇a1 · n, d2∇a2 · n, d3∇a3 · n, d4∇a4 · n, d5∇a5 · n)∗ = (0, 0, 0, 0, 0)∗,
+which yields that DMa = (0, 0, 0, 0, 0)∗, and then, that
+(λ − Aβ−1)DMa = (0, 0, 0, 0, 0)∗.
+Therefore, we need only to examine the terms Ki(t, ai, ∇a). That is, we have
+∇(Vi,maxai(1 −
+5
+�
+j=1
+aj)⃗ν(x)) = Vi,maxai(1 −
+5
+�
+j=1
+ai)∇(ν(x))
+for i = 2, 3.
+So, since by definition of ⃗ν, we have ∇(⃗ν(x)) ≤ 0, and then it follows that
+∇(Vi,maxai(1 −
+5
+�
+j=1
+aj)⃗ν(x)) ≥ 0
+by the fact that �5
+i=1 ai > 1. Hence
+If (ρ1, · · · , ρ4) = (a1, · · · , a4). Then, we have
+K1(t, a1, ∇a) = −(b1 + b2 + δ1)a1 + γ(t)a4 + b3a3 − F(a1, a3) − G(a1, a2) + b4a2
+= γ(t)a4 − (b1 + b2 + δ1)a1
+�
+��
+�
+C1
+1
++
+�
+b3 − αa→cξ(
+a3
+a1 + ε)a1
+�
+a3
+�
+��
+�
+C1
+2
++
+�
+b4 − αa→pξ(
+a2
+a1 + ε)a1
+�
+a2
+�
+��
+�
+C1
+3
+.
+Hence, C1
+1 ≤ 0 if a4 ≤ (b1 + b2 + δ1)a1. Moreover, b3 ≤ αa→cξ(
+a3
+a1+ε)a1 implies that C1
+2 ≤ 0, and
+C1
+3 ≤ 0 if b4 ≤ αa→pξ(
+a2
+a1+ε)a1. Furthermore,
+
+20
+K. KHALIL ET AL.
+K2(t, a2, ∇a) = −(b4 + c1 + δ2)a2 + b2a1 + c2a3 − ∇(V2,maxa2(1 −
+5
+�
+j=1
+aj)ν(x)) + G(a1, a2) − H(a2, a3)
+≤ −(b4 + c1 + δ2)a2 + b2a1 + αc→pξ(
+a2
+a3 + ε)a2a3
+�
+��
+�
+C2
+1
++ c2a3 + αa→pξ(
+a2
+a1 + ε)a1a2 − αp→cξ(
+a3
+a2 + ε)a2a3
+�
+��
+�
+C2
+2
+.
+So, C2
+1 ≤ 0 if (b4 + c1 + δ2)a2 ≥ b2a1 + αc→pξ(
+a2
+a3+ε)a2a3, and c2a3 + αa→pξ(
+a2
+a1+ε)a1 ≤
+αp→cξ(
+a3
+a2+ε)a3 implies that C2
+2 ≤ 0. Finally, we have
+K3(t, a3, ∇a) = −(b3 + c2 + δ3)a3 + b1a1 + c1a2 − ψ(t)a3 − ∇(V3,maxa3(1 −
+5
+�
+j=1
+aj)ν(x)) + F(a1, a3) + H(a2, a3)
+≤ −(b3 + c2 + δ3)a3 + b1a1 + αp→cξ(
+a3
+a2 + ε)a2a3
+�
+��
+�
+C3
+1
++ c1a2 + αa→cξ(
+a3
+a1 + ε)a1a3 − αc→pξ(
+a2
+a3 + ε)a2a3
+�
+��
+�
+C3
+2
+.
+That is, we obtain that (b3 + c2 + δ3)a3 ≥ b1a1 + αp→cξ(
+a3
+a2+ε)a2a3 implies C3
+1 ≤ 0 and c1a2 +
+αaξ(
+a3
+a1+ε)a1 ≤ αc→pξ(
+a2
+a3+ε)a2 implies C3
+2 ≤ 0. Then, (3.22) and so (3.21) holds too uniformly in
+x in a similar way as in the proof of Theorem 3.16. On the other hand, to show that 0 ≤ ρ5(t) ≤ a5
+for some a5 > 0 we use the variation of constant formula i.e., (3.14). In fact, we have
+ρ5(t) =
+� t
+0
+T5(t − s)ψ(s)ρ3(s)ds
+≤ T(u0)a3 := a5.
+This proves the result for a5 = T(u0)a3.
+□
+Remark 3.19. To verify conditions (3.20) and convince that they do not contradict each other,
+for simplicity and since the parameters ai > 0 for i = 1, · · · , 4, we may take for example ε = 0,
+a1 = m0a4 = m1a2 and a3 = a2 = ˜a where m0 > 0 is chosen such that 1 ≤ m0(b1 + b2 + δ1), and
+m1 > 0 is such that:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+b3 ≤
+m1
+1 + m2
+1
+αa→cβ;
+b4 ≤
+m1
+1 + m2
+1
+αa→p˜a
+m1b2 ≤ c1;
+αc→p
+2
+˜a ≤ b4;
+c2 +
+m1
+1 + m2
+1
+αa→p˜a ≤ αp→c
+2
+˜a
+m1b1 ≤ c2;
+αp→c
+2
+˜a ≤ b3;
+c1 +
+m1
+1 + m2
+1
+αa→c˜a ≤ αc→p
+2
+˜a.
+(3.23)
+
+ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL
+21
+Thus, for a special choice of the parameters m0, m1 and β, the condition (3.23) holds where we
+may have c1 < c2, b1 < b2, αc→p > αa→c and αa→p > αp→c corresponding to the case of a
+population with low risk culture studied in our simulation results in Section 4.
+Remark 3.20. Notice that the condition (3.20) is sufficient to obtain the pointwise subtangantial
+condition (3.22). That is, by construction, the different cases for ˜K(t, ai, 0) where the aj vanish
+for some (not all) j ̸= i hold true under the condition (4).
+Corollary 3.21. Equation (3.8) has a unique global bounded and positive solution u(·, u0) ∈
+C([0, T(u0)), X) ∩ C1((0, T(u0)), X).
+Proof. From Theorem 3.14 we obtain that equation (3.8) has a unique positive maximal solution
+u(·, u0) ∈ C([0, T(u0)), Xα)∩C1([0, T(u0)), Xβ) such that (3.15) holds. Therefore, from Theorem
+3.18 the solution u is bounded in Xα which yields from (3.15) that T(u0) = +∞.
+□
+4. Numerical Simulations
+In this section we present several numerical simulations for different scenarios for evacuation
+of populations in a catastrophic event. In order to highlight the behavior of the populations in
+such event, we study the case of no-back-to-daily population that corresponds to the case where
+φ(t) = 0
+for all t ≥ 0,
+and, for simplicity, we take γ(t) = 1 for all t ≥ 0. Here the population is supposed with low risk
+culture. All the parameters of the spatio-temporal APC model (2.14)-(2.16) are set as in Table
+1. In this case the system (3.8) is not time-dependent (it is autonomous) and all the results of
+Section 3 still hold.
+Parameters
+d1 = 0.001
+Diffusion
+d2 = 0.05
+d3 = 0.01
+d4 = 0.01
+V2,max = 0.3
+Advection
+V3,max = 0.2
+V1 = 0.2
+Speed at the
+V2 = 0.1
+boundary
+V3 = 0.3
+V4 = 0.2
+Parameters
+αa→c = 0.6
+Imitation
+αa→p = 0.7
+αp→c = 0.6
+αc→p = 0.7
+c1 = 0.1
+c2 = 0.4
+Intrinsic
+b1 = 0.1
+transitions
+b2 = 0.2
+b3 = 0.001
+b4 = 0.001
+Table 1. Table of parameter values. As explained in Section 2.3, we choose
+d1 < d3 = d4 < d2. Moreover, we are interested to consider a population with a
+low risk culture, so, for example, we take c2 > c1 and b2 > b1, as in [30].
+
+22
+K. KHALIL ET AL.
+We assume that the domain presents a target escape region (denoted by Γ2), thus the desired
+direction vector is defined by (2.12) where ⃗ν(x)|Ω = (νx1, νx2)∗ is given by:
+�
+�
+�
+�
+�
+�
+�
+�
+�
+νx1 = −
+x1 − xp
+1
+�
+(x1 − xp
+1)2 + (x2 − xp
+2)2 ,
+νx2 = −
+x2 − xp
+2
+�
+(x1 − xp
+1)2 + (x2 − xp
+2)2 ,
+(4.1)
+where (xp
+1, xp
+2) is a centered point in Γ2 localized out from Ω, see Figure 4.
+So, the desired
+direction in both situations of control and panic are supposed to be the same which is given by
+⃗ν(x). For more details about the desired direction of pedestrians, we refer to the works of Hughes
+[25] and also to the references [14, 43].
+Figure 4. The direction vector ⃗ν(x1, x2) given in (4.1) describing the desired
+direction of pedestrians to reach the point (xp
+1, xp
+2) which is located outside the
+domain Ω since the population looks to escape from the exit Γ2 towards this
+point.
+With regard to the diffusion process, we suppose that the crowd in a alert state hardly diffuses,
+since in a alert behavior, pedestrians are moving to look for information and to identify the
+hazard. Thus, the diffusion coefficient d1 should be considered small compared to the ones of the
+other populations. Moreover, here it is assumed that the most diffusive population is the panic
+population, since in panic behavior pedestrians move randomly in different directions, and thus
+d2 should be considered the largest diffusion coefficient.
+Thus, for the diffusivity coefficients of the control, daily, and return-to-day behaviors, we
+assume d1 < di < d2 for i = 3, 4, 5.
+In the following, we consider three different scenarios for the evacuation of a population whose
+aim is to escape by the unique exit denoted Γ2:
+Scenario 1: Evacuation of one centered cluster population. Here we consider a pedes-
+trian population located in a single group within the domain.
+Scenario 2: Evacuation of a population subdivided into three groups. Here we consider
+a population subdivided into three separated groups of pedestrians in different spatial localiza-
+tions.
+Scenario 3: Evacuation of a population with an obstacle in front of the exit. Here we
+
+1n0ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL
+23
+take into account the situation in which an obstacle is located between the exit and the popula-
+tion concentrated in a single group in the center of the domain.
+The initial distribution of population in the three scenarios are illustrated in Figure 5. Since
+(see Table 1) the population considered here is of low risk culture and then the dominated
+behavior is that of panic, for each scenario, we give simulation results describing the panic at
+different times t = 50, 100, 150, 200 and 250 respectively. This fact leads to compare between
+different scenarios (see Figure 6). Moreover, in order to highlight the time evolution of the other
+human behaviors, namely, alert, control and daily behavior, we present some simulations but
+only in Scenario 1, since the evolution is the same in other scenarios (see Figure 7).
+According to our numerical results, we notice that at the beginning of the simulation, for
+t = 50, there is a majority of daily and alert populations rather than population in a state
+of panic and control. This dynamic depends on the structure of the APC model, described in
+Section 2.1: at t = 0 everyone is in a daily behavior, then everyone goes through the state of
+alert before becoming panicked or controlled. Moreover, since the diffusion coefficients for ρ1 and
+ρ2 are low, the position of the populations is still more or less the same as the starting one, see
+(a1)-(a4) in Figure 7. For t = 250, for all scenarios, the dynamics of the APC is fully developed.
+Moreover, diffusion and advection phenomena are now visible: the whole population is in panic
+state (highest density of human behaviors) and control state, and they are concentrated near
+the exit, while the populations in alert state and in daily behavior are negligible, see (e1)-(e4) in
+Figure 7.
+Comparing the evacuation of population in panic in the three different scenarios, see in Figures
+6, (e1) for Scenario 1, (e2) for Scenario 2 and (e3) for Scenario 3 , one notices a strong congestion
+at the level of the exit in the first scenario. In the second scenario, splitting the initial population
+into three clusters reduces this congestion. Finally, the presence of an obstacle as in the third
+scenario further reduces congestion.
+Remark 4.1.
+(i) The more the color goes from light blue to dark red, the higher the popu-
+lation density is.
+(ii) For each figure, Scenario 1 is denoted by (a), Scenario 2 by (b) and Scenario 3 by (c),
+respectively.
+Remark 4.2. In the sequel, we give simulation results representing the other human behaviors,
+alert, control and daily behavior populations in order to show the time evolution of this popula-
+tions. But, we consider only Scenario 1, since the behavior of all populations is the same for the
+the other scenarios 1 and 2.
+
+24
+K. KHALIL ET AL.
+(a) Scenario 1
+(b) Scenario 2
+(c) Scenario 3
+Figure 5. Initial conditions: initial location of the population for each scenario:
+(a) the population is concentrated in a single group in the center of the domain; (b)
+the population is subdivided into three groups; (c) an obstacle is located between the
+exit and the population, which is concentrated in a single group within the domain.
+We recall that the exit is on the right of the domain, see Figure 4.
+
+1.0e+00
+0.8
+0.6
+0.4
+0.2
+0.0e+00LYANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL
+25
+t = 50
+(a1)
+t = 100
+(b1)
+t = 150
+(c1)
+t = 200
+(d1)
+t = 250
+(e1)
+Scenario 1
+(a2)
+(b2)
+(c2)
+(d2)
+(e2)
+Scenario 2
+(a3)
+(b3)
+(c3)
+(d3)
+(e3)
+Scenario 3
+Figure 6. Population in panic ρ2 over the three scenarios at the captures times
+t = 50, 100, 150, 200 and 250 respectively. Notice that, each row represent the (time)
+evolution of the population at each scenario: at time t = 50 of each scenario (column
+(a)), population in panic is small with respect to the initial population, since at the
+this time the majority of population is concentrated in alert, but over time, panicked
+ones become greater and greater, thanks to the APC dynamics, and the fact that the
+population has a low risk culture, in addition, the phenomena of diffusion and advection
+are now visible (the population in panic concentrated near the exit).
+Furthermore,
+by comparing captures at each column (b), (c), (d) and (e), we can observe that in
+Scenario 2 there is slightly less congestion near the exit than in Scenario 1, this is can
+be highlighted from time t = 100 to t = 250. Moreover, in Scenario 3, we observe that
+the congestion is less than in the previous cases. Indeed, the role of the obstacle is to
+facilitate the access to the exit.
+
+4.5e-01
+0.35
+0.25
+0.15
+0.05
+0.0e+004.5e-01
+0.35
+0.25
+0.15
+0.05
+0.0e+00C4.5e-01
+0.35
+0.25
+0.15
+0.05
+0.0e+0026
+K. KHALIL ET AL.
+t = 50
+(a1)
+t = 100
+(b1)
+t = 150
+(c1)
+t = 200
+(d1)
+t = 250
+(e1)
+Population in alert ρ1
+(a2)
+(b2)
+(c2)
+(d2)
+(e2)
+Population in panic ρ2
+(a3)
+(b3)
+(c3)
+(d3)
+(e3)
+Population in control ρ3
+(a4)
+(b4)
+(c4)
+(d4)
+(e4)
+Population in daily behavior ρ4
+Figure 7. In order to present the time evolution of the human behaviors (alert,
+control and daily behavior), we give simulations results of this populations in Scenario
+1 and with respect to captures times t = 50, 100, 150, 200 and 250 respectively. So, each
+row represent a human behavior. We notice that at the beginning of the simulation,
+there is a majority of daily and alert populations (row 1 and row 4 respectively) rather
+than population in a state of panic and control (row 2 and row 3 respectively). This
+dynamic depends on the structure of the APC model, described in Sections 2 and 3:
+at t = 0 everyone is in a daily behavior, then everyone goes through the state of alert
+before becoming panicked or controlled.
+
+4.5e-01
+0.35
+0.25
+0.15
+0.05
+0.0e+004.5e-01
+0.35
+0.25
+0.15
+0.05
+0.0e+004.5e-01
+0.35
+0.25
+0.15
+0.0e+004.5e-01
+0.35
+0.25
+0.15
+0.05
+0.0e+00ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL
+27
+5. Conclusion
+In this work, we introduce a new spatio-temporal APC (alert, panic and control) model describ-
+ing the evacuation of a population presented via different human behaviors during a catastrophic
+event. First, using the first-order macroscopic crowd theory, we derive a new spatio-temporal
+APC model. It is a system of advection-diffusion-reaction equations with nonlinear Robin bound-
+ary conditions. Then, using a semigroup approach and abstract evolution equations, we prove
+the local existence and a regularity result of the solutions of our model. Moreover, we establish
+the positivity of the solution and the existence of positively bounded invariant sets which leads
+to the global existence and the boundedness of the solutions. As far as we know, the theoretical
+results established in this work are new. Finally, to illustrate our results, we present different
+numerical simulations of the population evacuation, using three different scenarios.
+
+28
+K. KHALIL ET AL.
+Appendices
+A. Proofs of the preliminary results of Section 3
+Proof of Proposition 3.2. (i) It is well-known, from [32, Section 3.1.1], that the realization of
+the Laplacian operator di∆ on Lp(Ω) with Neumann boundary conditions generates a contraction
+holomorphic C0-semigroup (Ti(t))t≥0 of angle π/2, for i = 1, · · · , 5, see also [16, Sections 1.4] for
+a similar result. So A0 generates a contraction holomorphic C0-semigroup of angle π/2 on X as
+a diagonal matrix-valued operator.
+(ii) The compactness of the semigroup (T (t))t≥0 (i.e. when T(t) is a compact operator for each
+t > 0) follows from [16, Sections 1.6]. For the positivity of the semigroup (T (t))t≥0 on the Banach
+lattice X it suffices to prove that R(ω, A0) ∈ L(X) is a positive operator for ω ∈ ϱ(A0) large
+enough i.e., ϕ(x) ≥ 0 implies R(ω, A0)ϕ(x) ≥ 0 for all x ∈ Ω, see [20, Chapter VI, Section 1.8],
+see also the result of invariance under closed sets [35, Theorem 5.1]. This is equivalent to prove
+that ψ ∈ D(A0) := ker(L) which is the solution of ϕ(x) = (ω − A0)ψ(x) implies ψ is positive (it
+always exists, the question is about its positivity). That is, we have
+�
+(ω − A)ψ(x) = ϕ(x) ≥ 0,
+x ∈ Ω
+Lψ(x) = 0,
+x ∈ ∂Ω.
+Then, the result holds directly from the maximum principle.
+Proof of Proposition 3.5. Let 0 ≤ δ ≤ 1, the fact that the extension semigroup (Tδ−1(t))t≥0
+exists as a strongly continuous positive semigroup with generator (Aδ−1, D(Aδ−1) = Xδ) is due
+to [4]. The analycity and compactness of (Tδ−1(t))t≥0 follows from [21].
+Proof of Lemma 3.9. This holds by definition of the Dirichlet map D.
+Proof of Proposition 3.10. Since (λ − Aβ−1)D ∈ L(∂Xα, Xβ−1), it suffices to examine the
+operators K and M. Indeed, for K we show term by term. Let (ϕ1, · · · , ϕ5)∗ := ϕ, (υ1, · · · , υ5) :=
+υ ∈ Xα and R > 0 be such that ∥ϕ∥α, ∥υ∥α ≤ R. By construction, the functions F, G and H
+are (pointwisely) Lipschitzian in bounded sets i.e.,
+|F(ρ1, ρ3)(x) − F(υ1, υ3)(x)| ≤ L1
+R (|ρ1(x) − υ1(x)| + |ρ3(x) − υ3(x)|) ,
+x ∈ Ω
+|G(ρ1, ρ2)(x) − G(υ1, υ2)(x)| ≤ L2
+R (|ρ1(x) − υ1(x)| + |ρ2(x) − υ2(x)|) ,
+x ∈ Ω
+and
+|H(ρ2, ρ3)(x) − H(υ2, υ3)(x)| ≤ L3
+R (|ρ2(x) − υ2(x)| + |ρ3(x) − υ3(x)|) ,
+x ∈ Ω
+for some Li
+R ≥ 0, i = 1, 2, 3. So, by passing to the Lp-norm, and using the continuous embedding
+W 2α,p �→ C1(Ω) we obtain that
+∥F(ρ1, ρ3) − F(υ1, υ3)∥ ≤ |Ω|L1
+R (∥ρ1 − υ1∥0,α + ∥ρ3 − υ3∥0,α) ,
+(A.1)
+∥G(ρ1, ρ2) − G(υ1, υ2)∥ ≤ |Ω|L2
+R (∥ρ1 − υ1∥0,α + ∥ρ2 − υ2∥0,α) ,
+(A.2)
+
+29
+and
+∥H(ρ2, ρ3) − H(υ2, υ3)∥ ≤ |Ω|L3
+R (∥ρ2 − υ2∥0,α + ∥ρ3 − υ3∥0,α) .
+(A.3)
+Now, we show that the terms ∇ · (ϕ2v2(ϕ)ν)), ∇ · (ϕ3v3(ϕ)ν) are Lipschitzian in bounded sets in
+Xα. So, a straightforward calculus yields that
+ϕ2V2(ϕ)(x)−υ2V2(υ)(x) = V2,max
+�
+(1 − ˜ϕ(x)) [ϕ2(x) − υ2(x)] + υ2(x)
+5
+�
+i=1
+[ϕi(x) − υi(x)]
+�
+,
+x ∈ Ω.
+Furthermore, using the regularity of ϕ and υ and since the gradient operator ∇ is linear, we
+obtain that
+∇ · (ϕ2v2(ϕ)ν(x)) − ∇ · (υ2v2(υ)ν(x))
+=
+∇ · (ϕ2v2(ϕ)ν(x)) − ∇ · (ϕ2v2(υ)ν(x)) + ∇ · (ϕ2v2(υ)ν(x)) − ∇ · (υ2v2(υ)ν(x))
+=
+∇ϕ2(x) · (v2(ϕ)ν(x)) − ∇ϕ2(x) · (v2(υ)ν(x)) + ϕ2(x)∇ · (v2(ϕ)ν(x)) − ϕ2(x)∇ · (v2(υ)ν(x))
+�
+��
+�
+I1(x)
++ ∇ϕ2(x) · (v2(υ)ν(x)) − ∇υ2(x) · (v2(υ)ν(x)) + ϕ2(x)∇ · (v2(υ)ν(x)) − υ2(x)∇ · (v2(υ)ν(x))
+�
+��
+�
+I2(x)
+,
+x ∈ Ω.
+Then, we have
+| I1(x) | ≤ MV2
+�
+| ∇ϕ2(x) |
+5
+�
+i=1
+| ϕi(x) − υi(x) | + | ϕ2(x) |
+5
+�
+i=1
+| ∇(ϕi(x) − υi(x)) |
+�
+,
+x ∈ Ω,
+| I2(x) | ≤ MV2
+�
+| ∇(ϕ2(x) − υ2(x)) |
+5
+�
+i=1
+(1+ | ϕi(x) |)+ | ϕ2(x) − υ2(x) |
+5
+�
+i=1
+| ∇(υi(x) |
+�
+,
+x ∈ Ω,
+where MV2 = V2,max supx∈Ω |ν(x)|. Hence, we have
+| ∇ · (ϕ2v2(ϕ)ν(x)) − ∇ · (υ2v2(υ)ν(x)) |≤
+MV2
+�
+| ∇ϕ2(x) |
+5
+�
+i=1
+| ϕi(x) − υi(x) | + | ϕ2(x) |
+5
+�
+i=1
+| ∇(ϕi(x) − υi(x)) |
+�
++MV2
+�
+| ∇(ϕ2(x) − υ2(x)) |
+5
+�
+i=1
+(1+ | ϕi(x) |)+ | ϕ2(x) − υ2(x) |
+5
+�
+i=1
+| ∇(υi(x) |
+�
+,
+x ∈ Ω.
+Therefore, by passing to the norms, we have
+∥∇ · (ϕ2v2(ϕ)ν) − ∇ · (υ2v2(υ)ν)∥
+≤
+|Ω|V2,max ((1 + ∥ϕ∥) ∥∇(ϕ2 − υ2)∥∞ + ∥∇ϕν∥∥ϕ2 − υ2∥∞)
++|Ω|V2,max (∥ϕ − υ∥∥∇υ2ν∥∞ + ∥υ2∥∞∥∇(ϕ − υ)ν∥) .
+This leads to
+∥∇ · (ϕ2v2(ϕ)ν) − ∇ · (υ2v2(υ)ν∥
+≤
+|Ω|L4
+R∥ϕ − υ∥α
+(A.4)
+Similarly, we obtain that and
+∥∇ · ϕ3v3(ϕ)ν − ∇ · υ3v3(υ)ν∥
+≤
+|Ω|V3,max ((1 + ∥ϕ∥) ∥∇(ϕ3 − υ3)∥∞ + ∥∇ϕν∥∥ϕ3 − υ3∥∞)
++|Ω|V3,max (∥ϕ − υ∥∥∇υ3ν∥∞ + ∥υ3∥∞∥∇(ϕ − υ)ν∥) ,
+
+30
+and
+∥∇ · ϕ3v3(ϕ)ν − ∇ · υ3v3(υ)ν∥
+≤
+|Ω|L5
+R∥ϕ − υ∥α.
+(A.5)
+Now, we show that, there exists L9
+R ≥ 0 such that
+∥Mϕ − Mυ∥∂X ≤ L9
+R∥ϕ − υ∥α.
+To obtain that, we use the embedding W 1,p(∂Ω)5 �→ ∂X and we prove that
+∥Mϕ − Mυ∥1,p ≤ L9
+R∥ϕ − υ∥α.
+Indeed, we have,
+| ϕ2V2(ϕ)(x)−υ2V2(υ)(x) |≤ V2,max
+�
+(1+ | ˜ϕ(x) |) | ϕ2(x) − υ2(x) | + | υ2(x) |
+5
+�
+i=1
+| ϕi(x) − υi(x) |
+�
+, x ∈ Ω
+Hence, using the corresponding norms, we have
+|ϕ2V2(ϕ) − υ2V2(υ)|p ≤ |Ω|V2,max ((1 + ∥ϕ∥)∥ϕ2 − υ2∥∞ + ∥υ2∥∞∥ϕ − υ∥) ,
+That is using the embedding Xα �→ C1(Ω)5, we obtain that
+|ϕ2V2(ϕ) − υ2V2(υ)|p ≤ L6
+R∥ϕ − υ∥α.
+Arguing similarly, we obtain that
+|ϕ3V3(ϕ) − υ3V3(υ)|p ≤ L7
+R∥ϕ − υ∥α).
+In otherwise, by estimating, this time the gradient of the corresponding terms, we obtain that
+|∇ϕ2V2(ϕ) − ∇υ2V2(υ)|p ≤ L8
+R∥ϕ − υ∥α.
+Arguing similarly, we obtain that
+|∇ϕ3V3(ϕ) − ∇υ3V3(υ)|p ≤ L9
+R∥ϕ − υ∥α.
+Thus, Lemma 3.9 yields that,
+∥(ω − Aβ−1)D(Mϕ − Mυ)∥Xβ−1
+≤
+c∥Mϕ − Mυ∥∂X
+(A.6)
+≤
+cL10
+R ∥ϕ − υ∥α.
+(A.7)
+Furthermore, the fact that the functions γ and respectively φ are lγ-Lipschitzian and lφ-Lipschitzians
+with respect to t and 0 ≤ γ(t), φ(t) ≤ 1 yields
+∥γ(t)ϕ4 − γ(s)υ4∥ ≤ |Ω|lγ(∥ϕ4∥∞ | t − s | +∥ϕ4 − υ4∥α),
+t, s ≥ 0,
+(A.8)
+and through the same argument, we have
+∥φ(t)ϕ3 − φ(s)υ3∥ ≤ |Ω|lφ(∥ϕ3∥α | t − s | +∥ϕ3 − υ3∥α),
+t, s ≥ 0,
+(A.9)
+Consequently, from (A.1)-(A.9) we can find LR ≥ 0 such that
+∥ ˜K(t, ρ) − ˜K(s, υ)∥β−1 ≤ LR(| t − s | +∥ρ − υ∥α)
+for all t, s ≥ 0 and ϕ, υ ∈ Xα.
+This proves the result.
+
+31
+Proof of Lemma 3.12. Let us define
+v(t) =
+� t
+0
+Tβ−1(t − s)B(t)ds +
+� t
+0
+Tβ−1(t − s) [B(s) − B(t)] ds := v1(t) + v2(t),
+0 ≤ t ≤ T.
+It is clear that v1 ∈ C1((0, T], Xβ). So it suffices to prove that v2(t) ∈ Xβ and Aβ−1v2(·) is
+continuous. To this end, let ε > 0 and consider
+vε
+2(t) =
+�
+�
+�
+� t−ε
+0
+Tβ−1(t − s) [B(s) − B(t)] ds
+for t ≥ ε
+0
+for t < ε.
+So, the analyticity of the extension semigroup (Tβ−1(t))t≥0 on Xβ−1 yields that
+Tβ−1(t − s) [B(s) − B(t)] ∈ D(Aβ−1) = Xβ
+for 0 ≤ s ≤ t − ε.
+Then, vε
+2(t) ∈ D(Aβ−1). Moreover, vε
+2(t) converges to v2(t) as ε → 0. Since the operator Aβ−1 is
+closed, to conclude, we only need to show that Aβ−1vε
+2(t) =
+� t−ε
+0
+Aβ−1Tβ−1(t−s) [B(s) − B(t)] ds
+converges in Xβ−1 as ε → 0. That is, by the closedness of Aβ−1 we have
+Aβ−1vε
+2(t) −
+� t
+0
+Aβ−1Tβ−1(t − s) [B(s) − B(t)] ds =
+� t
+t−ε
+Aβ−1Tβ−1(t − s) [B(s) − B(t)] ds.
+Furthermore, the analytitcity of (Tβ−1(t))t≥0 yields that
+∥Aβ−1Tβ−1(t)∥L(Xβ−1) ≤ l0 t−1,
+t > 0
+(A.10)
+for some l0 ≥ 0. Hence, using (3.12)-(A.10), we conclude that
+∥Aβ−1vε
+2(t) −
+� t
+0
+Aβ−1Tβ−1(t − s) [B(s) − B(t)] ds∥β−1 ≤ ll0
+� ε
+0
+ση−1dσ → 0 as ε → 0.
+This proves that v2(t) ∈ Xβ and that Aβ−1v2(t) =
+� t
+0 Aβ−1Tβ−1(t − s) [B(s) − B(t)] ds for 0 <
+t ≤ T. So, it is clear that Aβ−1v is continuous.
+B. Tables of the functions and the parameters of the APC model
+In the sequel, we briefly recall the functions and the parameters of the APC model (2.1).
+Table 2. Functions in the APC model
+Functions
+Notation
+Beginning of the catastrophe
+γ(t)
+Return to a daily behavior
+φ(t)
+Imitation functions
+F, G, H
+
+32
+Table 3. Parameters of the APC model
+Parameters
+Notation
+Intrinsic evolution from alert to control
+b1
+Intrinsic evolution from alert to panic
+b2
+Intrinsic evolution from control to alert
+b3
+Intrinsic evolution from panic to alert
+b4
+Intrinsic evolution from panic to control
+c1
+Intrinsic evolution from control to panic
+c2
+Mortality rates for alert, panic and control populations respectively
+δ1, δ2, δ3
+Imitation from alert to control
+α13
+Imitation from alert to panic
+α12
+Imitation from panic to control
+α23
+Imitation from control to panic
+α32
+ACKNOWLEDGMENT
+This work has been supported by the French government, through the National Research
+Agency (ANR) under the Societal Challenge 9 “Freedom and security of Europe, its citizens
+and residents” with the reference number ANR- 17-CE39-0008, co-financed by French Defence
+Procurement Agency (DGA) and The General Secretariat for Defence and National Security
+(SGDSN).
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+Email address: kamal.khalil.00@gmail.com
+V. Lanza,
+LMAH, University of Le Havre Normandie, FR-CNRS-3335, ISCN, Le Havre 76600, France.
+Email address: valentina.lanza@univ-lehavre.fr
+D. Manceau,
+LMAH, University of Le Havre Normandie, FR-CNRS-3335, ISCN, Le Havre 76600, France.
+Email address: david.manceau@univ-lehavre.fr
+M-A. Alaoui,
+LMAH, University of Le Havre Normandie, FR-CNRS-3335, ISCN, Le Havre 76600, France.
+Email address: aziz.alaoui@univ-lehavre.fr
+D. Provitolo
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+bonne, France
+Email address: Damienne.provitolo@geoazur.unice.fr.
+
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+page_content=' KHALIL1, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' LANZA1, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' MANCEAU1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' AZIZ-ALAOUI1, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' PROVITOLO2  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In this work, using the theory of first-order macroscopic crowd models, we intro-  duce a compartmental advection-diffusion type model, describing the spatio-temporal dynamics  of a population in different behaviors (alert, panic and control behaviors) during a catastrophic  event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' For this model, we prove the local existence, regularity and uniqueness of a solution,  as well as the positivity and boundedness of this solution that allows the global existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Then, in order to study the spatio-temporal behavioural dynamics of a population during a  catastrophic event, we present several numerical simulations for different evacuation scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Contents  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Introduction  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  A spatio-temporal advection-diffusion model for human behaviors during a  catastrophic event  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  The temporal model of the human behavior of a population during a catastrophic  event  3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  First-order macroscopic pedestrian models  6  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  The spatio-temporal model corresponding to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1)  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Well-posedness, positivity and positively invariant sets of solutions for the  spatio-temporal model spatio-temporal model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='14), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='15) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='16)  10  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  The abstract formulation and the associated boundary value Cauchy problem  10  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Preliminary results  12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Local existence and regularity  14  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Positivity  16  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  L1-boundedness of the total population density  17  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Uniform boundedness and global existence  18  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Numerical Simulations  21  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Conclusion  27  Appendices  28  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Proofs of the preliminary results of Section 3  28  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Tables of the functions and the parameters of the APC model  31  ACKNOWLEDGMENT  32  References  32  1LMAH, University of Le Havre Normandie, FR-CNRS-3335, ISCN, Le Havre 76600, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  2000 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' 34G20, 47D06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' First-order macroscopic models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' pedestrians crowd models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' human behaviors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' mathe-  matical modeling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' panic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' semigroup theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  ∗Corresponding author: K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Khalil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' kamal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='khalil@univ-lehavre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='02520v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='AP]  6 Jan 2023  2  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' KHALIL ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Introduction  In the last decades the world has known some radical changes at almost all levels such as  technological developments, climatic changes and human evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Due to these factors, pop-  ulations (in both, developed or undeveloped countries) are aggressively facing many natural  disasters (tsunamis, earthquakes), technological events and terrorist attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In particular sit-  uations of sudden, unexpected and without alert disasters require high security measures and  strategies in order to predict and manage the movement and the behavior of a crowd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' During a  catastrophe, people may experience many different behaviors, but there is still few information  about the dynamics and the succession of such behaviors during the event, see [15, 36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Thus,  for the development of an efficient disaster management strategy, it becomes necessary not only  to take into account the disaster features but also the different psychological human behaviors  during the disaster event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Recently, several pedestrians crowd models have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Their main objective is to  predict the movements of a crowd in different environmental situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Mathematical models of  human crowds are mainly divided into two categories, namely, microscopic models and macro-  scopic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In the microscopic approach, individuals are treated separately as particles and the  evolution is determined using Newton’s second law and by considering physical and social forces  that describe the interaction among individuals as well as their interactions with the physical sur-  rounding (for more details we refer to the works of Helbing described in [33]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The macroscopic  approach, that we adopt in this paper, considers a crowd as a whole quantity, without recogniz-  ing individual differences, and it is therefore more suitable to the study of the movement of an  extremely large number of pedestrians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In particular, first-order macroscopic models introduced  by Hughes [25] (see also [14]) are based on a mass conservation equation and a density-velocity  closure equation with suitable boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Furthermore, several models are devoted  to study the dynamics of multiple pedestrian species in the context of macroscopic first-order  systems (see [8, 9, 13, 18, 19, 24, 25, 38, 40] and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In [25] Hughes studied  crowds with large density of multiple pedestrian classes with different walking characteristics and  destinations (identified by the index i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The system reads as  �  �  �  �  �  ∂tρi + ∇ · qi(ρ) = 0  in [0, T) × Ω,  qi · n = q0  i · n  in [0, T) × ∂Ω,  ρi(0) = ρ0  i  in Ω,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1)  for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' , N (N ≥ 2), where Ω ⊂ R2 is a bounded domain with smooth boundary ∂Ω and  qi(ρ) = ρiv(ρ)νi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The velocity is defined as v(ρ) = A − B˜ρ, thus it is linear with respect to the  total population ˜ρ and is the same for all populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' On the contrary, each population can  have a different direction of the movement νi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Finally, for each population i, q0  i is the outflow  from the boundary in the direction of the normal vector n and ρ0  i is the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Moreover,  in [8] authors studied a nonlinear drift-diffusion model with in-outflow boundary conditions for  the transport of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Notice that these systems are consisting of conservative equations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  with no reactions terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' A non-conservative system is proposed in [27] but the authors consider  ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL  3  only one population species and Neumann homogeneous boundary conditions, namely  �  �  �  �  �  ∂tρ + ∇ · q(ρ) = α(t, x)f(ρ) − β(t, x)ρ  in [0, T) × Ω,  ∇ρ · n = 0  in [0, T) × ∂Ω,  ρ(0) = ρ0  in Ω,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2)  where q(ρ) = −∇ρ + f(ρ)∇V (ρ) where f(ρ) = ρ(1 − ρ) and V : Rn −→ R is a potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In  all these papers, either there is no mention about the behaviors of the pedestrians or all the  pedestrians have the same emotional state (mainly panic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  In the recent years the RCP (Reflex-Panic-Control) [10] and the APC (Alert-Panic-Control)  [30] models have been proposed in order to describe the evolution in time of human behaviors  during a catastrophe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' They both consist in systems of nonlinear ODEs and have been devised  following the structure of the compartmental models in mathematical epidemiology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In [29] the  spatial dynamics has been integrated in the APC model, by considering the space as a discrete  variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In [11] the first system of reaction-diffusion equations describing a population with  several behaviors has been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  The aim of this present paper is to introduce a spatio-temporal macroscopic first-order non-  conservative pedestrians model describing the evolution of a population in a sudden, unexpected  and without warning signs disaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' For this purpose, starting from the nonlinear ODE APC  model proposed in [29, 30], we introduce a non-conservative first-order macroscopic model to  describe the spatio-temporal dynamics of a population exhibiting different behavioral states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Our model reads as �  �  �  �  �  ∂tρi + ∇ · qi(ρ) = fi(ρ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' , ρ5)  in [0, T) × Ω,  qi · n = q0  i · n  in [0, T) × ∂Ω,  ρi(0) = ρ0  i  in Ω,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='3)  with qi := −di∇ρi + ρi⃗vi(ρ), for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' , 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  We prove the well-posedness of this system and we present different numerical simulations for  several scenarios of evacuation of populations in emergency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  The organization of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In Section 3 we briefly present the APC model  equations, we recall the structure of a first-order macroscopic pedestrian model and we introduce  our advection-diffusion pedestrian APC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Section 3 is devoted to the mathematical analysis  of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Using the semigroup approach, we prove a local existence and uniqueness result and  the positivity of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Moreover, we prove the boundedness of this solution, under some  assumptions on the parameters, which yields the global existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Finally, Section 4 presents  numerical results about different scenarios of evacuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' A spatio-temporal advection-diffusion model for human behaviors during a  catastrophic event  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The temporal model of the human behavior of a population during a catastrophic  event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In this section, we briefly present a model describing the evolution of a population during  a sudden, rapid and unpredictable catastrophic event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  The model describes the evolution of  different human behaviors during a disaster, see [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Depending on the emotional charges  and their regulation, the different human reactions of a population in an emergency situation are  4  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' KHALIL ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Daily  population  density  Alert  population  density  Panic  population  density  Control  population  density  Back to Daily  population  density  Daily  population  density  b4  b3  c2  γ(t)  ϕ(t)  G  c1  b2  F  H  b1  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The transfer diagram of the APC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The arrows indicate the  transitions among the compartments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  here subdivided into three main categories, namely, alert, panic and control behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The APC  model considers the time evolution of the following five variables:  the density of individuals in an alert state ρ1(t),  the density of individuals that exhibit panic behaviors ρ2(t),  the density of individuals in a state of control ρ3(t),  the density of individuals in a daily behavior before the catastrophe ρ4(t),  the density of individuals in a behavior of everyday life after the disaster ρ5(t),  the density of individuals who die during the disaster ρ6(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  The corresponding model is given by the following nonlinear ODE system that matches with the  classical compartmental SIR models (t ≥ 0):  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  ρ′  1 =  −(b1 + b2 + δ1)ρ1 + γ(t)q + b3ρ3 + b4ρ2 − F(ρ1, ρ3) − G(ρ1, ρ2),  ρ′  2 =  −(b4 + c1 + δ2)ρ2 + b2ρ1 + c2ρ3 + G(ρ1, ρ2) − H(ρ2, ρ3),  ρ′  3 =  −(b3 + c2 + δ3)ρ3 + b1ρ1 + c1ρ5 − φ(t)ρ3 + F(ρ1, ρ3) + H(ρ2, ρ3),  ρ′  4 =  −γ(t)q,  ρ′  5 =  φ(t)ρ3,  ρ′  6 =  δ1ρ1 + δ2ρ2 + δ3ρ3,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1)  with the initial condition (ρ1, ρ2, ρ3, ρ4, ρ5, ρ6)(0) = (0, 0, 0, 1, 0, 0), since the population is sup-  posed to be in a daily behavior before the onset of the disaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The detailed description of all  the parameters is given in Table 3 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In particular, the transitions among the  compartments are of two types since they model two fundamental phenomena (see Figure 1):  The intrinsic transitions: They represent the behavioral transitions that depend on  the individual properties (past experiences, level of risk culture etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=') They are modeled  by linear terms in system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  The parameters of these transitions are bi > 0 for  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' , 4 and cj > 0 for j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL  5  2  4  6  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='0  ξ(w)  w  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The function ξ involved in the imitation terms: the imitation starts  very slowly, then it accelerates before slowing down and saturating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  The imitation phenomenon: Individuals have a tendency to imitate the behaviors of  people around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Here we follow the dominant behavior principle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' in the case of two  populations in interaction, the most adopted behavior is the most imitated one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Thus,  imitation between two behaviors depend on the ratio of the two populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Only alert  behaviors are not imitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1) the behavioral transitions due to imitation  are represented by nonlinear terms defined as:  F(ρ1, ρ3) := α13ξ  �  ρ3  ρ1 + ε  �  ρ1ρ3,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2)  G(ρ1, ρ2) := α12ξ  �  ρ2  ρ1 + ε  �  ρ1ρ2,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='3)  H(ρ2, ρ3) :=  �  α23ξ  �  ρ3  ρ2 + ε  �  − α32ξ  �  ρ2  ρ3 + ε  ��  ρ2ρ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='4)  The parameter 0 < ε ≪ 1 is considered here to avoid singularities, and the following  function  ξ(w) :=  w2  1 + w2 , w ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='5)  takes into account the dominant behavior principle, that is the fact that the rate of  imitation depends on the ratio of the corresponding populations (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  For  example, if we consider the imitation phenomenon from alert to panic, we remark that  if  ρ2  ρ1+ε < 1 is small, then ξ  �  ρ2  ρ1+ε  �  is almost equal to zero, so the imitation is weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Conversely, if  ρ2  ρ1+ε ≫ 1 is large, it means that we have a majority of individuals in panic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  In this case, ξ  �  ρ2  ρ1+ε  �  goes to 1 and alerted individuals would imitate the panic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  The same situation holds for the other imitation transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Finally, function γ describes the transition from the daily to the alert behavior at the beginning  of the catastrophic event, while function φ represents the transition from a control behavior to  an everyday life behavior at the end of the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' It is assumed that they are time-dependent  functions that depend on the nature of the disaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In [29] the authors consider the functions  6  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' KHALIL ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  2  4  6  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='0  t  γ(t)  20  40  60  80  100  120  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='0  t  (t)  ϕ  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Example of the functions γ and φ, which describe the transition from  the daily to the alert behaviors, and from the control to everyday life behaviors:  γ(t) = ζ(t, 1, 3) and φ(t) = ζ(t, 20, 70) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  φ, γ : [0, ∞) −→ [0, 1] defined as  φ(t) := ζ(t, τ0, τ1)  for  τ0 < τ1,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='6)  and  γ(t) := ζ(t, σ0, σ1)  for  σ0 < σ1,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='7)  where  ζ(t, z0, z1) :=  �  �  �  �  �  �  �  �  �  0  if t < z0,  1  2 − 1  2 cos  � t − z0  z1 − z0  π  �  if z0 ≤ t ≤ z1,  1  elswhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8)  Here τ0 is the time at which the daily population starts to be impacted by the event, and τ1 is  the time at which the total daily population becomes alert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Additionally, σ0 represents the time  at which the first individuals in a control state can go back to a pseudo-daily behavior, while σ1  is the time where this transition is highest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' See Figure 3 for an example of these two functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Summing up the equations of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1), we notice that  6  �  i=1  ρi(t) = 1 (the total population density), ∀t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='9)  For this reason we only need to solve the system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1) without considering the victims density ρ6,  since the last equation is a linear combination of the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' First-order macroscopic pedestrian models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In this section we give the mathematical  framework about first-order macroscopic pedestrian models [14, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' We recall that the continuity  principle indicates that the variation of the density ρ of a certain quantity, in Ω ⊂ R2, is given  by the balance of the flow q of this quantity across the boundary ∂Ω and the amount of quantity  produced or removed inside the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Mathematically, this can be expressed as follows:  ∂tρ + ∇ · q = S(t, x, ρ),  t ≥ 0, x ∈ Ω,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='10)  ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL  7  where S is the source term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' To be more precise, the flux q can be advective (qadv), proportional  to a velocity ⃗v(ρ) where ρ is the transported scalar quantity, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' qadv = ρ⃗v(ρ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' but it may also  be diffusive (qdiff), that is, it consists of a diffusion term qdiff = −d∇ρ corresponding to the  transportation of the scalar quantity according to its gradient, where d is the constant diffusion  rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Thus, we have  q = qadv + qdiff := −d∇ρ + ρ⃗v(ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Moreover, the source term S can be divided into a pure and a reaction source terms:  S = Sp + Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  The pure source term denoted by Sp represents the self-creation/destruction rate inside the  domain (using population dynamics terminology, it corresponds to the birth and death terms for  example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' This term will be denoted by Sp(t, x, ρ) = g(t, x, ρ) where g is a given linear function  with respect to ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The reaction term Sr describes the creation/destruction processes as a reaction  to this quantity (corresponding to the reaction and interaction terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' This term will be denoted  by Sr(t, x, ρ) = f(t, x, ρ), where f is a given nonlinear function with respect to ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Moreover, the  associated boundary conditions are expressed as follows  q · n = q0 · n,  where n is the outward unit normal vector to ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  These boundary conditions mean that the flux crossing the boundary part ∂Ω in the direction  of the normal vector n is given by an observed flux q0 in the same direction n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' According to that,  the complete first-order equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='10) in its non-conservative form is given by:  �  �  �  ∂tρ  = d∆ρ − ∇ · (ρ⃗v(ρ)) + g(t, x, ρ) + f(t, x, ρ),  t ≥ 0, in Ω,  q · n  = q0 · n,  t ≥ 0, on ∂Ω,  ρ(0)  = ρ0,  in Ω,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='11)  where ρ0 is the initial condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' For more details we refer to [3] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The spatio-temporal model corresponding to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In this section, we present our  advection-diffusion APC (Alert-Panic-Control) model using the first-order continuity equations  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='11) presented in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Notice that system (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='11) can be generalized to the case where  several populations are in interaction (as is the case for the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1), where each population  can for example have different directions of movement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Moreover, our new model takes into  account the transitions and the imitations characteristics which are not considered in the conser-  vative system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Let Ω ⊂ R2 be a non-empty bounded domain with Lipschitz boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Consider the local  population densities ρi : [0, +∞) × Ω −→ R for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' , 5 where  ρ1(t, x) is the local density of individuals in the alert situation,  ρ2(t, x) is the local density of individuals in the panic situation,  ρ3(t, x) is the local density of individuals in the control situation,  ρ4(t, x) is the local density of individuals in the daily behaviour before the disaster,  ρ5(t, x) is the local density of individuals corresponding to the daily situation after the  disaster,  8  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' KHALIL ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  and let ρ be given by  ρ := (ρ1, ρ2, ρ3, ρ4, ρ5)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  As for the model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='11) we will consider an advective flux and a diffusive flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The advection  phenomenon models the movement of a population in a chosen direction, typically to escape the  Ω domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' It is therefore natural to incorporate advection terms in our case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  We set  qi,adv := ρi⃗vi(ρ),  the advective flux, where ρ⃗vi(ρ), i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' , 5 is the corresponding velocity for each population  of density ρi, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' , 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The alert population corresponds to the set of behaviors such as  information seeking and hazard identification, and therefore here it is assumed that it cannot  undergo the advection phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Thus, it is assumed that only the populations in a situation  of panic or control are concerned by the advection phenomenon, hence  ⃗vi(ρ) = 0,  for i = 1, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  For i = 2, 3, we assume that the velocities ⃗vi satisfy  ⃗vi(ρ) = Vi(ρ)⃗ν,  for i = 2, 3,  where the vector ⃗ν : Ω → R2 that represents the direction of the movement, and satisfies certain  conditions that will specified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Moreover, V2, V3 are the scalar speed-density functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Several different type of speed-density functions are used in the literature (see a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' [7, 14, 41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Here, we choose a linear dependence:  V2(ρ) = V2,max (1 − ˜ρ)  and  V3(ρ) = V3,max (1 − ˜ρ) ,  where V2,max, V3,max are two positive constants and  ˜ρ :=  5  �  i=1  ρi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Similar assumptions on the panic and the control maximum speeds are used in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Moreover, each population diffuses in the spatial domain Ω according to the density gradient  qi,diff := −di∇ρi,  with constant diffusion coefficients di for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' , 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' It is assumed that the whole crowd diffuses  with different diffusion coefficients depending on the type of behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  With these assumptions and notations, the associated fluxes are given by  qi := −di∇ρi + ρi⃗vi(ρ),  for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' , 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  The source (pure and reaction) terms correspond to the intrinsic transitions and the imitation  terms described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Moreover, we divide the boundary ∂Ω of Ω in two parts, each of them corresponding to different  boundary conditions:  ∂Ω := Γ1 ∪ Γ2  with  Γ1 ∩ Γ2 = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL  9  Here Γ1 corresponds to the part of boundary that could not be crossed by the population, while Γ2  corresponds to an escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' We define the observed fluxes q0  i on the boundary ∂Ω by  q0  i (ρi) :=  �  0  on Γ1  −ρivi,out ⃗ν  on Γ2  where vi,out ≥ 0 is the constant speed at the boundary Γ2 and ⃗ν is the direction of the movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  This means that each population cannot cross along Γ1 and cross the escape Γ2 with speed vi,out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  We assume that the function ⃗ν satisfies:  ⃗ν(x) =  �  �  �  �  �  (0, 0)∗,  x ∈ Γ1  (νx1(x), νx2(x))∗,  x ∈ Ω  n(x),  x ∈ Γ2  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='12)  where n is the unit normal vector at the boundary part Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' This choice of vector ⃗ν means that,  at the part of the boundary where pedestrians cannot cross, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Γ1, the direction of movement  vanishes, but at the target exit, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=', Γ2, the pedestrians cross this part of the boundary in a  direction parallel to the normal vector n, while the desired direction of motion inside the domain  Ω is given in a suitable way that depends on the regularity of Ω, and it satisfies the following  assumption:  ⃗ν|Ω ∈ W 1,∞(Ω, R2),  such that ∇(ν(x)) ≤ 0 for all x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' For example, we can take ⃗ν|Ω to be normalized vectors  between any point x ∈ Ω and a centered (fixed) target point that lies outside Ω, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Thus,  the observed fluxes on the boundary ∂Ω are given by  q0  i (ρi) = ρivi,out ⃗ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  We assume that at the beginning t = 0, the whole population is in a daily behavior, so we consider  the following initial conditions: ρ(t = 0, 0) = ρ0 where ρ0 is given by  ρ0 := (0, 0, 0, θ, 0)∗  on Ω,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='13)  and θ : Ω → [0, ∞) is such that, the integral exists, and that  �  Ω  θ(x)dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Thus, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='11), we obtain the following system:  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  ∂tρ1 =  d1∆ρ1 − (b1 + b2 + δ1)ρ1 + γ(t)ρ4 + b3ρ3 + b4ρ2  −F(ρ1, ρ3) − G(ρ1, ρ2)  in Ω, t ≥ 0,  ∂tρ2 =  d2∆ρ2 − (b4 + c1 + δ2)ρ2 + b2ρ1 + c2ρ3  −∇ · (ρ2⃗v2(ρ)) + G(ρ1, ρ2) − H(ρ2, ρ3)  in Ω, t ≥ 0,  ∂tρ3 =  d3∆ρ3 − (b3 + c2 + δ3)ρ3 + b1ρ1 + c1ρ2  −φ(t)ρ3 − ∇ · (ρ3⃗v3(ρ)) + F(ρ1, ρ3) + H(ρ2, ρ3)  in Ω, t ≥ 0,  ∂tρ4 =  d4∆ρ4 − γ(t)ρ4  in Ω, t ≥ 0  ∂tρ5 =  d5∆ρ5 + φ(t)ρ3  in Ω, t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='14)  with the boundary conditions  di∇ρi · n = (ρi⃗vi(ρ) − ρivi,out⃗ν) · n  on ∂Ω, t ≥ 0,  i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' 5,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='15)  10  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' KHALIL ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  or more explicitly, using the definition of ⃗ν:  �  di∇ρi · n = 0  on Γ1 for i = 1, · · · , 5,  di∇ρi · n = −viρi + ρiVi(ρ)  on Γ2 for i = 1, · · · , 5,  and the initial condition  ρ(t = 0, ·) = ρ0  in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='16)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Well-posedness, positivity and positively invariant sets of solutions for the  spatio-temporal model spatio-temporal model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='14), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='15) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='16)  In this section, we prove the well-posedness of our spatio-temporal APC model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='14), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='15)  and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='16) introduced in Section .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Then we establish the positivity of the solutions, the L1-  boundedness of the sum of the population densities and the boundedness of the solution which  gives the global existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The abstract formulation and the associated boundary value Cauchy problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  To study the existence and uniqueness of solutions to the system spatio-temporal APC model  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='14)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='16) we use the abstract formulation and semigroup theory [20, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In order to do that,  for p > 2, we define the Banach space X := Lp(Ω)5, the product of the Lebesgue spaces of order  p, equipped with the following norm  ∥ϕ := (ϕ1, · · · , ϕ5)∗∥ :=  5  �  i=1  ∥ϕi∥,  where ∥ · ∥ is the usual norm in Lp(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' It is clear that X is a Banach lattice, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=',  |ϕi(x)| ≤ |ψi(x)| for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' x ∈ Ω for all i = 1, · · · , 5 implies that ∥ϕ∥ ≤ ∥ψ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Moreover, we define the linear closed operator (A, D(A)) on X by  �  A = diag(d1∆, · · · , d5∆)  D(A) = W 2,p(Ω)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1)  The nonlinear function K : [0, ∞) × Xα −→ X is defined by  K(t, ϕ) =  �  �  �  �  �  �  −(b1 + b2 + δ1)ϕ1 + γ(t)ϕ4 + b3ϕ3 + b4ϕ2 − F(ϕ1, ϕ3) − G(ϕ1, ϕ2)  −(b4 + c1 + δ2)ϕ2 + b2ϕ1 + c2ϕ3 − ∇ · (ϕ2vp(ϕ)ν) + G(ϕ1, ϕ2) − H(ϕ2, ϕ3)  −(b3 + c2 + δ3)ϕ3 + b1ϕ1 + c1ϕ2 − φ(t)ϕ3 − ∇ · (ϕ3v3(ϕ)ν) + F(ϕ1, ϕ3) + H(ϕ2, ϕ3)  −γ(t)ϕ4  φ(t)ϕ3  �  �  �  �  �  �  ,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2)  where we take  �  �  �  �  �  �  �  �  �  �  �  �  �  �  �  K1(t, ϕ1, ∇ϕ) = −(b1 + b2 + δ1)ϕ1 + γ(t)ϕ4 + b3ϕ3 + b4ϕ2 − F(ϕ1, ϕ3) − G(ϕ1, ϕ2),  K2(t, ϕ2, ∇ϕ) = −(b4 + c1 + δ2)ϕ2 + b2ϕ1 + c2ϕ3 − ∇ · (ϕ2vp(ϕ)ν) + G(ϕ1, ϕ2) − H(ϕ2, ϕ3),  K3(t, ϕ3, ∇ϕ) = −(b3 + c2 + δ3)ϕ3 + b1ϕ1 + c1ϕ2 − φ(t)ϕ3 − ∇ · (ϕ3v3(ϕ)ν) + F(ϕ1, ϕ3) + H(ϕ2, ϕ3),  K4(t, ϕ4, ∇ϕ) = −γ(t)ϕ4,  K5(t, ϕ5, ∇ϕ) = φ(t)ϕ3,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='3)  ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL  11  and Xα := {ϕ ∈ W 2α,p(Ω)5 : di∂nϕi|∂Ω = 0} for some (fixed) α ∈ (1/p + 1/2, 1) equipped with  the norm ∥ · ∥0,α := ∥ · ∥ + ∥∇ · ∥ + [·]ζ where  [ϕ]ζ :=  ��  Ω×Ω  |ϕ(x) − ϕ(y)|p  |x − y|2+pζ  dxdy  �1/p  ,  ζ = 2α − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Hence, ∥ϕ∥α := �5  i=1 ∥ϕi∥0,α defines a norm on Xα which make it a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Then from  the Sobolev embedding, we have  Xα �→ C1(Ω)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Define the boundary space ∂X := W 1−1/p,p(∂Ω)5 which is equipped with the norm  ∥ϕ∥∂X :=  5  �  i=1  |ϕi|p  where  |ϕ|p =  ��  ∂Ω  |ϕ(x)|pdx +  �  ∂Ω×∂Ω  |ϕ(x) − ϕ(y)|p  |x − y|p  dxdy  �1/p  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Since 1 − 2/p > 0, we obtain the continuous embedding ∂X �→ C(∂Ω)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Moreover, we define the  boundary operator L : Z −→ ∂X by  Lϕ = (d1∂nϕ1, · · · , d5∂nϕ5)∗  on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='4)  The nonlinear boundary term M : Xα −→ ∂X is given by  Mϕ =  �  (0, 0, 0, 0, 0)∗  on Γ1  (−v1ρ1, −v2ρ2 + ϕ2V2(ϕ), −v3ρ3 + ϕ3V3(ϕ), −v4ρ4, −v5ρ5)∗  on Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='5)  Let the Banach space Z := D(A) equipped with its usual norm, so the continuous embedding  Z �→ Xα holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Hence the linear operator A : Z −→ X is bounded and L : Z −→ ∂X is bounded  and surjective (see [1, 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The initial conditions are given by the following vector  ρ0 = (0, 0, 0, θ, 0)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='6)  Now we can write our boundary evolution system as  �  �  �  �  �  ut(t) =Au(t) + K(t, u(t)),  t ≥ 0,  Lu(t) =M(u(t)),  t ≥ 0,  u(0) =u0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='7)  where  u(t) := (ρ1(t, ·), ρ2(t, ·), ρ3(t, ·), ρ4(t, ·), ρ5(t, ·))∗,  and  u0 = ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  12  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' KHALIL ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In this section, we give our preliminary results, the proofs are given  for the sake of completeness for a curious reader, and will be available in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In the  following, we define A0 := A| ker(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' We recall that X is a Banach lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (i) A vector ϕ = (ϕ1, · · · , ϕ5)∗ ∈ X is said to be positive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=', ϕ(x) ≥ 0, if and only if, ϕi(x) ≥ 0  for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' x ∈ Ω for all i = 1, · · · , 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' So that, X+ denotes the positive cone of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (ii) A bounded operator T , in the Banach lattice X, is said to be positive if and only if, for every  ϕ ∈ X, ϕ(x) ≥ 0 implies T ϕ(x) ≥ 0 for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (iii) A semigroup (T (t))t≥0, in the Banach lattice X, is said to be positive if and only if, for  every ϕ ∈ X, ϕ(x) ≥ 0 implies T (t)ϕ(x) ≥ 0 for all t ≥ 0 for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The following assertions hold:  (i) The closed operator A0 generates a contraction holomorphic C0-semigroup (T (t))t≥0 on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (ii) The semigroup (T (t))t≥0 generated by A0 := A| ker(L) is compact and positive (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=', T (t)X+ ⊂  X+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Moreover, the semigroup (T (t))t≥0 is given by the following matrix-valued operators  T (t) = diag(T1(t), · · · , T5(t))∗,  t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Now, we present the inter- and extrapolation spaces associated to the generator A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' We define  on X the norm ∥x∥−1 = ∥R(λ, A0)x∥, for x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Then the completion of (X, ∥ · ∥−1) is called  the extrapolation space of X associated to A0 and will be denoted by X−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' This means that A0  has a unique extension A−1 : D(A−1) = X −→ X−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Since for every t ≥ 0, (T (t))t≥0 commutes  with the operator resolvent R(λ, A0), the extensions of (T (t))t≥0 to X−1 exists and defines an  analytic semigroup (T−1(t))t≥0 which is generated by A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Let α ∈ (0, 1), we define the following  interpolated extrapolation spaces by:  Xα−1 = X  ∥·∥α−1,  where  ∥x∥α−1 := sup  ω>0  ∥(ωαR(ω, A−1 − λ)x∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Then, we have the following continuous embeddings:  D(A0) �→ Xα �→ Xβ �→ X  X �→ Xα−1 �→ Xβ−1 �→ X−1,  for all 0 < β < α < 1, where D(A0) is equipped with the graph norm that makes it a Banach  space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' (i) Note that the extrapolated spaces introduced here do not depend on any choice  of λ ∈ ρ(A0), which means that, any other choice of λ gives the same extrapolated space with an  equivalent norm, this holds by virtue of the resolvent equation, see [4, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' We recall that, the  spectrum of A0 satisfies σ(A0) ⊂ (−∞, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' So that C \\ (−∞, 0] ⊂ ϱ(A0), see [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' It follows from [42, Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1], that the spaces Xα for 0 < α <  1 introduced here coincide with real interpolation spaces (of order α) between D(A0) and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Moreover, the embedding Z �→ Xα also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' For each 0 ≤ δ ≤ 1, (Tδ−1(t))t≥0 is the unique extension semigroup of  (T (t))t≥0 with the associated generator Aδ−1 satisfying D(Aδ−1) = Xδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Moreover, the semigroup  (Tδ−1(t))t≥0 inherits all the properties of (T (t))t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' That is, (Tδ−1(t))t≥0 is strongly continuous,  analytic, compact and positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL  13  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' We define the positive cone of Xα by  X+  α = X+ ∩ Xα,  Similarly, the positive cone of Xβ−1 is defined by  X+  β−1 = X+ ∩ Xβ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Let 0 < ai < +∞ for i = 1, · · · , 5 and Λa ⊂ R5 be such that  Λa := Π5  i=1[0, ai],  A function ϕ ∈ X belongs to Λa if and only if 0 ≤ ϕi(x) ≤ ai for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' x ∈ Ω, i = 1, · · · , 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Moreover,  Λ+∞ := Π4  i=1[0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  In particular, if only a5 = +∞, we give  Λa,+∞ := Π4  i=1[0, ai] × [0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Using Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='6, we may observe that  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The bounded positive cones of X, Xβ−1 and Xα are defined respectively by:  XΛa = {ϕ ∈ X : ϕ(x) ∈ Λa, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' x ∈ Ω},  XΛa  β−1 = {ϕ ∈ Xβ−1 : ϕ(x) ∈ Λa, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' x ∈ Ω},  and  XΛa  α  = {ϕ ∈ Xα : ϕ(x) ∈ Λa, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' x ∈ Ω}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' (i) Note that, since Xα �→ C1(Ω)5, the "a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' x ∈ Ω" in the definition of X+  α (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  of XΛa  α ) becomes "for all x ∈ Ω".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (ii) By definition, we have X+ = XΛ+∞, X+  α = XΛ+∞  α  and X+  β−1 = XΛ+∞  β−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  In order to study the boundary evolution equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='14), we proceed as in [2] and later [17]  (see [2, Section 12] and also [17, Section 4]), namely, the nonlinear boundary evolution problem  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='14) admits a solution, if and only if, the following semilinear Cauchy problem admits a solution  �  ut(t) =Aβ−1u(t) + ˜K(t, u(t)),  t ≥ 0,  u(0) =u0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8)  where ˜K := K + (λ − Aβ−1)DM with D is the Dirichlet map associated to the operator (λ − A)  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' v = Dw is the unique solution of the abstract boundary value problem  �(λ − A)v = 0  Lv = w  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='9)  for each w ∈ ∂X for some λ ∈ ϱ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In fact, let u ∈ X and w ∈ ∂X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Then, the equation  �(λ − A)v = u  Lv = w  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='10)  admits the solution v = R(λ, A)u + Dw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' This solution is unique in Z since λ − A is injective on  D(A0) := ker(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  14  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' KHALIL ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  This approach of studying the boundary evolution equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='7) by equivalently studying the  Cauchy problem (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8) was first introduced separately in [2, 21], and was later perfected in [17]  and others (see the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The conditions under which this approach is used are  cited in all the references [2, 17, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' However, for the sake of completeness, we will cite these  conditions as follows:  (C1) There exists a new norm | · |m which is finer than the norm of X, such that the space  Z := (D(A), | · |m) is complete, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e, Z is continuously embedded in X and A ∈ L(Z, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (C2) The restriction operator A0 = A|ker(L) generates a strongly continuous analytic semigroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (C3) The operator L : Z −→ ∂X is bounded and surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (C4) Z is continuously embedded in Xα, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=', Z �→ Xα for some 0 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (C5) The functions K : [0, +∞)×Xα −→ X and M : [0, +∞)×Xα −→ ∂X are locally integrable  in the first variable and continuous with respect to the second one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  We mention that all the conditions (C1)-(C5) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The operator (λ − A−1)D is bounded from ∂X to X−1 with its norm denoted by  ∥(λ − A−1)D∥∂X→X−1 ≤ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The function ˜K : [0, +∞) × Xα −→ Xβ−1 is Lipschitzian in bounded sets  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=', for all R > 0 there exists LR ≥ 0 such that  ∥ ˜K(t, ρ) − ˜K(s, υ)∥β−1 ≤ LR(| t − s | +∥ρ − υ∥α)  for all ρ, υ ∈ B(0, R) for all t, s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='11)  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Notice that if 1/p + 1/2 < β < α < 1, then Xβ �→ C1(Ω)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' So, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='11) holds also  in Xβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  The following Lemma is needed to show that, even equation in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Let 0 < β < 1 and B : [0, T] −→ Xβ−1 such that there exist 0 < η ≤ 1 and l ≥ 0  satisfying  ∥B(t) − B(s)∥β−1 ≤ l|t − s|η,  t, s ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='12)  Then,  v(t) =  � t  0  Tβ−1(t − s)B(s)ds ∈ D(Aβ−1) = Xβ  for 0 ≤ t ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Moreover, v ∈ C1((0, T], Xβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' [2, 17] Let u0 ∈ Xα and T > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' By a solution to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8), we mean  a function u ∈ C([0, T], Xα) ∩ C1([0, T], Xβ−1), such that u(t) ∈ Xβ for 0 ≤ t ≤ T and such  that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8) is pointwisely satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In particular, this solution must satisfy the following integral  formula:  u(t) = T (t)u0 +  � t  0  Tβ−1(t − s) (K(s, u(s)) + (ω − Aβ−1)DM(u(s))) ds,  t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='13)  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Local existence and regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In this Section, using the our we prove the local existence,  uniqueness and regularity of solutions to equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8) which yields the local well-posedness for  the model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='14)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL  15  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='14 (Local existence and regularity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' For each u0 ∈ Xα there exist a maximal time  T(u0) > 0 and a unique maximal solution u(·) := u(·, u0) ∈ C([0, T(u0)), Xα)∩C1([0, T(u0)), Xβ)  of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8) such that  u(t) = T (t)u0 +  � t  0  Tβ−1(t − s) (K(s, u(s)) + (ω − Aβ−1)DM(u(s)))  �  ��  �  = ˜K(s,u(s))  ds,  t ∈ [0, T(u0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='14)  Moreover the solution u satisfies the following blow-up property:  T(u0) = +∞  or  lim sup  t→T (u0)− ∥u(t)∥ = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='15)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Let u0 ∈ Xα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' So, using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='10, it yields from [32, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1] (by taking Xβ−1  instead of X and Tβ−1 instead of T ), that there exist T > 0 (small enough) and a unique solution  u ∈ C([0, T], Xα) ∩ C1([0, T], Xβ−1) of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8) satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Note that, in our case,  A0 is densely defined in X, so that the continuity at t = 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Now, to conclude, we use the  integral formula of our solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='14) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='12 to prove that u ∈ C1([0, T], Xβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  First, note that u0 ∈ Xα �→ Xβ implies that T (t)u0 ∈ C1([0, T], Xβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Then, it suffices to prove  that  t �→ v(t) =  � t  0  Tβ−1(t − s) ˜K(s, u(s))ds ∈ C1([0, T], Xβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Remark that, u is Hölder continuous in Xβ−1 (since it is C1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e, there exist ˜l ≥ 0 and 0 < ϑ ≤ 1,  such that  ∥u(t) − u(s)∥β−1 ≤ ˜l|t − s|ϑ,  t, s ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='16)  Moreover, since u0 ∈ Xα �→ Xβ, it yields, using Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='11, that  u ∈ C([0, T], Xβ) ∩ C1([0, T], Xβ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Hence, u is bounded in Xβ, since it is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Furthermore, using the reiteration theorem,  we obtain that Xβ = (Xα, Xβ−1)˜θ, with 0 < ˜θ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' That is,  ∥u(t) − u(s)∥β ≤ c(α, β)∥u(t) − u(s)∥1−˜θ  α  ∥u(t) − u(s)∥  ˜θ  1−β,  t, s ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Therefore, we have  ∥u(t) − u(s)∥β ≤ ˜c(α, β)|t − s|  ˜θϑ,  t, s ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Note that u is bounded in Xβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Hence, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='16) and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='11 (using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='10 for Xβ  instead of Xα), we obtain that  ∥ ˜K(t, u(t)) − ˜K(s, u(s))∥β−1 ≤ LR(| t − s | +∥u(t) − u(s)∥β)  ≤ ˜  LR(| t − s | +|t − s|  ˜θϑ),  t, s ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  This proves that ˜K(·, u(·)) is Hölder continuous in Xβ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Then, we conclude using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='12,  by taking B(·) = ˜K(·, u(·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Henceforth, we can argue similarly as in [32, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8] to prove that the solution u can  be extended continuously to a maximal interval [0, T(u0)), where T(u0) > 0 is the maximal time,  such that the property (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='15) is also satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  □  16  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' KHALIL ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' We mention that, in [32, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2], the result of existence of a solution  (without regularity) of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8) uses the fractional power space D(Aα  0 ) as an intermediate  space Xα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' However, this fact does not affect our existence result since the proof can be given in  a similar way for any intermediate Banach space, see the proof of [32, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Positivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' This section aims to show the positivity of the solution of our model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='14)-  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='16) obtained in Section (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Results and the proofs of the present section are inspired from  those in [28, Section 2], see also [12, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='3] in the case of homogeneous boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Let ϕ ∈ Xα, we define  [(λ − A−1)DM]ϕ(x) = ([(λ − A−1)DM]ϕ1(x), · · · , [(λ − A−1)DM]ϕ5(x))∗,  and, then  ˜K(t, ϕ)(x) := ˜K(t, ϕ(x))  =  �  K1(t, ϕ1(x), ∇ϕ(x)) + [(λ − A−1)DM]1ϕ1(x), · · · , K5(t, ϕ5(x), ∇ϕ(x)) + [(λ − A−1)DM]5ϕ5(x)  �∗  ,  for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Then, we have the following positivity result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='16 (Positivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' For each u0 ∈ X+  α equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8) has a unique maximal solution  u(·, u0) ∈ C([0, T(u0)), Xα) ∩ C1([0, T(u0)), Xβ) such that u(t) ∈ X+  α for all t ∈ [0, T(u0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2, it is clear that T (t)X+  α ⊂ X+  α for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Let ϕ ∈ X+  α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' So from  [28, Corollary 4] it suffices to show that  lim  h→0 h−1d(ϕ + h ˜K(t, ϕ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' X+  β−1) = 0  for each t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='17)  First, we prove (pointwisely) that  lim  h→0 h−1d(sup  ω>0  ωβ(R(ω, A−1 − λ)  �  ϕ(x) + h[ ˜K(t, ϕ)](x)  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Λ+∞) = 0  for each t ≥ 0, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='18)  Then, the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='18) holds since the transformation supω>0 ωβR(ω, A−1 − λ) preserves the  positivity, and due to [28, Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2] by the fact that Ki(t, 0) ≥ 0 and [(ω − A−1)DM]i0 = 0  for all t ≥ 0 which gives that ˜Ki(t, 0) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Note that the operator R(ω, A−1 − λ) is positive (see  [4]) which yields the positivity of supω>0 ωβR(ω, A−1 − λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Hence, we aim to prove that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='17)  holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Let | · |p be the p-norm in R5 defined as |(x1, · · · , x5)|p = (�5  k=1 |xi|p)  1  p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Then, for ϕ ∈ X,  the norm  ∥ϕ∥p := (  �  x∈Ω  | ϕ(x) |p dx)  1  p  is equivalent to the norm on X, this is due the fact that all the norms in R5 are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Similarly, using this new norm ∥ · ∥p, we can define an associated equivalent norm for Xα, and  the new equivalent norm on Xβ−1 which is given by  ∥ · ∥β−1,p = sup  ω>0  ∥ωβR(ω, λ − A−1) · ∥p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Let us define the Euclidean projection onto Λ+∞, πΛ : R5 −→ Λ0,+∞, by  | x − πΛx |= d(x, Λ+∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL  17  Notice that the mapping πΛ is well-defined and continuous on Rn (eventually it is 1-Lipschitzian).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Let ε > 0 such that there exists δ > 0 and define  ϕh(x) := πΛ(ϕ(x) + h[ ˜K(t, ϕ)](x))  for t ≥ 0, x ∈ Ω, h > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  So, ϕh ∈ XΛ+∞  β−1 and  d(ϕ + h ˜K(t, ϕ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' XΛ+∞  β−1 )p ≤ ∥ϕ + h ˜K(t, ϕ) − ϕh∥p  β−1  ≤ sup  ω>0  ωβ  �  x∈Ω  | R(ω, λ − A−1)  �  ϕ(x) + h[ ˜K(t, ϕ)](x) − ϕh(x)  �  |p  p dx  =  �  x∈Ω  d(sup  ω>0  ωβR(ω, λ − A−1)  �  ϕ(x) + h[ ˜K(t, ϕ)](x)  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Λ0,+∞)pdx  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='19)  Moreover, for 0 < h ≤ δ it follows in view of the convexity of the operator distance, by the  continuity of ˜K and using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='18), that  �  Ω  d(sup  ω>0  ωβ(R(ω, A−1 − λ)  �  ϕ(x) + h[ ˜K(t, ϕ)](x)  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Λ0,+∞)pdx ≤ |Ω|(hε)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Hence, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='19) we have  d(ϕ + h ˜K(t, ϕ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' X+  β−1) ≤ |Ω|  1  p hε  for all 0 < h ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Which proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' L1-boundedness of the total population density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In this Section, we show the L1-  boundedness of the total population density of our model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='14)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Let u0 ∈ X+  α and let  u(t, u0) = (ρ1(t, ·), · · · , ρ5(t, ·))∗ for all t ∈ [0, T(u0)) be the corresponding maximal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' It  is clear, from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='14, that u(·, u0) ∈ Xα �→ C1(Ω)5 �→ L1(Ω)5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' This means that each ρi  is bounded with respect to Ω and then it is L1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Hence, the following map  t ∈ [0, T(u0)) �−→ U(t) :=  �  Ω  [ρ1(t, x) + · · · + ρ5(t, x)] dx ∈ R,  is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Furthermore, we have  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='17 (L1-boundedness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  0 ≤ U(t) ≤ 1  for all t ∈ [0, T(u0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' By assumption, we have  U(0) =  �  Ω  θ(x)dx = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  In otherwise, the positivity result in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='16 gives  U(t) ≥ 0  for all t ∈ [0, T(u0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Moreover, the mapping U is well-defined and it is continuously differentiable on [0, T(u0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Hence,  we show that  d  dtU(t) ≤ 0,  t ∈ [0, T(u0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  18  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' KHALIL ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Indeed, using the Green-Ostrogradski formula, we obtain that  d  dtU(t) =  �  Ω  ∂t [ρ1 + ρ2 + ρ3 + ρ4 + ρ5] dx  =  5  �  i=1  di  �  Ω  ∆ρidx −  �  Ω  ∇(ρ2v(ρ))dx −  �  Ω  ∇(ρ3v(ρ))dx −  3  �  i=1  δi  �  Ω  ρi  =  5  �  i=1  di  �  ∂Ω  ∇ρi · ndx −  �  ∂Ω  ρ2v(ρ) · ndx −  �  ∂Ω  ρ3v(ρ) · ndx −  3  �  i=1  δi  �  Ω  ρidx  = −  5  �  i=1  vi  �  Γ2  ρidx −  3  �  i=1  δi  �  Ω  ρidx ≤ 0,  t ∈ [0, T(u0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  The last estimate is a consequence of the positivity of the terms ρi, i = 1, · · · , 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' That is, U is  decreasing and then, we have  U(t) ≤ U(0) = 1  for all t ∈ [0, T(u0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  □  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Uniform boundedness and global existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In this Section, using the method of pos-  itively invariant regions, we prove that the maximal solution of equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8) have a bounded  positive invariant region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' This fact guarantees the uniform boundedness of the solution (see The-  orem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='18) that yields the global existence (see Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' We recall also that the result  obtained in this section is new and generalize those in [28, Section 2] and [12, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='3] in the  case of inhomogeneous boundary type conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Moreover, we assume that  5  �  i=1  ai > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='18 (Uniform boundedness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' For each u0 ∈ XΛa equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8) has a unique max-  imal solution u(·, u0) ∈ C([0, T(u0)), Xα) ∩ C1([0, T(u0)), Xβ) such that u(t) ∈ XΛa for all  t ∈ [0, T(u0)) provided that,  �  �  �  �  �  �  �  �  �  �  �  �  �  a4 ≤ (b1 + b2 + δ1)a1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  b3 ≤ αa→cξ(  a3  a1 + ε)a1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  b4 ≤ αa→pξ(  a2  a1 + ε)a1,  b2a1 + αc→pξ(  a2  a3 + ε)a2a3 ≤ (b4 + c1 + δ2)a2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  c2a3 + αa→pξ(  a2  a1 + ε)a1a2 ≤ αp→cξ(  a3  a2 + ε)a2a3,  b1a1 + αp→cξ(  a3  a2 + ε)a2a3 ≤ (b3 + c2 + δ3)a3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  c1a2 + αa→cξ(  a3  a1 + ε)a1a3 ≤ αc→pξ(  a2  a3 + ε)a2a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='20)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Let ϕ ∈ XΛa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' First, we prove that T (t)XΛa ⊂ XΛa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' That is, from the invariance result  [35, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1], it suffices to show that R(λ, A0)XΛa ⊂ XΛa for some λ ∈ ρ(A0) large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  This last follows immediately since we have  (λ − A0)a = λa ≥ a,  for large λ ∈ ρ(A0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Remark that R(λ, A0) is a positive operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Furthermore, we show the invariance for the solution under the set  Λa,∞ := Π4  i=1[0, ai] × [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL  19  which leads to the invariance under the bounded region Π4  i=1[0, ai] for the vector components  (ρ1, · · · , ρ4)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' For the case ρ5 ≤ a5 we treat it separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' So it suffices to prove that  lim  h→0 h−1d(ϕ + h ˜K(t, ϕ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' XΛa,∞  −1  ) = 0,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='21)  which amounts, in view of the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='16, to proving that  lim  h→0 h−1d(sup  ω>0  ωβ(R(ω, A−1 − λ)  �  ϕ(x) + h[ ˜K(t, ϕ)](x)  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Λa,∞) = 0  for all t ≥ 0, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='22)  So, in view of [23, Proposition 12], formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='22) holds if we check that for ρ = a, we have  ˜Ki(t, ai, ∇a) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Thus, it suffices to prove that Ki(t, ai, ∇a) + [(λ − A−1)DM]iai ≤ 0 for i = 1, · · · , 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Note that,  by construction, ˜K4(t, a4, ∇β) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Moreover, since ∇ai · n = 0, for i = 1, · · · , 5, we obtain that  La = (d1∇a1 · n, d2∇a2 · n, d3∇a3 · n, d4∇a4 · n, d5∇a5 · n)∗ = (0, 0, 0, 0, 0)∗,  which yields that DMa = (0, 0, 0, 0, 0)∗, and then, that  (λ − Aβ−1)DMa = (0, 0, 0, 0, 0)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Therefore, we need only to examine the terms Ki(t, ai, ∇a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' That is, we have  ∇(Vi,maxai(1 −  5  �  j=1  aj)⃗ν(x)) = Vi,maxai(1 −  5  �  j=1  ai)∇(ν(x))  for i = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  So, since by definition of ⃗ν, we have ∇(⃗ν(x)) ≤ 0, and then it follows that  ∇(Vi,maxai(1 −  5  �  j=1  aj)⃗ν(x)) ≥ 0  by the fact that �5  i=1 ai > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Hence  If (ρ1, · · · , ρ4) = (a1, · · · , a4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Then, we have  K1(t, a1, ∇a) = −(b1 + b2 + δ1)a1 + γ(t)a4 + b3a3 − F(a1, a3) − G(a1, a2) + b4a2  = γ(t)a4 − (b1 + b2 + δ1)a1  �  ��  �  C1  1  +  �  b3 − αa→cξ(  a3  a1 + ε)a1  �  a3  �  ��  �  C1  2  +  �  b4 − αa→pξ(  a2  a1 + ε)a1  �  a2  �  ��  �  C1  3  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Hence, C1  1 ≤ 0 if a4 ≤ (b1 + b2 + δ1)a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Moreover, b3 ≤ αa→cξ(  a3  a1+ε)a1 implies that C1  2 ≤ 0, and  C1  3 ≤ 0 if b4 ≤ αa→pξ(  a2  a1+ε)a1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Furthermore,  20  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' KHALIL ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  K2(t, a2, ∇a) = −(b4 + c1 + δ2)a2 + b2a1 + c2a3 − ∇(V2,maxa2(1 −  5  �  j=1  aj)ν(x)) + G(a1, a2) − H(a2, a3)  ≤ −(b4 + c1 + δ2)a2 + b2a1 + αc→pξ(  a2  a3 + ε)a2a3  �  ��  �  C2  1  + c2a3 + αa→pξ(  a2  a1 + ε)a1a2 − αp→cξ(  a3  a2 + ε)a2a3  �  ��  �  C2  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  So, C2  1 ≤ 0 if (b4 + c1 + δ2)a2 ≥ b2a1 + αc→pξ(  a2  a3+ε)a2a3, and c2a3 + αa→pξ(  a2  a1+ε)a1 ≤  αp→cξ(  a3  a2+ε)a3 implies that C2  2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Finally, we have  K3(t, a3, ∇a) = −(b3 + c2 + δ3)a3 + b1a1 + c1a2 − ψ(t)a3 − ∇(V3,maxa3(1 −  5  �  j=1  aj)ν(x)) + F(a1, a3) + H(a2, a3)  ≤ −(b3 + c2 + δ3)a3 + b1a1 + αp→cξ(  a3  a2 + ε)a2a3  �  ��  �  C3  1  + c1a2 + αa→cξ(  a3  a1 + ε)a1a3 − αc→pξ(  a2  a3 + ε)a2a3  �  ��  �  C3  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  That is, we obtain that (b3 + c2 + δ3)a3 ≥ b1a1 + αp→cξ(  a3  a2+ε)a2a3 implies C3  1 ≤ 0 and c1a2 +  αaξ(  a3  a1+ε)a1 ≤ αc→pξ(  a2  a3+ε)a2 implies C3  2 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Then, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='22) and so (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='21) holds too uniformly in  x in a similar way as in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' On the other hand, to show that 0 ≤ ρ5(t) ≤ a5  for some a5 > 0 we use the variation of constant formula i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=', (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In fact, we have  ρ5(t) =  � t  0  T5(t − s)ψ(s)ρ3(s)ds  ≤ T(u0)a3 := a5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  This proves the result for a5 = T(u0)a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  □  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' To verify conditions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='20) and convince that they do not contradict each other,  for simplicity and since the parameters ai > 0 for i = 1, · · · , 4, we may take for example ε = 0,  a1 = m0a4 = m1a2 and a3 = a2 = ˜a where m0 > 0 is chosen such that 1 ≤ m0(b1 + b2 + δ1), and  m1 > 0 is such that:  �  �  �  �  �  �  �  �  �  �  �  �  �  b3 ≤  m1  1 + m2  1  αa→cβ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  b4 ≤  m1  1 + m2  1  αa→p˜a  m1b2 ≤ c1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  αc→p  2  ˜a ≤ b4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  c2 +  m1  1 + m2  1  αa→p˜a ≤ αp→c  2  ˜a  m1b1 ≤ c2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  αp→c  2  ˜a ≤ b3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  c1 +  m1  1 + m2  1  αa→c˜a ≤ αc→p  2  ˜a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='23)  ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL  21  Thus, for a special choice of the parameters m0, m1 and β, the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='23) holds where we  may have c1 < c2, b1 < b2, αc→p > αa→c and αa→p > αp→c corresponding to the case of a  population with low risk culture studied in our simulation results in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Notice that the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='20) is sufficient to obtain the pointwise subtangantial  condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' That is, by construction, the different cases for ˜K(t, ai, 0) where the aj vanish  for some (not all) j ̸= i hold true under the condition (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8) has a unique global bounded and positive solution u(·, u0) ∈  C([0, T(u0)), X) ∩ C1((0, T(u0)), X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' From Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='14 we obtain that equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8) has a unique positive maximal solution  u(·, u0) ∈ C([0, T(u0)), Xα)∩C1([0, T(u0)), Xβ) such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='15) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Therefore, from Theorem  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='18 the solution u is bounded in Xα which yields from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='15) that T(u0) = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  □  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Numerical Simulations  In this section we present several numerical simulations for different scenarios for evacuation  of populations in a catastrophic event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In order to highlight the behavior of the populations in  such event, we study the case of no-back-to-daily population that corresponds to the case where  φ(t) = 0  for all t ≥ 0,  and, for simplicity, we take γ(t) = 1 for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Here the population is supposed with low risk  culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' All the parameters of the spatio-temporal APC model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='14)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='16) are set as in Table  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In this case the system (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8) is not time-dependent (it is autonomous) and all the results of  Section 3 still hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Parameters  d1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='001  Diffusion  d2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='05  d3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='01  d4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='01  V2,max = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='3  Advection  V3,max = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2  V1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2  Speed at the  V2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1  boundary  V3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='3  V4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2  Parameters  αa→c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='6  Imitation  αa→p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='7  αp→c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='6  αc→p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='7  c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1  c2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='4  Intrinsic  b1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1  transitions  b2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2  b3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='001  b4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='001  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Table of parameter values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' As explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='3, we choose  d1 < d3 = d4 < d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Moreover, we are interested to consider a population with a  low risk culture, so, for example, we take c2 > c1 and b2 > b1, as in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  22  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' KHALIL ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  We assume that the domain presents a target escape region (denoted by Γ2), thus the desired  direction vector is defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='12) where ⃗ν(x)|Ω = (νx1, νx2)∗ is given by:  �  �  �  �  �  �  �  �  �  νx1 = −  x1 − xp  1  �  (x1 − xp  1)2 + (x2 − xp  2)2 ,  νx2 = −  x2 − xp  2  �  (x1 − xp  1)2 + (x2 − xp  2)2 ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1)  where (xp  1, xp  2) is a centered point in Γ2 localized out from Ω, see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  So, the desired  direction in both situations of control and panic are supposed to be the same which is given by  ⃗ν(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' For more details about the desired direction of pedestrians, we refer to the works of Hughes  [25] and also to the references [14, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The direction vector ⃗ν(x1, x2) given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1) describing the desired  direction of pedestrians to reach the point (xp  1, xp  2) which is located outside the  domain Ω since the population looks to escape from the exit Γ2 towards this  point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  With regard to the diffusion process, we suppose that the crowd in a alert state hardly diffuses,  since in a alert behavior, pedestrians are moving to look for information and to identify the  hazard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Thus, the diffusion coefficient d1 should be considered small compared to the ones of the  other populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Moreover, here it is assumed that the most diffusive population is the panic  population, since in panic behavior pedestrians move randomly in different directions, and thus  d2 should be considered the largest diffusion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Thus, for the diffusivity coefficients of the control, daily, and return-to-day behaviors, we  assume d1 < di < d2 for i = 3, 4, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  In the following, we consider three different scenarios for the evacuation of a population whose  aim is to escape by the unique exit denoted Γ2:  Scenario 1: Evacuation of one centered cluster population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Here we consider a pedes-  trian population located in a single group within the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Scenario 2: Evacuation of a population subdivided into three groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Here we consider  a population subdivided into three separated groups of pedestrians in different spatial localiza-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Scenario 3: Evacuation of a population with an obstacle in front of the exit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Here we  1n0ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL  23  take into account the situation in which an obstacle is located between the exit and the popula-  tion concentrated in a single group in the center of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  The initial distribution of population in the three scenarios are illustrated in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Since  (see Table 1) the population considered here is of low risk culture and then the dominated  behavior is that of panic, for each scenario, we give simulation results describing the panic at  different times t = 50, 100, 150, 200 and 250 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' This fact leads to compare between  different scenarios (see Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Moreover, in order to highlight the time evolution of the other  human behaviors, namely, alert, control and daily behavior, we present some simulations but  only in Scenario 1, since the evolution is the same in other scenarios (see Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  According to our numerical results, we notice that at the beginning of the simulation, for  t = 50, there is a majority of daily and alert populations rather than population in a state  of panic and control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' This dynamic depends on the structure of the APC model, described in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1: at t = 0 everyone is in a daily behavior, then everyone goes through the state of  alert before becoming panicked or controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Moreover, since the diffusion coefficients for ρ1 and  ρ2 are low, the position of the populations is still more or less the same as the starting one, see  (a1)-(a4) in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' For t = 250, for all scenarios, the dynamics of the APC is fully developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Moreover, diffusion and advection phenomena are now visible: the whole population is in panic  state (highest density of human behaviors) and control state, and they are concentrated near  the exit, while the populations in alert state and in daily behavior are negligible, see (e1)-(e4) in  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Comparing the evacuation of population in panic in the three different scenarios, see in Figures  6, (e1) for Scenario 1, (e2) for Scenario 2 and (e3) for Scenario 3 , one notices a strong congestion  at the level of the exit in the first scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In the second scenario, splitting the initial population  into three clusters reduces this congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Finally, the presence of an obstacle as in the third  scenario further reduces congestion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (i) The more the color goes from light blue to dark red, the higher the popu-  lation density is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (ii) For each figure, Scenario 1 is denoted by (a), Scenario 2 by (b) and Scenario 3 by (c),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In the sequel, we give simulation results representing the other human behaviors,  alert, control and daily behavior populations in order to show the time evolution of this popula-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' But, we consider only Scenario 1, since the behavior of all populations is the same for the  the other scenarios 1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  24  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' KHALIL ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (a) Scenario 1  (b) Scenario 2  (c) Scenario 3  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Initial conditions: initial location of the population for each scenario:  (a) the population is concentrated in a single group in the center of the domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' (b)  the population is subdivided into three groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' (c) an obstacle is located between the  exit and the population, which is concentrated in a single group within the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  We recall that the exit is on the right of the domain, see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='0e+00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='0e+00LYANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL  25  t = 50  (a1)  t = 100  (b1)  t = 150  (c1)  t = 200  (d1)  t = 250  (e1)  Scenario 1  (a2)  (b2)  (c2)  (d2)  (e2)  Scenario 2  (a3)  (b3)  (c3)  (d3)  (e3)  Scenario 3  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Population in panic ρ2 over the three scenarios at the captures times  t = 50, 100, 150, 200 and 250 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Notice that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' each row represent the (time)  evolution of the population at each scenario: at time t = 50 of each scenario (column  (a)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' population in panic is small with respect to the initial population,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' since at the  this time the majority of population is concentrated in alert,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' but over time,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' panicked  ones become greater and greater,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' thanks to the APC dynamics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' and the fact that the  population has a low risk culture,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' in addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' the phenomena of diffusion and advection  are now visible (the population in panic concentrated near the exit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Furthermore,  by comparing captures at each column (b), (c), (d) and (e), we can observe that in  Scenario 2 there is slightly less congestion near the exit than in Scenario 1, this is can  be highlighted from time t = 100 to t = 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Moreover, in Scenario 3, we observe that  the congestion is less than in the previous cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Indeed, the role of the obstacle is to  facilitate the access to the exit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='5e-01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='0e+004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='5e-01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='0e+00C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='5e-01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='0e+0026  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' KHALIL ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  t = 50  (a1)  t = 100  (b1)  t = 150  (c1)  t = 200  (d1)  t = 250  (e1)  Population in alert ρ1  (a2)  (b2)  (c2)  (d2)  (e2)  Population in panic ρ2  (a3)  (b3)  (c3)  (d3)  (e3)  Population in control ρ3  (a4)  (b4)  (c4)  (d4)  (e4)  Population in daily behavior ρ4  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' In order to present the time evolution of the human behaviors (alert,  control and daily behavior), we give simulations results of this populations in Scenario  1 and with respect to captures times t = 50, 100, 150, 200 and 250 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' So, each  row represent a human behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' We notice that at the beginning of the simulation,  there is a majority of daily and alert populations (row 1 and row 4 respectively) rather  than population in a state of panic and control (row 2 and row 3 respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' This  dynamic depends on the structure of the APC model, described in Sections 2 and 3:  at t = 0 everyone is in a daily behavior, then everyone goes through the state of alert  before becoming panicked or controlled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='5e-01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='0e+004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='5e-01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='0e+004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='5e-01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='0e+004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='5e-01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='0e+00ANALYSIS OF A SPATIO-TEMPORAL ADVECTION-DIFFUSION APC MODEL  27  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Conclusion  In this work, we introduce a new spatio-temporal APC (alert, panic and control) model describ-  ing the evacuation of a population presented via different human behaviors during a catastrophic  event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' First, using the first-order macroscopic crowd theory, we derive a new spatio-temporal  APC model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' It is a system of advection-diffusion-reaction equations with nonlinear Robin bound-  ary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Then, using a semigroup approach and abstract evolution equations, we prove  the local existence and a regularity result of the solutions of our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Moreover, we establish  the positivity of the solution and the existence of positively bounded invariant sets which leads  to the global existence and the boundedness of the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' As far as we know, the theoretical  results established in this work are new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Finally, to illustrate our results, we present different  numerical simulations of the population evacuation, using three different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  28  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' KHALIL ET AL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Appendices  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Proofs of the preliminary results of Section 3  Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' (i) It is well-known, from [32, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1], that the realization of  the Laplacian operator di∆ on Lp(Ω) with Neumann boundary conditions generates a contraction  holomorphic C0-semigroup (Ti(t))t≥0 of angle π/2, for i = 1, · · · , 5, see also [16, Sections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='4] for  a similar result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' So A0 generates a contraction holomorphic C0-semigroup of angle π/2 on X as  a diagonal matrix-valued operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (ii) The compactness of the semigroup (T (t))t≥0 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' when T(t) is a compact operator for each  t > 0) follows from [16, Sections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' For the positivity of the semigroup (T (t))t≥0 on the Banach  lattice X it suffices to prove that R(ω, A0) ∈ L(X) is a positive operator for ω ∈ ϱ(A0) large  enough i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=', ϕ(x) ≥ 0 implies R(ω, A0)ϕ(x) ≥ 0 for all x ∈ Ω, see [20, Chapter VI, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8],  see also the result of invariance under closed sets [35, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' This is equivalent to prove  that ψ ∈ D(A0) := ker(L) which is the solution of ϕ(x) = (ω − A0)ψ(x) implies ψ is positive (it  always exists, the question is about its positivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' That is, we have  �  (ω − A)ψ(x) = ϕ(x) ≥ 0,  x ∈ Ω  Lψ(x) = 0,  x ∈ ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Then, the result holds directly from the maximum principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Let 0 ≤ δ ≤ 1, the fact that the extension semigroup (Tδ−1(t))t≥0  exists as a strongly continuous positive semigroup with generator (Aδ−1, D(Aδ−1) = Xδ) is due  to [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' The analycity and compactness of (Tδ−1(t))t≥0 follows from [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' This holds by definition of the Dirichlet map D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Since (λ − Aβ−1)D ∈ L(∂Xα, Xβ−1), it suffices to examine the  operators K and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Indeed, for K we show term by term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Let (ϕ1, · · · , ϕ5)∗ := ϕ, (υ1, · · · , υ5) :=  υ ∈ Xα and R > 0 be such that ∥ϕ∥α, ∥υ∥α ≤ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' By construction, the functions F, G and H  are (pointwisely) Lipschitzian in bounded sets i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=',  |F(ρ1, ρ3)(x) − F(υ1, υ3)(x)| ≤ L1  R (|ρ1(x) − υ1(x)| + |ρ3(x) − υ3(x)|) ,  x ∈ Ω  |G(ρ1, ρ2)(x) − G(υ1, υ2)(x)| ≤ L2  R (|ρ1(x) − υ1(x)| + |ρ2(x) − υ2(x)|) ,  x ∈ Ω  and  |H(ρ2, ρ3)(x) − H(υ2, υ3)(x)| ≤ L3  R (|ρ2(x) − υ2(x)| + |ρ3(x) − υ3(x)|) ,  x ∈ Ω  for some Li  R ≥ 0, i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' So, by passing to the Lp-norm, and using the continuous embedding  W 2α,p �→ C1(Ω) we obtain that  ∥F(ρ1, ρ3) − F(υ1, υ3)∥ ≤ |Ω|L1  R (∥ρ1 − υ1∥0,α + ∥ρ3 − υ3∥0,α) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1)  ∥G(ρ1, ρ2) − G(υ1, υ2)∥ ≤ |Ω|L2  R (∥ρ1 − υ1∥0,α + ∥ρ2 − υ2∥0,α) ,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='2)  29  and  ∥H(ρ2, ρ3) − H(υ2, υ3)∥ ≤ |Ω|L3  R (∥ρ2 − υ2∥0,α + ∥ρ3 − υ3∥0,α) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='3)  Now, we show that the terms ∇ · (ϕ2v2(ϕ)ν)), ∇ · (ϕ3v3(ϕ)ν) are Lipschitzian in bounded sets in  Xα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' So, a straightforward calculus yields that  ϕ2V2(ϕ)(x)−υ2V2(υ)(x) = V2,max  �  (1 − ˜ϕ(x)) [ϕ2(x) − υ2(x)] + υ2(x)  5  �  i=1  [ϕi(x) − υi(x)]  �  ,  x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Furthermore, using the regularity of ϕ and υ and since the gradient operator ∇ is linear, we  obtain that  ∇ · (ϕ2v2(ϕ)ν(x)) − ∇ · (υ2v2(υ)ν(x))  =  ∇ · (ϕ2v2(ϕ)ν(x)) − ∇ · (ϕ2v2(υ)ν(x)) + ∇ · (ϕ2v2(υ)ν(x)) − ∇ · (υ2v2(υ)ν(x))  =  ∇ϕ2(x) · (v2(ϕ)ν(x)) − ∇ϕ2(x) · (v2(υ)ν(x)) + ϕ2(x)∇ · (v2(ϕ)ν(x)) − ϕ2(x)∇ · (v2(υ)ν(x))  �  ��  �  I1(x)  + ∇ϕ2(x) · (v2(υ)ν(x)) − ∇υ2(x) · (v2(υ)ν(x)) + ϕ2(x)∇ · (v2(υ)ν(x)) − υ2(x)∇ · (v2(υ)ν(x))  �  ��  �  I2(x)  ,  x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Then, we have  | I1(x) | ≤ MV2  �  | ∇ϕ2(x) |  5  �  i=1  | ϕi(x) − υi(x) | + | ϕ2(x) |  5  �  i=1  | ∇(ϕi(x) − υi(x)) |  �  ,  x ∈ Ω,  | I2(x) | ≤ MV2  �  | ∇(ϕ2(x) − υ2(x)) |  5  �  i=1  (1+ | ϕi(x) |)+ | ϕ2(x) − υ2(x) |  5  �  i=1  | ∇(υi(x) |  �  ,  x ∈ Ω,  where MV2 = V2,max supx∈Ω |ν(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Hence, we have  | ∇ · (ϕ2v2(ϕ)ν(x)) − ∇ · (υ2v2(υ)ν(x)) |≤  MV2  �  | ∇ϕ2(x) |  5  �  i=1  | ϕi(x) − υi(x) | + | ϕ2(x) |  5  �  i=1  | ∇(ϕi(x) − υi(x)) |  �  +MV2  �  | ∇(ϕ2(x) − υ2(x)) |  5  �  i=1  (1+ | ϕi(x) |)+ | ϕ2(x) − υ2(x) |  5  �  i=1  | ∇(υi(x) |  �  ,  x ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Therefore, by passing to the norms, we have  ∥∇ · (ϕ2v2(ϕ)ν) − ∇ · (υ2v2(υ)ν)∥  ≤  |Ω|V2,max ((1 + ∥ϕ∥) ∥∇(ϕ2 − υ2)∥∞ + ∥∇ϕν∥∥ϕ2 − υ2∥∞)  +|Ω|V2,max (∥ϕ − υ∥∥∇υ2ν∥∞ + ∥υ2∥∞∥∇(ϕ − υ)ν∥) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  This leads to  ∥∇ · (ϕ2v2(ϕ)ν) − ∇ · (υ2v2(υ)ν∥  ≤  |Ω|L4  R∥ϕ − υ∥α  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='4)  Similarly, we obtain that and  ∥∇ · ϕ3v3(ϕ)ν − ∇ · υ3v3(υ)ν∥  ≤  |Ω|V3,max ((1 + ∥ϕ∥) ∥∇(ϕ3 − υ3)∥∞ + ∥∇ϕν∥∥ϕ3 − υ3∥∞)  +|Ω|V3,max (∥ϕ − υ∥∥∇υ3ν∥∞ + ∥υ3∥∞∥∇(ϕ − υ)ν∥) ,  30  and  ∥∇ · ϕ3v3(ϕ)ν − ∇ · υ3v3(υ)ν∥  ≤  |Ω|L5  R∥ϕ − υ∥α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='5)  Now, we show that, there exists L9  R ≥ 0 such that  ∥Mϕ − Mυ∥∂X ≤ L9  R∥ϕ − υ∥α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  To obtain that, we use the embedding W 1,p(∂Ω)5 �→ ∂X and we prove that  ∥Mϕ − Mυ∥1,p ≤ L9  R∥ϕ − υ∥α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Indeed, we have,  | ϕ2V2(ϕ)(x)−υ2V2(υ)(x) |≤ V2,max  �  (1+ | ˜ϕ(x) |) | ϕ2(x) − υ2(x) | + | υ2(x) |  5  �  i=1  | ϕi(x) − υi(x) |  �  , x ∈ Ω  Hence, using the corresponding norms, we have  |ϕ2V2(ϕ) − υ2V2(υ)|p ≤ |Ω|V2,max ((1 + ∥ϕ∥)∥ϕ2 − υ2∥∞ + ∥υ2∥∞∥ϕ − υ∥) ,  That is using the embedding Xα �→ C1(Ω)5, we obtain that  |ϕ2V2(ϕ) − υ2V2(υ)|p ≤ L6  R∥ϕ − υ∥α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Arguing similarly, we obtain that  |ϕ3V3(ϕ) − υ3V3(υ)|p ≤ L7  R∥ϕ − υ∥α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  In otherwise, by estimating, this time the gradient of the corresponding terms, we obtain that  |∇ϕ2V2(ϕ) − ∇υ2V2(υ)|p ≤ L8  R∥ϕ − υ∥α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Arguing similarly, we obtain that  |∇ϕ3V3(ϕ) − ∇υ3V3(υ)|p ≤ L9  R∥ϕ − υ∥α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Thus, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='9 yields that,  ∥(ω − Aβ−1)D(Mϕ − Mυ)∥Xβ−1  ≤  c∥Mϕ − Mυ∥∂X  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='6)  ≤  cL10  R ∥ϕ − υ∥α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='7)  Furthermore, the fact that the functions γ and respectively φ are lγ-Lipschitzian and lφ-Lipschitzians  with respect to t and 0 ≤ γ(t), φ(t) ≤ 1 yields  ∥γ(t)ϕ4 − γ(s)υ4∥ ≤ |Ω|lγ(∥ϕ4∥∞ | t − s | +∥ϕ4 − υ4∥α),  t, s ≥ 0,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='8)  and through the same argument, we have  ∥φ(t)ϕ3 − φ(s)υ3∥ ≤ |Ω|lφ(∥ϕ3∥α | t − s | +∥ϕ3 − υ3∥α),  t, s ≥ 0,  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='9)  Consequently, from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1)-(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='9) we can find LR ≥ 0 such that  ∥ ˜K(t, ρ) − ˜K(s, υ)∥β−1 ≤ LR(| t − s | +∥ρ − υ∥α)  for all t, s ≥ 0 and ϕ, υ ∈ Xα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  This proves the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  31  Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Let us define  v(t) =  � t  0  Tβ−1(t − s)B(t)ds +  � t  0  Tβ−1(t − s) [B(s) − B(t)] ds := v1(t) + v2(t),  0 ≤ t ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  It is clear that v1 ∈ C1((0, T], Xβ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' So it suffices to prove that v2(t) ∈ Xβ and Aβ−1v2(·) is  continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' To this end, let ε > 0 and consider  vε  2(t) =  �  �  �  � t−ε  0  Tβ−1(t − s) [B(s) − B(t)] ds  for t ≥ ε  0  for t < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  So, the analyticity of the extension semigroup (Tβ−1(t))t≥0 on Xβ−1 yields that  Tβ−1(t − s) [B(s) − B(t)] ∈ D(Aβ−1) = Xβ  for 0 ≤ s ≤ t − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Then, vε  2(t) ∈ D(Aβ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Moreover, vε  2(t) converges to v2(t) as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Since the operator Aβ−1 is  closed, to conclude, we only need to show that Aβ−1vε  2(t) =  � t−ε  0  Aβ−1Tβ−1(t−s) [B(s) − B(t)] ds  converges in Xβ−1 as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' That is, by the closedness of Aβ−1 we have  Aβ−1vε  2(t) −  � t  0  Aβ−1Tβ−1(t − s) [B(s) − B(t)] ds =  � t  t−ε  Aβ−1Tβ−1(t − s) [B(s) − B(t)] ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Furthermore, the analytitcity of (Tβ−1(t))t≥0 yields that  ∥Aβ−1Tβ−1(t)∥L(Xβ−1) ≤ l0 t−1,  t > 0  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='10)  for some l0 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Hence, using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='12)-(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='10), we conclude that  ∥Aβ−1vε  2(t) −  � t  0  Aβ−1Tβ−1(t − s) [B(s) − B(t)] ds∥β−1 ≤ ll0  � ε  0  ση−1dσ → 0 as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  This proves that v2(t) ∈ Xβ and that Aβ−1v2(t) =  � t  0 Aβ−1Tβ−1(t − s) [B(s) − B(t)] ds for 0 <  t ≤ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' So, it is clear that Aβ−1v is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Tables of the functions and the parameters of the APC model  In the sequel, we briefly recall the functions and the parameters of the APC model (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Functions in the APC model  Functions  Notation  Beginning of the catastrophe  γ(t)  Return to a daily behavior  φ(t)  Imitation functions  F, G, H  32  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Parameters of the APC model  Parameters  Notation  Intrinsic evolution from alert to control  b1  Intrinsic evolution from alert to panic  b2  Intrinsic evolution from control to alert  b3  Intrinsic evolution from panic to alert  b4  Intrinsic evolution from panic to control  c1  Intrinsic evolution from control to panic  c2  Mortality rates for alert,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' panic and control populations respectively  δ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' δ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' δ3  Imitation from alert to control  α13  Imitation from alert to panic  α12  Imitation from panic to control  α23  Imitation from control to panic  α32  ACKNOWLEDGMENT  This work has been supported by the French government,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' through the National Research  Agency (ANR) under the Societal Challenge 9 “Freedom and security of Europe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' its citizens  and residents” with the reference number ANR- 17-CE39-0008,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' co-financed by French Defence  Procurement Agency (DGA) and The General Secretariat for Defence and National Security  (SGDSN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
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+page_content='  Email address: kamal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='khalil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='00@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='com  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Lanza,  LMAH, University of Le Havre Normandie, FR-CNRS-3335, ISCN, Le Havre 76600, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Email address: valentina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='lanza@univ-lehavre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='fr  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Manceau,  LMAH, University of Le Havre Normandie, FR-CNRS-3335, ISCN, Le Havre 76600, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Email address: david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='manceau@univ-lehavre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='fr  M-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Alaoui,  LMAH, University of Le Havre Normandie, FR-CNRS-3335, ISCN, Le Havre 76600, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='  Email address: aziz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='alaoui@univ-lehavre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='fr  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content=' Provitolo  Université Côte d’Azur, CNRS, Observatoire de la Côte d’Azur, IRD, Géoazur, UMR 7329, Val-  bonne, France  Email address: Damienne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='provitolo@geoazur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
+page_content='unice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/c9E0T4oBgHgl3EQfoAGA/content/2301.02520v1.pdf'}
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+LostNet: A smart way for lost and find 
+Meihua Zhou1, Ivan Fung2, Li Yang1, Nan Wan1, Keke Di1 
+Tingting Wang* 
+1 School of Medical Information, Wannan Medical Collage, Wuhu City, China 
+{mhzhou, 20120037, wannan, mhzhou0412}@wnmc.edu.cn 
+2 work.ivanfung@gmail.com 
+* School of Medical Information, Wannan Medical Collage, Wuhu City, China 
+wangtt@wnmc.edu.cn 
+Abstract:  
+Due to the enormous population growth of cities in recent years, objects are frequently lost 
+and unclaimed on public transportation, in restaurants, or any other public areas. While 
+services like Find My iPhone can easily identify lost electronic devices, more valuable 
+objects cannot be tracked in an intelligent manner, making it impossible for administrators to 
+reclaim a large number of lost and found items in a timely manner. We present a method that 
+significantly reduces the complexity of searching by comparing previous images of lost and 
+recovered things provided by the owner with photos taken when registered lost and found 
+items are received. In this research, we will primarily design a photo matching network by 
+combining the fine-tuning method of MobileNetv2 with CBAM Attention and using the 
+Internet framework to develop an online lost and found image identification system. Our 
+implementation gets a testing accuracy of 96.8% using only 665.12M GLFOPs and 3.5M 
+training parameters. It can recognize practice images and can be run on a regular laptop. 
+  
+Keyword: artificial intelligence, lost and found, intelligent city 
+ 
+Introduction: 
+The population density and the quantity of lost objects are both rising in the areas where 
+urban rail transit is located around the world at the present time; yet, the traditional manual 
+search service is ineffective. In this situation, there is an immediate need to speed up the 
+development of intelligent lost and found systems in order to lessen the difficulty that 
+transportation operators face when it comes to lost and found management. Deep learning can 
+be used to build recognition and classification models for lost and found items. This is a new 
+method that can reduce reliance on manual labour, quickly and accurately identify categories, 
+significantly reduce human service costs for transportation operators, and better practice 
+green development strategies. 
+The application of convolutional neural networks is widespread throughout many scientific 
+subfields pertaining to image recognition and classification (Sun et al., 2021). It is not a 
+passing fad that academics in a wide variety of fields have shifted their attention to the study 
+and practical use of image recognition. In the realm of trash classification, a method for 
+
+garbage image classification was devised. It was built on an enhanced version of MobileNet 
+v2 and paired with transfer learning to increase the real-time performance and accuracy of 
+garbage image classification models (Huang et al., 2021). 
+ 
+Research Background: 
+Many academics are also engaged in questioning the status quo in the many subfields that 
+make up the area of applied image recognition. (Yang et al., 2015) attempting to identify 
+plant leaves by the utilization of a hierarchical model that is based on CNN. A study on 
+establishing the optimal size of the training data set that is required to achieve high 
+classification accuracy with low variance in medical image classification systems is presented 
+by Cho et al. (2015). (Purnama et al., 2019) offer a method for the classification and 
+diagnosis of skin diseases that is suitable for use in teledermatology. 
+ 
+It has also been demonstrated that transfer learning is beneficial in a variety of contexts. 
+Convolutional neural networks are used in the methodology that Lee et al. (2016) propose as 
+a fine-grained classification method for large-scale plankton databases. The implementation 
+of transfer learning in CNN is one potential solution. (Liu et al., 2020) apply unsupervised 
+transfer learning to CNN training to address these problems. Specifically, they transform 
+similarity learning into deep ordinal classification with the assistance of several CNN experts 
+who were pretrained over large-scale-labelled everyday image sets. These CNN experts 
+jointly determine image similarities and provide pseudo labels for classification. (Purwar et 
+al., 2020) make use of some models that are related to convolutional neural networks (CNNs) 
+in order to identify mesangial hypercellularity in MEST-C. (Herzog et al., 2021) concentrate 
+on the classification of MRI for the diagnosis of early and progressive dementia by utilizing 
+transfer learning architectures that employ Convolutional Neural Networks-CNNs, as a base 
+model, and fully connected layers of SoftMax functions or Support Vector Machines-SVMs. 
+(Phankokkruad, 2021) propose the three CNN models for detecting lung cancer using 
+VGG16, ResNet50V2, and DenseNet201 architectures. The proposed method is based on 
+transfer learning. 
+ 
+Methodology: 
+In a procedure in which CNN is being used to address more and more components of the 
+problem, the lost and found problem has not been solved in any intelligent method as of yet. 
+As a result, we recommend taking a methodical approach by utilizing Mobilnet v2 and an 
+intuitive graphical user interface (GUI). In this study, we build on earlier research to further 
+investigate the detection and categorization of lost and discovered items, and we present an 
+approach that combines perceptual hashing with MobileNet v2 transfer learning. 
+In order to solve the relatively complex problems associated with the image dataset of lost 
+and found items, we carried out extensive research, classified the most common items that are 
+lost and found into ten categories using questionnaires and market research, and produced 
+private dataset images using crawlers, real-world photography, and research examples. After 
+that, the generated dataset is used to train the network and establish an intelligent recognition 
+
+classification model for lost and found images. This model solves the problems of labor cost 
+and time cost consumption that are associated with conventional methods and proposes an 
+accurate and complete solution with scientific and accurate experimental data. 
+I. 
+Mobilenet v2 
+An outstanding example of a good lightweight convolutional neural network is MobileNetV2 
+(Sandler et al., 2018). The network creates a reverse residual and linear bottlenecks, both of 
+which are helpful for feature extraction; the linear activation that is used in the final layer of 
+the inverted residual structure prevents the loss of low-latitude information; and the 
+traditional convolution is replaced by depth-separable convolution, which significantly 
+reduces the amount of the model's calculations as well as the number of the parameters that it 
+uses. It was developed specifically for pictures and has applications in image categorization 
+as well as the development of generic features. 
+Convolution on a depth-wise and point-wise scale are the two components that make up 
+depth-separable convolution. The depth separable convolution algorithm is not the same as 
+the standard convolution algorithm. During the process of convolution, each channel of the 
+feature map is covered by exactly one convolution nucleus, and the total number of 
+convolution nuclei is equal to the total number of channels. The following phrases can be 
+used to describe depth convolution: 
+𝑂𝑥,𝑦,𝑐 = ∑
+𝐾𝑤,ℎ,𝑐
+𝑊,𝐻
+𝑤,ℎ
+⋅ 𝐼𝑥+𝑤,𝑦+ℎ,𝑐（formula1） 
+In the equation presented above, the variable O stands for the output feature graph, c stands 
+for the channel of the feature graph, x and y stand for the coordinates of the output feature 
+graph on channel, K stands for the convolutional kernel with wide W and high H, I stand for 
+the input feature graph, and w and h stand for the convolutional kernel weight element 
+coordinates of channel. 
+The primary difference between point-by-point and standard convolution is the size of the 
+convolution kernel, which in point-by-point convolution is fixed at 1x1. The first step of 
+depth separable convolution is to employ depth convolution to extract the characteristics of 
+each channel. Next, point-by-point convolution is used to correlate the extracted channel 
+characteristics. The depth separable convolution is intended to reduce the number of 
+parameters and computations required by the conventional convolution. When compared to 
+the total number of calculations involved in the traditional convolution, the results are as 
+follows: 
+𝑅1
+𝑅2 =
+𝐷𝑓2𝐷𝑘2𝐼+𝐷𝑓2𝐼𝑂
+𝐷𝑓2𝐷𝑘2𝐼𝑂
+=
+1
+𝑁 +
+1
+𝐷𝑘2（formula2） 
+In the formula 2 shown above: R1 and R2 represent the calculations of depth separable 
+convolution and standard convolution, respectively; Df and Dk represent the height and width 
+of the input feature matrix; I represent the depth of the input feature matrix; and O represents 
+the depth of the output feature matrix. 
+In the process of feature extraction, MobileNetV2 makes use of a depth detachable 
+convolution with a size of 3x3, which means that the calculation cost is 1/9 of what it would 
+be for a standard convolution. However, the reduction in accuracy is very minimal, which is 
+also one of the most notable qualities of the model. 
+
+Figure 1 presents the organizational structure of the MobileNetV2 network. It is primarily 
+made up of three components: the front end is a convolutional neural network (CNN), which 
+is constructed by several layers of convolution, and then the average pooling of 1,280 7x7 
+blocks is utilized to generate 1,280 nerves. Element, which was afterwards completely 
+coupled with one thousand neurons. 
+ 
+Figure1 MobileNet v2 based classifier 
+MobileNet v2 borrows ResNet's residual connection notion and uses it to suggest a reverse 
+residual structure, as seen in the picture. The structure employs PW convolution to increase 
+dimension, 3x3 DW convolution to extract channel characteristics, then PW convolution once 
+again to decrease feature dimension. Notably, as seen in Figure 2, a residual link exists only 
+when the step duration is 1. In the second situation, when the step size is 2, the series 
+structure is employed directly, and the reverse residual structure is upgraded and then 
+lowered, so that the network permits smaller input and output dimensions, hence decreasing 
+the amount of computation and parameters of the network. Simultaneously, residual 
+connection may increase the effectiveness of gradient propagation and deepen the network 
+layer. 
+ 
+Figure2 illustrative example in graphical form of the MobileNetV2 architecture. 
+MobileNet v2 networks are well-suited for implementation on mobile terminals or embedded 
+systems because to their small model size, cheap processing capacity, and high computation 
+
+Bottlneck15
+Bottlneck16
+Bottlneck17
+convolution
+Conv2d
+Pooling
+Softmax
+Global
+1x1
+imag
+prediction
+vobi
+assAdd
+Conv 1 x 1,
+Linear
+Conv 1 x 1,
+Linear
+Dwise 3 x 3,
+Stride=2
+Dwise 3 x 3
+Relu6
+Relu6
+Conv 1 x 1,
+Conv 1 x 1
+Relu6
+Relu6
+Input
+Input
+trideeil block
+Stride=2 blockspeed in comparison to more conventional convolutional neural networks. Moreover, it can 
+keep up with the CPU's strict speed needs. 
+II. 
+Transfer learning 
+It takes a significant amount of time and resources to train a deep neural network model on ea
+ch job from the very beginning. The learning process is laborious and time consuming. In ord
+er for the network to begin its learning process, it must first texture the outline. Subsequently,
+ the network must extract the characteristics after increasing the number of network layers. H
+owever, in the majority of instances, academics will use the information that they have gained
+ in the past to apply to the current challenges that they face. Learning by transfer is the name 
+given to this kind of information transmission (Zhuang et al., 2021). In the field of deep learn
+ing, transfer learning is combined with it, and the pre-training weight obtained by the model i
+n similar tasks is used. This is done to avoid model training from scratch, reduce the cost con
+sumption caused by the network model due to new learning, accelerate the network converge
+nce speed, and further improve the stability and generalization ability of the model (Song et a
+l., 2021). 
+The ImageNet data collection includes one thousand categories and encompasses one million 
+two hundred thousand pictures. There is no shadow of a doubt that the utilization of pre-train
+ed network models on such a massive data collection is capable of being successfully ported t
+o a variety of picture categorization endeavors. 
+In this study, transfer learning is carried out by pre-training a deep CNN model using a portio
+n of the ImageNet dataset. This portion of the dataset contains image classification data. A cl
+assifier was added to the pre-trained Mobilenet v2 model, and then it was used to download t
+he private dataset and split it into 10 primary categories. The pre-trained model was imported 
+via making use of the torch vision library, and samples from the private dataset were used for 
+the training and testing of the model. By doing things this way, we were able to cut down on t
+he time spent training the features extractor, which is what takes the most time for the majorit
+y of models, and instead train the classifier directly. Because of this, the amount of money sp
+ent on training has decreased. 
+ 
+ 
+Figure3 transfer learning 
+ 
+
+Source domain
+Pre-
+Source
+ Training all MobileNet-v2
+Weight of the Pre-
+training
+dataset
+network parameters
+training model
+Feature
+MobileNet-v2 Weights on
+Transfer Learning
+transfer
+ImagNet
+Target domain
+Target
+New
+Fixed the new model
+New model
+Learning
+dataset
+parameters
+WeightIII. 
+ Convolutional block attention module 
+The Convolutional Block Attention Module, often known as CBAM, is a type of attention 
+mechanism module that integrates Channel and Spatial. This module may be embedded into 
+the Model module, trained end-to-end with Model, and only contributes a little amount of 
+additional processing (Woo et al, 2018). Figure 4 demonstrates the Channel attention module 
+in addition to the Spatial attention module. The Channel Attention Module is capable of 
+taking an Intermediate Feature Map as an input and outputting both a 1-D Channel Attention 
+Map and a 2-D Spatial Attention Map through the use of the CBAM. In order to aggregate 
+the spatial information, the average pooling and maximum pooling methods are utilized. The 
+approaches provide two descriptors, which are then passed to the same shared network in 
+order to produce the channel attention map. The complete channel attention map can then be 
+acquired by doing the following: 
+𝑀𝑐(𝐹) = 𝜎 (𝑀(𝑎(𝐹)) + 𝑀(𝑚(𝐹))) = 𝜎 (𝑊1 (𝑊0 (𝐹avg)) + 𝑊1 (𝑊0 (𝐹max))) (formula 
+3) 
+The spatial attention module can highlight the information region and generate two 2D maps, 
+which are then linked and convolved by a standard convolutional layer to produce a 2D 
+spatial attention map. This map is produced after first averaging and maximum pooling along 
+the channel to generate efficient feature descriptors. In conclusion, the formulation for the 
+output of the spatial attention mapping is as follows: 
+𝑀𝑠(𝐹) = 𝜎 (𝑓 (𝑓𝑐 (𝐹avg，𝐹max))) (formula4) 
+F is the input feature graph, is the sig-moid nonlinear activation function, M is the forward 
+calculation function of multi-layer perceptron without bias, a and m are the mean and 
+maximum pooling functions respectively, W0 and W1 are the weights of 2 linear layer, and 
+Favg and Fmax are the mean and maximum pooling functions respectively. In the equation 
+that was just presented, F represents the input feature graph, represents the sig-moid 
+nonlinear. 
+ 
+Figure4 Channel attention module and Spatial attention module  
+It is necessary to arrange the Channel attention module and the Spatial attention module in a 
+sequential fashion in order to generate the CBAM. This provides the model with the ability to 
+concentrate on important features in both the channel and spatial dimensions while 
+suppressing features that are not necessary. Figure 5 provides a high-level view of CBAM, 
+while the following provides a concise summary of the complete calculating process: 
+
+Channel Attention Module
+MaxPool
+AvgPool
+Channel Attention
+Shared MLP
+Mc
+Input feature F
+Spatial Attention Module
+conv
+layer
+Channel-refined [MaxPool, AvgPool]
+Spatial Attention
+feature F
+Ms𝐹′ = 𝑀𝑐(𝐹) ⊗ 𝐹 (formula5) 
+𝐹″ = 𝑀𝑠(𝐹′) ⊗ 𝐹 (formula6) 
+It is necessary to arrange the Channel attention module and the Spatial attention module in a 
+sequential fashion in order to generate the CBAM. This provides the model with the ability to 
+concentrate on important features in both the channel and spatial dimensions while 
+suppressing features that are not necessary. Figure 5 provides a high-level view of CBAM, 
+while the following provides a concise summary of the complete calculating process: 
+ 
+ 
+ 
+Figure5 The overview of CBAM 
+This algorithm integrates the attention mechanism in the dimension of the channel and the 
+space, strengthens the important features while suppressing the unimportant features, and 
+further highlights the required features in order to improve the classification accuracy of the 
+model. The CBAM structure is introduced in the first layer of the Mobilenet v2 network by 
+this algorithm. Because CBAM is a lightweight module, it does not have a significant impact 
+on the total number of model parameters, but it does improve emphasis on the model's 
+primary features. This effectively guarantees that the benefits of a lightweight model will be 
+realized. 
+ 
+IV. 
+Perceptual hash algorithm 
+The perceptual hash algorithm (Samanta & Jain, 2021) begins by reducing the size of the 
+picture and simplifying the colors. Next, it aggregates the decomposition frequency and 
+trapezoidal shape of the picture by using the discrete cosine transformation method. After 
+that, it reduces the DCT, keeps the 8*8 matrix in the upper right corner, calculates the 
+average value of all 64 values, further reduces the DCT, and finally calculates the hash value. 
+The algorithm creates a unique string that acts as a "fingerprint" for each image, and it then 
+analyzes the results by contrasting the various "fingerprint" strings. When the result is closer, 
+it indicates that the picture is more comparable. The advantage of using this approach is that 
+regardless of whether the height, width, brightness, or color of the image is modified, the 
+result value of the hash value will not remain intact. This allows one to prevent the impact 
+that would be caused by the modest modification of the image. An online image 
+identification system is constructed in this research using the perceptual hash method and the 
+concept of searching for images using photos. 
+V. 
+Measurement 
+There is currently no standardized dedicated data collection that can be used for the 
+categorization of photos of lost and found items. The study of confidential data sets is the 
+
+Convolutional Block Attention Module
+Channel
+InputFeature
+Attention
+Spatial
+Refined Feature
+Module
+Attention
+Modulefocus of this work. Using the results of the poll, streamline the complicated lost and found 
+categories into the top 10 categories that are the easiest to lose track of, with a total of 10499 
+photographs. In addition, in order to match the conditions of a practical application, we have 
+not performed any pre-processing on the dataset. Directly scaling the collected lost and found 
+data set pictures to the input size of the network model will directly lead to the loss of some 
+information in the picture, or even direct distortion, and will ultimately directly affect the 
+image classification effect and recognition accuracy. This is because the original image had 
+an uneven quality and a high resolution. As a result, prior to the pre-training, this article 
+automates the data set photos in batches by using the design action in Adobe Photoshop 
+software. Additionally, the article standardizes the image resolution to 300 x 300, which 
+helps to minimize the overall picture size. The data processing of the sample data has to be 
+improved in this work in order to increase the capacity of the model to generalize its findings. 
+The 10,499 common lost and lost photographs augmented by the data  were divided into the 
+training set and validation set with 70% and validation set with 30%. 
+Table 1. Sample distribution of the lost object Figure slice dataset used in 
+this paper 
+Class name 
+Number of images 
+bag 
+1036 
+book 
+1067 
+card 
+1043 
+earphone 
+1025 
+key 
+1070 
+lipstick 
+1066 
+Phone 
+1068 
+umbrella 
+1045 
+USBflashdisk 
+1020 
+vacuumcup 
+1059 
+ 
+ 
+Figure6 The data inside the dataset 
+
+045
+实习生This article employs Accuracy, AgePrecision (AP), Recall, Precision, and Loss as model 
+performance evaluation indicators in order to analyse the advantages and disadvantages of the 
+model.  
+Calculating values using typical complex matrix approaches like as accuracy, recall, and 
+precision may be difficult. When it comes to image classification, the confusion matrix is 
+most commonly employed to make a comparison between the classification and the actual 
+measurement value. This allows for a more understandable and accurate description of the 
+correctness of model categorization. The formula for the computation is as follows: 
+Accuracy =
+𝑇𝑁+𝑇𝑃
+𝑇𝑁+𝐹𝑁+𝐹𝑃+𝑇𝑃（formula7） 
+Recall =
+𝑇𝑃
+𝑇𝑃+𝐹𝑁（formula8） 
+𝑃recision =
+𝑇𝑃
+𝑇𝑃+𝐹𝑃（formula9） 
+The number of successfully predicted positive samples is denoted by the letter TP in the 
+formula 7,8,9, the number of correctly anticipated negative samples is denoted by the letter 
+TN. FN is for the number of samples that had errors predicted for them when they were 
+positive, and FP stands for the number of samples that had errors predicted for them when 
+they were negative. 
+The performance of the model recognition accuracy of the selected model may be evaluated 
+in an objective manner by using the average accuracy rate. The mathematical formula for the 
+calculation is as follows: 
+𝐴𝑃 =
+∑ 𝐸
+1Accuracy
+𝐸
+（formula10） 
+The term AP, which is utilized in the aforementioned formula 10, stands for the average 
+accuracy of the model that was picked, while E stands for the total number of iterations, and 
+Accuracy stands for the accuracy of each iteration of the model that was selected. 
+The degree to which the model's predicted value and the actual value deviate from one 
+another may be estimated using the loss value. If the loss value is lower, then the model's 
+predicted result will be closer to the actual outcome. The formulae for the calculations are as 
+follows: 
+𝐿𝑂𝑆𝑆 =
+1
+𝑁 ∑ 𝐿𝑖
+𝑖
+=
+1
+𝑁 ∑ − ∑
+𝑦𝑖𝑐 𝑙𝑜𝑔( 𝑝𝑖𝑐)
+𝐾
+𝑐=1
+𝑖
+（formula11） 
+In the previous formula number 11, Loss is a numerical representation of the loss value 
+associated with the model that was chosen, and K is the total number of categories. pic is the 
+anticipated probability that sample I belongs to category c. If the category is the same as the 
+category of sample I it has a value of 1, and otherwise, it has a value of 0. The cross-entropy 
+loss function is a convex function, and it is possible to calculate the value that is best for the 
+whole world.  
+ 
+
+Experimental results: 
+I. 
+Training 
+In the course of our research, we looked at the problem of lost-object picture categorization. 
+In order to classify images, we made use of Convolutional block attention module in 
+conjunction with the well-known framework MobileNet v2. We utilized this architecture 
+(transfer learning) to speed up the learning process while also reducing the amount of time 
+needed for training. In order to evaluate the effectiveness of transfer deep learning, we 
+examined the outcomes of three different optimizer functions: the Adaptive Moment 
+Estimation Algorithm (ADAM), the Root Mean Square Propagation (RMSprop), and the 
+Stochastic gradient descent (SGD) algorithm. We noticed that the optimizer SGD function 
+functioned in a better method and got an accuracy of 96.8 percent, while the loss detected 
+was 0.0047. We figured this out by looking at the table. The results of the SGD function's 
+performance are presented in the form of a linear graph of the accuracy and loss parameters 
+in the figure. When compared to the other optimizers, SGD's training time for the model was 
+significantly shorter, despite the fact that all other model parameters remained same. 
+ 
+Table2. Comparative chart of results using optimizer function 
+ 
+Components 
+SGD 
+RMSprop 
+ADAM 
+Accuracy/%  
+96.8 
+89.5 
+94.7                                                                                                                                                                                                                                                                                             
+Loss  
+0.0047 
+0.0050 
+0.0048 
+Freeze Epoch 
+50 
+50 
+50 
+Freeze batch size 
+32 
+32 
+32 
+ UnFreeze Epoch 
+400 
+400 
+400 
+ Unfreeze batch size 
+64 
+64 
+64 
+Init lr 
+1e-2 
+1e-2 
+1e-2 
+Lr decay type 
+cos 
+cos 
+cos 
+ momentum 
+0.9 
+0.9 
+0.9 
+Execution Time/ minutes 
+5.38 
+14.28 
+10.71 
+ 
+ 
+Figure7a(left) Training and validation curve of the model 
+Figure7b(right) the valuation index of different measure index 
+ 
+It is abundantly evident, after reviewing the preceding explanation complete with graphs and 
+table, that the optimizer function SGD performed far better than the performance of the other 
+functions when applied to the issue of lost-object picture classification. 
+EfficientNet, Inception v4, ViT-B. 32, DenseNet201, VGGNet19, ShuffleNet v2, ResNet152 
+and MobileNet v3 were the eight transfer learning algorithm models of the same kind that 
+
+Acccurve
+1.0
+60
+0.8
+0.7
+0.6
+train_acc
+val_acc
+0
+50
+100
+150
+200
+250
+OOE
+350
+400
+EpochModel evaluation value
+0.991140
+0.991135
+0.991130
+0.991125
+0.991120
+0.991115
+0.991110
+0.991105
+0.991100
+0.991095
+0.991090
+Accuracy
+Recall
+Precision
+F1-Scorewere chosen at random for the purpose of doing a comparison study in order to test the 
+efficiency and superiority of the research approach that was provided in this work.The 
+optimal parameter values of the model convergence were selected as the respective 
+parameters.This study's private data set is used to train and verify a total of nine models using 
+these settings. The outcomes of the training are presented in Table 3. 
+Table3. Results of the selected model training 
+ 
+Figure8 The performance, accuracy and parameter amount compare of difference models 
+The algorithm that is suggested in this article is transfer learning, which is based on the 
+improved MobileNetv2. This can be seen in Table 3. The results of the experiments 
+demonstrate that the method under study has an average accuracy of 96.2% when applied to 
+Model 
+Accura
+cy/% 
+AP/% 
+Total 
+parameters/M 
+EfficientNet 
+（Koonce et al，2021） 
+93.9 
+89.0 
+66.348 
+Inception v4 
+（Szegedy et al，2017） 
+89.5 
+80.3 
+41.158 
+ViT-B/32 
+（Dosovitskiy et al， 
+2010） 
+94.9 
+94.3 
+104.766 
+DenseNet201 
+（Huang et al， 
+2017） 
+95.9 
+95.4 
+20.014 
+VGGNet19 
+（SIMONYAN et al， 
+2021） 
+95.6 
+94.7 
+139.611 
+ShuffleNet v2 
+（Ma et al，2018） 
+88.6 
+82.3 
+2.279 
+ResNet152 
+（He et al，2016） 
+87.8 
+81.9 
+60.193 
+MobileNet v3 
+（Koonce et al，2021） 
+95.7 
+94.9 
+5.483 
+us 
+96.8 
+96.2 
+3.505 
+
+10
+98-
+96
+9-1-
+AccuTacy(%)
+92-
+06
+EllicientNel
+Trceplion 4
+WiT-B:32
+88
+DenseNet201
+VGGNet19
+ShulmleNet 2 x1 0
+86-
+ResNet152
+MobileNet v3
+Ours
+64
+111
+1
+101
+GLFOPs(G)data sets that were self-built. This is a higher accuracy rate than EfficientNet, Inception v4, 
+ViT-B. 32, DenseNet201, VGGNet19, ShuffleNet v2, ResNet152 and MobileNet v3 
+correspondingly. 7.2%, 15.9%, 1.9%,0.8%,1.5%, 13.9%, 14.3% and 1.3% of the total 
+learning was transferred from other models. The suggested technique has obtained the 
+greatest accuracy rate in private data sets, which is 96.8%, along with strong generalization 
+ability and resilience. When compared to the similar kind of transfer learning model that has 
+been developed, this is how it stands out. In correlation with the information shown in Figure 
+8, The ordinate in the figure shows the test accuracy, the abscissa represents GFLOPs, the 
+circle colors reflect distinct transfer learning models, and the figure size denotes total 
+parameters. The suggested method gets the maximum accuracy on private datasets, and it is 
+robust and has a decent ability to generalize. When Figure 8 and Table 3 are considered 
+together, a further complete evaluation of the performance of each model is carried out. This 
+model is only second to MobileNet v3 and ShuffleNet v2, but it is worth recognizing that the 
+Total parameters of the proposed model are almost half that of MobileNet v3 Total 
+parameters, and the accuracy is higher than ShuffleNet v2 8.2%. MobileNet v3 and 
+ShuffleNet v2 are the only models that are ahead of this model. In comprehensive 
+comparison, the model that is being offered is a fantastic lightweight network, which 
+provides the possibility that the model might be transplanted to mobile devices. 
+ 
+ 
+Figure9a(left) Confusion matrix of the model 
+Figure9b(right) ROC curve of the model 
+We decided to combine the confusion matrix from the model single-picture test with the 
+ROC curve for the study so that we could have a more in-depth look at the data. The 
+confusion matrix is the method for measuring the accuracy of classification models that is the 
+simplest and most straightforward. Calculating the confusion matrix of the suggested model 
+requires using equation (7) and equation (9). This matrix represents the identification result of 
+the different types of lost object photographs and is derived from these equations. The 
+confusion matrix is shown in Figure 9a, with the rows representing the expected category for 
+the item that was lost, and the columns representing the actual category for the object that 
+was lost. Figure 9b illustrates the characteristic curves of the class 10 lost item categories for 
+the enhanced MobileNetv2 transfer learning model. ROC stands for receiver operating 
+characteristic. The fact that the curves for all ten classes are located in the top left corner is 
+evidence that the model is accurate, and the fact that the area under the ROC curve for each 
+class is equal to one further demonstrates the superiority and efficiency of the model that was 
+provided. 
+ 
+ 
+
+book
+Confusion Matrix
+USBflashdisk
+bag
+card
+earphone
+lipstick
+phone
+USBflashdisk
+315
+0
+0
+：
+o
+0
+30C
+bag
+309
+0
+2
+250
+book
+0
+313
+0
+0
+card
+313
+0
+200
+earphone
+0
+312
+0
+0
+0
+anul
+150
+key
+0
+314
+lipstick 
+0
+314
+100
+phone
+0
+1
+0
+0
+0
+314
+0
+umbrella
+E
+LOE
+50
+vacuumcup
+0
+1
+0
+1
+0
+311
+Predicted label1.0
+0.8
+True Positive Rate
+macro-average RoC curve (area = 1.oo)
+0.6
+ROC curve of USBflashdisk (area = 1.0o)
+RoC curve of bag (area=1.oo)
+RoC curve of book (area = 1.0o)
+0.4 
+ROC curve of card (area = 1.0o)
+RoC curve of earphone (area = 1.oo)
+RoC curve of key (area = 1.00)
+ROC curveof lipstick(area=1.0o)
+0.2 -
+RoC curve of phone (area =1.oo)
+RoCcurve of umbrella(area=1.oo)
+RoC curveof vacuumcup (area= 1.oo)
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+False Positive RateII. 
+Inference 
+In order to successfully apply the model to the actual world, it needs to be simple enough that 
+it can be executed on most laptops. In order to accomplish this, we implement this model as 
+well as a model that is very close to it on a regular computer that has a CPU, and we 
+comprehensively consider the inference speed of the computer as the evaluation index. It 
+takes the model 1.5 seconds to reason per picture on the CPU, which is significantly longer 
+than the standard migration model of this type or even 1/16 of the EfficientNet's processing 
+time. This can be seen in Figure 10a. Because it is such an exceptionally lightweight network, 
+it is not hard to imagine that the concept will eventually be adapted to work with other mobile 
+devices in the not too distant future. In light of this, we provide an engineering system that 
+integrates the perceptual hashing method. 
+ 
+This paper, which is based on the model training using the pytorch and torchvision machine 
+learning frameworks, uses the Spring Boot framework as the back-end service of the web 
+page in order to realize the online recognition of pictures. The front-end web page uses the 
+Layui framework in order to port the trained model to the Web end in order to realize the 
+online recognition of images that are uploaded by users. The steps involved in the online 
+identification procedure are as follows: the user launches the browser, navigates to the 
+system's website, navigates to the system's home page, clicks the button to upload the local 
+image, the front end sends the POST request to the back end, and the image is transmitted to 
+the back end. After the back end has received the picture that was uploaded by the user, it 
+will first make a request to the trained and upgraded MobileNet v2 migration learning model 
+for prediction and recognition, and it will then return a category. After it has determined the 
+category, it will next submit a request to the database, asking it to deliver an array containing 
+the picture address associated with the category it has just determined. After obtaining the 
+address, the image is then downloaded by using the image address, and then the perceptual 
+hash algorithm is used to compare the downloaded image to the image that was uploaded by 
+the user. The alignment will return a number for the similarity error, and the array will be 
+used to send the few photos that have the least significant value for the similarity error to the 
+front end. Figure 10b depicts the engineering interface after the model has been applied to the 
+situation. 
+ 
+Figure10a(left) The speed of inference of each model in cpu 
+Figure10b(right) The improved model is applied to the actual engineering interface diagram 
+ 
+
+100-
+-86
+96
+94
+92
+F06
+EfficientNet
+Inception_v4
+88-
+★
+ViT-B/32
+DenseNet201
+86
+VGGNet19
+ShuffleNet_v2_x1_0
+84
+★
+ResNet152
+MobileNet_v3
+Ours
+82
++08
+0
+5
+10
+15
+20
+Inference speed using CPU(second)LostNet:AsmartwayforlostandfindConcludes: 
+This research presents a novel design plan in light of the difficulties that are associated 
+with the management of urban traffic operators. Even while the method of merging 
+convolutional neural networks with transfer learning has caused quite a stir in the world of 
+image recognition, it is not widely utilized when it comes to the automatic recognition of 
+photos that have been misplaced or found. This study offers a new technique of intelligent 
+picture identification based on hash algorithm to give full play to the advantages of hash 
+algorithm. The approach makes use of the concept of "search by map," and it is based on 
+hash algorithm. Because there are a significant number of missing images in the library and a 
+great number of feature points that have been extracted, the CBAM structure has been 
+introduced in the first layer of the network. This has been done in order to integrate the 
+attention mechanism in the dimension of channel and space in order to further highlight the 
+required features, which will ultimately lead to an improvement in the classification accuracy 
+of the model. Improved MobileNetv2 is used to establish a transfer learning training model to 
+identify common lost objects. This is done in order to reduce the number of ways in which 
+lost objects and database pictures can be compared to one another, thereby making it possible 
+to search for objects in a more convenient manner. Extensive experimental arguments on 
+photos taken from the loss-object dataset, employing a variety of transfer learning models. 
+The findings indicate that the proposed model has a high recognition accuracy in the GFLOPs 
+and Total parameters extremes, and the features derived by this model significantly beat those 
+extracted by the other approaches when it comes to the classification job. In order to enhance 
+the precision with which missing object categories may be identified, the proposed model is 
+being developed and put into practice in the search sector. 
+However, further solutions are required in order to further increase the accuracy of the 
+model and the segmentation of the various kinds of lost object. In order to do this, the 
+following stage will involve further subdividing each category so that it can be utilized in a 
+wider variety of search circumstances. 
+ 
+Acknowledgement: 
+This work was supported by Provincial College Students' Innovation and Entrepreneurship 
+Project Project for College Students [Grant numbers S202110368112]; University 
+Humanities and Social Science Research Program of Anhui Province [Grant numbers 
+SK2020A0380]; School level Project of Key Humanities and Social Sciences Research Base 
+of Anhui Province, Center for Mental Health Education of College Students [Grant numbers 
+SJD202001]; and School level Project of the Young and Middle-Aged Natural Science 
+Foundation of Wannan Medical College[Grant numbers WK202115] 
+ 
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf,len=565
+page_content='LostNet: A smart way for lost and find  Meihua Zhou1, Ivan Fung2, Li Yang1, Nan Wan1, Keke Di1  Tingting Wang*  1 School of Medical Information, Wannan Medical Collage, Wuhu City, China  {mhzhou, 20120037, wannan, mhzhou0412}@wnmc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='cn  2 work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='ivanfung@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='com  School of Medical Information, Wannan Medical Collage, Wuhu City, China  wangtt@wnmc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='cn  Abstract:  Due to the enormous population growth of cities in recent years, objects are frequently lost  and unclaimed on public transportation, in restaurants, or any other public areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' While  services like Find My iPhone can easily identify lost electronic devices, more valuable  objects cannot be tracked in an intelligent manner, making it impossible for administrators to  reclaim a large number of lost and found items in a timely manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' We present a method that  significantly reduces the complexity of searching by comparing previous images of lost and  recovered things provided by the owner with photos taken when registered lost and found  items are received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' In this research, we will primarily design a photo matching network by  combining the fine-tuning method of MobileNetv2 with CBAM Attention and using the  Internet framework to develop an online lost and found image identification system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Our  implementation gets a testing accuracy of 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='8% using only 665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='12M GLFOPs and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='5M  training parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' It can recognize practice images and can be run on a regular laptop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Keyword: artificial intelligence, lost and found, intelligent city  Introduction:  The population density and the quantity of lost objects are both rising in the areas where  urban rail transit is located around the world at the present time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' yet, the traditional manual  search service is ineffective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' In this situation, there is an immediate need to speed up the  development of intelligent lost and found systems in order to lessen the difficulty that  transportation operators face when it comes to lost and found management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Deep learning can  be used to build recognition and classification models for lost and found items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' This is a new  method that can reduce reliance on manual labour, quickly and accurately identify categories,  significantly reduce human service costs for transportation operators, and better practice  green development strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  The application of convolutional neural networks is widespread throughout many scientific  subfields pertaining to image recognition and classification (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' It is not a  passing fad that academics in a wide variety of fields have shifted their attention to the study  and practical use of image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' In the realm of trash classification, a method for  garbage image classification was devised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' It was built on an enhanced version of MobileNet  v2 and paired with transfer learning to increase the real-time performance and accuracy of  garbage image classification models (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Research Background:  Many academics are also engaged in questioning the status quo in the many subfields that  make up the area of applied image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=', 2015) attempting to identify  plant leaves by the utilization of a hierarchical model that is based on CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' A study on  establishing the optimal size of the training data set that is required to achieve high  classification accuracy with low variance in medical image classification systems is presented  by Cho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' (Purnama et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=', 2019) offer a method for the classification and  diagnosis of skin diseases that is suitable for use in teledermatology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  It has also been demonstrated that transfer learning is beneficial in a variety of contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Convolutional neural networks are used in the methodology that Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' (2016) propose as  a fine-grained classification method for large-scale plankton databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The implementation  of transfer learning in CNN is one potential solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=', 2020) apply unsupervised  transfer learning to CNN training to address these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Specifically, they transform  similarity learning into deep ordinal classification with the assistance of several CNN experts  who were pretrained over large-scale-labelled everyday image sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' These CNN experts  jointly determine image similarities and provide pseudo labels for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' (Purwar et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=', 2020) make use of some models that are related to convolutional neural networks (CNNs)  in order to identify mesangial hypercellularity in MEST-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' (Herzog et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=', 2021) concentrate  on the classification of MRI for the diagnosis of early and progressive dementia by utilizing  transfer learning architectures that employ Convolutional Neural Networks-CNNs, as a base  model, and fully connected layers of SoftMax functions or Support Vector Machines-SVMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  (Phankokkruad, 2021) propose the three CNN models for detecting lung cancer using  VGG16, ResNet50V2, and DenseNet201 architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The proposed method is based on  transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Methodology:  In a procedure in which CNN is being used to address more and more components of the  problem, the lost and found problem has not been solved in any intelligent method as of yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  As a result, we recommend taking a methodical approach by utilizing Mobilnet v2 and an  intuitive graphical user interface (GUI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' In this study, we build on earlier research to further  investigate the detection and categorization of lost and discovered items, and we present an  approach that combines perceptual hashing with MobileNet v2 transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  In order to solve the relatively complex problems associated with the image dataset of lost  and found items, we carried out extensive research, classified the most common items that are  lost and found into ten categories using questionnaires and market research, and produced  private dataset images using crawlers, real-world photography, and research examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' After  that, the generated dataset is used to train the network and establish an intelligent recognition  classification model for lost and found images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' This model solves the problems of labor cost  and time cost consumption that are associated with conventional methods and proposes an  accurate and complete solution with scientific and accurate experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Mobilenet v2  An outstanding example of a good lightweight convolutional neural network is MobileNetV2  (Sandler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The network creates a reverse residual and linear bottlenecks, both of  which are helpful for feature extraction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' the linear activation that is used in the final layer of  the inverted residual structure prevents the loss of low-latitude information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=" and the  traditional convolution is replaced by depth-separable convolution, which significantly  reduces the amount of the model's calculations as well as the number of the parameters that it  uses." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' It was developed specifically for pictures and has applications in image categorization  as well as the development of generic features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Convolution on a depth-wise and point-wise scale are the two components that make up  depth-separable convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The depth separable convolution algorithm is not the same as  the standard convolution algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' During the process of convolution, each channel of the  feature map is covered by exactly one convolution nucleus, and the total number of  convolution nuclei is equal to the total number of channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The following phrases can be  used to describe depth convolution:  𝑂𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='𝑐 = ∑  𝐾𝑤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='𝑐  𝑊,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='𝐻  𝑤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='ℎ  ⋅ 𝐼𝑥+𝑤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='𝑦+ℎ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='𝑐（formula1）  In the equation presented above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' the variable O stands for the output feature graph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' c stands  for the channel of the feature graph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' x and y stand for the coordinates of the output feature  graph on channel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' K stands for the convolutional kernel with wide W and high H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' I stand for  the input feature graph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' and w and h stand for the convolutional kernel weight element  coordinates of channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  The primary difference between point-by-point and standard convolution is the size of the  convolution kernel, which in point-by-point convolution is fixed at 1x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The first step of  depth separable convolution is to employ depth convolution to extract the characteristics of  each channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Next, point-by-point convolution is used to correlate the extracted channel  characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The depth separable convolution is intended to reduce the number of  parameters and computations required by the conventional convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' When compared to  the total number of calculations involved in the traditional convolution, the results are as  follows:  𝑅1  𝑅2 =  𝐷𝑓2𝐷𝑘2𝐼+𝐷𝑓2𝐼𝑂  𝐷𝑓2𝐷𝑘2𝐼𝑂  =  1  𝑁 +  1  𝐷𝑘2（formula2）  In the formula 2 shown above: R1 and R2 represent the calculations of depth separable  convolution and standard convolution, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Df and Dk represent the height and width  of the input feature matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' I represent the depth of the input feature matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' and O represents  the depth of the output feature matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  In the process of feature extraction, MobileNetV2 makes use of a depth detachable  convolution with a size of 3x3, which means that the calculation cost is 1/9 of what it would  be for a standard convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' However, the reduction in accuracy is very minimal, which is  also one of the most notable qualities of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Figure 1 presents the organizational structure of the MobileNetV2 network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' It is primarily  made up of three components: the front end is a convolutional neural network (CNN), which  is constructed by several layers of convolution, and then the average pooling of 1,280 7x7  blocks is utilized to generate 1,280 nerves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Element, which was afterwards completely  coupled with one thousand neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content="  Figure1 MobileNet v2 based classifier  MobileNet v2 borrows ResNet's residual connection notion and uses it to suggest a reverse  residual structure, as seen in the picture." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The structure employs PW convolution to increase  dimension, 3x3 DW convolution to extract channel characteristics, then PW convolution once  again to decrease feature dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Notably, as seen in Figure 2, a residual link exists only  when the step duration is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' In the second situation, when the step size is 2, the series  structure is employed directly, and the reverse residual structure is upgraded and then  lowered, so that the network permits smaller input and output dimensions, hence decreasing  the amount of computation and parameters of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Simultaneously, residual  connection may increase the effectiveness of gradient propagation and deepen the network  layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Figure2 illustrative example in graphical form of the MobileNetV2 architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  MobileNet v2 networks are well-suited for implementation on mobile terminals or embedded  systems because to their small model size,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' cheap processing capacity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' and high computation  Bottlneck15  Bottlneck16  Bottlneck17  convolution  Conv2d  Pooling  Softmax  Global  1x1  imag  prediction  vobi  assAdd  Conv 1 x 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Linear  Conv 1 x 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Linear  Dwise 3 x 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Stride=2  Dwise 3 x 3  Relu6  Relu6  Conv 1 x 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Conv 1 x 1  Relu6  Relu6  Input  Input  trideeil block  Stride=2 blockspeed in comparison to more conventional convolutional neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=" Moreover, it can  keep up with the CPU's strict speed needs." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Transfer learning  It takes a significant amount of time and resources to train a deep neural network model on ea  ch job from the very beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The learning process is laborious and time consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' In ord  er for the network to begin its learning process, it must first texture the outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Subsequently,  the network must extract the characteristics after increasing the number of network layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' H  owever, in the majority of instances, academics will use the information that they have gained  in the past to apply to the current challenges that they face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Learning by transfer is the name  given to this kind of information transmission (Zhuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' In the field of deep learn  ing, transfer learning is combined with it, and the pre-training weight obtained by the model i  n similar tasks is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' This is done to avoid model training from scratch, reduce the cost con  sumption caused by the network model due to new learning, accelerate the network converge  nce speed, and further improve the stability and generalization ability of the model (Song et a  l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  The ImageNet data collection includes one thousand categories and encompasses one million  two hundred thousand pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' There is no shadow of a doubt that the utilization of pre-train  ed network models on such a massive data collection is capable of being successfully ported t  o a variety of picture categorization endeavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  In this study, transfer learning is carried out by pre-training a deep CNN model using a portio  n of the ImageNet dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' This portion of the dataset contains image classification data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' A cl  assifier was added to the pre-trained Mobilenet v2 model, and then it was used to download t  he private dataset and split it into 10 primary categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The pre-trained model was imported  via making use of the torch vision library, and samples from the private dataset were used for  the training and testing of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' By doing things this way, we were able to cut down on t  he time spent training the features extractor, which is what takes the most time for the majorit  y of models, and instead train the classifier directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Because of this, the amount of money sp  ent on training has decreased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Figure3 transfer learning  Source domain  Pre-  Source  Training all MobileNet-v2  Weight of the Pre-  training  dataset  network parameters  training model  Feature  MobileNet-v2 Weights on  Transfer Learning  transfer  ImagNet  Target domain  Target  New  Fixed the new model  New model  Learning  dataset  parameters  WeightIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Convolutional block attention module  The Convolutional Block Attention Module, often known as CBAM, is a type of attention  mechanism module that integrates Channel and Spatial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' This module may be embedded into  the Model module, trained end-to-end with Model, and only contributes a little amount of  additional processing (Woo et al, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Figure 4 demonstrates the Channel attention module  in addition to the Spatial attention module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The Channel Attention Module is capable of  taking an Intermediate Feature Map as an input and outputting both a 1-D Channel Attention  Map and a 2-D Spatial Attention Map through the use of the CBAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' In order to aggregate  the spatial information, the average pooling and maximum pooling methods are utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The  approaches provide two descriptors, which are then passed to the same shared network in  order to produce the channel attention map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The complete channel attention map can then be  acquired by doing the following:  𝑀𝑐(𝐹) = 𝜎 (𝑀(𝑎(𝐹)) + 𝑀(𝑚(𝐹))) = 𝜎 (𝑊1 (𝑊0 (𝐹avg)) + 𝑊1 (𝑊0 (𝐹max))) (formula  3)  The spatial attention module can highlight the information region and generate two 2D maps,  which are then linked and convolved by a standard convolutional layer to produce a 2D  spatial attention map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' This map is produced after first averaging and maximum pooling along  the channel to generate efficient feature descriptors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' In conclusion, the formulation for the  output of the spatial attention mapping is as follows:  𝑀𝑠(𝐹) = 𝜎 (𝑓 (𝑓𝑐 (𝐹avg，𝐹max))) (formula4)  F is the input feature graph, is the sig-moid nonlinear activation function, M is the forward  calculation function of multi-layer perceptron without bias, a and m are the mean and  maximum pooling functions respectively, W0 and W1 are the weights of 2 linear layer, and  Favg and Fmax are the mean and maximum pooling functions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' In the equation  that was just presented, F represents the input feature graph, represents the sig-moid  nonlinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Figure4 Channel attention module and Spatial attention module  It is necessary to arrange the Channel attention module and the Spatial attention module in a  sequential fashion in order to generate the CBAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' This provides the model with the ability to  concentrate on important features in both the channel and spatial dimensions while  suppressing features that are not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Figure 5 provides a high-level view of CBAM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  while the following provides a concise summary of the complete calculating process:  Channel Attention Module  MaxPool  AvgPool  Channel Attention  Shared MLP  Mc  Input feature F  Spatial Attention Module  conv  layer  Channel-refined [MaxPool,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' AvgPool]  Spatial Attention  feature F  Ms𝐹′ = 𝑀𝑐(𝐹) ⊗ 𝐹 (formula5)  𝐹″ = 𝑀𝑠(𝐹′) ⊗ 𝐹 (formula6)  It is necessary to arrange the Channel attention module and the Spatial attention module in a  sequential fashion in order to generate the CBAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' This provides the model with the ability to  concentrate on important features in both the channel and spatial dimensions while  suppressing features that are not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Figure 5 provides a high-level view of CBAM,  while the following provides a concise summary of the complete calculating process:  Figure5 The overview of CBAM  This algorithm integrates the attention mechanism in the dimension of the channel and the  space, strengthens the important features while suppressing the unimportant features, and  further highlights the required features in order to improve the classification accuracy of the  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The CBAM structure is introduced in the first layer of the Mobilenet v2 network by  this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=" Because CBAM is a lightweight module, it does not have a significant impact  on the total number of model parameters, but it does improve emphasis on the model's  primary features." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' This effectively guarantees that the benefits of a lightweight model will be  realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Perceptual hash algorithm  The perceptual hash algorithm (Samanta & Jain, 2021) begins by reducing the size of the  picture and simplifying the colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Next, it aggregates the decomposition frequency and  trapezoidal shape of the picture by using the discrete cosine transformation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' After  that, it reduces the DCT, keeps the 8*8 matrix in the upper right corner, calculates the  average value of all 64 values, further reduces the DCT, and finally calculates the hash value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  The algorithm creates a unique string that acts as a "fingerprint" for each image, and it then  analyzes the results by contrasting the various "fingerprint" strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' When the result is closer,  it indicates that the picture is more comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The advantage of using this approach is that  regardless of whether the height, width, brightness, or color of the image is modified, the  result value of the hash value will not remain intact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' This allows one to prevent the impact  that would be caused by the modest modification of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' An online image  identification system is constructed in this research using the perceptual hash method and the  concept of searching for images using photos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Measurement  There is currently no standardized dedicated data collection that can be used for the  categorization of photos of lost and found items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The study of confidential data sets is the  Convolutional Block Attention Module  Channel  InputFeature  Attention  Spatial  Refined Feature  Module  Attention  Modulefocus of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Using the results of the poll, streamline the complicated lost and found  categories into the top 10 categories that are the easiest to lose track of, with a total of 10499  photographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' In addition, in order to match the conditions of a practical application, we have  not performed any pre-processing on the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Directly scaling the collected lost and found  data set pictures to the input size of the network model will directly lead to the loss of some  information in the picture, or even direct distortion, and will ultimately directly affect the  image classification effect and recognition accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' This is because the original image had  an uneven quality and a high resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' As a result, prior to the pre-training, this article  automates the data set photos in batches by using the design action in Adobe Photoshop  software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Additionally, the article standardizes the image resolution to 300 x 300, which  helps to minimize the overall picture size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The data processing of the sample data has to be  improved in this work in order to increase the capacity of the model to generalize its findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  The 10,499 common lost and lost photographs augmented by the data  were divided into the  training set and validation set with 70% and validation set with 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Sample distribution of the lost object Figure slice dataset used in  this paper  Class name  Number of images  bag  1036  book  1067  card  1043  earphone  1025  key  1070  lipstick  1066  Phone  1068  umbrella  1045  USBflashdisk  1020  vacuumcup  1059  Figure6 The data inside the dataset  045  实习生This article employs Accuracy, AgePrecision (AP), Recall, Precision, and Loss as model  performance evaluation indicators in order to analyse the advantages and disadvantages of the  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Calculating values using typical complex matrix approaches like as accuracy, recall, and  precision may be difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' When it comes to image classification, the confusion matrix is  most commonly employed to make a comparison between the classification and the actual  measurement value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' This allows for a more understandable and accurate description of the  correctness of model categorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The formula for the computation is as follows:  Accuracy =  𝑇𝑁+𝑇𝑃  𝑇𝑁+𝐹𝑁+𝐹𝑃+𝑇𝑃（formula7）  Recall =  𝑇𝑃  𝑇𝑃+𝐹𝑁（formula8）  𝑃recision =  𝑇𝑃  𝑇𝑃+𝐹𝑃（formula9）  The number of successfully predicted positive samples is denoted by the letter TP in the  formula 7,8,9, the number of correctly anticipated negative samples is denoted by the letter  TN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' FN is for the number of samples that had errors predicted for them when they were  positive, and FP stands for the number of samples that had errors predicted for them when  they were negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  The performance of the model recognition accuracy of the selected model may be evaluated  in an objective manner by using the average accuracy rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The mathematical formula for the  calculation is as follows:  𝐴𝑃 =  ∑ 𝐸  1Accuracy  𝐸  （formula10）  The term AP, which is utilized in the aforementioned formula 10, stands for the average  accuracy of the model that was picked, while E stands for the total number of iterations, and  Accuracy stands for the accuracy of each iteration of the model that was selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content="  The degree to which the model's predicted value and the actual value deviate from one  another may be estimated using the loss value." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=" If the loss value is lower, then the model's  predicted result will be closer to the actual outcome." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The formulae for the calculations are as  follows:  𝐿𝑂𝑆𝑆 =  1  𝑁 ∑ 𝐿𝑖  𝑖  =  1  𝑁 ∑ − ∑  𝑦𝑖𝑐 𝑙𝑜𝑔( 𝑝𝑖𝑐)  𝐾  𝑐=1  𝑖  （formula11）  In the previous formula number 11, Loss is a numerical representation of the loss value  associated with the model that was chosen, and K is the total number of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' pic is the  anticipated probability that sample I belongs to category c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' If the category is the same as the  category of sample I it has a value of 1, and otherwise, it has a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The cross-entropy  loss function is a convex function, and it is possible to calculate the value that is best for the  whole world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Experimental results:  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Training  In the course of our research, we looked at the problem of lost-object picture categorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  In order to classify images, we made use of Convolutional block attention module in  conjunction with the well-known framework MobileNet v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' We utilized this architecture  (transfer learning) to speed up the learning process while also reducing the amount of time  needed for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' In order to evaluate the effectiveness of transfer deep learning, we  examined the outcomes of three different optimizer functions: the Adaptive Moment  Estimation Algorithm (ADAM), the Root Mean Square Propagation (RMSprop), and the  Stochastic gradient descent (SGD) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' We noticed that the optimizer SGD function  functioned in a better method and got an accuracy of 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='8 percent, while the loss detected  was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='0047.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' We figured this out by looking at the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=" The results of the SGD function's  performance are presented in the form of a linear graph of the accuracy and loss parameters  in the figure." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=" When compared to the other optimizers, SGD's training time for the model was  significantly shorter, despite the fact that all other model parameters remained same." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Table2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Comparative chart of results using optimizer function  Components  SGD  RMSprop  ADAM  Accuracy/%  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='8  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='5  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='7  Loss  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='0047  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='0050  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='0048  Freeze Epoch  50  50  50  Freeze batch size  32  32  32  UnFreeze Epoch  400  400  400  Unfreeze batch size  64  64  64  Init lr  1e-2  1e-2  1e-2  Lr decay type  cos  cos  cos  momentum  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='9  Execution Time/ minutes  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='38  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='28  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='71  Figure7a(left) Training and validation curve of the model  Figure7b(right) the valuation index of different measure index  It is abundantly evident, after reviewing the preceding explanation complete with graphs and  table, that the optimizer function SGD performed far better than the performance of the other  functions when applied to the issue of lost-object picture classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  EfficientNet, Inception v4, ViT-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' 32, DenseNet201, VGGNet19, ShuffleNet v2, ResNet152  and MobileNet v3 were the eight transfer learning algorithm models of the same kind that  Acccurve  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='0  60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='6  train_acc  val_acc  0  50  100  150  200  250  OOE  350  400  EpochModel evaluation value  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='991140  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
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+page_content='991125  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='991120  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='991115  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
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+page_content='991105  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='991100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='991095  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='991090  Accuracy  Recall  Precision  F1-Scorewere chosen at random for the purpose of doing a comparison study in order to test the  efficiency and superiority of the research approach that was provided in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='The  optimal parameter values of the model convergence were selected as the respective  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content="This study's private data set is used to train and verify a total of nine models using  these settings." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The outcomes of the training are presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Table3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Results of the selected model training  Figure8 The performance, accuracy and parameter amount compare of difference models  The algorithm that is suggested in this article is transfer learning, which is based on the  improved MobileNetv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' This can be seen in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The results of the experiments  demonstrate that the method under study has an average accuracy of 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='2% when applied to  Model  Accura  cy/%  AP/%  Total  parameters/M  EfficientNet  （Koonce et al，2021）  93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='9  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='0  66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='348  Inception v4  （Szegedy et al，2017）  89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='5  80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='3  41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='158  ViT-B/32  （Dosovitskiy et al，  2010）  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='9  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='3  104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='766  DenseNet201  （Huang et al，  2017）  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='9  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='4  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='014  VGGNet19  （SIMONYAN et al，  2021）  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='6  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='7  139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='611  ShuffleNet v2  （Ma et al，2018）  88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='6  82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='3  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='279  ResNet152  （He et al，2016）  87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='8  81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='9  60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='193  MobileNet v3  （Koonce et al，2021）  95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='7  94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='483  us  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='8  96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='505  10  98-  96  9-1-  AccuTacy(%)  92-  06  EllicientNel  Trceplion 4  WiT-B:32  88  DenseNet201  VGGNet19  ShulmleNet 2 x1 0  86-  ResNet152  MobileNet v3  Ours  64  111  1  101  GLFOPs(G)data sets that were self-built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' This is a higher accuracy rate than EfficientNet, Inception v4,  ViT-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' 32, DenseNet201, VGGNet19, ShuffleNet v2, ResNet152 and MobileNet v3  correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='2%, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='9%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='9%,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='8%,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='5%, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='9%, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='3% and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='3% of the total  learning was transferred from other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The suggested technique has obtained the  greatest accuracy rate in private data sets, which is 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='8%, along with strong generalization  ability and resilience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' When compared to the similar kind of transfer learning model that has  been developed, this is how it stands out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' In correlation with the information shown in Figure  8, The ordinate in the figure shows the test accuracy, the abscissa represents GFLOPs, the  circle colors reflect distinct transfer learning models, and the figure size denotes total  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The suggested method gets the maximum accuracy on private datasets, and it is  robust and has a decent ability to generalize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' When Figure 8 and Table 3 are considered  together, a further complete evaluation of the performance of each model is carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' This  model is only second to MobileNet v3 and ShuffleNet v2, but it is worth recognizing that the  Total parameters of the proposed model are almost half that of MobileNet v3 Total  parameters, and the accuracy is higher than ShuffleNet v2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' MobileNet v3 and  ShuffleNet v2 are the only models that are ahead of this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' In comprehensive  comparison, the model that is being offered is a fantastic lightweight network, which  provides the possibility that the model might be transplanted to mobile devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Figure9a(left) Confusion matrix of the model  Figure9b(right) ROC curve of the model  We decided to combine the confusion matrix from the model single-picture test with the  ROC curve for the study so that we could have a more in-depth look at the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The  confusion matrix is the method for measuring the accuracy of classification models that is the  simplest and most straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Calculating the confusion matrix of the suggested model  requires using equation (7) and equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' This matrix represents the identification result of  the different types of lost object photographs and is derived from these equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The  confusion matrix is shown in Figure 9a, with the rows representing the expected category for  the item that was lost, and the columns representing the actual category for the object that  was lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Figure 9b illustrates the characteristic curves of the class 10 lost item categories for  the enhanced MobileNetv2 transfer learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' ROC stands for receiver operating  characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The fact that the curves for all ten classes are located in the top left corner is  evidence that the model is accurate, and the fact that the area under the ROC curve for each  class is equal to one further demonstrates the superiority and efficiency of the model that was  provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  book  Confusion Matrix  USBflashdisk  bag  card  earphone  lipstick  phone  USBflashdisk  315  0  0  ：  o  0  30C  bag  309  0  2  250  book  0  313  0  0  card  313  0  200  earphone  0  312  0  0  0  anul  150  key  0  314  lipstick  0  314  100  phone  0  1  0  0  0  314  0  umbrella  E  LOE  50  vacuumcup  0  1  0  1  0  311  Predicted label1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='8  True Positive Rate  macro-average RoC curve (area = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='oo)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='6  ROC curve of USBflashdisk (area = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='0o)  RoC curve of bag (area=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='oo)  RoC curve of book (area = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='0o)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='4  ROC curve of card (area = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='0o)  RoC curve of earphone (area = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='oo)  RoC curve of key (area = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='00)  ROC curveof lipstick(area=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='0o)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='2 -  RoC curve of phone (area =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='oo)  RoCcurve of umbrella(area=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='oo)  RoC curveof vacuumcup (area= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='oo)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='0  False Positive RateII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  Inference  In order to successfully apply the model to the actual world, it needs to be simple enough that  it can be executed on most laptops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' In order to accomplish this, we implement this model as  well as a model that is very close to it on a regular computer that has a CPU, and we  comprehensively consider the inference speed of the computer as the evaluation index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' It  takes the model 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content="5 seconds to reason per picture on the CPU, which is significantly longer  than the standard migration model of this type or even 1/16 of the EfficientNet's processing  time." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' This can be seen in Figure 10a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Because it is such an exceptionally lightweight network,  it is not hard to imagine that the concept will eventually be adapted to work with other mobile  devices in the not too distant future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' In light of this, we provide an engineering system that  integrates the perceptual hashing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  This paper, which is based on the model training using the pytorch and torchvision machine  learning frameworks, uses the Spring Boot framework as the back-end service of the web  page in order to realize the online recognition of pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The front-end web page uses the  Layui framework in order to port the trained model to the Web end in order to realize the  online recognition of images that are uploaded by users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=" The steps involved in the online  identification procedure are as follows: the user launches the browser, navigates to the  system's website, navigates to the system's home page, clicks the button to upload the local  image, the front end sends the POST request to the back end, and the image is transmitted to  the back end." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' After the back end has received the picture that was uploaded by the user, it  will first make a request to the trained and upgraded MobileNet v2 migration learning model  for prediction and recognition, and it will then return a category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' After it has determined the  category, it will next submit a request to the database, asking it to deliver an array containing  the picture address associated with the category it has just determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' After obtaining the  address, the image is then downloaded by using the image address, and then the perceptual  hash algorithm is used to compare the downloaded image to the image that was uploaded by  the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The alignment will return a number for the similarity error, and the array will be  used to send the few photos that have the least significant value for the similarity error to the  front end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Figure 10b depicts the engineering interface after the model has been applied to the  situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='Figure10a(left) The speed of inference of each model in cpu  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='Figure10b(right) The improved model is applied to the actual engineering interface diagram  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='100-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='86  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='96  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='94  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='92  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='F06  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='EfficientNet  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='Inception_v4  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='88-  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='★  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='ViT-B/32  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='DenseNet201  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='86  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='VGGNet19  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='ShuffleNet_v2_x1_0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='84  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='★  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='ResNet152  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='MobileNet_v3  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='Ours  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='82  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='+08  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='0  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='5  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='15  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='Inference speed using CPU(second)LostNet:AsmartwayforlostandfindConcludes:  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='This research presents a novel design plan in light of the difficulties that are associated  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='with the management of urban traffic operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Even while the method of merging  convolutional neural networks with transfer learning has caused quite a stir in the world of  image recognition, it is not widely utilized when it comes to the automatic recognition of  photos that have been misplaced or found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' This study offers a new technique of intelligent  picture identification based on hash algorithm to give full play to the advantages of hash  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' The approach makes use of the concept of "search by map," and it is based on  hash algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Because there are a significant number of missing images in the library and a  great number of feature points that have been extracted, the CBAM structure has been  introduced in the first layer of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' This has been done in order to integrate the  attention mechanism in the dimension of channel and space in order to further highlight the  required features, which will ultimately lead to an improvement in the classification accuracy  of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Improved MobileNetv2 is used to establish a transfer learning training model to  identify common lost objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' This is done in order to reduce the number of ways in which  lost objects and database pictures can be compared to one another, thereby making it possible  to search for objects in a more convenient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' Extensive experimental arguments on  photos taken from the loss-object dataset, employing a variety of transfer learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  The findings indicate that the proposed model has a high recognition accuracy in the GFLOPs  and Total parameters extremes, and the features derived by this model significantly beat those  extracted by the other approaches when it comes to the classification job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' In order to enhance  the precision with which missing object categories may be identified, the proposed model is  being developed and put into practice in the search sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content='  However, further solutions are required in order to further increase the accuracy of the  model and the segmentation of the various kinds of lost object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
+page_content=' In order to do this, the  following stage will involve further subdividing each category so that it can be utilized in a  wider variety of search circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE0T4oBgHgl3EQfWQDo/content/2301.02277v1.pdf'}
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+arXiv:2301.13168v1  [math.AG]  30 Jan 2023
+THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.
+DANIEL HALPERN-LEISTNER
+Abstract. This note aims to clarify the deep relationship between birational
+modifications of a variety and semiorthogonal decompositions of its derived
+category of coherent sheaves.
+The result is a conjecture on the existence
+and properties of canonical semiorthogonal decompositions, which is a non-
+commutative analog of the minimal model program. We identify a mechanism
+for constructing semiorthogonal decompositions using Bridgeland stability con-
+ditions, and we propose that through this mechanism the quantum differential
+equation of the variety controls the conjectured semiorthogonal decompositions.
+We establish several implications of the conjectures: one direction of Dubrovin’s
+conjecture on the existence of full exceptional collections; the D-equivalence
+conjecture; the existence of new categorical birational invariants for varieties
+of positive genus; and the existence of minimal noncommutative resolutions
+of singular varieties. Finally, we verify the conjectures for smooth projective
+curves by establishing a previously conjectured description of the stability
+manifold of P1.
+Contents
+The Key Lemma
+2
+Noncommutative birational geometry
+5
+Related work and author’s note
+5
+1.
+The NMMP conjectures
+6
+1.1.
+The truncated quantum differential equation
+7
+1.2.
+The Hodge-theoretic MMP
+14
+2.
+Applications
+15
+2.1.
+Minimal models and the D-equivalence conjecture
+15
+2.2.
+Minimal resolutions
+16
+2.3.
+Example: simple flips and flops
+17
+2.4.
+Dubrovin’s conjecture
+19
+3.
+Example of curves
+21
+3.1.
+The projective line
+21
+3.2.
+Higher genus curves
+27
+References
+28
+The derived category of coherent sheaves Db(X) of a smooth projective variety
+X often reveals hidden structure in the geometry of X. We recall two examples of
+this phenomenon:
+In their seminal preprint [BO], Bondal and Orlov first glimpsed a deep relationship
+between birational modifications of varieties and the structure of their derived
+categories.
+A birational transformation X ��� X′ that preserves the canonical
+1
+
+2
+DANIEL HALPERN-LEISTNER
+bundle is expected to induce an equivalence of derived categories Db(X) ∼= Db(X′),
+which is now known as the D-equivalence conjecture [K1].
+More generally, for
+a KX-negative birational contraction π : X ��� X′, meaning there is a smooth
+projective W with birational morphisms f : W → X and g : W → X′ resolving π
+such that g∗(KX′) − f ∗(KX) is effective, then Db(X′) is expected to be a factor in
+a semiorthogonal decomposition of Db(X).
+In a different context, Beilinson observed in [B1] that Pn admits a full exceptional
+collection, which is the extreme form of a semiorthogonal decomposition in which
+all of the factors are equivalent to Db(pt). Many other examples of Fano manifolds
+admit large semiorthogonal decompositions of Db(X) that do not directly come
+from birational geometry, such as Lefschetz decompositions [K1]. Motivated by
+mirror symmetry, Dubrovin conjectured in [D2, Conj. 4.2.2] that a Fano manifold
+admits a full exceptional collection if and only if the quantum cohomology of X is
+generically semisimple.
+In this note, we attempt to clarify these phenomena, and to propose a mechanism
+for obtaining canonical semiorthogonal decompositions of derived categories. We
+formulate a conjecture, the noncommutative minimal model program (MMP), that
+implies the D-equivalence conjecture (see Corollary 18) as well as a version of
+Dubrovin’s conjecture (see Proposition 23). Although the conjectures are inspired
+by homological mirror symmetry, our mechanism is independent of mirror symmetry
+and, we believe, well-motivated by the classical MMP.
+The space of Bridgeland stability conditions on Db(X), which we refer to as
+Stab(X), plays a central role. We will shortly observe in Lemma 1 that an unbounded
+path in Stab(X) satisfying certain conditions naturally determines a semiorthogonal
+decomposition of Db(X). The noncommutative MMP, Conjecture II, proposes the
+existence of certain canonical paths of this kind, and hence canonical semiorthogonal
+decompositions of Db(X), with nice formal properties. These formal properties
+alone are enough to imply the D-equivalence conjecture, and more generally the
+existence of canonical admissible subcategories MX ⊂ Db(X) that are preserved
+under birational modifications of X (see Proposition 17).
+Then in Proposal III, we give a more precise proposal for the canonical paths in
+Stab(X). Specifically, one can view solutions of the quantum differential equation as
+defining paths in the dual of the algebraic cohomology H∗
+alg(X) ⊂ Heven(X; C), and
+we propose that these paths should be the central charges for the canonical paths
+in Stab(X). To avoid convergence issues, and motivated by the classical MMP,
+we will truncate the quantum differential equation by only counting sufficiently
+KX-negative curves.
+For the purposes of illustration, in Section 3 we describe these canonical paths on
+Stab(X) when X is a smooth projective curve. In order to do this, we establish the
+description of Stab(P1) predicted by homological mirror symmetry in Proposition 26,
+which to our knowledge has not been explicitly proved before.
+We fix a base field k ⊂ C, and use the term “variety” to refer to a reduced and
+geometrically irreducible finite type k-scheme.
+The Key Lemma. Let C be a pre-triangulated dg-category, fix a “Chern character”
+homomorphism v : K0(C) → Λ to a finite rank free abelian group Λ, and fix a
+reference norm on Λ. Recall that a Bridgeland stability condition on (C, Λ) consists
+of [B3]:
+
+THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.
+3
+i) A collection of full subcategories P = {Pφ ⊂ C}φ∈R, known as the categories
+of semistable objects of phase φ, such that Pφ[1] = Pφ+1, Hom(Pφ, Pϕ) = 0
+if φ > ϕ, and every object E ∈ C admits a finite R-weighted descending
+filtration such that grφ(E) ∈ Pφ; and
+ii) A central charge homomorphism Z : Λ → C, which we regard as a function
+on K0(C) via v, such that Z(Pφ \ 0) ⊂ R>0 · eiπφ and
+inf
+φ∈R
+E∈Pφ\0
+|Z(E)|
+∥v(E)∥ > 0.
+The set of stability conditions Stab(C) admits a metric topology such that the
+forgetful map Stab(C) → Λ∗
+C = Hom(Λ, C) that takes (Z, P•) �→ Z is a local
+homeomorphism [B3, Thm. 1.2].
+On the other hand, a semiorthogonal decomposition C = ⟨C1, . . . , Cn⟩ consists of a
+collection of full subcategories C1, . . . , Cn ⊂ C such that Ci[1] = Ci, Hom(Ci, Cj) = 0
+for i > j, and every E ∈ C admits a descending filtration indexed by i = 1, . . . , n
+such that gri(E) ∈ Ci. Despite the formal similarities in these notions, the difference
+in how Pφ and Ci behave under homological shift causes these structures to behave
+very differently.
+Now consider a continuous path σt = (Zt, Pt
+•) ∈ Stab(C) for t ∈ [t0, ∞) that
+satisfies:
+(1) For any E ∈ C the σt Harder-Narasimhan (HN) filtration of E stabilizes
+for t ≫ 0. We call this the eventual HN filtration, and we call an object
+eventually semistable if its eventual HN filtration has length 1.
+(2) For any eventually semistable object E there are αE, βE ∈ C such that
+logZt(E) = αEt + βE + o(1)
+as t → ∞,
+where the real part ℜ(βE) > C+ln∥v(E)∥ for some constant C independent
+of E, and
+(3) If the imaginary part ℑ(αE − αF ) = 0, then αE = αF .
+This is a special case of the concept of a quasi-convergent path we will develop with
+Alekos Robotis in [HLR]. For the purpose completeness here, we will explain the
+key property of such a path:
+Lemma 1. A path σt satisfying conditions (1),(2), and (3) above determines:
+• A finite set of complex constants such that αE ∈ {α1, . . . , αn} for any
+eventually semistable E, indexed so that ℑ(αi) is increasing in i;
+• A semiorthogonal decomposition C = ⟨C1, . . . , Cn⟩, where Ci is the subcategory
+of objects whose eventual HN subquotients all have αE = αi;
+• A direct sum decomposition Image(v) = �
+i Λi, where Λi = v(Ci); and
+
+4
+DANIEL HALPERN-LEISTNER
+• A stability condition on each Ci (with respect to v : K0(Ci) → Λi) whose
+semistable objects are precisely the eventually semistable objects E with
+αE = αi, and whose central charge is Zi(E) = limt→∞ e−αitZt(E).
+We sketch the proof, leaving some of the details to [HLR].
+Proof idea. We begin by showing that the categories Cβ generated by eventually
+semistable E with αE = β define a semiorthogonal decomposition indexed by the
+set {αE|E eventually semistable} ⊂ C, ordered by imaginary part. The condition
+Hom(Cβ, Cγ) = 0 for ℑ(β) > ℑ(γ) holds because for eventually semistable objects
+E, F with ℑ(αE) > ℑ(αF ), for t large enough both E and F are σt-semistable
+with E having a larger phase. The condition Ci[1] = Ci follows from the fact that
+Harder-Narasimhan filtrations are preserved by homological shift. Finally, consider
+an object E ∈ C, and let gri(E) denote the ith associated graded object for the
+eventual HN filtration of E with respect to σt.
+Because the phase of gri(E) is
+increasing in i, we must have ℑ(αgr1(E)) ≤ · · · ≤ ℑ(αgrm(E)). We then coarsen
+this filtration by grouping all associated graded pieces that have the same value of
+ℑ(αgri(E)) to obtain a descending filtration of E with gri(E) ∈ Ci.
+The fact that the images v(Ci) are linearly independent, and thus only finitely
+many values of αE can appear, follows from the observation that condition (2)
+implies that for any collection of Ei ∈ Ci with Zt(Ei) ̸= 0 for all i and t ≫ 0, the
+functions Zt(Ei) ∈ C0([t0, ∞)) are linearly independent.
+□
+By Bridgeland’s theorem, a path in Stab(C) is uniquely determined by its starting
+point (Z0, P0
+•) ∈ Stab(C) and a path Zt in Hom(Λ, C) extending Z0.
+So, the
+remarkable thing about Lemma 1 is that it suggests that once you fix a reference
+stability condition on C, one can study semiorthogonal decompositions of C simply
+by studying interesting paths in the complex vector space Hom(Λ, C).
+Remark 2. The following picture informs Lemma 1, but is not strictly necessary
+for this paper: The conclusion of Lemma 1 only depends on the image of the
+path σt in Stab(C)/Ga, and the stability conditions on the factors in Lemma 1 are
+only naturally defined up to the action of Ga on Stab(Ci). In [HLR] we construct
+a partial compactification Stab(C)/Ga ⊂ PStab(C) whose points correspond to
+semiorthogonal decompositions C = ⟨C1, . . . , Cn⟩ for some n along with a point
+in Stab(Ci)/Ga for each i = 1, . . . , n, along with some additional continuous data.
+The paths studied in Lemma 1 are convergent in this bordification.
+The notion of a quasi-convergent path in [HLR] is much more general than the
+one in Lemma 1. For one thing, the proof of Lemma 1 holds verbatim for a more
+general asymptotic estimate, such as
+logZt(E) =
+:=αE(t)
+�
+��
+�
+αpt + αp−1t(p−1)/p + · · · + α1t1/p + α−1 ln(t) +βE + o(1).
+In addition, the genericity condition (3) can be removed entirely, and this is the
+version of “quasi-convergent” that we will refer to in Proposal III below.
+For
+such a path one obtains a slightly weaker structure on C: still only finitely many
+functions appear as αE(t) for some eventually semistable E, but now Ci consists of
+objects whose eventual HN subquotients have ℑ(αE(t)) = fi(t) for some set of real
+functions f1(t), . . . , fn(t). Then, each Ci admits a filtration by thick triangulated
+subcategories, generated by eventually semistable objects with ℜ(αE(t)) ≤ g(t)
+
+THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.
+5
+as t → ∞ for certain functions g(t), and the associated subquotient categories
+canonically admit stability conditions. We refer to [HLR] for more details.1
+Noncommutative birational geometry. In addition to generalizing Lemma 1,
+[HLR] also establishes a converse: If C is smooth and proper [KS, §8], then any semi-
+orthogonal decomposition C = ⟨C1, . . . , Cn⟩ such that the Ci all admit Bridgeland
+stability conditions arises from a suitable quasi-convergent path in Stab(C)/Ga via
+Lemma 1. So, perhaps it is reasonable to restrict our focus to these semiorthogonal
+decompositions.
+On a more philosophical note, in the minimal model program K¨ahler structures,
+in the form of ample divisor classes, are essential to making sense of birational
+geometry. For a smooth projective variety X, the categorical analogue of a K¨ahler
+structure on X is a Bridgeland stability condition on Db(X).
+(See [B4, §7.1].)
+We argue that stability conditions are as essential to studying semiorthogonal
+decompositions of Db(X) as K¨ahler classes are to studying birational geometry.
+This suggests the following:
+Principle. Noncommutative birational geometry is the study of semiorthogonal
+decompositions of smooth and proper pre-triangulated dg-categories in which every
+factor admits a stability condition.
+We hope this principle helps to explain some recent failures of folk expectations
+in the field, such as the failure of the Jordan-H¨older property [BGvBS]. Perhaps
+restricting to semiorthogonal decompositions of C that are polarizable, in the sense
+that every factor admits a stability condition, may rehabilitate some of these
+predictions.
+Example 3. The paper [BGvBKS] constructs surfaces X with a semiorthogonal
+decomposition Db(X) = ⟨L1, . . . , L11, C⟩, where Li are exceptional line bundles and
+C is a phantom, meaning K0(C) and HH∗(C) vanish. A non-zero phantom does not
+admit a Bridgeland stability condition, so this semiorthogonal decomposition could
+not arise from Lemma 1. More is true, though: We will see in Lemma 24 that if
+a semiorthogonal decomposition appears to come from a full exceptional collection
+on the level of K-theory, and every factor admits a stability condition, then it
+does come from a full exceptional collection. So if one lets C′ be the subcategory
+generated by C and L11, then Db(X) = ⟨L1, . . . , L10, C′⟩ can not arise from a quasi-
+convergent path in Stab(X)/Ga.
+Related work and author’s note. As we will discuss in Section 2.4, the form-
+ulation of Proposal III is closely related to and inspired by the Gamma II conjecture
+of [GGI], which predicts the existence of full exceptional collections whose Chern
+characters give solutions of the quantum differential equation with special asymptotic
+properties.
+The paper [SS] generalizes Dubrovin’s conjecture and the Gamma
+conjectures to predict the existence of canonical semiorthogonal decompositions
+for Db(X) for any Fano manifold X, whose factors again correspond to solutions of
+the quantum differential equation with prescribed asymptotic properties. Maxim
+Kontsevich has also given several talks [K3] in which he conjectures the existence
+of canonical semiorthogonal decompositions of Db(X), whose factors correspond
+1The condition (1) on stabilization of HN filtrations is also significantly relaxed in [HLR], but
+this is not essential to our discussion here.
+
+6
+DANIEL HALPERN-LEISTNER
+to the eigenspaces of quantum multiplication by c1(X), and speculates about the
+implications.
+The main contributions of this paper are: 1) to suggest an underlying mechanism
+for the conjectures above; 2) to extend the conjectures to non-Fano X in a way that
+avoids convergence issues for the quantum differential equation; and 3) to propose
+a specific conjecture on the compatibility of these semiorthogonal decompositions
+with birational morphisms and to prove some interesting implications.
+I thank Jeffrey Jiang and Alekos Robotis for many enlightening discussions about
+Bridgeland stability conditions.
+In addition, I thank Tom Bridgeland, Davesh
+Maulik, Tudor P˘adurariu, Daniel Pomerleano, Claude Sabbah, and Nicolas Templier
+for helpful suggestions on this project.
+1. The NMMP conjectures
+We will formulate a noncommutative minimal model program (NMMP) associated
+to a contraction of a smooth projective variety X, meaning a surjective morphism
+to a projective variety X → Y with connected fibers. (Y is not necessarily smooth,
+but it must be normal by the uniqueness of Stein factorization.)
+The NMMP
+predicts canonical semiorthogonal decompositions of Db(X). The first piece of our
+program is the following difficult folk conjecture:
+Conjecture I. Db(X) admits stability conditions for any smooth projective variety
+X.
+We define the lattice ΛX = H∗
+alg(X) as the image of the twisted Chern character
+homomorphism v := (2πi)deg /2 ch : K0(X) → H∗(X; C). We only consider stability
+conditions on Db(X) defined with respect to v.2
+To simplify notation, we let
+Stab(X) := Stab(Db(X)) for a scheme X.
+In our first and most flexible formulation of the conjectures, we will use ψ to
+denote a generic, unspecified, parameter. Below we will specify more precisely what
+ψ might be. We formulate the conjectures relative to a fixed normal variety Y that
+need not be smooth. (The most interesting case might be Y = pt.)
+Conjecture II. Let π : X → Y be a contraction of a smooth projective variety X.
+(A) One can associate to π a canonical class of quasi-convergent paths {σπ,ψ
+t
+}
+in Stab(X)/Ga. Generic values of the parameter ψ give rise to a semi-
+orthogonal decomposition of Db(X), e.g., via Lemma 1, and different generic
+values of ψ give mutation-equivalent semiorthogonal decompositions.3
+(B) For a generic value of ψ, the semiorthogonal factors of Db(X) are closed
+under tensor product with complexes of the form π∗(E) for E ∈ Perf(Y ).
+(C) Given a further contraction Y → Y ′, for some values of the parameters,
+the semiorthogonal decomposition of Db(X) associated to the composition
+2In general if v : K0(C) → Λ has image Λ′ ̸= Λ, then after choosing a splitting ΛC ∼
+= Λ′
+C ⊕ W
+one can identify StabΛ(C) ∼
+= StabΛ′(C) × W ∗. Also the quantum differential equation preserves
+H∗
+alg(X). So, our entire discussion would work just as well using � (2πi)n
+n!
+H2n(X; Z)/{torsion}
+instead of H∗
+alg(X), but it is a bit simpler to assume that v is surjective.
+3Because X is smooth and proper, any semiorthogonal decomposition of Db(X) is admissible,
+meaning arbitrary mutations exist. See [BK].
+
+THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.
+7
+X → Y ′ refines the semiorthogonal decomposition associated to X → Y .
+To formulate the final conjecture, we recall that if π : X → X′ is a morphism
+of smooth varieties and Rπ∗(OX) = OX′, then π∗ is fully faithful and we have a
+semiorthogonal decomposition
+Db(X) = ⟨ker(π∗), π∗(Db(X′))⟩.
+(1)
+We again consider a composition of contractions X → X′ → Y .
+(D) If X′ is smooth and Rπ∗(OX) = OX′, then for some values of the parameters,
+the semiorthogonal decomposition of Db(X) associated to X → Y refines
+the semiorthogonal decomposition obtained by combining the semiorthogonal
+decomposition of π∗(Db(X′)) ∼= Db(X′) associated to X′ → Y with (1).
+We expect several of the most basic examples of semiorthogonal decompositions
+in geometry to arise in this way.
+The following examples may be regarded as
+extensions of Conjecture II.
+Example 4. In the special case where X is the blowup of Y along a smooth
+subvariety S ֒→ Y of codimension n + 1, we expect the canonical semiorthogonal
+decomposition associated to X → Y to agree with the semiorthogonal decomp-
+osition from [BO, Prop. 3.4]
+Db(X) = ⟨Db(S)(−n), . . . , Db(S)(−1), π∗(Db(Y ))⟩.
+Example 5. If X = P(E) for some locally free sheaf E on Y of rank n and π :
+X → Y is the projection, then we expect that for a suitable choice of parameter
+the semiorthogonal decomposition in (A) is
+Db(X) =
+�
+π∗(Db(Y )), π∗(Db(Y )) ⊗ O(1), . . . , π∗(Db(Y )) ⊗ O(n)
+�
+.
+1.1. The truncated quantum differential equation. We now formulate a more
+precise proposal for the canonical quasi-convergent paths in Stab(Db(X)) conjectured
+in (A).
+The small quantum product ⋆τ on H∗
+alg(X), parameterized by τ ∈ H2(X; C), is
+defined by the formula
+(α1 ⋆τ α2, α3)X =
+�
+d∈NE(X)Z
+⟨α1, α2, α3⟩X
+0,3,deτ·d,
+(2)
+where α1, α2, α3 ∈ H∗
+alg(X; C), (−, −)X denotes the Poincar´e pairing on H∗
+alg(X),
+NE(X)Z denotes the numerical equivalence classes of 1-cycles with nonnegative
+integer coefficients, and ⟨α1, α2, α3⟩X
+0,3,d denotes the Gromov-Witten invariant that
+counts curves of class d on X.
+Let us consider a function of a single complex
+parameter ζ = ζ(u) ∈ H∗
+alg(X; C). The quantum differential equation is
+0 = u dζ
+du + c1(X) ⋆ln(u)c1(X) ζ.
+In general, if neither c1(X) := −c1(KX) or −c1(X) is ample, then the sum in (2)
+is infinite and thus this is only a formal differential equation in u. We propose to
+modify this differential equation by replacing c1(X) ⋆ln(u)c1(X) (−) by an operator
+Eψ(u) whose definition involves only finite sums.
+
+8
+DANIEL HALPERN-LEISTNER
+Definition 6. For d ∈ NE(X)Z with c1(X) · d ≥ 0, let Td ∈ End(H∗
+alg(X; Q)) be
+defined by the identity (Tdα1, α2)X = ⟨α1, α2⟩X
+0,2,d for all α1, α2 ∈ H∗
+alg(X; Q).
+When α1 has degree 2, the divisor equation ⟨α1, α2, α3⟩X
+0,3,d = (α1·d)⟨α2, α3⟩X
+0,2,d
+allows one to express
+α1 ⋆τ (−) = α1 ∪ (−) +
+�
+d∈NE(X)Z\{0}
+(α1 · d)eτ·dTd.
+We let NE(X/Y )Z denote the numerical equivalence classes of effective integral 1-
+cycles spanned by curves that are contracted by π, and note the natural injective
+map NE(X/Y )Z → NE(X)Z. We make the following key observation:
+Lemma 7. If ω ∈ H2(X; R) is the Chern class of a relatively ample R-divisor
+for a contraction π : X → Y , then Td is homogeneous of degree 2(1 − c1(X) · d)
+with respect to the cohomological grading. As a result, there are only finitely many
+classes d ∈ NE(X/Y )Z such that: 1) d · (c1(X) − ω) > 0; and 2) Td ̸= 0.
+Proof. First let H be an ample Cartier divisor on Y that is large enough such
+that ω + π∗(H) is ample on X. Because the moduli space M 0,2,d(X) has virtual
+dimension c1(X)·d+dim(X)−1, one has (Tdα1, α2)X = ⟨α1, α2⟩X
+0,2,d = 0 whenever
+1
+2(deg(α1) + deg(α2)) ̸= c1(X) · d + dim(X) − 1. This implies that deg(Tdα) −
+deg(α) = 2(1 − c1(X) · d), so Td = 0 for degree reasons unless c1(X) · d − 1 ∈
+[− dim(X), dim(X)]. Combining this with the constraint (1) gives
+(ω + π∗(H)) · d = ω · d < c1(X) · d ≤ dim(X) + 1
+(3)
+There are finitely many numerical equivalence classes of cycles satisfying this bound,
+by [KM, Cor.1.19].
+□
+Definition 8. Let ψ := ω + iB ∈ NS(X)C/2πi NS(X) be a class whose real part
+ω is the Chern class of a relatively ample R-divisor for the contraction π : X → Y .
+We define the truncated quantum endomorphism Eψ(u) : H∗
+alg(X; C) → H∗
+alg(X; C)
+by the formula
+Eψ(u) = c1(X) ∪ (−) +
+�
+d∈NE(X/Y )Z s.t.
+(c1(X)−ω)·d>0
+(c1(X) · d)uc1(X)·de−ψ·dTd.
+The restriction d · (c1(X) − ω) > 0, is motivated by the Cone Theorem, which
+states that the (c1(X) − ω)-positive piece of the cone of curves is polyhedral, and
+its rays are generated by rational curves with c1(X) · d ∈ (0, dim(X) + 1]. This is
+precisely the bound in (3), outside of which Td = 0 for degree reasons.
+Without this restriction, and when Y = pt, the sum defining Eψ(u) would agree
+with the definition of c1(X) ⋆−ψ+ln(u)c1(X) (−). Eψ(u) keeps terms that dominate
+the sum as |u| → ∞. We therefore regard Eψ(u) as a polynomial approximation
+Eψ(u) ≈ c1(X) ⋆rel
+−ψ+ln(u)c1(X) (−)
+(4)
+that is valid when ω is close to 0 and |u| ≫ 0, and where ⋆rel denotes a relative
+quantum product for the morphism X → Y that only counts classes of contracted
+curves. In fact, if c1(X) is relatively ample for the contraction π : X → Y , such as
+when Y = pt and X is Fano, and ω is small enough that c1(X)−ω is still relatively
+ample, then (4) becomes an equality.
+
+THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.
+9
+Now, a path in Stab(X) is uniquely determined by a starting point (Z1, P1)
+and a path Z• : [1, ∞) → Hom(ΛX, C) starting at Z1. We will construct paths in
+Hom(Λ, C) by studying solutions of the truncated quantum differential equation:
+0 = tdζ(t)
+dt
++ 1
+z Eψ(t)ζ(t),
+(5)
+where z ∈ C is a parameter, ψ the parameter in Definition 8, and ζ(t) ∈ H∗
+alg(X)C.
+Note that this agrees with the usual quantum differential equation when X is Fano,
+Y = pt, and ω is sufficiently small.
+Following [GGI], we will analyze the differential equation (5) by making the
+change of variables ˜ζ(t) := tµζ(t), where µ := (deg − dim(X))/2 is the grading
+operator on H∗
+alg(X). Lemma 7 implies that tEψ(1)tµ = tµEψ(t), so (5) becomes
+d˜ζ
+dt + 1
+z Eψ(1)˜ζ − 1
+t µ˜ζ = 0,
+(6)
+which is much simpler because it has only three terms, with a regular singularity
+at t = 0 and a pole of order ≤ 2 at t = ∞.
+The Hukuhara-Turritin theorem [W, Thm. 19.1] says that a differential equation
+of the form (6) has a fundamental solution of the form
+Φt = A(t1/p)eD(t1/p)+ln(t)C,
+(7)
+where D(s) is a diagonal matrix with polynomial entries, C is a constant matrix
+that commutes with D(s) for all s, and A(s) is a holomorphic invertible matrix-
+valued function that converges as s → ∞. In the special case of (6), we can be
+more precise.
+Proposition 9. The differential equation (6) has a holomorphic fundamental solution
+of the form Φt = Y (t)etD+B(t) for |t| > t0 in some sector S ⊂ C centered at the
+origin and containing R>0, where
+(1) Y (t) is an invertible matrix that admits a uniform asymptotic expansion
+Y (t) ∼ Y0 + Y1t−1/p + Y2t−2/p + · · · on S, for some p ∈ Z>0, such that the
+columns of Y0 are a basis of generalized eigenvectors of −1
+z Eψ(1),
+(2) D is the diagonal matrix of eigenvalues of −1
+z Eψ(1) corresponding to the
+columns of Y0, and
+(3) B(t) = Dp−1t(p−1)/p + · · · + D2t2/p + D1t1/p + C ln(t) for certain constant
+diagonal matrices D1, . . . , Dp−1 and a constant matrix C, all of which
+commute with D. In particular, ∥B(t)∥ = O(t(p−1)/p).
+Furthermore, if Eψ(1) is semisimple, then one can arrange that Dp−1 = Dp−2 =
+· · · = D1 = 0, and if the eigenvalues of Eψ(1) are distinct, then one can arrange
+B(t) = 0.
+We will apply Proposition 9 here, and postpone its proof to the end of this
+subsection. It implies that for any solution ζ(t) of (5), we have
+lim sup
+t→∞
+ln∥ζ(t)∥
+t
+= r,
+
+10
+DANIEL HALPERN-LEISTNER
+where r is the real part of an eigenvalue of −1
+z Eψ(1).Using this, we can state a more
+precise elaboration on what the canonical quasi-convergent paths in Conjecture II(A)
+should look like:
+Proposal III. There are quasi-convergent paths in Stab(X)/Ga whose central
+charges have the form Zt(α) =
+�
+X Φt(α), where Φt ∈ End(H∗
+alg(X)C) is a fun-
+damental solution of the truncated quantum differential equation (5) with parameters
+z ∈ C and ψ = ω+iB ∈ NS(X)C, where ω is small and relatively ample for X → Y .
+Furthermore, the following spanning condition holds: for any r ∈ R that is the
+real part of an eigenvalue of −1
+z Eψ(1), the subspace
+F rΛC := {α ∈ ΛC s.t. ln∥Φt(α)∥ ≤ rt + o(t) as t → ∞}
+should be spanned over C by the classes of eventually semistable E ∈ Db(X) with
+lim inft→∞ |Zt(E)|/∥Φt(E)∥ > 0.
+The proposal is inspired by Iritani’s quantum cohomology central charge [I],
+which has previously been conjectured to be the central charge of a family of
+stability conditions on Db(X) [D1]. The main innovations here: 1) the modification
+of the quantum differential equation to reflect the relative geometry of X → Y and
+to only count sufficiently KX-negative curves, 2) the assertion that the resulting
+paths in Stab(X)/Ga are quasi-convergent, and 3) the spanning condition, which
+is analogous to the Gamma II conjecture [GGI, Conj. 4.6.1].
+We will see in Proposition 23 that the spanning condition is crucial, because
+it guarantees that when the eigenvalues of −1
+z Eψ(1) have distinct real parts, the
+semiorthogonal factors coming from the canonical quasi-convergent paths are in
+bijection with the eigenvalues of −1
+z Eψ(1). These eigenvalues are precisely the αj
+that arise in the key lemma, Lemma 1.
+Without the spanning condition, the proposal is nearly a tautology. Indeed, the
+central charge Zt in Proposal III always converges in the projective space P(Λ∗
+C) as
+t → ∞ to a point Z∞. If Z∞ lifts to a point in Stab(X)/Ga, then for sufficiently
+large t the central charges Zt will also lift to Stab(X)/Ga, and the resulting path
+is quasi-convergent in the tautological sense that it converges in Stab(X)/Ga.
+Remark 10. Because
+�
+X tµ(−) = tdim(X)/2 �
+X(−), and the path in Stab(X)/Ga
+only depends on the central charge Zt up to scale, Proposal III is unchanged if we
+assert instead that Φt is a solution of (6) rather than (5). We have used (5) to be
+compatible with [I].
+Remark 11 (The meaning of small ample classes). If ω ∈ NS(X)R is relatively
+ample for X → Y , then for r ≫ 0, there will be no classes in d ∈ NE(X/Y )Z such
+that (c1(X) − rω) · d > 0. So if ω were large, one would have Eψ(1) = c1(X) ∪ (−),
+and Proposal III could not produce an interesting semiorthogonal decomposition.
+On the other hand, as r → 0+, the condition (c1(X) − rω) · d > 0 will include
+more and more terms in Eψ(1). The intuition behind requiring ω to be “small”
+in Proposal III is that as r → 0+, the resulting semiorthogonal decomposition of
+Db(X) should stabilize. Then Conjecture II(A) predicts that in this stable range,
+the semiorthogonal decomposition is independent of ω up to mutation.
+Remark 12 (Refined proposal). The spanning condition in Proposal III only
+addresses the leading order asymptotics of solutions.
+A more precise spanning
+condition is that one can arrange in Proposition 9 that for each function ϕ(t)
+
+THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.
+11
+appearing as a diagonal entry of tD + t(p−1)/pDp−1 + · · · + t1/pD1, the solution
+space with exponential factor eϕ(t) is spanned by Φt(v(E)) for some collection of
+eventually semistable E. In this case the quasi-convergent path in Stab(X)/Ga
+would lead to a semiorthogonal decomposition indexed by the ϕ(t) that appear,
+and Proposal III would only see the coarser decomposition that merges categories
+corresponding to ϕ with the same leading coefficient. We have not emphasized
+this for the following reason: In situations where (6) agrees with the quantum
+differential equation, such as when X is Fano, it is conjectured in [KKP, Conj. 3.4]
+that the connection ∇∂t = d+( 1
+zEψ(1)− 1
+t µ)dt on H∗(X; C)[t±1] is of non-ramified
+exponential type. In that case, one can take B(t) = 0 in Proposition 9, and if z is
+chosen generically so that the eigenvalues of 1
+zEψ(1) have distinct real parts, this
+refined formulation agrees with that in Proposal III.
+Remark 13 (Canonical fundamental solution). We have left some flexibility as
+to which fundamental solution Φt to use in Proposal III. In [GGI, Prop. 2.3.1], it
+is shown that when X is Fano, so that (5) agrees with the quantum differential
+equation, there is a unique fundamental solution Φt ∈ End(H∗
+alg(X)C) of (5) of the
+form T (t)t−c1(X) such that both T (t) and S (t) := tµT (t)t−µ are holomorphic in t
+and regular at t = 0, with T (0) = S (0) = idΛC. In fact, the proof applies verbatim
+to the truncated quantum differential equation (5) in general. The canonical fun-
+damental solution is defined to be
+Φt(α) = T (t)t−c1(X)�ΓX ∪ α,
+(8)
+where α ∈ H∗
+alg(X), ˆΓX = �dim X
+i=1
+Γ(1 + δi), and δi are the Chern roots of the
+tangent bundle TX. Iritani’s quantum cohomology central charge [I] is then
+Zt(E) ∝
+�
+X
+T (t)t−c1(X)�ΓX ∪ v(E).
+(9)
+It is tempting to use the canonical fundamental solution (8) in Proposal III.
+However, outside of the Fano situation, more investigation is needed to settle on a
+final interpretation of Proposal III:
+For varieties such that Db(X) admits no semiorthogonal decompositions, one
+natural interpretation is that the quasi-convergent paths in Proposal III should
+converge to a point in Stab(X)/Ga itself.
+We will see in Section 3.2 that for
+higher genus curves one can arrange this, but the fundamental solutions needed
+do not appear to be canonical. A second natural interpretation is that the paths in
+Proposal III are quasi-convergent in the more general sense studied in [HLR], but
+the filtration that they induce on Db(X) is not admissible. This is the behavior one
+sees for the canonical fundamental solution in the case of curves of higher genus.
+Remark 14. We do not have a specific prediction as to a starting point for the
+canonical paths σπ,ψ
+t
+. In many examples, Stab(X) has a “geometric” region in which
+all skyscraper sheaves of points are stable of the same phase. It would be satisfying
+if one could start with a stability condition (Z1, P1) in the geometric region, and
+show that the path in Hom(ΛX, C) defined by (5) lifts to a quasi-convergent path
+in Stab(X)/Ga. In this sense the truncated quantum differential equation would
+“discover” semiorthogonal decompositions that were not already known.
+Remark 15 (Alternative differential equations). Eψ(1) is meant to approximate
+c1(X) ⋆−ψ (−) by an a priori convergent expression. However, when c1(X) ⋆−ψ
+
+12
+DANIEL HALPERN-LEISTNER
+(−) is known to converge for ψ in a neighborhood of ψ0 this approximation is
+not necessary. In this case, the equation (6) admits a well-known isomonodromic
+deformation where c1(X)⋆−ψ (−) is replaced with the “big” quantum product E ⋆τ
+(−) where τ ∈ Heven(X; C) rather than H2 and E is the Euler vector field (see [GGI,
+2.2.3]). There are known examples of varieties with full exceptional collections for
+which this full isomonodromic deformation is needed to get an operator with distinct
+eigenvalues [GMS]. So the full deformation is needed for the converse implication of
+Dubrovin’s conjecture, or its refinement as the Gamma II conjecture [GGI] to hold.
+In these situations, we would expect the semiorthogonal decomposition arising from
+the full isomonodromic deformation to refine the semiorthogonal decomposition
+arising from Proposal III.
+1.1.1. Proof of Proposition 9. The first part of the analysis works for any vector-
+valued differential equation of the form X′(t) = A(t)X(t), where A(t) is a holomorphic
+matrix-valued function that admits an asymptotic expansion A(t) ∼ A0 + A1t−1 +
+A2t−2 + · · · as t → ∞ in some sector S. As in the proof of the Hukuhara-Turritin
+theorem, we begin by using [W, §11 and Thm. 12.2] to construct a holomorphic
+change of variables X(t) = P(t)Z(t) such that the equation for X(t) becomes
+Z′(t) = Q(t)Z(t), where: i) P(t) admits an asymptotic expansion P(t) ∼ �
+n≥0 Pnt−n
+on S with P0 a matrix of generalized eigenvectors for A0; ii) there is an asymptotic
+expansion Q(t) ∼ �
+n≥0 Qnt−n with A0 = P0Q0P −1
+0
+; and iii) Q(t) = R1(t) ⊕ · · · ⊕
+Rk(t) is block diagonal, where the leading term of each Ri(t) as t → ∞ has a single
+eigenvalue. Therefore, the entire differential equation for Z splits as a direct sum
+of differential equations of the original form in which A0 has a single eigenvalue,
+and it suffices to prove the claim in this case.
+So let us return to the original notation and assume that A0 has a single
+eigenvalue λ.
+Making the substitution X(t) = eλtZ(t), the equation for X(t)
+becomes Z′(t) = P(t)Z(t), where P(t) := A(t)−λI admits an asymptotic expansion
+in t−1 whose leading term is nilpotent. At this point, if A0 = λI, then the resulting
+differential equation has a pole of order 1 at ∞, and the result follows. Otherwise,
+we apply the general Hukuhara-Turritin theorem [W, Thm. 19.1] to conclude that
+the equation for Z has a fundamental solution of the form
+Z(t) = Y (t)eDmtm/p+···+D1t1/p+ln(t)C,
+(10)
+where: i) Y (t) is holomorphic on a (potentially smaller) sector S′ ⊂ S containing
+R>0 and admits an asymptotic expansion on S′ in powers of t−1/p with invertible
+leading term; ii) Dj are diagonal constant matrices and commute with the constant
+matrix C. The proof of the first part of the Proposition will be complete once we
+show that Dj = 0 for j ≥ p.
+Let Z(t) be a particular solution of Z′(t) = P(t)Z(t). Choose a Hermitian norm
+∥−∥ on ΛC and fix a small ǫ > 0. We compute
+d∥Z(t)∥2
+dt
+= 2ℜ⟨Z(t), Z′(t)⟩ = 2ℜ⟨Z(t), P(t)Z(t)⟩.
+If ∥P(t)∥ denotes the operator norm, then because P(t) converges to P0 as t → ∞,
+we can choose a t0 such that for any t ≥ t0, we have ∥P(t)∥ ≤ N := (1 + ǫ)∥P0∥
+for all t ≥ t0. Now applying the Cauchy-Schwartz inequality to the computation
+above gives
+−2N∥Z(t)∥2 ≤ d∥Z(t)∥2
+dt
+≤ 2N∥Z(t)∥2.
+
+THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.
+13
+Now y(t) := ∥Z(t)∥2 is a smooth nonnegative real-valued function of t ∈ R such
+that f(t) := 2Ny(t)−y′(t) ≥ 0 and g(t) := 2Ny(t)+y′(t) ≥ 0 for all t ≥ t0. Solving
+these first order ODE’s for y(t) gives
+y(t) = e2Nt
+�
+e−2Nt0y(t0) −
+� t
+t0
+e2Nsf(s)ds
+�
+= e−2Nt
+�
+e2Nt0y(t0) +
+� t
+t0
+e2Nsg(s)ds
+�
+.
+It follows from the nonnegativity of f and g that letting c1 := e−Nt0�
+y(t0) ≤ c2 :=
+eNt0�
+y(t0), we have
+c2e−Nt ≤ ∥Z(t)∥ ≤ c1eNt
+(11)
+for all t ≥ t0.
+Observe that, after adjusting the constants c1 and c2, the bounds in (11) continue
+to hold if we replace ∥Z(t)∥ with ∥Z(t)∥ref for some other Hermitian norm ∥−∥ref.
+Thus if we fix a reference norm ∥−∥ref, we have shown that for any Hermitian norm
+∥−∥, there are constants c1, c2, t0 > 0 such that
+c2e−(1+ǫ)∥P0∥t ≤ ∥Z(t)∥ref ≤ c1e(1+ǫ)∥P0∥t
+for all t ≥ t0. On the other hand, because P0 is nilpotent, one can choose Hermitian
+norms in which ∥P0∥ is arbitrary small. Indeed, one can choose a basis in which P0
+is r times a sum of nilpotent Jordan matrices, and in the norm in which this basis
+is orthonormal one has ∥P0∥ ≤ (rank(P0) − 1)r.
+It follows that for any r > 0, there are constants c1, c2, t0 > 0 such that c2e−rt ≤
+∥Z(t)∥ref ≤ c1ert for all t ≥ t0. If ℜ(Dj) ̸= 0 for any j ≥ p in the fundamental
+solution (10), then one of the columns of this matrix would violate this bound for
+some r. Hence we conclude that ℜ(Dj) = 0 for all j ≥ p. However, an analysis
+identical to the one above gives the same bounds for the function ∥Z(eiθt)∥, where
+θ is any angle close enough to 0 that the ray eiθR>0 lies in the sector S′ on which
+Y (t) and P(t) are defined and satisfy the desired asymptotic estimate. It follows
+that ℜ(Djeiθj/p) = 0 for all sufficiently small θ and j ≥ p, and hence Dj = 0 for
+all j ≥ p. This completes the proof of the main claim.
+For the further claim when the eigenvalues of Eψ(1) are distinct, we use a
+different argument.
+It follows from the symmetry of the two-point function in
+Definition 6 that Td and hence Eψ(1) is symmetric with respect to the Poincar´e
+pairing (−, −)X on H∗
+alg(X)C, and it is easy to show that the grading operator µ is
+anti-symmetric with respect to (−, −)X. Now, an endomorphism that is symmetric
+with respect to a non-degenerate complex bilinear form need not be diagonalizable,
+but its generalized eigenspaces are orthogonal to one another, and the restriction
+of the form to each generalized eigenspace is still non-degenerate.
+It follows that if the eigenvalues of Eψ(1) are distinct, then this endomorphism
+admits an orthonormal eigenbasis. In this basis, the matrix D of Eψ(1) is diagonal
+with distinct diagonal entries, and the matrix M for µ satisfies M T = −M. In
+particular the diagonal entries of M are all 0. After a change in variables u = 1/t,
+our differential equation (6) becomes
+dζ
+du +
+� D
+u2 + M
+u
+�
+ζ = 0.
+
+14
+DANIEL HALPERN-LEISTNER
+We are now in the setting of [BTL, Sect.8].
+The vanishing of the diagonal of
+M implies the conditions (D) and (F), and the fundamental solution near u = 0
+described in [BTL, Sect.8] gives the claim of Proposition 9.
+□
+1.2. The Hodge-theoretic MMP. For a smooth projective complex variety X,
+the topological K-theory Ktop
+i
+(X) admits a canonical weight-i pure Hodge structure
+induced by the twisted Chern character Ch : Ktop
+i
+(X)⊗C ∼= Hi+2∗(X; C). Concretely,
+after tensoring with Q we have an isomorphism of Hodge structures Ktop
+i
+(X)⊗Q ∼=
+� Hi+2n(X; Q)(n), where (n) denotes the Tate twist.
+In fact, this Hodge structure can be reconstructed entirely from Db(X): The
+paper [B2] constructs a topological K-theory spectrum for a dg-category over C,
+and a canonical isomorphism with periodic cyclic homology
+Ch : Ktop(X) ⊗ C ∼= Ktop(Db(X)) ⊗ C
+∼
+=
+−→ HP(Db(X)),
+which takes the Bott element to the periodic parameter in periodic cyclic homology.
+The degeneration of the noncommutative Hodge-de Rham spectral sequence for
+HP(Db(X)) induces the Hodge filtration on Ktop(X), and this is enough to reconstruct
+the Hodge structure.
+Both Ktop(C) and the noncommutative Hodge-de Rham sequence for HP(C)
+are additive invariants of dg-categories, and therefore take finite semiorthogonal
+decompositions to direct sum decompositions.
+Thus an immediate consequence
+of Conjecture II is the following de-categorified variant, which can be investigated
+independently:
+Conjecture IV (Hodge-theoretic MMP). Let X → Y be a contraction of a smooth
+projective variety.
+(A/B) There is a canonical direct sum decomposition of Hodge structures
+Ktop(X)Q ∼= H1,ψ ⊕ · · · ⊕ Hn,ψ
+(12)
+that is upper triangular with respect to the Euler pairing and closed under
+multiplication by classes from Y . This decomposition depends on a parameter
+ψ, but different values of ψ give mutation-equivalent decompositions.4
+(C) Given another contraction Y → Y ′, the decomposition of Ktop(X)Q associated
+to X → Y ′ refines the decomposition associated to X → Y for suitable
+parameters.
+(D) If π : X → X′ is a morphism of smooth varieties with Rπ∗(OX) = OX′,
+then for suitable values of the parameters, the decomposition of Ktop(X)Q
+associated to X → Y refines the decomposition obtained by combining
+the canonical decomposition Ktop(X)Q ∼= Ktop(X′)Q
+� ker(π∗) with the
+decomposition of Ktop(X′)Q associated to X′ → Y .
+4Consider a direct sum decomposition of a finite rank free abelian group Λ = Λ1 ⊕ · · · Λn that
+is upper-triangular with respect to a non-degenerate bilinear pairing [−, −) on Λ. To any braid
+on n-strands, with underlying permutation s, the mutation along this braid is a new direct sum
+decomposition Λ = Λ′
+s(1) ⊕. . .⊕Λ′
+s(n), and it is equipped with canonical isomorphisms Λi ∼
+= Λ′
+s(i).
+See [SS, §2.2] for a discussion.
+
+THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.
+15
+In fact in (D), if π : X → Y is a blowup of Y along a smooth center S of
+codimension n + 1, then one expects the decomposition of Ktop(X)Q associated to
+X → Y ′ to refine the canonical decomposition Ktop(X)Q ∼= Ktop(Y )Q⊕(Ktop(S)Q)n
+combined with the canonical decompositions associated to Y → Y ′ and S → π(S) ⊂
+Y ′.
+We expect the decomposition in (12) to arise in the same way as in Proposal III.
+Namely, under a suitable fundamental solution of (6), the lattice in each Hi,ψ
+should span the space of solutions with exponential growth rate eαit as t → ∞,
+where α1, . . . , αn are the eigenvalues of −1
+z Eψ(1).
+Remark 16. The decategorification Conjecture IV (specifically part (D)) is a
+variant of the blowup formula conjectured and investigated by Katzarkov, Kontsevich,
+Pantev, and Yu [K3]. Although our conjecture deals with decompositions of rational
+Hodge structures rather than (formal) Frobenius manifolds, we expect that these
+conjectures would have many of the same applications to rationality questions that
+have been announced for the blowup formula.
+2. Applications
+2.1. Minimal models and the D-equivalence conjecture. Our first application
+defines a dg-category MX/Y that is a relative birational invariant of a contraction
+X → Y , where X has positive geometric genus. We call MX/Y the noncommutative
+minimal model of X relative to Y .
+Proposition 17. Let X → Y be a contraction of projective varieties with X smooth
+and pg(X) > 0. Assuming Conjecture I and Conjecture II(A,D) for varieties over
+Y , there is an admissible subcategory MX/Y ⊂ Db(X) that contains an object whose
+support is X and that has the following property:
+For any other contraction X′ → Y such that X′ is birationally equivalent
+to X relative to Y , there is an admissible embedding MX/Y ֒→ Db(X′) as
+well.
+Furthermore, assuming Conjecture II(B) for varieties over Y , MX/Y has a canonical
+Perf(Y )⊗-module structure such that the embeddings MX/Y ֒→ Db(X′) are Perf(Y )⊗-
+linear.
+As the proof will show, MX/Y arises as one of the semiorthogonal factors coming
+from the NMMP for some birational cover of X.
+Proof. Let Db(X) = ⟨C1, . . . , Cn⟩ denote the semiorthogonal decomposition that
+Conjecture II(A) associates to π and a generic choice of parameter ψ.
+Because
+pg > 0, KX has a non-vanishing section, i.e., the base locus of |KX| has positive
+codimension. It follows from [KO, Thm. 1.2] that exactly one of the categories Ci
+contains an object whose support is all of X. Let us call this category CX,ψ.
+Conjecture II(A) asserts that different choices of ψ give mutation equivalent
+semiorthogonal decompositions, but any mutation of ⟨C1, . . . , Cn⟩ gives a canonical
+equivalence between the subcategories containing a densely supported object, so
+CX,ψ ∼= CX,ψ′ for different generic values of the parameter. We therefore denote
+CX = CX,ψ for any fixed choice of ψ and suppress ψ from the notation below.
+Conjecture II(B) implies that CX,ψ is a module category for Perf(Y )⊗, and the
+mutation equivalences respect this structure, so CX has a well-defined Perf(Y )⊗-
+module structure.
+
+16
+DANIEL HALPERN-LEISTNER
+Now let f : Z → X be a projective birational morphism, with Z smooth, and
+consider the NMMP of Z relative to Y . Then Conjecture II(D) implies that for
+suitable choices of parameter, CZ ⊂ f ∗(CX) ∼= CX is an admissible subcategory.
+Furthermore, by hypothesis CZ corresponds to a direct summand of the charge
+lattice of CX. Because this charge lattice is finite dimensional, there must be a
+birational morphism Z → X such that for any further birational morphism Z′ →
+Z → X, CZ ∼= CZ′. For any other contraction X′ → Y that is birational to X
+relative to Y , one can find a smooth projective Z′ with birational maps Z′ → Z
+and Z′ → X′ that are compatible with the given birational equivalence over Y . It
+follows that CZ = CZ′ ⊂ CX′ ⊂ Db(X′) are admissible inclusions.
+□
+Proposition 17 explains why birationally equivalent Calabi-Yau manifolds should
+have equivalent derived categories. In fact, we have the stronger statement:
+Corollary 18. Assuming Conjecture I and Conjecture II(A,D) hold for varieties
+over Spec(k), if X and X′ are birationally equivalent smooth projective varieties and
+|KX| is base-point free, then there is a canonical admissible embedding Db(X) ֒→
+Db(X′), which is an equivalence if |KX′| is also base-point free.
+Proof. If KX is base-point free, then Db(X) admits no semiorthogonal decompositions
+[KO, Thm. 1.2]. By Proposition 17, it suffices to show that MX/ Spec(k) = Db(X).
+To see this, consider a birational morphism f : Z → X with Z smooth and
+projective.
+If we apply the NMMP for Z → Spec(k), Conjecture II(D) implies
+that for a suitable choice of parameter, the unique semiorthogonal factor CZ ⊂
+Db(Z) that is densely supported must lie in f ∗(Db(X)) and thus must be equal to
+f ∗(Db(X)).
+□
+Remark 19. In Corollary 18, it suffices to verify 1) Conjecture II(A) only for X
+in the birational equivalence class of interest; and 2) Conjecture II(D) holds when
+X → X′ is the blowup of the smooth variety X′ along a smooth center, but with the
+stronger requirement that the semiorthogonal decomposition of Db(X′) obtained
+as a piece of the semiorthogonal decomposition of Db(X) agrees with (rather than
+refines) the decomposition associated to X′ → Spec(k).
+Proof. By the weak factorization theorem, the birational morphism X ��� X′
+can be factored as a sequence of birational maps X = X1 ��� X2 ��� · · · ���
+Xn = X′, where each morphism or its inverse is a blowup of a smooth variety
+along a smooth center. One can then use Conjecture II(D) for blowups to argue
+by induction that in the decomposition of Db(Xi) associated to Xi → Spec(k)
+by Conjecture II(A), the unique generically supported semiorthogonal factor is
+indecomposable and equivalent to Db(X).
+□
+2.2. Minimal resolutions. A similar application of Conjecture II is to define for
+any variety Y a dg-category RY that we regard as the minimal noncommutative
+resolution of Y .5
+Below we will use the monoidal structure on the ∞-category
+of small idempotent complete module categories over a small idempotent complete
+symmetric monoidal stable ∞-category A⊗, which is induced from that on presentable
+stable module categories over Ind(A⊗). Namely M⊗AN is the category of compact
+objects in Ind(M) ⊗Ind(A) Ind(N).
+The key fact is that for a Tor-independent
+5Several different notions of noncommutative resolution of singularities exist in the literature,
+but we are not aware of one that agrees with what we establish here.
+
+THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.
+17
+cartesian diagram of schemes, Y ′ ∼= X′×X Y , one has Perf(Y ′) ∼= Perf(X′)⊗Perf(X)
+Perf(Y ) by [BZFN, Thm. 1.2].
+Proposition 20. Let Y be a reduced variety (possibly singular), and assume that
+Conjecture II(A,B,D) holds for any birational morphism from a smooth projective
+variety X → Y . There is a canonical smooth and proper dg-category RY equipped
+with a Perf(Y )⊗-module structure such that:
+(1) If U ⊂ Y is the smooth locus, then Perf(U) ⊗Perf(Y ) RY ∼= Perf(U); and
+(2) For any resolution of singularities X → Y , there is a Perf(Y )⊗-linear
+admissible embedding RY ֒→ Db(X).
+Proof. The proof is identical to that of Proposition 17, except that we use the
+following definition for the admissible subcategory CX ⊂ Db(X) associated to a
+resolution π : X → Y : Conjecture II(A,B) gives a Perf(Y )⊗-linear semiorthogonal
+decomposition Db(X) = ⟨C1,ψ, . . . , Cn,ψ⟩ associated to π and a parameter ψ. For
+any U ⊂ Y such that π−1(U) → U is an isomorphism, Perf(U) ∼= Perf(π−1(U)) ∼=
+Perf(X) ⊗Perf(Y ) Perf(U). Using base change for Perf(Y )⊗-linear semiorthogonal
+decompositions [K2], one obtains a Perf(U)⊗-linear semiorthogonal decomposition
+Perf(U) = ⟨Perf(U) ⊗Perf(Y ) C1,ψ, . . . , Perf(U) ⊗Perf(Y ) Cn,ψ⟩.
+In particular, each Perf(U) ⊗Perf(Y ) Cj,ψ is a thick ⊗-ideal of Perf(U), and thus by
+[T, Thm. 3.15] is the category of complexes supported on some subspace Zi ⊂ |U|
+that is a union of closed subspaces (with quasi-compact complement, but that is
+automatic here). U is irreducible, so Perf(U) = Perf(U)⊗Perf(Y ) Ci,ψ for the unique
+index i such that Zi contains the generic point of U, and thus Perf(U)⊗Perf(Y )Cj,ψ =
+0 for all j ̸= i. The identification of this distinguished index i does not depend on
+the specific choice of U. Also, as in the proof of Proposition 17, Conjecture II(A,B)
+implies that up to a canonical Perf(Y )⊗-linear equivalence, the category Ci,ψ does
+not depend on ψ, so we define CX := Ci,ψ for this i.
+The rest of the proof of
+Proposition 17 now applies verbatim.
+□
+Remark 21. In the lectures [K3], Maxim Kontsevich has also speculated about
+the existence of canonical noncommutative resolutions for varieties with canonical
+singularities in the context of the blowup formula. Proposition 20 explains how a
+version of this follows from the formal properties laid out in Conjecture II.
+2.3. Example: simple flips and flops. It might not stand out in Conjecture II(A),
+but the key idea behind Corollary 18 is that canonical semiorthogonal decompositions
+associated to different birational morphisms X → Y and X → Y + should be related
+via mutation. For example, let Y be a smooth projective variety with a smooth
+embedding Pn ֒→ Y with normal bundle OPn(1)⊕m+1, where m ≤ n. Then one has
+a diagram
+Pn × Pm� �
+j
+�
+p
+�✉✉✉✉✉✉✉✉✉
+X
+π
+�✈✈✈✈✈✈✈✈✈✈
+π+
+�❇
+❇
+❇
+❇
+❇
+❇
+❇
+❇
+Pn� �
+� Y
+Y +
+,
+(13)
+where π+ is the blow up, with exceptional divisor Pn×Pm, of the smooth projective
+variety Y + along an embedded Pm ֒→ Y + with normal bundle OPm(1)⊕n+1. It is
+shown in [BO] that the composition of derived functors π∗(π+)∗ : Db(Y +) → Db(Y )
+is fully faithful, and an equivalence when m = n.
+
+18
+DANIEL HALPERN-LEISTNER
+In the simple case where m = 1, we can recover this fact using mutations of semi-
+orthogonal decompositions. We let Eq
+p denote the exceptional object j∗(OPn×Pm(p, q))
+in Db(X).
+Example 22 (Atiyah flops). Consider the above set up with n = m = 1, so that
+(13) is a flop of 3-folds. The semiorthogonal decomposition of Example 4 combined
+with the semiorthogonal decomposition Db(P1) = ⟨O(−1), O⟩ gives semiorthogonal
+decompositions
+Db(X) = ⟨E−1
+−1, E−1
+0 , Db(Y )⟩
+= ⟨E−2
+−1, E−1
+−1, Db(Y +)⟩
+Using the fact that ωX|Pn×Pm ∼= O(−n, −m), one sees that the right orthogonal
+complement of E−1
+0
+agrees with the left orthogonal complement of E−2
+−1, and both
+objects are left orthogonal to E−1
+−1. It follows that we have the following mutations
+E−1
+−1
+E−1
+0
+Db(Y )
+E0
+−1
+E−1
+−1
+E−2
+−1
+Db(Y +)
+Composing mutation equivalence functors gives an equivalence Db(Y ) ∼= Db(Y +).
+In the more general situation where m = 1 and n ≥ 1, the semiorthogonal
+decomposition of Example 4 applied to π+ combined with the Beilinson exceptional
+collections on P1 gives
+Db(X) = ⟨
+Db(P1)(−n)
+�
+��
+�
+E−2
+−n, E−1
+−n,
+Db(P1)(−n+1)
+�
+��
+�
+E−1
+−n+1, E0
+−n+1,
+Db(P1)(−n+2)
+�
+��
+�
+E−1
+−n+2, E0
+−n+2, . . . ,
+Db(P1)(−1)
+�
+��
+�
+E−1
+−1, E0
+−1, Db(Y +)⟩.
+We first mutate this to
+Db(X) = ⟨E−1
+−n,
+Db(P1)(−n+1)
+�
+��
+�
+E−1
+−n+1, E0
+−n+1,
+Db(P1)(−n+2)
+�
+��
+�
+E−1
+−n+2, E0
+−n+2, . . . ,
+Db(P1)(−1)
+�
+��
+�
+E−1
+−1, E0
+−1, Db(Y +), E−1
+0 ⟩.
+If we mutate the objects E−1
+−n, . . . , E−1
+0
+to the left over the other summands, one
+obtains a collection of exceptional objects A−n+1, . . . , A−1 fitting into a semi-
+orthogonal decomposition
+Db(X) = ⟨
+Db(Pn)(−1)
+�
+��
+�
+E−1
+−n, . . . , E−1
+0 , A−n+1, . . . , A−1, B⟩,
+
+THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.
+19
+where mutation gives a canonical equivalence B ∼= Db(Y +). This last semiorthogonal
+decomposition refines Db(X) = ⟨E−1
+−n, . . . , E−1
+0 , Db(Y )⟩ coming from the morphism
+π, hence we have
+Db(Y ) = ⟨A−n+1, . . . , A−1, Db(Y +)⟩.
+More precisely, because the right projection onto Db(Y ) ⊂ Db(X) is π∗π∗, the fully
+faithful functor Db(Y +) ֒→ Db(Y ) coming from this mutation agrees with π∗(π+)∗,
+and one has Ai = π∗π∗(E0
+i ), which as an object of Db(Y ) corresponds to OPn(i).
+2.4. Dubrovin’s conjecture. Dubrovin conjectured [D2] that for a Fano manifold
+X, Db(X) admits a full exceptional collection if and only if the quantum cohomology
+QH∗(X) is generically semisimple. Here we observe that the NMMP conjectures
+imply one direction, that generic semisimplicity implies the existence of a full
+exceptional collection.
+Proposition 23. Let X be a smooth projective variety for which Proposal III holds
+for generic z. If in addition ch : K0(Db(X)) ⊗ Q → H∗(X; Q) is an isomorphism
+and there is a ψ such that Eψ(1) ∈ End(H∗
+alg(X) ⊗ C) is semisimple with distinct
+eigenvalues, then Db(X) admits a full exceptional collection consisting of eventually
+semistable objects.
+The condition that ch is an isomorphism can often be checked in practice. For
+instance, it holds for compact homogeneous spaces of reductive groups, smooth and
+proper toric varieties, and more generally any variety that admits an affine paving.
+Note that the result above does not require X to be Fano, and does not explicitly
+require QH∗(X) to be generically semisimple.
+Lemma 24. Let C be a regular proper idempotent complete pre-triangulated dg-
+category, and let C = ⟨C1, . . . , Cn⟩ be a semiorthogonal decomposition such that,
+dim(K0(Ci) ⊗ Q) = 1 for all i, and each Ci admits a stability condition. Then each
+Ci is generated by a single exceptional object, i.e., the semiorthogonal decomposition
+arises from a full exceptional collection in C.
+Proof. Because the property of being regular and proper is inherited by semi-
+orthogonal factors, it suffices to prove this for the trivial semiorthogonal decomp-
+osition, i.e., in the case n = 1. If C admits a Bridgeland stability condition and
+dim(K0(C)⊗Q) = 1, then all semistable objects in the heart have the same phase. It
+follows that the heart P(0, 1] is Artinian. We will prove that if dim(K0(C)⊗Q) = 1
+and C is regular, proper and admits a bounded t-structure with Artinian heart,
+then C ∼= Db(k).
+Let A ⊂ C be the heart of the t-structure. Then K0(A) = K0(C), and the former
+has a basis given by the classes of simple objects in A. Because K0(A) has rank
+1, there is a unique simple module E. Because every object in A has a Jordan-
+Holder filtration, whose graded pieces must be isomorphic to E, we see that C is
+the smallest triangulated category containing E. We will complete the proof by
+showing that E is an exceptional object.
+Let A = RHom(E, E) be the dg-algebra of endomorphisms of E. We claim that
+there is a A-module M such that H∗(M) = k. Indeed, because Hi(A) = 0 for
+i < 0 and H0(A) = k, by [K2, Lem. 3.5] there is a dg-subalgebra B ⊂ A that
+admits a strictly unital A∞ morphism f : B → k. Then f ∗(k) is a B-module whose
+homology is k in degree 0, and pullback induces an equivalence A -Mod → B -Mod
+that preserves homology of modules [K2, Prop. 6.2].
+
+20
+DANIEL HALPERN-LEISTNER
+Because E generates C and C is idempotent complete, the functor RHom(E, −) :
+C → Perf(A) is an equivalence of dg-categories. Also, because C is regular and
+proper, an A-module is perfect if and only if its underlying complex of k-vector
+spaces is perfect [O2, Thm. 3.18]. In particular, M ∈ Perf(A), and so there is an
+object E′ ∈ C such that RHom(E, E′) ∼= k as complexes. We will show that E′ ∼= E
+to conclude the proof.
+Let n ≥ 0 be the largest i such that Hi(RHom(E, E)) ̸= 0. Then examining the
+long exact cohomology sequence for RHom(E, −) of an extension of objects shows
+that for F ∈ A, n = max{i|Hi(RHom(E, F))} as well. Likewise, if Hi(−) denotes
+the cohomology object with respect to the t-structure on C, then for any F ∈ C,
+max{i|Hi(RHom(E, F)) ̸= 0} = n + max{i|Hi(F) ̸= 0}.
+This is proved by inductively by examining the long exact cohomology sequence for
+RHom(E, −) applied to the exact triangle τ <m(F) → F → Hm(F)[−m], where m
+is the highest non-vanishing cohomology object of F. The same reasoning shows
+that
+min{i|Hi(RHom(E, F)) ̸= 0} = min{i|Hi(F) ̸= 0}
+for any F ∈ C. Applying this to F = E′, we see that
+min{i|Hi(E′) ̸= 0} = n + max{i|Hi(E′) ̸= 0}.
+This can only hold if n = 0, hence E is exceptional.
+□
+Proof of Proposition 23. Let us fix a z close to 1 such that −1
+z Eψ(1) has distinct
+eigenvalues u1, . . . , un with distinct real parts. Proposition 9 then implies that for
+any solution Φt of (6), ∥Φt∥ ∼ eℜ(uj)t for some j.
+Also, if E is an eventually
+semistable object such that |Zt(E)| ∼ ∥Φt(E)∥ ∼ eℜ(uj)t, then Zt(E) ∼ eujt.
+Proposal III then implies that every eigenvalue uj must occur for some eventually
+semistable E with |Zt(E)| ∼ ∥Φt(E)∥.
+This implies that the semiorthogonal
+decomposition C = ⟨C1, . . . , Cn⟩ associated to the quasi-convergent path stipulated
+in Proposal III has length n = dim(ΛQ). The assumption that K0(C)Q → ΛQ is an
+isomorphism then implies that dim(K0(Ci)⊗ Q) = 1 for all i. Hence C admits a full
+exceptional collection consisting of eventually semistable objects by Lemma 24
+□
+Finally, let us make the following connection with the Gamma II conjecture of
+[GGI]. This conjecture states that, in the situation where X is a Fano manifold with
+a full exceptional collection and generically semisimple quantum cohomology, there
+is a full exceptional collection Db(X) = ⟨E1, . . . , En⟩ such that for the canonical
+fundamental solution Φt of the quantum differential equation (see Remark 13),
+Φt(v(Ei)) give the “asymptotically exponential” basis of solutions, characterized
+uniquely in [GGI, Prop. 2.5.1].6
+We observe that, in a sense, the Gamma II
+conjecture is equivalent to Proposal III in this context, where we let Φt be the
+canonical fundamental solution.
+6Some notational comments are in order: our variable t corresponds to the variable 1/z in the
+terminology of [GGI]. On the other hand, the parameter z in our (6) is set to one in [GGI, Eq. 2.2.3].
+Instead, they introduce an auxiliary parameter φ and consider solutions in a sector centered on
+the ray R>0eiφ, which is equivalent to setting z = eiφ in (6) and considering solutions centered
+on the positive real axis, as we do.
+
+THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.
+21
+Proposition 25. Let X be a Fano manifold with generically semisimple quantum
+cohomology that admits a full exceptional collection Db(X) = ⟨E1, . . . , En⟩ such
+that Φt(v(Ei)) give the asymptotically exponential basis of solutions of the quantum
+differential equation.
+Then this full exceptional collection arises from a quasi-
+convergent path as in Proposal III.
+Proof. After a homological shift, we may assume that the collection Ei is “Ext-
+exceptional” in the language of [M, Def. 3.10], meaning Hom≤0(Ei, Ej) = 0 for all
+i ̸= j. Let u1, . . . , un ∈ C be the eigenvalues of − 1
+z Eψ(1), ordered so that ℑ(ui)
+is strictly increasing in i, after fixing a generic choice of z (see [GGI, Rem. 2.6.5]).
+It is observed in [GGI, §4.7] that if the Gamma II conjecture holds, then up to
+an overall scalar multiple, the quantum cohomology central charges Zt(Ei) have
+asymptotic estimates in our notation Zt(Ei) ∼ cietui as t → ∞ for some constants
+ci ̸= 0.
+Let t0 ∈ R be such that Zt(Ei) ̸= 0 for all i and all t ≥ t0. Then after choosing
+a branch for ln(Zt0(Ei)), we can lift Zt(Ei) uniquely to a function ln(Zt(Ei)) that
+is continuous in t ≥ t0 for each i. The asymptotic estimate above implies that
+ln(Zt(Ei)) = tui + βi + o(1) as t → ∞. We can therefore choose t1 ≥ t0 such that
+for all t ≥ t1, φi,t := ℑ(ln(Zt(Ei)))/π is strictly increasing in i. The collection
+Ei[−⌊φi,t⌋] is still Ext-exceptional, so by the discussion following [M, Prop. 3.17],
+there is a unique stability condition with central charge Zt such that the Ei[−⌊φi,t⌋]
+are all in the heart and semistable, and hence Ei is semistable of phase φi,t for all
+i. This defines a quasi-convergent path σt ∈ Stab(X) for t ≥ t1 such that the
+Ei are eventually semistable, and it recovers the semiorthogonal decomposition
+Db(X) = ⟨E1, . . . , En⟩.
+□
+The quasi-convergent paths constructed in Proposition 25 lie entirely in the
+region of stability conditions that are glued from the given full exceptional collection.
+While these paths technically satisfy Proposal III, due to its flexible formulation,
+it would be more satisfying to give a description of the paths that does not make
+a priori use of the full exceptional collection.
+For instance, it is an interesting
+question as to whether the paths in Proposition 25 extend in the t → 0 direction
+into a geometric region in Stab(X)/Ga. (See Remark 14.)
+3. Example of curves
+3.1. The projective line.
+The stability manifold of P1. In [O1], it is shown that
+Stab(P1)/Ga =
+�
+k∈Z
+Xk ∼= C,
+where Xk ⊂ Stab(P1)/Ga is the open submanifold of stability conditions in which
+O(k) and O(k − 1) are stable. The map ϕk : Xk → C defined by
+ϕk(σ) = logZσ(O(k)) − logZσ(O(k − 1))
+defines an isomorphism between Xk and the open upper half plane H ⊂ C. Under
+this isomorphism, the strip {x+iy|y ∈ (0, iπ)} lies in Xk for all k, and these stability
+conditions coincide with slope stability, up to the canonical action of �
+GL
++(2, R) on
+Stab(X) [B3, Lem. 8.2]. The short exact sequence 0 → O(k − 1) → O(k)⊕2 →
+
+22
+DANIEL HALPERN-LEISTNER
+O(k + 1) → 0 implies that on this common strip, the coordinate functions ϕk are
+related by the equation
+eϕk+1 = 2 −
+1
+eϕk .
+(14)
+The isomorphism Stab(P1)/Ga ∼= C is not explicit in [O1], so let us give an
+explicit parameterization. The central charge will be described in terms of modified
+Bessel functions of the first and second kind I0(u) and K0(u), which are a basis of
+solutions to the modified Bessel differential equation
+(u d
+du)2Z(u) = u2Z(u).
+(15)
+The function I0 is entire, but the function K0 has a branch point at 0, and we will
+use the principal branch with branch cut along i(−∞, 0]. These are characterized
+among solutions of (15) by the following asymptotic estimates as u → 0
+I0(u) = 1 + O(|u|2)
+K0(u) = − ln( u
+2 ) − Ceu + O(|u|2| ln(u)|) ,
+(16)
+where Ceu = 0.57721... is Euler’s constant, and we also take the principle branch
+of ln with branch cut along i(−∞, 0].
+Proposition 26. For any k ∈ Z and τ ∈ R + iπ[k − 1, k], there is a unique (up
+to homological shift [2]) stability condition on Db(P1) such that O(k − 1) and O(k)
+are stable, and the central charge is determined by
+Zτ(Op) = iπI0((−1)k−1eτ), and
+Zτ(O(k − 1)) = K0((−1)k−1eτ)
+The resulting maps Bk : R+iπ[k−1, k] → Stab(P1)/Ga glue to give an isomorphism
+of complex manifolds B : C ∼= Stab(P1)/Ga, such that the action of O(1) ⊗ (−) on
+Stab(P1)/Ga is identified with the shift τ �→ τ + iπ on C.
+Note that (−1)k−1eτ ∈ H ∪ R \ 0 in the formulas above.
+Lemma 27. For x ∈ C with ℜ(x) > 0, we have K0(−x) = K0(x) − iπI0(x) and
+I0(−x) = I0(x).
+Proof. K0(−x) and I0(−x) are also solutions of the modified Bessel differential
+equation, so they are expressible as linear combinations of K0(x) and I0(x). The
+coefficients are determined by the asymptotic estimates (16) as x → 0.
+□
+Lemma 28. ℑ(xK0(x)(K0(x) + iπI0(x))) > 0 for x ̸= 0 with ℑ(x) ≥ 0.
+We thank Nicolas Templier for his assistance in proving this Lemma.
+Proof. Let g(x) = xK0(x)(K0(x) + iπI0(x)), and observe that because of the
+asymptotics (16), g(x) extends continuously over the origin by letting g(0) = 0.
+We will need the following asymptotic estimates as |u| → ∞ with −π/2 + δ ≤
+arg(u) ≤ 3π/2 − δ for some small δ > 0, from [L, 10.40.2 and 10.40.5]:
+K0(u) = � π
+2ue−u(1 −
+1
+8u +
+9
+128u2 + O(
+1
+|u|3 ))
+I0(u) =
+eu
+√
+2πu(1 +
+1
+8u +
+9
+128u2 + O(
+1
+|u|3 )) + i e−u
+√
+2πu(1 −
+1
+8u +
+9
+128u2 + O(
+1
+|u|3 ))
+.
+(17)
+
+THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.
+23
+These imply the following estimate for x ∈ H ∪ R with |x| large:
+g(x) = iπ
+2 (1 +
+1
+8x2 ) + O( 1
+|x|3 ).
+It follows that there is some r0 > 0 such that ℑ(g(x)) > 0 for all |x| > r0. For
+x ∈ R>0, both K0(x) > 0 and I0(x) > 0, so ℑ(g(x)) = πxK0(x)I0(x) > 0.
+Using the identity of Lemma 27, one can compute for x < 0 that ℑ(g(x)) =
+πxI0(−x)K0(−x) > 0 as well.
+We now apply the maximum principle to the non-constant continuous real-valued
+function ℑ(g(x)) on the closed half disc {x ∈ C||x| ≤ r and ℑ(x) ≥ 0} for some
+r > r0. This function is harmonic on the interior {|x| < r and ℑ(x) > 0} and
+we have shown that ℑ(g(x)) ≥ 0 on the boundary, with strict inequality except at
+x = 0. We conclude ℑ(g(x)) > 0 on the interior and on the boundary away from
+x = 0. Because r was arbitrary, the claim follows.
+□
+Proof of Proposition 26. As X1 ⊂ Stab(P1)/Ga is the open subset where O and
+O(1) are stable, and X1 is identified with H ⊂ C by the coordinate ϕ1, we first
+describe the map B1 : R + i[0, π] → X1 via the formula τ �→ f(eτ), where
+f(x) := ln
+�Zln x(O(1))
+Zln x(O)
+�
+= ln
+�K0(x) + iπI0(x)
+K0(x)
+�
+.
+The ln(−) is defined because Lemma 28 implies that K0(x) ̸= 0 and K0(x) +
+iπI0(x) ̸= 0 if ℑ(x) ≥ 0 and x ̸= 0. For x ∈ R \ 0, we interpret ln(−) above as
+the principle branch of the logarithm with branch cut along (−∞, 0]. But as x
+varies in H ∪ R \ 0, (K0(x) + iπI0(x))/K0(x) crosses this branch cut many times,
+so we define the ln(−) as the unique lift along the exponential covering C → C∗.
+The asymptotics (16) imply that f extends continuously over the origin by letting
+f(0) = 0, and we use this convention.
+Claim 1: f(x) maps R \ 0 to the curve
+C := {a + ib|0 < b < π/2 and e|a| cos(b) = 1} :
+For x ∈ R>0, K0(x) > 0 and I0(x) > 0, so ln(1 + iπI0(x)/K0(x)) ∈ ln(1 + iR>0),
+which lies on the curve {a + ib|ea+ib − 1 ∈ iR, 0 < b < π/2, 0 < a}. On the other
+hand, for x < 0, Lemma 27 implies that ℜ(K0(−x)) > 0 and I0(−x) > 0, and this
+implies
+f(x) = ln
+�
+K0(−x)
+K0(−x) − iπI0(−x)
+�
+= − ln
+�
+1 − iπ I0(−x)
+K0(−x)
+�
+.
+This lies on the curve {a + ib|e−a−ib − 1 ∈ iR, 0 < b < π/2, a < 0}. C is the union
+of these two curves.
+Claim 2: f maps H ∪ R \ 0 injectively to the region of C lying on or above C ∪ {0}:
+Abel’s identity allows one to compute the Wronskian K0(x)I′
+0(x)−I0(x)K′
+0(x) =
+1/x, and using this we compute
+f ′(x) =
+iπ
+xK0(x)(K0(x) + iπI0(x)).
+
+24
+DANIEL HALPERN-LEISTNER
+Lemma 28 implies that ℜ(f ′(x)) > 0 for all x ∈ H ∪ R \ 0. It follows that for any
+x, v ∈ H ∪ R with v ̸= 0,
+ℜ(¯v(f(x + v) − f(x))) =
+� 1
+0
+|v|2ℜ(f ′(x + tv))dt > 0.
+Therefore, f(x + v) ̸= f(x), and so f is injective.
+If we apply this inequality
+specifically to x = a ∈ R and v = ib ∈ iR>0, then it implies ℑ(f(a+ib))−ℑ(f(a)) >
+0. So Claim 1 implies that f(a + ib) lies above the curve C ∪ {0}.
+Verifying that the maps Bk glue:
+The description of the map Bk from the strip Sk := R + iπ[k − 1, k] to Xk
+in the proposition simply results from applying O(1) ⊗ (−) on the one hand, and
+τ �→ τ + iπ on the other, so these maps are well-defined. We must show that they
+glue to a holomorphic map C → Stab(P1)/Ga, which will then automatically be
+Z-equivariant.
+It suffices, by Z-equivariance, to show that B0 and B1 glue to a holomorphic
+map in a neighborhood of the boundary S1 ∩ S0 = R.
+The first map is given
+in coordinates by ϕ0 = f(−ex) for x ∈ R, whereas the second map is given by
+ϕ1 = f(ex). Using the relation (14), showing that these maps agree amounts to
+showing that
+K0(ex) + iπI0(ex)
+K0(ex)
+= 2 −
+K0(−ex)
+K0(−ex) + iπI0(−ex).
+This follows from the identities proved in Lemma 27. Both sides of this identity
+are holomorphic and agree in an open neighborhood of R, which implies the claim.
+Showing B is an isomorphism:
+Claim 2 above shows that the image of Bk is contained in the region in Xk ∼= H
+on or above the curve C ∪ {0}.
+In [O1], it is shown that this region in Xk is
+a fundamental domain for the action of Z on Stab(P1)/Ga.
+It follows from Z-
+equivariance that B is injective, because it is injective on Sk and no other strip Sk′
+can map to this region in Xk (except for the two boundary components Sk−1 ∩ Sk
+and Sk∩Sk+1). We also know from [O1] that Stab(P1)/Ga ∼= C, so B is an injective
+entire function, which implies it is an isomorphism. Alternatively, the little Picard
+theorem implies that B is surjective, because by Z-equivariance, if there is one
+point that is not in the image of B, then there are infinitely many.
+□
+Mirror symmetry and the noncommutative MMP. We begin by studying the quantum
+differential equation.
+(See [FLZ] for a very thorough discussion.)
+Let us use
+the standard basis 1, H := c1(O(1)) for the cohomology of P1.
+The quantum
+cohomology ring is C[H, q]/(H2 − q), and the quantum differential equation in the
+basis {1, H} and parameters ψ = 2aH and z = e−b is
+tdΦt
+dt = −eb(2H) ⋆2(ln(t)−a)H Φt = −2eb
+�0
+e−2at2
+1
+0
+�
+Φt.
+(18)
+For any solution of (18), the function Zt(E) =
+�
+P1 Φt(E) satisfies the second order
+equation
+(t d
+dt)2Zt(E) =
+�
+2eb−a�2 t2Zt(E).
+(19)
+
+THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.
+25
+⟨O, O(1)⟩
+⟨O(−1), O⟩
+⟨O(−2), O(−1)⟩
+Figure 1. A visualization of Stab(P1)/Ga
+∼=
+C.
+The red
+region is the �
+GL
++(2, R)-orbit of slope stability. The blue regions
+are stability conditions that are glued from the full exceptional
+collections shown, which correspond to the regions with imaginary
+part > π in each of the coordinate charts Xk. The black path is
+determined by a particular solution to the quantum differential
+equation.
+The green vertical line represents the line added at
+infinity in the partial compactification of Stab(P1)/Ga. The dotted
+horizontal lines differ by integer multiples of πi.
+Choose a k such that ℑ(b−a) ∈ π[k−1, k], which guarantees that κ := (−1)k−12eb−a
+lies in H. After a change of variables u = κt, (19) is the modified Bessel differential
+equation (15), so we have Zt(E) = c1(E)I0(κt) + c2(E)K0(κt) for some constants
+c1 and c2 depending on E.
+We have shown in Proposition 26 that the paths
+σt := B(ln(2t) + b − a) ∈ Stab(P1)/Ga
+for t ∈ [1, ∞) have central charges of the form above. Let us show that for generic
+values of b − a, σt is quasi-convergent as t → ∞:
+For any value of b − a ∈ C, the path σt lies entirely in Xk. In the coordinate ϕk
+on Xk, the paths have the form
+ϕk = ln
+�K0(κt) + iπI0(κt)
+K0(κt)
+�
+.
+Using the asymptotic estimates (17), we have
+ϕk = ln
+
+
+eκt
+√
+2πκt(1 + O( 1
+|κt|))
+� π
+2κte−κt(1 + O( 1
+|κt|))
+
+ = 2κt − ln(π) + O( 1
+|κt|).
+If ℑ(κ) > 0 this path therefore eventually enters the region of Xk where ℑ(ϕk) > π.
+In this region, the only stable objects are O(k−1) and O(k), and they remain stable
+
+26
+DANIEL HALPERN-LEISTNER
+for all t ≫ 0. Therefore, these paths satisfy the conditions of Lemma 1, and the
+only eventually semistable objects are of the form O(k − 1)⊕m[n] and O(k)⊕m[n].
+Remark 29. For non-generic values of b − a, meaning those for which κ ∈ R, the
+paths above also converge in the partial compactification of Stab(P1)/Ga constructed
+in [HLR], but recovering the semiorthogonal decomposition is a bit more complicated
+than applying Lemma 1.
+In order to verify Proposal III, we must describe a fundamental solution of the
+quantum differential equation (18) whose integral is the path of central charges Zt
+underlying σt. We will do this with SYZ mirror symmetry:
+The mirror of P1 is the Landau-Ginzburg model (C∗, Wt(x) = x +
+t2
+e2ax), where
+t is regarded as a parameter. It is straightforward to check that
+1
+2
+�
+L
+e−ebWt(x)(H · dx
+x + 1 · t2dx
+e2ax2 ) ∈ H∗(P1; C)
+solves the differential equation (18) whenever L ⊂ C∗ is a contour such that
+ℜ(ebWt(x)) → ∞ at the ends.
+For simplicity, let us assume L lies on the ray
+R>0e−b in a neighborhood of ∞, and lies on the ray R>0eb−2a in a neighborhood
+of 0. Under Proposal III, up to a constant multiple one has (after substituting ebx
+for x)
+Zt(E) =
+�
+P1 Φt(E) = 1
+2
+�
+L(E)
+e
+−
+�
+x+ (κt)2
+4x
+�
+dx
+x ,
+(20)
+where Lκ(E) is the Lagrangian, or formal sum of Lagrangians, in C∗ that is SYZ-
+dual to E. Because of the substitution of variables, the Lagrangian Lκ(E) ⊂ C∗
+lies along R>0 in a neighborhood of ∞, and along R>0e2(b−a) in a neighborhood of
+0.
+We can define Lκ(E) so that Zt(E) matches the central charges of the paths
+σt = B(ln(2t) + b − a) as follows:
+• For any closed point p ∈ P1, we let Lκ(Op) be the closed contour Lκ(Op) :=
+{eiθ}2π
+θ=0. Because this contour is compact, the resulting function Zt(Op)
+is a solution of (19) that extends holomorphically to t = 0, and the residue
+theorem implies Z0(Op) = 1
+2
+�
+e−xdx/x = πi. The aymptotics (16) then
+imply that Zt(Op) = πiI0(κt).
+• For any θ ∈ R, let Cθ ⊂ C∗ be the image under exp : C → C∗ of the contour
+{t + iθ}0
+t=−∞ ∪ {it}0
+t=θ ∪ {t}∞
+t=0. Note that as long as ℜ(κ2eiθ) > 0, the
+contour integral (20) over Cθ is convergent, and it is independent of θ in
+this range by Cauchy’s theorem. This implies that if we let Lκ(O(k−1)) be
+the contour C2ℑ(b−a)−2π(k−1), then the formula for Zt(O(k − 1)) in (20) is
+holomorphic in κ. When κ ∈ R>0, (20) recovers a known integral formula
+for K0(κt) [L, 10.32.10], which can be proven using the method of steepest
+descent. We therefore conclude that with this choice of Lκ(O(k − 1)), we
+have Zt(O(k − 1)) = K0(κt) for all κ ∈ H ∪ R \ 0.
+These values of Zt(Op) and Zt(O(k − 1)) precisely match the characterization of
+B(ln(2t) + b − a) in Proposition 26, and Lκ(Op) and Lκ(O(k − 1)) determine a
+fundamental solution of (18) because [O(k − 1)] and [Op] are a basis for K0(P1).
+
+THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.
+27
+3.2. Higher genus curves. If X is a smooth projective curve of genus g > 1, then
+the set of stable objects for any stability condition consists of shifts of line bundles
+and structure sheaves of points (see [M, Thm. 2.7]). Choosing some point p ∈ X,
+the map
+Stab(X)/Ga → C
+taking
+(P, Z) �→ Z(Op)/Z(OX)
+is injective and identifies Stab(X)/Ga with the upper half space H. Therefore, a
+path in the space of central charges lifts to Stab(X)/Ga if and only if it is contained
+in H.
+Let us use the standard basis 1, H for H∗
+alg(X), where H ∈ H2(X; Z) is the
+generator of degree 1. Because there are no non-trivial maps from P1 to X, the
+equation (5) has a particularly simple form:
+tΦt
+dt = −1
+z c1(X) ∪ Φt = 2g − 2
+z
+�0
+0
+1
+0
+�
+Φt.
+When g = 1, this equation is trivial, so we assume that g > 1. Any fundamental
+solution has the form
+Φt = t
+−1
+z c1(X)A =
+�
+1 + 2g − 2
+z
+ln(t)
+�
+0
+0
+1
+0
+��
+A,
+(21)
+for some invertible 2 × 2 matrix A.
+Remark 30. This phenomenon is more general: For any smooth projective variety
+X that is minimal in the sense that KX is nef, Eψ(u) = c1(X) ∪ (−) is nilpotent
+and independent of both ψ and u, and every fundamental solution of (5) has the
+form t
+−1
+z c1(X)A for some invertible matrix A.
+Now let us imagine a family of stability conditions satisfying Proposal III, i.e.,
+such that for any E ∈ Db(X),
+Zt(E) =
+�
+X
+Φt(E) =
+�0, 1�
+×
+�
+1 + 2g − 2
+z
+ln(t)
+�0
+0
+1
+0
+��
+× A × v(E).
+Here v : K0(X) → H∗(X; C) is the twisted Chern character, so v(OX) = [1, 0]T and
+v(Op) = [0, 2πi]T. (The twist by (2πi)deg /2 appears in [I], and is justified by the fact
+that the twisted Chern character gives an isomorphism Ktop
+0
+(X) ⊗ C ∼= H∗(X; C)
+that is compatible with the natural Hodge structures on each group.) Because we
+are in Stab(X)/Ga, we only need to determine A up to scalar multiple. Let us
+write
+A =
+�a
+aτ∞
+2πi
+1
+τ0
+2πi
+�
+for some constants a, τ0, τ∞ ∈ C. It is convenient to reparameterize the path above
+using the parameters eiθ = ¯z/|z| and s = (2g − 2) ln(t)/|z| ∈ (−∞, ∞), and we are
+interested in paths starting at s = 0 and going towards s = ∞. We compute
+τ(s) := Zs(Op)
+Zs(OX) = aeiθsτ∞ + τ0
+aeiθs + 1
+This is a linear fractional transformation applied to the line eiθR ⊂ C, so the
+resulting path traces out a generalized circle, starting at τ0 when s = 0 and limiting
+to τ∞ as s → ±∞. For the path τ(s) to stay in H for all s ∈ (0, ∞), and thus lift
+uniquely to Stab(X)/Ga it is necessary to have τ0, τ∞ ∈ H, but not sufficient. One
+choice of a that works for all τ0 and τ∞ is the following:
+
+28
+DANIEL HALPERN-LEISTNER
+Lemma 31. Let a = e−iθ. Then for any τ0, τ∞ ∈ H the formula above gives a
+solution of the quantum differential equation such that the associated central charge
+Zs lifts uniquely to Stab(X)/Ga. The associated paths begin at τ0 when s = 0 and
+limit to τ∞ as s → ∞.
+Proof. If one substitutes a = e−iθ above, it is clear that τ(s) remains on the line
+segment connecting τ0 and τ∞ for s > 0 and thus remains in H.
+□
+In fact, it is not hard to show that the conclusion of the lemma only holds if
+a ∈ R>0e−iθ.
+This shows that Proposal III can be carried out in the case where X is a higher
+genus curve. In this case, the path in Stab(X)/Ga is quasi-convergent for a trivial
+reason – it converges to a point in Stab(X)/Ga itself. Rather than being canonical,
+the limit point τ∞ ∈ Stab(X)/Ga depends on the choice of fundamental solution
+of the quantum differential equation.
+The canonical fundamental solution. In contrast to the solution constructed above,
+the quantum cohomology central charge (9), corresponding to the canonical fun-
+damental solution of the quantum differential equation Φt = T (t)t−c1(X)�ΓX from
+[GGI] (see Remark 13), does not lift to a convergent path in Stab(X)/Ga. For any
+minimal variety X, one has T (t) = idΛC. Because dim(X) = 1 in our situation,
+one has �ΓX = exp(−Ceuc1(X)) [GGI, §3.4]. Therefore, the canonical fundamental
+solution corresponds to (21) above with matrix
+A =
+�
+1
+0
+2(g − 1)Ceu
+1
+�
+Up to rescaling, this is the solution with a = 1/(2(g − 1)Ceu), τ∞ = 0, and τ0 =
+πi/((g − 1)Ceu), and the path of this solution in C is parameterized by
+τ(s) =
+2πi
+eiθs + 2(g − 1)Ceu
+.
+If θ ∈ (−π/2, π/2), then this path stays in H for s ≥ 0 and thus lifts to Stab(X)/Ga.
+Regardless of θ, this path in H ∼= Stab(X)/Ga always limits to 0 as s → ∞, and
+hence does not have a limit in Stab(X)/Ga. On the other hand, this path is still
+quasi-convergent in the most general sense studied in [HLR]. Rather than a semi-
+orthogonal decomposition, it recovers the two-step filtration by thick triangulated
+subcategories 0 ⊂ {torsion complexes} ⊂ Db(X), along with stability conditions on
+the associated graded categories. This provides another possible interpretation of
+Proposal III for minimal varieties.
+References
+[B1] A. A. Be˘ılinson, Coherent sheaves on Pn and problems in linear algebra, Funktsional.
+Anal. i Prilozhen. 12 (1978), no. 3, 68–69. MR509388
+[B2] Anthony Blanc, Topological K-theory of complex noncommutative spaces, Compos.
+Math. 152 (2016), no. 3, 489–555. MR3477639
+[B3] Tom Bridgeland, Stability conditions on triangulated categories, Ann. of Math. (2)
+166 (2007), no. 2, 317–345. MR2373143
+[B4]
+, Spaces of stability conditions, Algebraic geometry—Seattle 2005. Part 1, 2009,
+pp. 1–21. MR2483930
+[BGvBKS] Christian B¨ohning, Hans-Christian Graf von Bothmer, Ludmil Katzarkov, and Pawel
+Sosna, Determinantal Barlow surfaces and phantom categories, J. Eur. Math. Soc.
+(JEMS) 17 (2015), no. 7, 1569–1592. MR3361723
+
+THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.
+29
+[BGvBS] Christian B¨ohning, Hans-Christian Graf von Bothmer, and Pawel Sosna, On the
+Jordan-H¨older property for geometric derived categories, Adv. Math. 256 (2014), 479–
+492. MR3177299
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+reconstructions, Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1183–1205, 1337.
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+[BO] A. Bondal and D. Orlov, Semiorthogonal decomposition for algebraic varieties, arXiv,
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+Reine Angew. Math. 682 (2013), 89–128. MR3181500
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+centers in derived algebraic geometry, J. Amer. Math. Soc. 23 (2010), no. 4, 909–966.
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+[D1] Michael R. Douglas, Dirichlet branes, homological mirror symmetry, and stability,
+Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002),
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+no. 4, 2049–2092. MR3654104
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+cohomology of Fano manifolds: gamma conjectures, Duke Math. J. 165 (2016), no. 11,
+2005–2077. MR3536989
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+Int. Math. Res. Not. IMRN 18 (2015), 8847–8859. MR3417694
+[HLR] Daniel Halpern-Leistner and Antonios-Alexandros Robotis, Stability conditions and
+semiorthogonal decompositions.
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+for toric orbifolds, Adv. Math. 222 (2009), no. 3, 1016–1079. MR2553377
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+no. 1, 147–171. MR1949787
+[K2] Bernhard
+Keller,
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+February 1, 2020: https://schms.math.berkeley.edu/events/miami2020/.
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+DANIEL HALPERN-LEISTNER
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+
diff --git a/cdFPT4oBgHgl3EQfxjU-/content/tmp_files/load_file.txt b/cdFPT4oBgHgl3EQfxjU-/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..3a541ddf48bec9aa0de126018d6f9381759cf839
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@@ -0,0 +1,969 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf,len=968
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='13168v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='AG]  30 Jan 2023  THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  DANIEL HALPERN-LEISTNER  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' This note aims to clarify the deep relationship between birational  modifications of a variety and semiorthogonal decompositions of its derived  category of coherent sheaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The result is a conjecture on the existence  and properties of canonical semiorthogonal decompositions, which is a non-  commutative analog of the minimal model program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We identify a mechanism  for constructing semiorthogonal decompositions using Bridgeland stability con-  ditions, and we propose that through this mechanism the quantum differential  equation of the variety controls the conjectured semiorthogonal decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  We establish several implications of the conjectures: one direction of Dubrovin’s  conjecture on the existence of full exceptional collections;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' the D-equivalence  conjecture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' the existence of new categorical birational invariants for varieties  of positive genus;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' and the existence of minimal noncommutative resolutions  of singular varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Finally, we verify the conjectures for smooth projective  curves by establishing a previously conjectured description of the stability  manifold of P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Contents  The Key Lemma  2  Noncommutative birational geometry  5  Related work and author’s note  5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The NMMP conjectures  6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The truncated quantum differential equation  7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The Hodge-theoretic MMP  14  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Applications  15  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Minimal models and the D-equivalence conjecture  15  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Minimal resolutions  16  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Example: simple flips and flops  17  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Dubrovin’s conjecture  19  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Example of curves  21  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The projective line  21  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Higher genus curves  27  References  28  The derived category of coherent sheaves Db(X) of a smooth projective variety  X often reveals hidden structure in the geometry of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We recall two examples of  this phenomenon:  In their seminal preprint [BO], Bondal and Orlov first glimpsed a deep relationship  between birational modifications of varieties and the structure of their derived  categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  A birational transformation X ��� X′ that preserves the canonical  1  2  DANIEL HALPERN-LEISTNER  bundle is expected to induce an equivalence of derived categories Db(X) ∼= Db(X′),  which is now known as the D-equivalence conjecture [K1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  More generally, for  a KX-negative birational contraction π : X ��� X′, meaning there is a smooth  projective W with birational morphisms f : W → X and g : W → X′ resolving π  such that g∗(KX′) − f ∗(KX) is effective, then Db(X′) is expected to be a factor in  a semiorthogonal decomposition of Db(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  In a different context, Beilinson observed in [B1] that Pn admits a full exceptional  collection, which is the extreme form of a semiorthogonal decomposition in which  all of the factors are equivalent to Db(pt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Many other examples of Fano manifolds  admit large semiorthogonal decompositions of Db(X) that do not directly come  from birational geometry, such as Lefschetz decompositions [K1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Motivated by  mirror symmetry, Dubrovin conjectured in [D2, Conj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2] that a Fano manifold  admits a full exceptional collection if and only if the quantum cohomology of X is  generically semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  In this note, we attempt to clarify these phenomena, and to propose a mechanism  for obtaining canonical semiorthogonal decompositions of derived categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We  formulate a conjecture, the noncommutative minimal model program (MMP), that  implies the D-equivalence conjecture (see Corollary 18) as well as a version of  Dubrovin’s conjecture (see Proposition 23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Although the conjectures are inspired  by homological mirror symmetry, our mechanism is independent of mirror symmetry  and, we believe, well-motivated by the classical MMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The space of Bridgeland stability conditions on Db(X), which we refer to as  Stab(X), plays a central role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We will shortly observe in Lemma 1 that an unbounded  path in Stab(X) satisfying certain conditions naturally determines a semiorthogonal  decomposition of Db(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The noncommutative MMP, Conjecture II, proposes the  existence of certain canonical paths of this kind, and hence canonical semiorthogonal  decompositions of Db(X), with nice formal properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' These formal properties  alone are enough to imply the D-equivalence conjecture, and more generally the  existence of canonical admissible subcategories MX ⊂ Db(X) that are preserved  under birational modifications of X (see Proposition 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Then in Proposal III, we give a more precise proposal for the canonical paths in  Stab(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Specifically, one can view solutions of the quantum differential equation as  defining paths in the dual of the algebraic cohomology H∗  alg(X) ⊂ Heven(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' C), and  we propose that these paths should be the central charges for the canonical paths  in Stab(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' To avoid convergence issues, and motivated by the classical MMP,  we will truncate the quantum differential equation by only counting sufficiently  KX-negative curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  For the purposes of illustration, in Section 3 we describe these canonical paths on  Stab(X) when X is a smooth projective curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In order to do this, we establish the  description of Stab(P1) predicted by homological mirror symmetry in Proposition 26,  which to our knowledge has not been explicitly proved before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  We fix a base field k ⊂ C, and use the term “variety” to refer to a reduced and  geometrically irreducible finite type k-scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The Key Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Let C be a pre-triangulated dg-category, fix a “Chern character”  homomorphism v : K0(C) → Λ to a finite rank free abelian group Λ, and fix a  reference norm on Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Recall that a Bridgeland stability condition on (C, Λ) consists  of [B3]:  THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  3  i) A collection of full subcategories P = {Pφ ⊂ C}φ∈R, known as the categories  of semistable objects of phase φ, such that Pφ[1] = Pφ+1, Hom(Pφ, Pϕ) = 0  if φ > ϕ, and every object E ∈ C admits a finite R-weighted descending  filtration such that grφ(E) ∈ Pφ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' and  ii) A central charge homomorphism Z : Λ → C, which we regard as a function  on K0(C) via v, such that Z(Pφ \\ 0) ⊂ R>0 · eiπφ and  inf  φ∈R  E∈Pφ\\0  |Z(E)|  ∥v(E)∥ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The set of stability conditions Stab(C) admits a metric topology such that the  forgetful map Stab(C) → Λ∗  C = Hom(Λ, C) that takes (Z, P•) �→ Z is a local  homeomorphism [B3, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  On the other hand, a semiorthogonal decomposition C = ⟨C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , Cn⟩ consists of a  collection of full subcategories C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , Cn ⊂ C such that Ci[1] = Ci, Hom(Ci, Cj) = 0  for i > j, and every E ∈ C admits a descending filtration indexed by i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , n  such that gri(E) ∈ Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Despite the formal similarities in these notions, the difference  in how Pφ and Ci behave under homological shift causes these structures to behave  very differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Now consider a continuous path σt = (Zt, Pt  ) ∈ Stab(C) for t ∈ [t0, ∞) that  satisfies:  (1) For any E ∈ C the σt Harder-Narasimhan (HN) filtration of E stabilizes  for t ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We call this the eventual HN filtration, and we call an object  eventually semistable if its eventual HN filtration has length 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  (2) For any eventually semistable object E there are αE, βE ∈ C such that  logZt(E) = αEt + βE + o(1)  as t → ∞,  where the real part ℜ(βE) > C+ln∥v(E)∥ for some constant C independent  of E, and  (3) If the imaginary part ℑ(αE − αF ) = 0, then αE = αF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  This is a special case of the concept of a quasi-convergent path we will develop with  Alekos Robotis in [HLR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' For the purpose completeness here, we will explain the  key property of such a path:  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' A path σt satisfying conditions (1),(2), and (3) above determines:  A finite set of complex constants such that αE ∈ {α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , αn} for any  eventually semistable E, indexed so that ℑ(αi) is increasing in i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  A semiorthogonal decomposition C = ⟨C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , Cn⟩, where Ci is the subcategory  of objects whose eventual HN subquotients all have αE = αi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  A direct sum decomposition Image(v) = �  i Λi, where Λi = v(Ci);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' and  4  DANIEL HALPERN-LEISTNER  A stability condition on each Ci (with respect to v : K0(Ci) → Λi) whose  semistable objects are precisely the eventually semistable objects E with  αE = αi, and whose central charge is Zi(E) = limt→∞ e−αitZt(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  We sketch the proof, leaving some of the details to [HLR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Proof idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We begin by showing that the categories Cβ generated by eventually  semistable E with αE = β define a semiorthogonal decomposition indexed by the  set {αE|E eventually semistable} ⊂ C, ordered by imaginary part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The condition  Hom(Cβ, Cγ) = 0 for ℑ(β) > ℑ(γ) holds because for eventually semistable objects  E, F with ℑ(αE) > ℑ(αF ), for t large enough both E and F are σt-semistable  with E having a larger phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The condition Ci[1] = Ci follows from the fact that  Harder-Narasimhan filtrations are preserved by homological shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Finally, consider  an object E ∈ C, and let gri(E) denote the ith associated graded object for the  eventual HN filtration of E with respect to σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Because the phase of gri(E) is  increasing in i, we must have ℑ(αgr1(E)) ≤ · · · ≤ ℑ(αgrm(E)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We then coarsen  this filtration by grouping all associated graded pieces that have the same value of  ℑ(αgri(E)) to obtain a descending filtration of E with gri(E) ∈ Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The fact that the images v(Ci) are linearly independent, and thus only finitely  many values of αE can appear, follows from the observation that condition (2)  implies that for any collection of Ei ∈ Ci with Zt(Ei) ̸= 0 for all i and t ≫ 0, the  functions Zt(Ei) ∈ C0([t0, ∞)) are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  □  By Bridgeland’s theorem, a path in Stab(C) is uniquely determined by its starting  point (Z0, P0  ) ∈ Stab(C) and a path Zt in Hom(Λ, C) extending Z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  So, the  remarkable thing about Lemma 1 is that it suggests that once you fix a reference  stability condition on C, one can study semiorthogonal decompositions of C simply  by studying interesting paths in the complex vector space Hom(Λ, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The following picture informs Lemma 1, but is not strictly necessary  for this paper: The conclusion of Lemma 1 only depends on the image of the  path σt in Stab(C)/Ga, and the stability conditions on the factors in Lemma 1 are  only naturally defined up to the action of Ga on Stab(Ci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In [HLR] we construct  a partial compactification Stab(C)/Ga ⊂ PStab(C) whose points correspond to  semiorthogonal decompositions C = ⟨C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , Cn⟩ for some n along with a point  in Stab(Ci)/Ga for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , n, along with some additional continuous data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The paths studied in Lemma 1 are convergent in this bordification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The notion of a quasi-convergent path in [HLR] is much more general than the  one in Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' For one thing, the proof of Lemma 1 holds verbatim for a more  general asymptotic estimate, such as  logZt(E) =  :=αE(t)  �  ��  �  αpt + αp−1t(p−1)/p + · · · + α1t1/p + α−1 ln(t) +βE + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  In addition, the genericity condition (3) can be removed entirely, and this is the  version of “quasi-convergent” that we will refer to in Proposal III below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  For  such a path one obtains a slightly weaker structure on C: still only finitely many  functions appear as αE(t) for some eventually semistable E, but now Ci consists of  objects whose eventual HN subquotients have ℑ(αE(t)) = fi(t) for some set of real  functions f1(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , fn(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Then, each Ci admits a filtration by thick triangulated  subcategories, generated by eventually semistable objects with ℜ(αE(t)) ≤ g(t)  THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  5  as t → ∞ for certain functions g(t), and the associated subquotient categories  canonically admit stability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We refer to [HLR] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='1  Noncommutative birational geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In addition to generalizing Lemma 1,  [HLR] also establishes a converse: If C is smooth and proper [KS, §8], then any semi-  orthogonal decomposition C = ⟨C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , Cn⟩ such that the Ci all admit Bridgeland  stability conditions arises from a suitable quasi-convergent path in Stab(C)/Ga via  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' So, perhaps it is reasonable to restrict our focus to these semiorthogonal  decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  On a more philosophical note, in the minimal model program K¨ahler structures,  in the form of ample divisor classes, are essential to making sense of birational  geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' For a smooth projective variety X, the categorical analogue of a K¨ahler  structure on X is a Bridgeland stability condition on Db(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  (See [B4, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=')  We argue that stability conditions are as essential to studying semiorthogonal  decompositions of Db(X) as K¨ahler classes are to studying birational geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  This suggests the following:  Principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Noncommutative birational geometry is the study of semiorthogonal  decompositions of smooth and proper pre-triangulated dg-categories in which every  factor admits a stability condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  We hope this principle helps to explain some recent failures of folk expectations  in the field, such as the failure of the Jordan-H¨older property [BGvBS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Perhaps  restricting to semiorthogonal decompositions of C that are polarizable, in the sense  that every factor admits a stability condition, may rehabilitate some of these  predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The paper [BGvBKS] constructs surfaces X with a semiorthogonal  decomposition Db(X) = ⟨L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , L11, C⟩, where Li are exceptional line bundles and  C is a phantom, meaning K0(C) and HH∗(C) vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' A non-zero phantom does not  admit a Bridgeland stability condition, so this semiorthogonal decomposition could  not arise from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' More is true, though: We will see in Lemma 24 that if  a semiorthogonal decomposition appears to come from a full exceptional collection  on the level of K-theory, and every factor admits a stability condition, then it  does come from a full exceptional collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' So if one lets C′ be the subcategory  generated by C and L11, then Db(X) = ⟨L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , L10, C′⟩ can not arise from a quasi-  convergent path in Stab(X)/Ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Related work and author’s note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' As we will discuss in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='4, the form-  ulation of Proposal III is closely related to and inspired by the Gamma II conjecture  of [GGI], which predicts the existence of full exceptional collections whose Chern  characters give solutions of the quantum differential equation with special asymptotic  properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The paper [SS] generalizes Dubrovin’s conjecture and the Gamma  conjectures to predict the existence of canonical semiorthogonal decompositions  for Db(X) for any Fano manifold X, whose factors again correspond to solutions of  the quantum differential equation with prescribed asymptotic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Maxim  Kontsevich has also given several talks [K3] in which he conjectures the existence  of canonical semiorthogonal decompositions of Db(X), whose factors correspond  1The condition (1) on stabilization of HN filtrations is also significantly relaxed in [HLR], but  this is not essential to our discussion here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  6  DANIEL HALPERN-LEISTNER  to the eigenspaces of quantum multiplication by c1(X), and speculates about the  implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The main contributions of this paper are: 1) to suggest an underlying mechanism  for the conjectures above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 2) to extend the conjectures to non-Fano X in a way that  avoids convergence issues for the quantum differential equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' and 3) to propose  a specific conjecture on the compatibility of these semiorthogonal decompositions  with birational morphisms and to prove some interesting implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  I thank Jeffrey Jiang and Alekos Robotis for many enlightening discussions about  Bridgeland stability conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  In addition, I thank Tom Bridgeland, Davesh  Maulik, Tudor P˘adurariu, Daniel Pomerleano, Claude Sabbah, and Nicolas Templier  for helpful suggestions on this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The NMMP conjectures  We will formulate a noncommutative minimal model program (NMMP) associated  to a contraction of a smooth projective variety X, meaning a surjective morphism  to a projective variety X → Y with connected fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' (Y is not necessarily smooth,  but it must be normal by the uniqueness of Stein factorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=')  The NMMP  predicts canonical semiorthogonal decompositions of Db(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The first piece of our  program is the following difficult folk conjecture:  Conjecture I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Db(X) admits stability conditions for any smooth projective variety  X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  We define the lattice ΛX = H∗  alg(X) as the image of the twisted Chern character  homomorphism v := (2πi)deg /2 ch : K0(X) → H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We only consider stability  conditions on Db(X) defined with respect to v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2  To simplify notation, we let  Stab(X) := Stab(Db(X)) for a scheme X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  In our first and most flexible formulation of the conjectures, we will use ψ to  denote a generic, unspecified, parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Below we will specify more precisely what  ψ might be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We formulate the conjectures relative to a fixed normal variety Y that  need not be smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' (The most interesting case might be Y = pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=')  Conjecture II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Let π : X → Y be a contraction of a smooth projective variety X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  (A) One can associate to π a canonical class of quasi-convergent paths {σπ,ψ  t  }  in Stab(X)/Ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Generic values of the parameter ψ give rise to a semi-  orthogonal decomposition of Db(X), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=', via Lemma 1, and different generic  values of ψ give mutation-equivalent semiorthogonal decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='3  (B) For a generic value of ψ, the semiorthogonal factors of Db(X) are closed  under tensor product with complexes of the form π∗(E) for E ∈ Perf(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  (C) Given a further contraction Y → Y ′, for some values of the parameters,  the semiorthogonal decomposition of Db(X) associated to the composition  2In general if v : K0(C) → Λ has image Λ′ ̸= Λ, then after choosing a splitting ΛC ∼  = Λ′  C ⊕ W  one can identify StabΛ(C) ∼  = StabΛ′(C) × W ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Also the quantum differential equation preserves  H∗  alg(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' So, our entire discussion would work just as well using � (2πi)n  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  H2n(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Z)/{torsion}  instead of H∗  alg(X), but it is a bit simpler to assume that v is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  3Because X is smooth and proper, any semiorthogonal decomposition of Db(X) is admissible,  meaning arbitrary mutations exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' See [BK].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  7  X → Y ′ refines the semiorthogonal decomposition associated to X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  To formulate the final conjecture, we recall that if π : X → X′ is a morphism  of smooth varieties and Rπ∗(OX) = OX′, then π∗ is fully faithful and we have a  semiorthogonal decomposition  Db(X) = ⟨ker(π∗), π∗(Db(X′))⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  (1)  We again consider a composition of contractions X → X′ → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  (D) If X′ is smooth and Rπ∗(OX) = OX′, then for some values of the parameters,  the semiorthogonal decomposition of Db(X) associated to X → Y refines  the semiorthogonal decomposition obtained by combining the semiorthogonal  decomposition of π∗(Db(X′)) ∼= Db(X′) associated to X′ → Y with (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  We expect several of the most basic examples of semiorthogonal decompositions  in geometry to arise in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The following examples may be regarded as  extensions of Conjecture II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In the special case where X is the blowup of Y along a smooth  subvariety S ֒→ Y of codimension n + 1, we expect the canonical semiorthogonal  decomposition associated to X → Y to agree with the semiorthogonal decomp-  osition from [BO, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='4]  Db(X) = ⟨Db(S)(−n), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , Db(S)(−1), π∗(Db(Y ))⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' If X = P(E) for some locally free sheaf E on Y of rank n and π :  X → Y is the projection, then we expect that for a suitable choice of parameter  the semiorthogonal decomposition in (A) is  Db(X) =  �  π∗(Db(Y )), π∗(Db(Y )) ⊗ O(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , π∗(Db(Y )) ⊗ O(n)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The truncated quantum differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We now formulate a more  precise proposal for the canonical quasi-convergent paths in Stab(Db(X)) conjectured  in (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The small quantum product ⋆τ on H∗  alg(X), parameterized by τ ∈ H2(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' C), is  defined by the formula  (α1 ⋆τ α2, α3)X =  �  d∈NE(X)Z  ⟨α1, α2, α3⟩X  0,3,deτ·d,  (2)  where α1, α2, α3 ∈ H∗  alg(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' C), (−, −)X denotes the Poincar´e pairing on H∗  alg(X),  NE(X)Z denotes the numerical equivalence classes of 1-cycles with nonnegative  integer coefficients, and ⟨α1, α2, α3⟩X  0,3,d denotes the Gromov-Witten invariant that  counts curves of class d on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Let us consider a function of a single complex  parameter ζ = ζ(u) ∈ H∗  alg(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The quantum differential equation is  0 = u dζ  du + c1(X) ⋆ln(u)c1(X) ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  In general, if neither c1(X) := −c1(KX) or −c1(X) is ample, then the sum in (2)  is infinite and thus this is only a formal differential equation in u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We propose to  modify this differential equation by replacing c1(X) ⋆ln(u)c1(X) (−) by an operator  Eψ(u) whose definition involves only finite sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  8  DANIEL HALPERN-LEISTNER  Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' For d ∈ NE(X)Z with c1(X) · d ≥ 0, let Td ∈ End(H∗  alg(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Q)) be  defined by the identity (Tdα1, α2)X = ⟨α1, α2⟩X  0,2,d for all α1, α2 ∈ H∗  alg(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  When α1 has degree 2, the divisor equation ⟨α1, α2, α3⟩X  0,3,d = (α1·d)⟨α2, α3⟩X  0,2,d  allows one to express  α1 ⋆τ (−) = α1 ∪ (−) +  �  d∈NE(X)Z\\{0}  (α1 · d)eτ·dTd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  We let NE(X/Y )Z denote the numerical equivalence classes of effective integral 1-  cycles spanned by curves that are contracted by π, and note the natural injective  map NE(X/Y )Z → NE(X)Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We make the following key observation:  Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' If ω ∈ H2(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' R) is the Chern class of a relatively ample R-divisor  for a contraction π : X → Y , then Td is homogeneous of degree 2(1 − c1(X) · d)  with respect to the cohomological grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' As a result, there are only finitely many  classes d ∈ NE(X/Y )Z such that: 1) d · (c1(X) − ω) > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' and 2) Td ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' First let H be an ample Cartier divisor on Y that is large enough such  that ω + π∗(H) is ample on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Because the moduli space M 0,2,d(X) has virtual  dimension c1(X)·d+dim(X)−1, one has (Tdα1, α2)X = ⟨α1, α2⟩X  0,2,d = 0 whenever  1  2(deg(α1) + deg(α2)) ̸= c1(X) · d + dim(X) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' This implies that deg(Tdα) −  deg(α) = 2(1 − c1(X) · d), so Td = 0 for degree reasons unless c1(X) · d − 1 ∈  [− dim(X), dim(X)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Combining this with the constraint (1) gives  (ω + π∗(H)) · d = ω · d < c1(X) · d ≤ dim(X) + 1  (3)  There are finitely many numerical equivalence classes of cycles satisfying this bound,  by [KM, Cor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  □  Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Let ψ := ω + iB ∈ NS(X)C/2πi NS(X) be a class whose real part  ω is the Chern class of a relatively ample R-divisor for the contraction π : X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  We define the truncated quantum endomorphism Eψ(u) : H∗  alg(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' C) → H∗  alg(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' C)  by the formula  Eψ(u) = c1(X) ∪ (−) +  �  d∈NE(X/Y )Z s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  (c1(X)−ω)·d>0  (c1(X) · d)uc1(X)·de−ψ·dTd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The restriction d · (c1(X) − ω) > 0, is motivated by the Cone Theorem, which  states that the (c1(X) − ω)-positive piece of the cone of curves is polyhedral, and  its rays are generated by rational curves with c1(X) · d ∈ (0, dim(X) + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' This is  precisely the bound in (3), outside of which Td = 0 for degree reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Without this restriction, and when Y = pt, the sum defining Eψ(u) would agree  with the definition of c1(X) ⋆−ψ+ln(u)c1(X) (−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Eψ(u) keeps terms that dominate  the sum as |u| → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We therefore regard Eψ(u) as a polynomial approximation  Eψ(u) ≈ c1(X) ⋆rel  −ψ+ln(u)c1(X) (−)  (4)  that is valid when ω is close to 0 and |u| ≫ 0, and where ⋆rel denotes a relative  quantum product for the morphism X → Y that only counts classes of contracted  curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In fact, if c1(X) is relatively ample for the contraction π : X → Y , such as  when Y = pt and X is Fano, and ω is small enough that c1(X)−ω is still relatively  ample, then (4) becomes an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  9  Now, a path in Stab(X) is uniquely determined by a starting point (Z1, P1)  and a path Z• : [1, ∞) → Hom(ΛX, C) starting at Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We will construct paths in  Hom(Λ, C) by studying solutions of the truncated quantum differential equation:  0 = tdζ(t)  dt  + 1  z Eψ(t)ζ(t),  (5)  where z ∈ C is a parameter, ψ the parameter in Definition 8, and ζ(t) ∈ H∗  alg(X)C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Note that this agrees with the usual quantum differential equation when X is Fano,  Y = pt, and ω is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Following [GGI], we will analyze the differential equation (5) by making the  change of variables ˜ζ(t) := tµζ(t), where µ := (deg − dim(X))/2 is the grading  operator on H∗  alg(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Lemma 7 implies that tEψ(1)tµ = tµEψ(t), so (5) becomes  d˜ζ  dt + 1  z Eψ(1)˜ζ − 1  t µ˜ζ = 0,  (6)  which is much simpler because it has only three terms, with a regular singularity  at t = 0 and a pole of order ≤ 2 at t = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The Hukuhara-Turritin theorem [W, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='1] says that a differential equation  of the form (6) has a fundamental solution of the form  Φt = A(t1/p)eD(t1/p)+ln(t)C,  (7)  where D(s) is a diagonal matrix with polynomial entries, C is a constant matrix  that commutes with D(s) for all s, and A(s) is a holomorphic invertible matrix-  valued function that converges as s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In the special case of (6), we can be  more precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The differential equation (6) has a holomorphic fundamental solution  of the form Φt = Y (t)etD+B(t) for |t| > t0 in some sector S ⊂ C centered at the  origin and containing R>0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' where  (1) Y (t) is an invertible matrix that admits a uniform asymptotic expansion  Y (t) ∼ Y0 + Y1t−1/p + Y2t−2/p + · · · on S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' for some p ∈ Z>0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' such that the  columns of Y0 are a basis of generalized eigenvectors of −1  z Eψ(1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  (2) D is the diagonal matrix of eigenvalues of −1  z Eψ(1) corresponding to the  columns of Y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' and  (3) B(t) = Dp−1t(p−1)/p + · · · + D2t2/p + D1t1/p + C ln(t) for certain constant  diagonal matrices D1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , Dp−1 and a constant matrix C, all of which  commute with D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In particular, ∥B(t)∥ = O(t(p−1)/p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Furthermore, if Eψ(1) is semisimple, then one can arrange that Dp−1 = Dp−2 =  · · = D1 = 0, and if the eigenvalues of Eψ(1) are distinct, then one can arrange  B(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  We will apply Proposition 9 here, and postpone its proof to the end of this  subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' It implies that for any solution ζ(t) of (5), we have  lim sup  t→∞  ln∥ζ(t)∥  t  = r,  10  DANIEL HALPERN-LEISTNER  where r is the real part of an eigenvalue of −1  z Eψ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='Using this, we can state a more  precise elaboration on what the canonical quasi-convergent paths in Conjecture II(A)  should look like:  Proposal III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' There are quasi-convergent paths in Stab(X)/Ga whose central  charges have the form Zt(α) =  �  X Φt(α), where Φt ∈ End(H∗  alg(X)C) is a fun-  damental solution of the truncated quantum differential equation (5) with parameters  z ∈ C and ψ = ω+iB ∈ NS(X)C, where ω is small and relatively ample for X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Furthermore, the following spanning condition holds: for any r ∈ R that is the  real part of an eigenvalue of −1  z Eψ(1), the subspace  F rΛC := {α ∈ ΛC s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' ln∥Φt(α)∥ ≤ rt + o(t) as t → ∞}  should be spanned over C by the classes of eventually semistable E ∈ Db(X) with  lim inft→∞ |Zt(E)|/∥Φt(E)∥ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The proposal is inspired by Iritani’s quantum cohomology central charge [I],  which has previously been conjectured to be the central charge of a family of  stability conditions on Db(X) [D1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The main innovations here: 1) the modification  of the quantum differential equation to reflect the relative geometry of X → Y and  to only count sufficiently KX-negative curves, 2) the assertion that the resulting  paths in Stab(X)/Ga are quasi-convergent, and 3) the spanning condition, which  is analogous to the Gamma II conjecture [GGI, Conj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  We will see in Proposition 23 that the spanning condition is crucial, because  it guarantees that when the eigenvalues of −1  z Eψ(1) have distinct real parts, the  semiorthogonal factors coming from the canonical quasi-convergent paths are in  bijection with the eigenvalues of −1  z Eψ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' These eigenvalues are precisely the αj  that arise in the key lemma, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Without the spanning condition, the proposal is nearly a tautology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Indeed, the  central charge Zt in Proposal III always converges in the projective space P(Λ∗  C) as  t → ∞ to a point Z∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' If Z∞ lifts to a point in Stab(X)/Ga, then for sufficiently  large t the central charges Zt will also lift to Stab(X)/Ga, and the resulting path  is quasi-convergent in the tautological sense that it converges in Stab(X)/Ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Remark 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Because  �  X tµ(−) = tdim(X)/2 �  X(−), and the path in Stab(X)/Ga  only depends on the central charge Zt up to scale, Proposal III is unchanged if we  assert instead that Φt is a solution of (6) rather than (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We have used (5) to be  compatible with [I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Remark 11 (The meaning of small ample classes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' If ω ∈ NS(X)R is relatively  ample for X → Y , then for r ≫ 0, there will be no classes in d ∈ NE(X/Y )Z such  that (c1(X) − rω) · d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' So if ω were large, one would have Eψ(1) = c1(X) ∪ (−),  and Proposal III could not produce an interesting semiorthogonal decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  On the other hand, as r → 0+, the condition (c1(X) − rω) · d > 0 will include  more and more terms in Eψ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The intuition behind requiring ω to be “small”  in Proposal III is that as r → 0+, the resulting semiorthogonal decomposition of  Db(X) should stabilize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Then Conjecture II(A) predicts that in this stable range,  the semiorthogonal decomposition is independent of ω up to mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Remark 12 (Refined proposal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The spanning condition in Proposal III only  addresses the leading order asymptotics of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  A more precise spanning  condition is that one can arrange in Proposition 9 that for each function ϕ(t)  THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  11  appearing as a diagonal entry of tD + t(p−1)/pDp−1 + · · · + t1/pD1, the solution  space with exponential factor eϕ(t) is spanned by Φt(v(E)) for some collection of  eventually semistable E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In this case the quasi-convergent path in Stab(X)/Ga  would lead to a semiorthogonal decomposition indexed by the ϕ(t) that appear,  and Proposal III would only see the coarser decomposition that merges categories  corresponding to ϕ with the same leading coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We have not emphasized  this for the following reason: In situations where (6) agrees with the quantum  differential equation, such as when X is Fano, it is conjectured in [KKP, Conj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='4]  that the connection ∇∂t = d+( 1  zEψ(1)− 1  t µ)dt on H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' C)[t±1] is of non-ramified  exponential type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In that case, one can take B(t) = 0 in Proposition 9, and if z is  chosen generically so that the eigenvalues of 1  zEψ(1) have distinct real parts, this  refined formulation agrees with that in Proposal III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Remark 13 (Canonical fundamental solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We have left some flexibility as  to which fundamental solution Φt to use in Proposal III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In [GGI, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='1], it  is shown that when X is Fano, so that (5) agrees with the quantum differential  equation, there is a unique fundamental solution Φt ∈ End(H∗  alg(X)C) of (5) of the  form T (t)t−c1(X) such that both T (t) and S (t) := tµT (t)t−µ are holomorphic in t  and regular at t = 0, with T (0) = S (0) = idΛC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In fact, the proof applies verbatim  to the truncated quantum differential equation (5) in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The canonical fun-  damental solution is defined to be  Φt(α) = T (t)t−c1(X)�ΓX ∪ α,  (8)  where α ∈ H∗  alg(X), ˆΓX = �dim X  i=1  Γ(1 + δi), and δi are the Chern roots of the  tangent bundle TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Iritani’s quantum cohomology central charge [I] is then  Zt(E) ∝  �  X  T (t)t−c1(X)�ΓX ∪ v(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  (9)  It is tempting to use the canonical fundamental solution (8) in Proposal III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  However, outside of the Fano situation, more investigation is needed to settle on a  final interpretation of Proposal III:  For varieties such that Db(X) admits no semiorthogonal decompositions, one  natural interpretation is that the quasi-convergent paths in Proposal III should  converge to a point in Stab(X)/Ga itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  We will see in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2 that for  higher genus curves one can arrange this, but the fundamental solutions needed  do not appear to be canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' A second natural interpretation is that the paths in  Proposal III are quasi-convergent in the more general sense studied in [HLR], but  the filtration that they induce on Db(X) is not admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' This is the behavior one  sees for the canonical fundamental solution in the case of curves of higher genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Remark 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We do not have a specific prediction as to a starting point for the  canonical paths σπ,ψ  t  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In many examples, Stab(X) has a “geometric” region in which  all skyscraper sheaves of points are stable of the same phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' It would be satisfying  if one could start with a stability condition (Z1, P1) in the geometric region, and  show that the path in Hom(ΛX, C) defined by (5) lifts to a quasi-convergent path  in Stab(X)/Ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In this sense the truncated quantum differential equation would  “discover” semiorthogonal decompositions that were not already known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Remark 15 (Alternative differential equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Eψ(1) is meant to approximate  c1(X) ⋆−ψ (−) by an a priori convergent expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' However, when c1(X) ⋆−ψ  12  DANIEL HALPERN-LEISTNER  (−) is known to converge for ψ in a neighborhood of ψ0 this approximation is  not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In this case, the equation (6) admits a well-known isomonodromic  deformation where c1(X)⋆−ψ (−) is replaced with the “big” quantum product E ⋆τ  (−) where τ ∈ Heven(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' C) rather than H2 and E is the Euler vector field (see [GGI,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' There are known examples of varieties with full exceptional collections for  which this full isomonodromic deformation is needed to get an operator with distinct  eigenvalues [GMS].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' So the full deformation is needed for the converse implication of  Dubrovin’s conjecture, or its refinement as the Gamma II conjecture [GGI] to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  In these situations, we would expect the semiorthogonal decomposition arising from  the full isomonodromic deformation to refine the semiorthogonal decomposition  arising from Proposal III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Proof of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The first part of the analysis works for any vector-  valued differential equation of the form X′(t) = A(t)X(t), where A(t) is a holomorphic  matrix-valued function that admits an asymptotic expansion A(t) ∼ A0 + A1t−1 +  A2t−2 + · · · as t → ∞ in some sector S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' As in the proof of the Hukuhara-Turritin  theorem, we begin by using [W, §11 and Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2] to construct a holomorphic  change of variables X(t) = P(t)Z(t) such that the equation for X(t) becomes  Z′(t) = Q(t)Z(t), where: i) P(t) admits an asymptotic expansion P(t) ∼ �  n≥0 Pnt−n  on S with P0 a matrix of generalized eigenvectors for A0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' ii) there is an asymptotic  expansion Q(t) ∼ �  n≥0 Qnt−n with A0 = P0Q0P −1  0  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' and iii) Q(t) = R1(t) ⊕ · · · ⊕  Rk(t) is block diagonal, where the leading term of each Ri(t) as t → ∞ has a single  eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Therefore, the entire differential equation for Z splits as a direct sum  of differential equations of the original form in which A0 has a single eigenvalue,  and it suffices to prove the claim in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  So let us return to the original notation and assume that A0 has a single  eigenvalue λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Making the substitution X(t) = eλtZ(t), the equation for X(t)  becomes Z′(t) = P(t)Z(t), where P(t) := A(t)−λI admits an asymptotic expansion  in t−1 whose leading term is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' At this point, if A0 = λI, then the resulting  differential equation has a pole of order 1 at ∞, and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Otherwise,  we apply the general Hukuhara-Turritin theorem [W, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='1] to conclude that  the equation for Z has a fundamental solution of the form  Z(t) = Y (t)eDmtm/p+···+D1t1/p+ln(t)C,  (10)  where: i) Y (t) is holomorphic on a (potentially smaller) sector S′ ⊂ S containing  R>0 and admits an asymptotic expansion on S′ in powers of t−1/p with invertible  leading term;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' ii) Dj are diagonal constant matrices and commute with the constant  matrix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The proof of the first part of the Proposition will be complete once we  show that Dj = 0 for j ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Let Z(t) be a particular solution of Z′(t) = P(t)Z(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Choose a Hermitian norm  ∥−∥ on ΛC and fix a small ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We compute  d∥Z(t)∥2  dt  = 2ℜ⟨Z(t), Z′(t)⟩ = 2ℜ⟨Z(t), P(t)Z(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  If ∥P(t)∥ denotes the operator norm, then because P(t) converges to P0 as t → ∞,  we can choose a t0 such that for any t ≥ t0, we have ∥P(t)∥ ≤ N := (1 + ǫ)∥P0∥  for all t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Now applying the Cauchy-Schwartz inequality to the computation  above gives  −2N∥Z(t)∥2 ≤ d∥Z(t)∥2  dt  ≤ 2N∥Z(t)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  13  Now y(t) := ∥Z(t)∥2 is a smooth nonnegative real-valued function of t ∈ R such  that f(t) := 2Ny(t)−y′(t) ≥ 0 and g(t) := 2Ny(t)+y′(t) ≥ 0 for all t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Solving  these first order ODE’s for y(t) gives  y(t) = e2Nt  �  e−2Nt0y(t0) −  � t  t0  e2Nsf(s)ds  �  = e−2Nt  �  e2Nt0y(t0) +  � t  t0  e2Nsg(s)ds  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  It follows from the nonnegativity of f and g that letting c1 := e−Nt0�  y(t0) ≤ c2 :=  eNt0�  y(t0), we have  c2e−Nt ≤ ∥Z(t)∥ ≤ c1eNt  (11)  for all t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Observe that, after adjusting the constants c1 and c2, the bounds in (11) continue  to hold if we replace ∥Z(t)∥ with ∥Z(t)∥ref for some other Hermitian norm ∥−∥ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Thus if we fix a reference norm ∥−∥ref, we have shown that for any Hermitian norm  ∥−∥, there are constants c1, c2, t0 > 0 such that  c2e−(1+ǫ)∥P0∥t ≤ ∥Z(t)∥ref ≤ c1e(1+ǫ)∥P0∥t  for all t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' On the other hand, because P0 is nilpotent, one can choose Hermitian  norms in which ∥P0∥ is arbitrary small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Indeed, one can choose a basis in which P0  is r times a sum of nilpotent Jordan matrices, and in the norm in which this basis  is orthonormal one has ∥P0∥ ≤ (rank(P0) − 1)r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  It follows that for any r > 0, there are constants c1, c2, t0 > 0 such that c2e−rt ≤  ∥Z(t)∥ref ≤ c1ert for all t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' If ℜ(Dj) ̸= 0 for any j ≥ p in the fundamental  solution (10), then one of the columns of this matrix would violate this bound for  some r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Hence we conclude that ℜ(Dj) = 0 for all j ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' However, an analysis  identical to the one above gives the same bounds for the function ∥Z(eiθt)∥, where  θ is any angle close enough to 0 that the ray eiθR>0 lies in the sector S′ on which  Y (t) and P(t) are defined and satisfy the desired asymptotic estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' It follows  that ℜ(Djeiθj/p) = 0 for all sufficiently small θ and j ≥ p, and hence Dj = 0 for  all j ≥ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' This completes the proof of the main claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  For the further claim when the eigenvalues of Eψ(1) are distinct, we use a  different argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  It follows from the symmetry of the two-point function in  Definition 6 that Td and hence Eψ(1) is symmetric with respect to the Poincar´e  pairing (−, −)X on H∗  alg(X)C, and it is easy to show that the grading operator µ is  anti-symmetric with respect to (−, −)X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Now, an endomorphism that is symmetric  with respect to a non-degenerate complex bilinear form need not be diagonalizable,  but its generalized eigenspaces are orthogonal to one another, and the restriction  of the form to each generalized eigenspace is still non-degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  It follows that if the eigenvalues of Eψ(1) are distinct, then this endomorphism  admits an orthonormal eigenbasis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In this basis, the matrix D of Eψ(1) is diagonal  with distinct diagonal entries, and the matrix M for µ satisfies M T = −M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In  particular the diagonal entries of M are all 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' After a change in variables u = 1/t,  our differential equation (6) becomes  dζ  du +  � D  u2 + M  u  �  ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  14  DANIEL HALPERN-LEISTNER  We are now in the setting of [BTL, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The vanishing of the diagonal of  M implies the conditions (D) and (F), and the fundamental solution near u = 0  described in [BTL, Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='8] gives the claim of Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  □  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The Hodge-theoretic MMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' For a smooth projective complex variety X,  the topological K-theory Ktop  i  (X) admits a canonical weight-i pure Hodge structure  induced by the twisted Chern character Ch : Ktop  i  (X)⊗C ∼= Hi+2∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Concretely,  after tensoring with Q we have an isomorphism of Hodge structures Ktop  i  (X)⊗Q ∼=  � Hi+2n(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Q)(n), where (n) denotes the Tate twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  In fact, this Hodge structure can be reconstructed entirely from Db(X): The  paper [B2] constructs a topological K-theory spectrum for a dg-category over C,  and a canonical isomorphism with periodic cyclic homology  Ch : Ktop(X) ⊗ C ∼= Ktop(Db(X)) ⊗ C  ∼  =  −→ HP(Db(X)),  which takes the Bott element to the periodic parameter in periodic cyclic homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The degeneration of the noncommutative Hodge-de Rham spectral sequence for  HP(Db(X)) induces the Hodge filtration on Ktop(X), and this is enough to reconstruct  the Hodge structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Both Ktop(C) and the noncommutative Hodge-de Rham sequence for HP(C)  are additive invariants of dg-categories, and therefore take finite semiorthogonal  decompositions to direct sum decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Thus an immediate consequence  of Conjecture II is the following de-categorified variant, which can be investigated  independently:  Conjecture IV (Hodge-theoretic MMP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Let X → Y be a contraction of a smooth  projective variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  (A/B) There is a canonical direct sum decomposition of Hodge structures  Ktop(X)Q ∼= H1,ψ ⊕ · · · ⊕ Hn,ψ  (12)  that is upper triangular with respect to the Euler pairing and closed under  multiplication by classes from Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' This decomposition depends on a parameter  ψ, but different values of ψ give mutation-equivalent decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='4  (C) Given another contraction Y → Y ′, the decomposition of Ktop(X)Q associated  to X → Y ′ refines the decomposition associated to X → Y for suitable  parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  (D) If π : X → X′ is a morphism of smooth varieties with Rπ∗(OX) = OX′,  then for suitable values of the parameters, the decomposition of Ktop(X)Q  associated to X → Y refines the decomposition obtained by combining  the canonical decomposition Ktop(X)Q ∼= Ktop(X′)Q  � ker(π∗) with the  decomposition of Ktop(X′)Q associated to X′ → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  4Consider a direct sum decomposition of a finite rank free abelian group Λ = Λ1 ⊕ · · · Λn that  is upper-triangular with respect to a non-degenerate bilinear pairing [−, −) on Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' To any braid  on n-strands, with underlying permutation s, the mutation along this braid is a new direct sum  decomposition Λ = Λ′  s(1) ⊕.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='⊕Λ′  s(n), and it is equipped with canonical isomorphisms Λi ∼  = Λ′  s(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  See [SS, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2] for a discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  15  In fact in (D), if π : X → Y is a blowup of Y along a smooth center S of  codimension n + 1, then one expects the decomposition of Ktop(X)Q associated to  X → Y ′ to refine the canonical decomposition Ktop(X)Q ∼= Ktop(Y )Q⊕(Ktop(S)Q)n  combined with the canonical decompositions associated to Y → Y ′ and S → π(S) ⊂  Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  We expect the decomposition in (12) to arise in the same way as in Proposal III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Namely, under a suitable fundamental solution of (6), the lattice in each Hi,ψ  should span the space of solutions with exponential growth rate eαit as t → ∞,  where α1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , αn are the eigenvalues of −1  z Eψ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Remark 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The decategorification Conjecture IV (specifically part (D)) is a  variant of the blowup formula conjectured and investigated by Katzarkov, Kontsevich,  Pantev, and Yu [K3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Although our conjecture deals with decompositions of rational  Hodge structures rather than (formal) Frobenius manifolds, we expect that these  conjectures would have many of the same applications to rationality questions that  have been announced for the blowup formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Applications  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Minimal models and the D-equivalence conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Our first application  defines a dg-category MX/Y that is a relative birational invariant of a contraction  X → Y , where X has positive geometric genus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We call MX/Y the noncommutative  minimal model of X relative to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Proposition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Let X → Y be a contraction of projective varieties with X smooth  and pg(X) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Assuming Conjecture I and Conjecture II(A,D) for varieties over  Y , there is an admissible subcategory MX/Y ⊂ Db(X) that contains an object whose  support is X and that has the following property:  For any other contraction X′ → Y such that X′ is birationally equivalent  to X relative to Y , there is an admissible embedding MX/Y ֒→ Db(X′) as  well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Furthermore, assuming Conjecture II(B) for varieties over Y , MX/Y has a canonical  Perf(Y )⊗-module structure such that the embeddings MX/Y ֒→ Db(X′) are Perf(Y )⊗-  linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  As the proof will show, MX/Y arises as one of the semiorthogonal factors coming  from the NMMP for some birational cover of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Let Db(X) = ⟨C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , Cn⟩ denote the semiorthogonal decomposition that  Conjecture II(A) associates to π and a generic choice of parameter ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Because  pg > 0, KX has a non-vanishing section, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=', the base locus of |KX| has positive  codimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' It follows from [KO, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2] that exactly one of the categories Ci  contains an object whose support is all of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Let us call this category CX,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Conjecture II(A) asserts that different choices of ψ give mutation equivalent  semiorthogonal decompositions, but any mutation of ⟨C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , Cn⟩ gives a canonical  equivalence between the subcategories containing a densely supported object, so  CX,ψ ∼= CX,ψ′ for different generic values of the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We therefore denote  CX = CX,ψ for any fixed choice of ψ and suppress ψ from the notation below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Conjecture II(B) implies that CX,ψ is a module category for Perf(Y )⊗, and the  mutation equivalences respect this structure, so CX has a well-defined Perf(Y )⊗-  module structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  16  DANIEL HALPERN-LEISTNER  Now let f : Z → X be a projective birational morphism, with Z smooth, and  consider the NMMP of Z relative to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Then Conjecture II(D) implies that for  suitable choices of parameter, CZ ⊂ f ∗(CX) ∼= CX is an admissible subcategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Furthermore, by hypothesis CZ corresponds to a direct summand of the charge  lattice of CX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Because this charge lattice is finite dimensional, there must be a  birational morphism Z → X such that for any further birational morphism Z′ →  Z → X, CZ ∼= CZ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' For any other contraction X′ → Y that is birational to X  relative to Y , one can find a smooth projective Z′ with birational maps Z′ → Z  and Z′ → X′ that are compatible with the given birational equivalence over Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' It  follows that CZ = CZ′ ⊂ CX′ ⊂ Db(X′) are admissible inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  □  Proposition 17 explains why birationally equivalent Calabi-Yau manifolds should  have equivalent derived categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In fact, we have the stronger statement:  Corollary 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Assuming Conjecture I and Conjecture II(A,D) hold for varieties  over Spec(k), if X and X′ are birationally equivalent smooth projective varieties and  |KX| is base-point free, then there is a canonical admissible embedding Db(X) ֒→  Db(X′), which is an equivalence if |KX′| is also base-point free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' If KX is base-point free, then Db(X) admits no semiorthogonal decompositions  [KO, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' By Proposition 17, it suffices to show that MX/ Spec(k) = Db(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  To see this, consider a birational morphism f : Z → X with Z smooth and  projective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  If we apply the NMMP for Z → Spec(k), Conjecture II(D) implies  that for a suitable choice of parameter, the unique semiorthogonal factor CZ ⊂  Db(Z) that is densely supported must lie in f ∗(Db(X)) and thus must be equal to  f ∗(Db(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  □  Remark 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In Corollary 18, it suffices to verify 1) Conjecture II(A) only for X  in the birational equivalence class of interest;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' and 2) Conjecture II(D) holds when  X → X′ is the blowup of the smooth variety X′ along a smooth center, but with the  stronger requirement that the semiorthogonal decomposition of Db(X′) obtained  as a piece of the semiorthogonal decomposition of Db(X) agrees with (rather than  refines) the decomposition associated to X′ → Spec(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' By the weak factorization theorem, the birational morphism X ��� X′  can be factored as a sequence of birational maps X = X1 ��� X2 ��� · · · ���  Xn = X′, where each morphism or its inverse is a blowup of a smooth variety  along a smooth center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' One can then use Conjecture II(D) for blowups to argue  by induction that in the decomposition of Db(Xi) associated to Xi → Spec(k)  by Conjecture II(A), the unique generically supported semiorthogonal factor is  indecomposable and equivalent to Db(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  □  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Minimal resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' A similar application of Conjecture II is to define for  any variety Y a dg-category RY that we regard as the minimal noncommutative  resolution of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='5  Below we will use the monoidal structure on the ∞-category  of small idempotent complete module categories over a small idempotent complete  symmetric monoidal stable ∞-category A⊗, which is induced from that on presentable  stable module categories over Ind(A⊗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Namely M⊗AN is the category of compact  objects in Ind(M) ⊗Ind(A) Ind(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The key fact is that for a Tor-independent  5Several different notions of noncommutative resolution of singularities exist in the literature,  but we are not aware of one that agrees with what we establish here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  17  cartesian diagram of schemes, Y ′ ∼= X′×X Y , one has Perf(Y ′) ∼= Perf(X′)⊗Perf(X)  Perf(Y ) by [BZFN, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Proposition 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Let Y be a reduced variety (possibly singular), and assume that  Conjecture II(A,B,D) holds for any birational morphism from a smooth projective  variety X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' There is a canonical smooth and proper dg-category RY equipped  with a Perf(Y )⊗-module structure such that:  (1) If U ⊂ Y is the smooth locus, then Perf(U) ⊗Perf(Y ) RY ∼= Perf(U);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' and  (2) For any resolution of singularities X → Y , there is a Perf(Y )⊗-linear  admissible embedding RY ֒→ Db(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The proof is identical to that of Proposition 17, except that we use the  following definition for the admissible subcategory CX ⊂ Db(X) associated to a  resolution π : X → Y : Conjecture II(A,B) gives a Perf(Y )⊗-linear semiorthogonal  decomposition Db(X) = ⟨C1,ψ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , Cn,ψ⟩ associated to π and a parameter ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' For  any U ⊂ Y such that π−1(U) → U is an isomorphism, Perf(U) ∼= Perf(π−1(U)) ∼=  Perf(X) ⊗Perf(Y ) Perf(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Using base change for Perf(Y )⊗-linear semiorthogonal  decompositions [K2], one obtains a Perf(U)⊗-linear semiorthogonal decomposition  Perf(U) = ⟨Perf(U) ⊗Perf(Y ) C1,ψ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , Perf(U) ⊗Perf(Y ) Cn,ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  In particular, each Perf(U) ⊗Perf(Y ) Cj,ψ is a thick ⊗-ideal of Perf(U), and thus by  [T, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='15] is the category of complexes supported on some subspace Zi ⊂ |U|  that is a union of closed subspaces (with quasi-compact complement, but that is  automatic here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' U is irreducible, so Perf(U) = Perf(U)⊗Perf(Y ) Ci,ψ for the unique  index i such that Zi contains the generic point of U, and thus Perf(U)⊗Perf(Y )Cj,ψ =  0 for all j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The identification of this distinguished index i does not depend on  the specific choice of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Also, as in the proof of Proposition 17, Conjecture II(A,B)  implies that up to a canonical Perf(Y )⊗-linear equivalence, the category Ci,ψ does  not depend on ψ, so we define CX := Ci,ψ for this i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The rest of the proof of  Proposition 17 now applies verbatim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  □  Remark 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In the lectures [K3], Maxim Kontsevich has also speculated about  the existence of canonical noncommutative resolutions for varieties with canonical  singularities in the context of the blowup formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Proposition 20 explains how a  version of this follows from the formal properties laid out in Conjecture II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Example: simple flips and flops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' It might not stand out in Conjecture II(A),  but the key idea behind Corollary 18 is that canonical semiorthogonal decompositions  associated to different birational morphisms X → Y and X → Y + should be related  via mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' For example, let Y be a smooth projective variety with a smooth  embedding Pn ֒→ Y with normal bundle OPn(1)⊕m+1, where m ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Then one has  a diagram  Pn × Pm� �  j  �  p  �✉✉✉✉✉✉✉✉✉  X  π  �✈✈✈✈✈✈✈✈✈✈  π+  �❇  ❇  ❇  ❇  ❇  ❇  ❇  ❇  Pn� �  � Y  Y +  ,  (13)  where π+ is the blow up, with exceptional divisor Pn×Pm, of the smooth projective  variety Y + along an embedded Pm ֒→ Y + with normal bundle OPm(1)⊕n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' It is  shown in [BO] that the composition of derived functors π∗(π+)∗ : Db(Y +) → Db(Y )  is fully faithful, and an equivalence when m = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  18  DANIEL HALPERN-LEISTNER  In the simple case where m = 1, we can recover this fact using mutations of semi-  orthogonal decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We let Eq  p denote the exceptional object j∗(OPn×Pm(p, q))  in Db(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Example 22 (Atiyah flops).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Consider the above set up with n = m = 1, so that  (13) is a flop of 3-folds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The semiorthogonal decomposition of Example 4 combined  with the semiorthogonal decomposition Db(P1) = ⟨O(−1), O⟩ gives semiorthogonal  decompositions  Db(X) = ⟨E−1  −1, E−1  0 , Db(Y )⟩  = ⟨E−2  −1, E−1  −1, Db(Y +)⟩  Using the fact that ωX|Pn×Pm ∼= O(−n, −m), one sees that the right orthogonal  complement of E−1  0  agrees with the left orthogonal complement of E−2  −1, and both  objects are left orthogonal to E−1  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' It follows that we have the following mutations  E−1  −1  E−1  0  Db(Y )  E0  −1  E−1  −1  E−2  −1  Db(Y +)  Composing mutation equivalence functors gives an equivalence Db(Y ) ∼= Db(Y +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  In the more general situation where m = 1 and n ≥ 1, the semiorthogonal  decomposition of Example 4 applied to π+ combined with the Beilinson exceptional  collections on P1 gives  Db(X) = ⟨  Db(P1)(−n)  �  ��  �  E−2  −n, E−1  −n,  Db(P1)(−n+1)  �  ��  �  E−1  −n+1, E0  −n+1,  Db(P1)(−n+2)  �  ��  �  E−1  −n+2, E0  −n+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' ,  Db(P1)(−1)  �  ��  �  E−1  −1, E0  −1, Db(Y +)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  We first mutate this to  Db(X) = ⟨E−1  −n,  Db(P1)(−n+1)  �  ��  �  E−1  −n+1, E0  −n+1,  Db(P1)(−n+2)  �  ��  �  E−1  −n+2, E0  −n+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' ,  Db(P1)(−1)  �  ��  �  E−1  −1, E0  −1, Db(Y +), E−1  0 ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  If we mutate the objects E−1  −n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , E−1  0  to the left over the other summands, one  obtains a collection of exceptional objects A−n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , A−1 fitting into a semi-  orthogonal decomposition  Db(X) = ⟨  Db(Pn)(−1)  �  ��  �  E−1  −n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , E−1  0 , A−n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , A−1, B⟩,  THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  19  where mutation gives a canonical equivalence B ∼= Db(Y +).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' This last semiorthogonal  decomposition refines Db(X) = ⟨E−1  −n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , E−1  0 , Db(Y )⟩ coming from the morphism  π, hence we have  Db(Y ) = ⟨A−n+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , A−1, Db(Y +)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  More precisely, because the right projection onto Db(Y ) ⊂ Db(X) is π∗π∗, the fully  faithful functor Db(Y +) ֒→ Db(Y ) coming from this mutation agrees with π∗(π+)∗,  and one has Ai = π∗π∗(E0  i ), which as an object of Db(Y ) corresponds to OPn(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Dubrovin’s conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Dubrovin conjectured [D2] that for a Fano manifold  X, Db(X) admits a full exceptional collection if and only if the quantum cohomology  QH∗(X) is generically semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Here we observe that the NMMP conjectures  imply one direction, that generic semisimplicity implies the existence of a full  exceptional collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Proposition 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Let X be a smooth projective variety for which Proposal III holds  for generic z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' If in addition ch : K0(Db(X)) ⊗ Q → H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Q) is an isomorphism  and there is a ψ such that Eψ(1) ∈ End(H∗  alg(X) ⊗ C) is semisimple with distinct  eigenvalues, then Db(X) admits a full exceptional collection consisting of eventually  semistable objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The condition that ch is an isomorphism can often be checked in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' For  instance, it holds for compact homogeneous spaces of reductive groups, smooth and  proper toric varieties, and more generally any variety that admits an affine paving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Note that the result above does not require X to be Fano, and does not explicitly  require QH∗(X) to be generically semisimple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Lemma 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Let C be a regular proper idempotent complete pre-triangulated dg-  category, and let C = ⟨C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , Cn⟩ be a semiorthogonal decomposition such that,  dim(K0(Ci) ⊗ Q) = 1 for all i, and each Ci admits a stability condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Then each  Ci is generated by a single exceptional object, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=', the semiorthogonal decomposition  arises from a full exceptional collection in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Because the property of being regular and proper is inherited by semi-  orthogonal factors, it suffices to prove this for the trivial semiorthogonal decomp-  osition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=', in the case n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' If C admits a Bridgeland stability condition and  dim(K0(C)⊗Q) = 1, then all semistable objects in the heart have the same phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' It  follows that the heart P(0, 1] is Artinian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We will prove that if dim(K0(C)⊗Q) = 1  and C is regular, proper and admits a bounded t-structure with Artinian heart,  then C ∼= Db(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Let A ⊂ C be the heart of the t-structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Then K0(A) = K0(C), and the former  has a basis given by the classes of simple objects in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Because K0(A) has rank  1, there is a unique simple module E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Because every object in A has a Jordan-  Holder filtration, whose graded pieces must be isomorphic to E, we see that C is  the smallest triangulated category containing E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We will complete the proof by  showing that E is an exceptional object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Let A = RHom(E, E) be the dg-algebra of endomorphisms of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We claim that  there is a A-module M such that H∗(M) = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Indeed, because Hi(A) = 0 for  i < 0 and H0(A) = k, by [K2, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='5] there is a dg-subalgebra B ⊂ A that  admits a strictly unital A∞ morphism f : B → k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Then f ∗(k) is a B-module whose  homology is k in degree 0, and pullback induces an equivalence A -Mod → B -Mod  that preserves homology of modules [K2, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  20  DANIEL HALPERN-LEISTNER  Because E generates C and C is idempotent complete, the functor RHom(E, −) :  C → Perf(A) is an equivalence of dg-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Also, because C is regular and  proper, an A-module is perfect if and only if its underlying complex of k-vector  spaces is perfect [O2, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In particular, M ∈ Perf(A), and so there is an  object E′ ∈ C such that RHom(E, E′) ∼= k as complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We will show that E′ ∼= E  to conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Let n ≥ 0 be the largest i such that Hi(RHom(E, E)) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Then examining the  long exact cohomology sequence for RHom(E, −) of an extension of objects shows  that for F ∈ A, n = max{i|Hi(RHom(E, F))} as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Likewise, if Hi(−) denotes  the cohomology object with respect to the t-structure on C, then for any F ∈ C,  max{i|Hi(RHom(E, F)) ̸= 0} = n + max{i|Hi(F) ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  This is proved by inductively by examining the long exact cohomology sequence for  RHom(E, −) applied to the exact triangle τ <m(F) → F → Hm(F)[−m], where m  is the highest non-vanishing cohomology object of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The same reasoning shows  that  min{i|Hi(RHom(E, F)) ̸= 0} = min{i|Hi(F) ̸= 0}  for any F ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Applying this to F = E′, we see that  min{i|Hi(E′) ̸= 0} = n + max{i|Hi(E′) ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  This can only hold if n = 0, hence E is exceptional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  □  Proof of Proposition 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Let us fix a z close to 1 such that −1  z Eψ(1) has distinct  eigenvalues u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , un with distinct real parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Proposition 9 then implies that for  any solution Φt of (6), ∥Φt∥ ∼ eℜ(uj)t for some j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Also, if E is an eventually  semistable object such that |Zt(E)| ∼ ∥Φt(E)∥ ∼ eℜ(uj)t, then Zt(E) ∼ eujt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Proposal III then implies that every eigenvalue uj must occur for some eventually  semistable E with |Zt(E)| ∼ ∥Φt(E)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  This implies that the semiorthogonal  decomposition C = ⟨C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , Cn⟩ associated to the quasi-convergent path stipulated  in Proposal III has length n = dim(ΛQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The assumption that K0(C)Q → ΛQ is an  isomorphism then implies that dim(K0(Ci)⊗ Q) = 1 for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Hence C admits a full  exceptional collection consisting of eventually semistable objects by Lemma 24  □  Finally, let us make the following connection with the Gamma II conjecture of  [GGI].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' This conjecture states that, in the situation where X is a Fano manifold with  a full exceptional collection and generically semisimple quantum cohomology, there  is a full exceptional collection Db(X) = ⟨E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , En⟩ such that for the canonical  fundamental solution Φt of the quantum differential equation (see Remark 13),  Φt(v(Ei)) give the “asymptotically exponential” basis of solutions, characterized  uniquely in [GGI, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='6  We observe that, in a sense, the Gamma II  conjecture is equivalent to Proposal III in this context, where we let Φt be the  canonical fundamental solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  6Some notational comments are in order: our variable t corresponds to the variable 1/z in the  terminology of [GGI].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' On the other hand, the parameter z in our (6) is set to one in [GGI, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Instead, they introduce an auxiliary parameter φ and consider solutions in a sector centered on  the ray R>0eiφ, which is equivalent to setting z = eiφ in (6) and considering solutions centered  on the positive real axis, as we do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  21  Proposition 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Let X be a Fano manifold with generically semisimple quantum  cohomology that admits a full exceptional collection Db(X) = ⟨E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , En⟩ such  that Φt(v(Ei)) give the asymptotically exponential basis of solutions of the quantum  differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Then this full exceptional collection arises from a quasi-  convergent path as in Proposal III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' After a homological shift, we may assume that the collection Ei is “Ext-  exceptional” in the language of [M, Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='10], meaning Hom≤0(Ei, Ej) = 0 for all  i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Let u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , un ∈ C be the eigenvalues of − 1  z Eψ(1), ordered so that ℑ(ui)  is strictly increasing in i, after fixing a generic choice of z (see [GGI, Rem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  It is observed in [GGI, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='7] that if the Gamma II conjecture holds, then up to  an overall scalar multiple, the quantum cohomology central charges Zt(Ei) have  asymptotic estimates in our notation Zt(Ei) ∼ cietui as t → ∞ for some constants  ci ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Let t0 ∈ R be such that Zt(Ei) ̸= 0 for all i and all t ≥ t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Then after choosing  a branch for ln(Zt0(Ei)), we can lift Zt(Ei) uniquely to a function ln(Zt(Ei)) that  is continuous in t ≥ t0 for each i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The asymptotic estimate above implies that  ln(Zt(Ei)) = tui + βi + o(1) as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We can therefore choose t1 ≥ t0 such that  for all t ≥ t1, φi,t := ℑ(ln(Zt(Ei)))/π is strictly increasing in i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The collection  Ei[−⌊φi,t⌋] is still Ext-exceptional, so by the discussion following [M, Prop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='17],  there is a unique stability condition with central charge Zt such that the Ei[−⌊φi,t⌋]  are all in the heart and semistable, and hence Ei is semistable of phase φi,t for all  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' This defines a quasi-convergent path σt ∈ Stab(X) for t ≥ t1 such that the  Ei are eventually semistable, and it recovers the semiorthogonal decomposition  Db(X) = ⟨E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' , En⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  □  The quasi-convergent paths constructed in Proposition 25 lie entirely in the  region of stability conditions that are glued from the given full exceptional collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  While these paths technically satisfy Proposal III, due to its flexible formulation,  it would be more satisfying to give a description of the paths that does not make  a priori use of the full exceptional collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  For instance, it is an interesting  question as to whether the paths in Proposition 25 extend in the t → 0 direction  into a geometric region in Stab(X)/Ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' (See Remark 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=')  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Example of curves  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The projective line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The stability manifold of P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In [O1], it is shown that  Stab(P1)/Ga =  �  k∈Z  Xk ∼= C,  where Xk ⊂ Stab(P1)/Ga is the open submanifold of stability conditions in which  O(k) and O(k − 1) are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The map ϕk : Xk → C defined by  ϕk(σ) = logZσ(O(k)) − logZσ(O(k − 1))  defines an isomorphism between Xk and the open upper half plane H ⊂ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Under  this isomorphism, the strip {x+iy|y ∈ (0, iπ)} lies in Xk for all k, and these stability  conditions coincide with slope stability, up to the canonical action of �  GL  +(2, R) on  Stab(X) [B3, Lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The short exact sequence 0 → O(k − 1) → O(k)⊕2 →  22  DANIEL HALPERN-LEISTNER  O(k + 1) → 0 implies that on this common strip, the coordinate functions ϕk are  related by the equation  eϕk+1 = 2 −  1  eϕk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  (14)  The isomorphism Stab(P1)/Ga ∼= C is not explicit in [O1], so let us give an  explicit parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The central charge will be described in terms of modified  Bessel functions of the first and second kind I0(u) and K0(u), which are a basis of  solutions to the modified Bessel differential equation  (u d  du)2Z(u) = u2Z(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  (15)  The function I0 is entire, but the function K0 has a branch point at 0, and we will  use the principal branch with branch cut along i(−∞, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' These are characterized  among solutions of (15) by the following asymptotic estimates as u → 0  I0(u) = 1 + O(|u|2)  K0(u) = − ln( u  2 ) − Ceu + O(|u|2| ln(u)|) ,  (16)  where Ceu = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='57721.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' is Euler’s constant, and we also take the principle branch  of ln with branch cut along i(−∞, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Proposition 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' For any k ∈ Z and τ ∈ R + iπ[k − 1, k], there is a unique (up  to homological shift [2]) stability condition on Db(P1) such that O(k − 1) and O(k)  are stable, and the central charge is determined by  Zτ(Op) = iπI0((−1)k−1eτ), and  Zτ(O(k − 1)) = K0((−1)k−1eτ)  The resulting maps Bk : R+iπ[k−1, k] → Stab(P1)/Ga glue to give an isomorphism  of complex manifolds B : C ∼= Stab(P1)/Ga, such that the action of O(1) ⊗ (−) on  Stab(P1)/Ga is identified with the shift τ �→ τ + iπ on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Note that (−1)k−1eτ ∈ H ∪ R \\ 0 in the formulas above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Lemma 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' For x ∈ C with ℜ(x) > 0, we have K0(−x) = K0(x) − iπI0(x) and  I0(−x) = I0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' K0(−x) and I0(−x) are also solutions of the modified Bessel differential  equation, so they are expressible as linear combinations of K0(x) and I0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The  coefficients are determined by the asymptotic estimates (16) as x → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  □  Lemma 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' ℑ(xK0(x)(K0(x) + iπI0(x))) > 0 for x ̸= 0 with ℑ(x) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  We thank Nicolas Templier for his assistance in proving this Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Let g(x) = xK0(x)(K0(x) + iπI0(x)), and observe that because of the  asymptotics (16), g(x) extends continuously over the origin by letting g(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  We will need the following asymptotic estimates as |u| → ∞ with −π/2 + δ ≤  arg(u) ≤ 3π/2 − δ for some small δ > 0, from [L, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='5]:  K0(u) = � π  2ue−u(1 −  1  8u +  9  128u2 + O(  1  |u|3 ))  I0(u) =  eu  √  2πu(1 +  1  8u +  9  128u2 + O(  1  |u|3 )) + i e−u  √  2πu(1 −  1  8u +  9  128u2 + O(  1  |u|3 ))  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  (17)  THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  23  These imply the following estimate for x ∈ H ∪ R with |x| large:  g(x) = iπ  2 (1 +  1  8x2 ) + O( 1  |x|3 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  It follows that there is some r0 > 0 such that ℑ(g(x)) > 0 for all |x| > r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' For  x ∈ R>0, both K0(x) > 0 and I0(x) > 0, so ℑ(g(x)) = πxK0(x)I0(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Using the identity of Lemma 27, one can compute for x < 0 that ℑ(g(x)) =  πxI0(−x)K0(−x) > 0 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  We now apply the maximum principle to the non-constant continuous real-valued  function ℑ(g(x)) on the closed half disc {x ∈ C||x| ≤ r and ℑ(x) ≥ 0} for some  r > r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' This function is harmonic on the interior {|x| < r and ℑ(x) > 0} and  we have shown that ℑ(g(x)) ≥ 0 on the boundary, with strict inequality except at  x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We conclude ℑ(g(x)) > 0 on the interior and on the boundary away from  x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Because r was arbitrary, the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  □  Proof of Proposition 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' As X1 ⊂ Stab(P1)/Ga is the open subset where O and  O(1) are stable, and X1 is identified with H ⊂ C by the coordinate ϕ1, we first  describe the map B1 : R + i[0, π] → X1 via the formula τ �→ f(eτ), where  f(x) := ln  �Zln x(O(1))  Zln x(O)  �  = ln  �K0(x) + iπI0(x)  K0(x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The ln(−) is defined because Lemma 28 implies that K0(x) ̸= 0 and K0(x) +  iπI0(x) ̸= 0 if ℑ(x) ≥ 0 and x ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' For x ∈ R \\ 0, we interpret ln(−) above as  the principle branch of the logarithm with branch cut along (−∞, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' But as x  varies in H ∪ R \\ 0, (K0(x) + iπI0(x))/K0(x) crosses this branch cut many times,  so we define the ln(−) as the unique lift along the exponential covering C → C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The asymptotics (16) imply that f extends continuously over the origin by letting  f(0) = 0, and we use this convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Claim 1: f(x) maps R \\ 0 to the curve  C := {a + ib|0 < b < π/2 and e|a| cos(b) = 1} :  For x ∈ R>0, K0(x) > 0 and I0(x) > 0, so ln(1 + iπI0(x)/K0(x)) ∈ ln(1 + iR>0),  which lies on the curve {a + ib|ea+ib − 1 ∈ iR, 0 < b < π/2, 0 < a}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' On the other  hand, for x < 0, Lemma 27 implies that ℜ(K0(−x)) > 0 and I0(−x) > 0, and this  implies  f(x) = ln  �  K0(−x)  K0(−x) − iπI0(−x)  �  = − ln  �  1 − iπ I0(−x)  K0(−x)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  This lies on the curve {a + ib|e−a−ib − 1 ∈ iR, 0 < b < π/2, a < 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' C is the union  of these two curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Claim 2: f maps H ∪ R \\ 0 injectively to the region of C lying on or above C ∪ {0}:  Abel’s identity allows one to compute the Wronskian K0(x)I′  0(x)−I0(x)K′  0(x) =  1/x, and using this we compute  f ′(x) =  iπ  xK0(x)(K0(x) + iπI0(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  24  DANIEL HALPERN-LEISTNER  Lemma 28 implies that ℜ(f ′(x)) > 0 for all x ∈ H ∪ R \\ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' It follows that for any  x, v ∈ H ∪ R with v ̸= 0,  ℜ(¯v(f(x + v) − f(x))) =  � 1  0  |v|2ℜ(f ′(x + tv))dt > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Therefore, f(x + v) ̸= f(x), and so f is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  If we apply this inequality  specifically to x = a ∈ R and v = ib ∈ iR>0, then it implies ℑ(f(a+ib))−ℑ(f(a)) >  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' So Claim 1 implies that f(a + ib) lies above the curve C ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Verifying that the maps Bk glue:  The description of the map Bk from the strip Sk := R + iπ[k − 1, k] to Xk  in the proposition simply results from applying O(1) ⊗ (−) on the one hand, and  τ �→ τ + iπ on the other, so these maps are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We must show that they  glue to a holomorphic map C → Stab(P1)/Ga, which will then automatically be  Z-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  It suffices, by Z-equivariance, to show that B0 and B1 glue to a holomorphic  map in a neighborhood of the boundary S1 ∩ S0 = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The first map is given  in coordinates by ϕ0 = f(−ex) for x ∈ R, whereas the second map is given by  ϕ1 = f(ex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Using the relation (14), showing that these maps agree amounts to  showing that  K0(ex) + iπI0(ex)  K0(ex)  = 2 −  K0(−ex)  K0(−ex) + iπI0(−ex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  This follows from the identities proved in Lemma 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Both sides of this identity  are holomorphic and agree in an open neighborhood of R, which implies the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Showing B is an isomorphism:  Claim 2 above shows that the image of Bk is contained in the region in Xk ∼= H  on or above the curve C ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  In [O1], it is shown that this region in Xk is  a fundamental domain for the action of Z on Stab(P1)/Ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  It follows from Z-  equivariance that B is injective, because it is injective on Sk and no other strip Sk′  can map to this region in Xk (except for the two boundary components Sk−1 ∩ Sk  and Sk∩Sk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We also know from [O1] that Stab(P1)/Ga ∼= C, so B is an injective  entire function, which implies it is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Alternatively, the little Picard  theorem implies that B is surjective, because by Z-equivariance, if there is one  point that is not in the image of B, then there are infinitely many.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  □  Mirror symmetry and the noncommutative MMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We begin by studying the quantum  differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  (See [FLZ] for a very thorough discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=')  Let us use  the standard basis 1, H := c1(O(1)) for the cohomology of P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The quantum  cohomology ring is C[H, q]/(H2 − q), and the quantum differential equation in the  basis {1, H} and parameters ψ = 2aH and z = e−b is  tdΦt  dt = −eb(2H) ⋆2(ln(t)−a)H Φt = −2eb  �0  e−2at2  1  0  �  Φt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  (18)  For any solution of (18), the function Zt(E) =  �  P1 Φt(E) satisfies the second order  equation  (t d  dt)2Zt(E) =  �  2eb−a�2 t2Zt(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  (19)  THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  25  ⟨O, O(1)⟩  ⟨O(−1), O⟩  ⟨O(−2), O(−1)⟩  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' A visualization of Stab(P1)/Ga  ∼=  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The red  region is the �  GL  +(2, R)-orbit of slope stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The blue regions  are stability conditions that are glued from the full exceptional  collections shown, which correspond to the regions with imaginary  part > π in each of the coordinate charts Xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The black path is  determined by a particular solution to the quantum differential  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The green vertical line represents the line added at  infinity in the partial compactification of Stab(P1)/Ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The dotted  horizontal lines differ by integer multiples of πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Choose a k such that ℑ(b−a) ∈ π[k−1, k], which guarantees that κ := (−1)k−12eb−a  lies in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' After a change of variables u = κt, (19) is the modified Bessel differential  equation (15), so we have Zt(E) = c1(E)I0(κt) + c2(E)K0(κt) for some constants  c1 and c2 depending on E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  We have shown in Proposition 26 that the paths  σt := B(ln(2t) + b − a) ∈ Stab(P1)/Ga  for t ∈ [1, ∞) have central charges of the form above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Let us show that for generic  values of b − a, σt is quasi-convergent as t → ∞:  For any value of b − a ∈ C, the path σt lies entirely in Xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In the coordinate ϕk  on Xk, the paths have the form  ϕk = ln  �K0(κt) + iπI0(κt)  K0(κt)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Using the asymptotic estimates (17), we have  ϕk = ln  \uf8eb  \uf8ed  eκt  √  2πκt(1 + O( 1  |κt|))  � π  2κte−κt(1 + O( 1  |κt|))  \uf8f6  \uf8f8 = 2κt − ln(π) + O( 1  |κt|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  If ℑ(κ) > 0 this path therefore eventually enters the region of Xk where ℑ(ϕk) > π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  In this region, the only stable objects are O(k−1) and O(k), and they remain stable  26  DANIEL HALPERN-LEISTNER  for all t ≫ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Therefore, these paths satisfy the conditions of Lemma 1, and the  only eventually semistable objects are of the form O(k − 1)⊕m[n] and O(k)⊕m[n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Remark 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' For non-generic values of b − a, meaning those for which κ ∈ R, the  paths above also converge in the partial compactification of Stab(P1)/Ga constructed  in [HLR], but recovering the semiorthogonal decomposition is a bit more complicated  than applying Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  In order to verify Proposal III, we must describe a fundamental solution of the  quantum differential equation (18) whose integral is the path of central charges Zt  underlying σt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We will do this with SYZ mirror symmetry:  The mirror of P1 is the Landau-Ginzburg model (C∗, Wt(x) = x +  t2  e2ax), where  t is regarded as a parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' It is straightforward to check that  1  2  �  L  e−ebWt(x)(H · dx  x + 1 · t2dx  e2ax2 ) ∈ H∗(P1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' C)  solves the differential equation (18) whenever L ⊂ C∗ is a contour such that  ℜ(ebWt(x)) → ∞ at the ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  For simplicity, let us assume L lies on the ray  R>0e−b in a neighborhood of ∞, and lies on the ray R>0eb−2a in a neighborhood  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Under Proposal III, up to a constant multiple one has (after substituting ebx  for x)  Zt(E) =  �  P1 Φt(E) = 1  2  �  L(E)  e  −  �  x+ (κt)2  4x  �  dx  x ,  (20)  where Lκ(E) is the Lagrangian, or formal sum of Lagrangians, in C∗ that is SYZ-  dual to E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Because of the substitution of variables, the Lagrangian Lκ(E) ⊂ C∗  lies along R>0 in a neighborhood of ∞, and along R>0e2(b−a) in a neighborhood of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  We can define Lκ(E) so that Zt(E) matches the central charges of the paths  σt = B(ln(2t) + b − a) as follows:  For any closed point p ∈ P1, we let Lκ(Op) be the closed contour Lκ(Op) :=  {eiθ}2π  θ=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Because this contour is compact, the resulting function Zt(Op)  is a solution of (19) that extends holomorphically to t = 0, and the residue  theorem implies Z0(Op) = 1  2  �  e−xdx/x = πi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The aymptotics (16) then  imply that Zt(Op) = πiI0(κt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  For any θ ∈ R, let Cθ ⊂ C∗ be the image under exp : C → C∗ of the contour  {t + iθ}0  t=−∞ ∪ {it}0  t=θ ∪ {t}∞  t=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Note that as long as ℜ(κ2eiθ) > 0, the  contour integral (20) over Cθ is convergent, and it is independent of θ in  this range by Cauchy’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' This implies that if we let Lκ(O(k−1)) be  the contour C2ℑ(b−a)−2π(k−1), then the formula for Zt(O(k − 1)) in (20) is  holomorphic in κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' When κ ∈ R>0, (20) recovers a known integral formula  for K0(κt) [L, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='10], which can be proven using the method of steepest  descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We therefore conclude that with this choice of Lκ(O(k − 1)), we  have Zt(O(k − 1)) = K0(κt) for all κ ∈ H ∪ R \\ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  These values of Zt(Op) and Zt(O(k − 1)) precisely match the characterization of  B(ln(2t) + b − a) in Proposition 26, and Lκ(Op) and Lκ(O(k − 1)) determine a  fundamental solution of (18) because [O(k − 1)] and [Op] are a basis for K0(P1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  THE NONCOMMUTATIVE MINIMAL MODEL PROGRAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  27  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Higher genus curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' If X is a smooth projective curve of genus g > 1, then  the set of stable objects for any stability condition consists of shifts of line bundles  and structure sheaves of points (see [M, Thm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Choosing some point p ∈ X,  the map  Stab(X)/Ga → C  taking  (P, Z) �→ Z(Op)/Z(OX)  is injective and identifies Stab(X)/Ga with the upper half space H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Therefore, a  path in the space of central charges lifts to Stab(X)/Ga if and only if it is contained  in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Let us use the standard basis 1, H for H∗  alg(X), where H ∈ H2(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Z) is the  generator of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Because there are no non-trivial maps from P1 to X, the  equation (5) has a particularly simple form:  tΦt  dt = −1  z c1(X) ∪ Φt = 2g − 2  z  �0  0  1  0  �  Φt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  When g = 1, this equation is trivial, so we assume that g > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Any fundamental  solution has the form  Φt = t  −1  z c1(X)A =  �  1 + 2g − 2  z  ln(t)  �  0  0  1  0  ��  A,  (21)  for some invertible 2 × 2 matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Remark 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' This phenomenon is more general: For any smooth projective variety  X that is minimal in the sense that KX is nef, Eψ(u) = c1(X) ∪ (−) is nilpotent  and independent of both ψ and u, and every fundamental solution of (5) has the  form t  −1  z c1(X)A for some invertible matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Now let us imagine a family of stability conditions satisfying Proposal III, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=',  such that for any E ∈ Db(X),  Zt(E) =  �  X  Φt(E) =  �0, 1�  ×  �  1 + 2g − 2  z  ln(t)  �0  0  1  0  ��  × A × v(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Here v : K0(X) → H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' C) is the twisted Chern character, so v(OX) = [1, 0]T and  v(Op) = [0, 2πi]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' (The twist by (2πi)deg /2 appears in [I], and is justified by the fact  that the twisted Chern character gives an isomorphism Ktop  0  (X) ⊗ C ∼= H∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' C)  that is compatible with the natural Hodge structures on each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=') Because we  are in Stab(X)/Ga, we only need to determine A up to scalar multiple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Let us  write  A =  �a  aτ∞  2πi  1  τ0  2πi  �  for some constants a, τ0, τ∞ ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' It is convenient to reparameterize the path above  using the parameters eiθ = ¯z/|z| and s = (2g − 2) ln(t)/|z| ∈ (−∞, ∞), and we are  interested in paths starting at s = 0 and going towards s = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' We compute  τ(s) := Zs(Op)  Zs(OX) = aeiθsτ∞ + τ0  aeiθs + 1  This is a linear fractional transformation applied to the line eiθR ⊂ C, so the  resulting path traces out a generalized circle, starting at τ0 when s = 0 and limiting  to τ∞ as s → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' For the path τ(s) to stay in H for all s ∈ (0, ∞), and thus lift  uniquely to Stab(X)/Ga it is necessary to have τ0, τ∞ ∈ H, but not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' One  choice of a that works for all τ0 and τ∞ is the following:  28  DANIEL HALPERN-LEISTNER  Lemma 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Let a = e−iθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Then for any τ0, τ∞ ∈ H the formula above gives a  solution of the quantum differential equation such that the associated central charge  Zs lifts uniquely to Stab(X)/Ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' The associated paths begin at τ0 when s = 0 and  limit to τ∞ as s → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' If one substitutes a = e−iθ above, it is clear that τ(s) remains on the line  segment connecting τ0 and τ∞ for s > 0 and thus remains in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  □  In fact, it is not hard to show that the conclusion of the lemma only holds if  a ∈ R>0e−iθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  This shows that Proposal III can be carried out in the case where X is a higher  genus curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In this case, the path in Stab(X)/Ga is quasi-convergent for a trivial  reason – it converges to a point in Stab(X)/Ga itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Rather than being canonical,  the limit point τ∞ ∈ Stab(X)/Ga depends on the choice of fundamental solution  of the quantum differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  The canonical fundamental solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' In contrast to the solution constructed above,  the quantum cohomology central charge (9), corresponding to the canonical fun-  damental solution of the quantum differential equation Φt = T (t)t−c1(X)�ΓX from  [GGI] (see Remark 13), does not lift to a convergent path in Stab(X)/Ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' For any  minimal variety X, one has T (t) = idΛC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Because dim(X) = 1 in our situation,  one has �ΓX = exp(−Ceuc1(X)) [GGI, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Therefore, the canonical fundamental  solution corresponds to (21) above with matrix  A =  �  1  0  2(g − 1)Ceu  1  �  Up to rescaling, this is the solution with a = 1/(2(g − 1)Ceu), τ∞ = 0, and τ0 =  πi/((g − 1)Ceu), and the path of this solution in C is parameterized by  τ(s) =  2πi  eiθs + 2(g − 1)Ceu  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  If θ ∈ (−π/2, π/2), then this path stays in H for s ≥ 0 and thus lifts to Stab(X)/Ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content='  Regardless of θ, this path in H ∼= Stab(X)/Ga always limits to 0 as s → ∞, and  hence does not have a limit in Stab(X)/Ga.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' On the other hand, this path is still  quasi-convergent in the most general sense studied in [HLR].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' Rather than a semi-  orthogonal decomposition, it recovers the two-step filtration by thick triangulated  subcategories 0 ⊂ {torsion complexes} ⊂ Db(X), along with stability conditions on  the associated graded categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
+page_content=' This provides another possible interpretation of  Proposal III for minimal varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdFPT4oBgHgl3EQfxjU-/content/2301.13168v1.pdf'}
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+1 
+ 
+Quantum Feasibility Labeling for NP-complete Vertex Coloring Problem 
+Junpeng Zhan1 
+ 
+Abstract: Many important science and engineering problems can be converted into NP-complete 
+problems which are of significant importance in computer science and mathematics. Currently, 
+neither existing classical nor quantum algorithms can solve these problems in polynomial time. To 
+overcome this difficulty, this paper proposes a quantum feasibility labeling (QFL) algorithm to 
+label all possible solutions to the vertex coloring problem, which is a well-known NP-complete 
+problem. The variational quantum search (VQS) algorithm proposed in my previous work has been 
+demonstrated, up to 26 qubits, to achieve an exponential speedup in finding good element(s) from 
+an unstructured database. Using the labels and the associated possible solutions as input, the VQS 
+can find all feasible solutions to the vertex coloring problem. The number of qubits and the circuit 
+depth required by the QFL each is a polynomial function of the number of vertices, the number of 
+edges, and the number of colors of a vertex coloring problem. The QFL together with the VQS 
+could be the first algorithm to solve an NP-complete problem in polynomial time, provided that 
+the VQS is proved to be efficient for any number of qubits.  
+ 
+1. Introduction 
+NP-complete problems [1], [2] are a type of problem for which it is believed that there is no 
+efficient algorithm for solving them. Many science and engineering problems can be converted 
+into NP-complete problems. These problems are typically characterized by the need to find the 
+optimal solution among a large number of possible solutions.  
+Here are a few examples of the many science and engineering problems that can be converted 
+into NP-complete problems: Traveling Salesman Problem [3]–[5] (of interest in transportation and 
+logistics), Knapsack Problem [6]–[9] (of interest in resource allocation and inventory 
+management), Satisfiability Problem [10]–[14] (SAT, of interest in computer science, artificial 
+intelligence, and logic), Subset Sum Problem [1] (of interest in cryptography and computer 
+security), Graph Coloring Problem [15], [16] (of interest in scheduling and resource allocation), 
+Hamiltonian Cycle Problem [15]–[17] (of interest in network design), and the Steiner Tree 
+Problem [17] (of interest in telecommunications and computer networks). 
+Despite the many efforts [3]–[17] to develop classical algorithms for solving NP-complete 
+problems, it is generally believed that there is no classical algorithm that can solve NP-complete 
+problems in polynomial time, which means that the time and resources required to solve these 
+problems increase faster than a polynomial function of the size of the problem. Whether NP-
+complete problems can be solved efficiently is an extremely important but unsolved question. The 
+Clay Mathematics Institute has offered a prize of $1 million for a solution to the problem. 
+Quantum computing has been the subject of intense research and development in recent years 
+due to its potential to solve certain types of problems much faster than classical computers [18]–
+[29]. The progress in quantum computer hardware development has been particularly impressive 
+in recent years. Researchers have made significant advances in the design and construction of 
+quantum computer hardware [28]–[33], and several companies have released commercial quantum 
+computers that are available for use by researchers and businesses [34]–[36]. These quantum 
+computers have demonstrated impressive performance on a variety of tasks [18], [28], [29], and 
+their capabilities are expected to continue to improve in the coming years [37], [38]. 
+ 
+1 Department of Renewable Energy Engineering, Alfred University, Alfred, NY, USA. E-mail: zhanj@alfred.edu 
+
+2 
+ 
+In addition to the progress in quantum computer hardware development, there has also been 
+significant progress in the development of a wide range of quantum algorithms for a variety of 
+different types of problems [20], [21], [25], [39]–[49]. Some of these algorithms have 
+demonstrated impressive performance on tasks that are difficult or impossible to solve using 
+classical algorithms [18], [28], and it is expected that the development of quantum algorithms will 
+continue to be an active area of research in the coming years. 
+Depending on whether classical computers are used, we can divide quantum algorithms into 
+1) pure quantum algorithms such as Shor's algorithm for factorization [24] and Grover's algorithm 
+for searching unordered lists [22], [23], and 2) variational quantum algorithms (VQAs) [50]–[52] 
+which involve both classical and quantum computers. VQAs have been successfully used in areas 
+such as optimization problems [53]–[55], quantum chemistry [32], [50], [52], [56], [57], and 
+machine learning [42], [58]–[63], and are expected to help achieve systematic quantum supremacy 
+over classical algorithms using hundreds of qubits which are already available [34]. 
+Given the promising potential and recent success of quantum computing, it is natural to ask 
+whether it can provide a quantum exponential speedup in solving NP-complete problems. 
+Unfortunately, it is widely believed (though not proven) that the answer is no [64], [65]. However, 
+a recent paper [25] proposed a variational quantum search (VQS) algorithm, which has been shown 
+to have an exponential advantage over Grover’s search algorithm in searching an unstructured 
+database, up to a limit of 26 qubits (the limit is due to the memory constraints of the 48-GB GPU 
+used in the demonstration). According to that paper, a depth-10 quantum circuit can amplify the 
+total probability of k (k≥1) good elements, out of 2n elements represented by n+1 qubits, from 
+k/2n to nearly 1 for n up to 26. Additionally, the maximum depth of quantum circuits in the VQS 
+increases linearly with the number of qubits. Given that Grover’s algorithm can provide quadratic 
+speedup in solving NP-complete problems, the VQS could solve them in polynomial time. 
+This paper focuses on the vertex coloring problem [15], [16] which is a well-known NP-
+complete problem in graph theory. It involves assigning colors to the vertices of a graph (Fig. 1 
+shows a graph with 7 vertices) such that no two adjacent vertices have the same color. It has many 
+practical applications, including scheduling, resource allocation, and network design, and has been 
+the subject of much research in both classical and quantum computing. Note that solving one NP-
+complete is equivalent to solving all hundreds of NP-complete problems. 
+Paper [66] proposed a quantum method that has a complexity of O(1.9140n) in solving a k-
+colorable n-vertex graph coloring problem. Paper [67] presented a simple method based on 
+Grover’s algorithm to obtain a quadratic speedup in solving NP-complete problems. 
+To efficiently solve the vertex coloring problem, this paper proposes a quantum feasibility 
+labeling (QFL) algorithm, which assigns a feasibility label to every possible feasible/infeasible 
+solution. By using the labels and the associated solutions as input, the VQS can then be directly 
+used to find the feasible solutions in polynomial time. If the VQS is proven to be efficient for more 
+than 26 qubits [25], the combination of the QFL and VQS will be the first quantum algorithm to 
+solve an NP-complete problem in polynomial time and will be a significant milestone and a major 
+achievement in the field of quantum computing.  
+Section 2 describes the vertex coloring problem. Section 3 details the QFL algorithm. Results 
+are presented in Section 4 followed by conclusions given in Section 5. 
+  
+1
+2
+3
+4
+6
+5
+7
+
+3 
+ 
+Fig. 1. A graph with 7 vertices and 7 edges. A line between two vertices indicates the two vertices 
+are directly connected, and we also say that the two vertices are adjacent. 
+2. Vertex Coloring Problem 
+The constraints for a vertex coloring problem are: any two directly connected vertices should 
+not have the same color. Here a solution in a ������������-vertex graph is represented as a vector of k elements, 
+where each element is a color and is associated with a vertex of the graph. The solution is feasible 
+if all the constraints are satisfied. The solution is infeasible if at least one constraint is violated. 
+3. Quantum Feasibility Labeling Algorithm 
+The QFL generates a feasibility label for each solution. The feasibility label is represented by 
+a single qubit which is in either |0⟩ or |1⟩ state, indicating that the solution is infeasible or feasible, 
+respectively. The number of data qubits is set to the product of the number of vertices and the 
+number of qubits per vertex.  
+The number of possible colors for a vertex is ������������. Then, the total number of possible solutions 
+to the vertex coloring with ������������ vertices is equal to ������������������������. Let ������������ represent the minimal integer number 
+such that 2������������ ≥ ������������ . Then ������������ × ������������  qubits represent 2������������������������  states which can represent all the ������������������������ 
+solutions as 2������������������������ ≥ ������������������������. Therefore, a circuit of ������������ × ������������ qubits is used to represent all the possible 
+solutions. The QFL uses a single qubit to represent the feasibility label of all possible solutions, 
+which is possible because 2������������������������+1 ≥ ������������������������ + ������������������������ , i.e., adding one more qubit can represent the 
+feasibility label for every possible solution. 
+Note that each color is uniquely represented by a unique value of a state. Take m=2 for example, 
+a 2-qubit state has four possible measurement values (i.e., |00〉, |01〉, |10〉, and |11〉) and we can let 
+them represent four different colors, respectively.  
+3.1 Quantum SO/SOR Module 
+Before detailing the QFL, we detail its core module, SO/SOR. First, a quantum subtraction 
+circuit compares two groups of m-qubit data (the blue part of Fig. 2 shows an example with m=2). 
+The quantum subtraction results are stored in m+1 Ancilla qubits. Then, a quantum OR circuit 
+(the pink part of Fig. 2 shows an example with three input qubits and one output qubit) obtains a 
+one-qubit feasibility label from the subtraction results. At last, a reset circuit (an example for the 
+2-qubit subtraction [68] and 3-input OR module is shown in Fig. 3) resets the state of each Ancilla 
+qubit to |0⟩ such that every subtraction can use the same Ancilla qubits without requesting new 
+qubits.  
+For the convenience of expression, the quantum subtraction and quantum OR circuits are 
+collectively referred to as an SO module. The SO module and the reset circuit are collectively 
+referred to as an SOR module. Fig. 2 shows an SOR module with data input on the far left. 
+To better understand the SO module, Table 1 provides its truth table. Note that the subtraction 
+circuit does not change the states of input data, which can be easily verified from Figs. 2 and 5. 
+From Table 1, we can conclude that output o7=0 when q3q2 is equal to q1q0 and o7=1 when q3q2 is 
+different from q1q0. This is what we need, i.e., the feasibility label is 1 when the colors of two 
+directly connected vertices are different and 0 otherwise. 
+
+4 
+ 
+ 
+Fig. 2. Data input (left-hand side of the red dashed line) and an SOR module (right-hand side of 
+the red dashed line) consisting of a 2-qubit subtraction, a 3-input OR, and a reset circuits. 
+ 
+Fig. 3. The reset circuit used in Fig. 2. 
+ 
+Fig. 4.  The quantum circuit for the Quantum Feasibility Labeling algorithm, where FLQ is short 
+for feasibility label qubit. 
+ 
+ 
+
+2-qubit Subtraction
+0)
+H
+q1
+H
+0
+data
+qubits
+q2
+0)
+H
+q3
+<0
+reset
+H
+3-input OR
+q4
+04
+<0
+Ancilla
+q5
+05
+10)
+qubits
+q6
+06
+<0l
+q7
+07
+feasibility label qubit l0)ob
+q1
+q2
+q3
+04
+05
+06
+(T
+Hon
+Hon
+SO/SOR
+n+1
+n+1
+Hon
+Hon
+AND
+SO/SOR
+SO/SOR
+n+1
+n+1
+n+1
+tu
+FLQ 10)
+L2
+AND
+FLQ 10)
+L1
+FLQ 10)
+L4
+L3
+0)
+<0
+L55 
+ 
+Table 1. The truth table for the circuit given in Fig. 2, where ������������7 = ������������6|������������5|������������4 and ������������6������������5������������4 = ������������1������������0 −
+������������3������������2. Note: ������������6 = ������������6, ������������5 = ������������5
+���, ������������4 = ������������4
+���, where the quantum states given in the top row of the table 
+are indicated in Fig. 2 (on the right-hand sides of the blue and pink blocks). 
+row o7 o6 o5 o4 q6 q5 q4 q3 q2 q1 q0 
+1 0 0 1 1 0 0 0 0 0 0 0 
+2 0 0 1 1 0 0 0 0 1 0 1 
+3 0 0 1 1 0 0 0 1 0 1 0 
+4 0 0 1 1 0 0 0 1 1 1 1 
+5 1 0 0 0 0 1 1 0 0 1 1 
+6 1 0 0 1 0 1 0 0 0 1 0 
+7 1 0 0 1 0 1 0 0 1 1 1 
+8 1 0 1 0 0 0 1 0 0 0 1 
+9 1 0 1 0 0 0 1 0 1 1 0 
+10 1 0 1 0 0 0 1 1 0 1 1 
+11 1 1 0 0 1 1 1 0 1 0 0 
+12 1 1 0 0 1 1 1 1 0 0 1 
+13 1 1 0 0 1 1 1 1 1 1 0 
+14 1 1 0 1 1 1 0 1 0 0 0 
+15 1 1 0 1 1 1 0 1 1 0 1 
+16 1 1 1 0 1 0 1 1 1 0 0 
+ 
+3.2 Quantum Reset Circuit 
+Each m-qubit SO/SOR module needs m+1 Ancilla qubits that have initial states of |0〉. Here 
+we use a reset circuit to return the state of each Ancilla qubit to |0〉 after passing through the 
+SO/SOR module such that the Ancilla qubits can be reused for the next SO/SOR module. The 
+basic idea is to use multi-controlled multi-target CNOT to change the state of each Ancilla qubit 
+to |0〉 according to the relationship between the input data and the output of the OR circuit. For 
+example, for the circuit given in Fig. 2, the relationship (or truth table) between o4-o6 and q0-q3 is 
+given in Table 1. We can then use the values of q0-q3 as controls and let the qubits associated with 
+o4-o6 be the target(s) if they are in state |1〉, which can reset the outputs of o4-o6 to |0〉. Fig. 3 
+provides an example reset circuit for the reset block (rightmost, in red) in Fig. 2. Fig. 3 has 15 
+layers, each associated with a row in Table 1. Note that row 5 of Table 1 does not need a layer in 
+Fig. 3 because its o4-o6 are already in state |0〉. 
+3.3 Construction of the Entire Quantum Circuit 
+Each SO/SOR module corresponds to a constraint of the vertex coloring problem. Each 
+SO/SOR module has a one-qubit feasibility label, denoted as Li (shown in the lower-right corner 
+of each SO/SOR block in Fig. 4), indicating whether the constraint is feasible. All constraints need 
+to be satisfied for a feasible solution. Therefore, we use an AND circuit to combine two SO/SOR 
+together in a sequential way: starting from the third SO/SOR block, the output of the previous 
+AND circuit is used as the second input of the current AND circuit. 
+For the data input, each vertex has m qubits. If a vertex is involved in multiple SO/SOR blocks, 
+all blocks should use the same m qubits for that vertex, which is possible because the states of the 
+data qubits do not change before and after the SO/SOR block. Also, the reset circuit does not 
+change the state of any data qubits. 
+The construction of the entire circuit of QFL is described in the following 4 steps. 
+
+6 
+ 
+Step 1 - Feasibility label: Use two SO/SOR modules (see the left and middle parts of Fig. 4) 
+to obtain two feasibility labels (i.e., L1 and L2 in Fig. 4) for the first two constraints, respectively.  
+Step 2 - Quantum AND: The two feasibility labels are converted into one feasibility label 
+using a quantum AND circuit (see the red part in the middle of Fig. 4). The output of this circuit 
+is a one-qubit feasibility label (L3 in Fig. 4). The logical relationship for this AND circuit can be 
+represented as L3=L1&L2. 
+Step 3 - Incremental step: Then, a new SOR circuit (see the right side of Fig. 4) is added for 
+a new constraint of the vertex coloring problem. The output at the feasibility label qubit of this 
+circuit (L4) and the feasibility label (L3) obtained from the previous AND circuit are used as inputs 
+of a new quantum AND circuit (see the rightmost red part of Fig. 4). The output of the new 
+quantum AND circuit is a one-qubit feasibility label (L5). The  logical relationship for this circuit 
+can be expressed as L5=L3&L4.  
+Step 4 - Repeat the incremental step for each remaining constraint of the vertex coloring 
+problem. 
+Now, the feasibility label qubit from the last quantum AND circuit and the data qubits together 
+represent all solutions and their feasibility labels. 
+3.4 Quantum Subtraction with More Qubits 
+This section presents the m-qubit quantum subtraction circuit, shown in Fig. 2 (blue block) and 
+Fig. 5 for m=2 and m=3, respectively. In Fig. 5, the logical relationship of the 3-qubit quantum 
+subtraction can be expressed as ������������9������������8������������7������������6 = ������������2������������1������������0 − ������������5������������4������������3 . Then, an m-qubit quantum 
+subtraction circuit can be derived, i.e., m SUBT blocks are connected in series such that the indices 
+of the four qubits connected to the four wires of a block differ from those of its adjacent block by 
+exactly 1, respectively, where m≥2. 
+ 
+ 
+ 
+ 
+Fig. 5. Quantum subtraction circuits. The top left panel in blue is a 1-qubit quantum subtraction 
+circuit having four qubits indicated by I, II, III, and IV, respectively. Its abstract form (denoted as 
+
+SUBT
+II
+II
+TT
+SUBT
+III
+III
+III
+IV
+IV
+IV
+IVq0
+1
+1
+q1
+I
+q2
+1
+I
+SUBT
+q3
+II
+II
+q4
+II
+II
+SUBT
+q5
+II
+II
+q6
+III
+III
+SUBT
+q7
+IV
+IV
+III
+I
+q8
+IV
+IV
+IHI
+III
+6b
+IV
+IV7 
+ 
+SUBT) is shown in the top right panel. The bottom panel shows a 3-qubit quantum subtraction 
+circuit consisting of three 1-qubit quantum subtraction circuits. The I, II, III, and IV indicate where 
+the four qubits of a SUBT block are connected to q0~q9. For example, the four qubits of the 
+leftmost SUBT block are connected to q0, q3, q6, and q7, respectively. 
+3.5 Quantum OR Circuits with More Qubits 
+This section presents the m-qubit quantum OR circuit, shown in Fig. 2 (pink block) and Fig. 6 
+for m=3, and m=4, respectively. In Fig. 6, the logical relationship of the 4-qubit quantum OR can 
+be expressed as ������������4 = ������������0|������������1|������������2|������������3. Then, an m-input quantum OR circuit can be derived and it can 
+be constructed using the pseudo code given in Table 2. 
+ 
+Fig. 6. A 4-input quantum OR circuit. 
+ 
+Table 2. Pseudo code for generating the m-input quantum OR circuit. 
+ 
+ 
+3.6 Complexity Analysis 
+This section calculates the number of qubits and the depth of the circuit required by the QFL 
+for solving a ������������-colorable n-vertex g-edge coloring problem.  
+In the calculation given in the following two paragraphs, the reset circuit (the red block on the 
+far right in Fig. 2) is used, i.e., each SO/SOR module in Fig. 4 uses SOR. The total depth of the 
+entire circuit is ������������ �5������������ + 1 + �∑
+�������������
+������������ �
+������������
+������������=1
++ 2� + (22������������ − 1)� where the four terms in the square 
+brackets are the depths of the subtraction, AND, OR, and reset circuits, respectively, and �������������
+������������ � 
+Input: m+1 qubits 
+Output: m-input quantum OR circuit 
+Add m−1 CNOT gates with the control qubits 
+located at qubits q0~qm−2, respectively, and 
+the target qubit for each gate is located at 
+qubit qm+1. 
+Let u=2 
+while u≤m−1 
+Add a ������������������������(������������) gate with u control qubits 
+located at set ℒ and a target qubit located 
+at qubit qm+1. 
+Repeat the previous step for ℒ being any 
+combinations of u qubits from q0 to qm−2. 
+u ← u+1. 
+Add a Pauli X gate to each of qubits q0~qm-2. 
+Add a ������������������������(������������) gate with a target qubit located 
+at qubit qm+1 and m control qubits located at 
+qubits q0~qm−1. 
+ 
+
+qo
+q1
+q2
+q3
+948 
+ 
+denotes the number of i-combinations from m elements. Note that ∑
+�������������
+������������ �
+������������
+������������=1
+= 2������������ − 1. Then, the 
+total depth can be simplified to ������������(22������������ + 2������������ + 5������������ + 1). Note that m is related to the number of 
+possible colors k, i.e., ������������ = ⌈log2 ������������⌉ where the symbol ⌈⋅⌉ means rounding up to the nearest integer 
+number. Then the total depth can be written as ������������(������������2������������).  
+The total number of qubits is ������������������������ + (������������ + 1) + ������������ + (������������ − 1) where the four terms correspond 
+to the numbers of data qubits, Ancilla qubits, feasibility label qubits, output qubits of quantum 
+AND circuits, respectively. It can be simplified as ������������������������ + ������������ + 2������������. 
+In the calculation given in this paragraph, the reset circuit is not used, i.e., each SO/SOR 
+module in Fig. 4 uses SO. The total depth of the entire circuit is ������������ �5������������ + 1 + �∑
+�������������
+������������ �
+������������
+������������=1
++ 2�� 
+which can be simplified to ������������[5������������ + 2������������ + 2]. Then the total depth can be written as ������������(������������������������). As the 
+reset circuit is not used, each SO module requires new Ancilla qubits. Therefore, the total number 
+of qubits is ������������������������ + (������������ + 1)������������ + ������������ + (������������ − 1) which can be rewritten as ������������������������ + (������������ + 3)������������ − 1.  
+Table 3 lists the circuit depths and the numbers of qubits in both situations, showing that using 
+the reset circuit reduces the required number of qubits at the expense of deeper circuit depth. When 
+k (the number of colors) is large, we recommend not using the reset circuit. On the other hand, 
+when ������������ (the number of edges) is large and k is small, it is better to use the reset circuit. 
+ 
+Table 3. The circuit depths and the numbers of qubits required by the QFL with and without 
+using the reset circuit. 
+With Reset Circuit Circuit Depth 
+# of qubits 
+Yes 
+������������(������������2������������) 
+������������������������ + ������������ + 2������������ 
+No 
+������������(������������������������) 
+������������������������ + (������������ + 3)������������ − 1 
+ 
+4. Result 
+The QFL has been implemented on an IBM Qiskit [34] simulator to solve a 4-colorable 4-
+vertex 3-edge coloring problem. The three edges are 1-2, 1-3, and 1-4, i.e., vertex 1 connects to 
+vertices 2, 3, and 4, respectively. Based on the principles of permutation and combination, vertex 
+1 can be colored with any of the four available colors, and then vertices 2, 3, and 4 can each be 
+colored with any of the remaining three colors. As a result, the number of possible feasible 
+solutions is 4 × 3 × 3 × 3 = 108. The circuit used by the QFL is given in Fig. 4, where the three 
+SOR modules are for the three constraints, respectively. These modules use the same qubits for 
+vertex 1, i.e., the output of one SOR block at these qubits is used as input for the next SOR.  
+The circuit is run and measured 20,000 times with the results shown in Table 4. The table 
+shows that 256 quantum states are obtained, which exactly corresponds to all possible 
+combinations of 8 qubits (counted from the far right of each state). The feasibility label qubit is 
+the 9th qubit counted from the right. The labels of the states shown in the first 37 rows are 0, while 
+those in rows 38-64 are 1. A label of 0 represents an infeasible solution, while a label of 1 indicates 
+a feasible solution. Considering that each row has 4 states, there are 148 infeasible states and 108 
+feasible states, which matches the number calculated in the previous paragraph and validates the 
+QFL.   
+ 
+
+9 
+ 
+5. Conclusion 
+We have developed a QFL algorithm that provides a feasibility label for every possible solution 
+to the vertex coloring problem. The output of this algorithm can be directly input into the VQS 
+algorithm, which can then find all feasible solutions in polynomial time. The next step is to prove 
+that the VQS is efficient for any number of qubits, which would be a significant milestone, as it 
+would mean that NP-complete problems can be efficiently solved using quantum algorithms. 
+ 
+Table 4. Measurement results. The strings in single quotes represent measured quantum states, 
+and the number after a colon is the number of times that the state shown in the same cell is 
+obtained from measurement. 
+1 
+'0000000000000000': 86 
+'0000000000000100': 80 
+'0000000000000101': 64 
+'0000000000001000': 83 
+2 
+'0000000000001010': 83 
+'0000000000001100': 84 
+'0000000000001111': 72 
+'0000000000010000': 69 
+3 
+'0000000000010001': 69 
+'0000000000010100': 68 
+'0000000000010101': 93 
+'0000000000011000': 71 
+4 
+'0000000000011001': 77 
+'0000000000011010': 69 
+'0000000000011100': 80 
+'0000000000011101': 65 
+5 
+'0000000000011111': 78 
+'0000000000100000': 86 
+'0000000000100010': 81 
+'0000000000100100': 75 
+6 
+'0000000000100101': 68 
+'0000000000100110': 90 
+'0000000000101000': 68 
+'0000000000101010': 75 
+7 
+'0000000000101100': 84 
+'0000000000101110': 60 
+'0000000000101111': 68 
+'0000000000110000': 69 
+8 
+'0000000000110011': 89 
+'0000000000110100': 80 
+'0000000000110101': 77 
+'0000000000110111': 71 
+9 
+'0000000000111000': 84 
+'0000000000111010': 72 
+'0000000000111011': 77 
+'0000000000111100': 73 
+10 
+'0000000000111111': 81 
+'0000000001000000': 68 
+'0000000001000001': 80 
+'0000000001000100': 89 
+11 
+'0000000001000101': 90 
+'0000000001001000': 78 
+'0000000001001001': 81 
+'0000000001001010': 69 
+12 
+'0000000001001100': 85 
+'0000000001001101': 84 
+'0000000001001111': 75 
+'0000000001010000': 72 
+13 
+'0000000001010001': 72 
+'0000000001010101': 83 
+'0000000001011001': 70 
+'0000000001011010': 79 
+14 
+'0000000001011101': 76 
+'0000000001011111': 89 
+'0000000001100000': 88 
+'0000000001100001': 74 
+15 
+'0000000001100010': 67 
+'0000000001100101': 68 
+'0000000001100110': 74 
+'0000000001101001': 82 
+16 
+'0000000001101010': 65 
+'0000000001101101': 61 
+'0000000001101110': 84 
+'0000000001101111': 84 
+17 
+'0000000001110000': 87 
+'0000000001110001': 74 
+'0000000001110011': 72 
+'0000000001110101': 77 
+18 
+'0000000001110111': 81 
+'0000000001111001': 80 
+'0000000001111010': 76 
+'0000000001111011': 79 
+19 
+'0000000001111101': 94 
+'0000000001111111': 88 
+'0000000010000000': 86 
+'0000000010000010': 78 
+20 
+'0000000010000100': 79 
+'0000000010000101': 82 
+'0000000010000110': 88 
+'0000000010001000': 69 
+21 
+'0000000010001010': 67 
+'0000000010001100': 74 
+'0000000010001110': 78 
+'0000000010001111': 84 
+22 
+'0000000010010000': 87 
+'0000000010010001': 89 
+'0000000010010010': 89 
+'0000000010010101': 74 
+23 
+'0000000010010110': 63 
+'0000000010011001': 66 
+'0000000010011010': 75 
+'0000000010011101': 70 
+24 
+'0000000010011110': 80 
+'0000000010011111': 78 
+'0000000010100000': 66 
+'0000000010100010': 80 
+25 
+'0000000010100101': 89 
+'0000000010100110': 73 
+'0000000010101010': 81 
+'0000000010101110': 80 
+26 
+'0000000010101111': 73 
+'0000000010110000': 84 
+'0000000010110010': 69 
+'0000000010110011': 86 
+27 
+'0000000010110101': 83 
+'0000000010110110': 92 
+'0000000010110111': 73 
+'0000000010111010': 88 
+28 
+'0000000010111011': 73 
+'0000000010111110': 76 
+'0000000010111111': 83 
+'0000000011000000': 71 
+29 
+'0000000011000011': 86 
+'0000000011000100': 79 
+'0000000011000101': 75 
+'0000000011000111': 67 
+30 
+'0000000011001000': 80 
+'0000000011001010': 70 
+'0000000011001011': 89 
+'0000000011001100': 68 
+31 
+'0000000011001111': 71 
+'0000000011010000': 68 
+'0000000011010001': 84 
+'0000000011010011': 70 
+32 
+'0000000011010101': 80 
+'0000000011010111': 70 
+'0000000011011001': 93 
+'0000000011011010': 71 
+33 
+'0000000011011011': 84 
+'0000000011011101': 81 
+'0000000011011111': 67 
+'0000000011100000': 97 
+34 
+'0000000011100010': 83 
+'0000000011100011': 87 
+'0000000011100101': 73 
+'0000000011100110': 80 
+35 
+'0000000011100111': 86 
+'0000000011101010': 78 
+'0000000011101011': 70 
+'0000000011101110': 68 
+36 
+'0000000011101111': 63 
+'0000000011110000': 81 
+'0000000011110011': 68 
+'0000000011110101': 79 
+37 
+'0000000011110111': 72 
+'0000000011111010': 89 
+'0000000011111011': 87 
+'0000000011111111': 83 
+38 
+'0000000100000001': 67 
+'0000000100000010': 104 
+'0000000100000011': 93 
+'0000000100000110': 66 
+39 
+'0000000100000111': 83 
+'0000000100001001': 66 
+'0000000100001011': 58 
+'0000000100001101': 66 
+40 
+'0000000100001110': 72 
+'0000000100010010': 77 
+'0000000100010011': 89 
+'0000000100010110': 91 
+41 
+'0000000100010111': 73 
+'0000000100011011': 66 
+'0000000100011110': 90 
+'0000000100100001': 92 
+42 
+'0000000100100011': 80 
+'0000000100100111': 78 
+'0000000100101001': 80 
+'0000000100101011': 82 
+43 
+'0000000100101101': 77 
+'0000000100110001': 70 
+'0000000100110010': 81 
+'0000000100110110': 73 
+44 
+'0000000100111001': 105 
+'0000000100111101': 84 
+'0000000100111110': 82 
+'0000000101000010': 69 
+45 
+'0000000101000011': 94 
+'0000000101000110': 74 
+'0000000101000111': 76 
+'0000000101001011': 84 
+46 
+'0000000101001110': 83 
+'0000000101010010': 83 
+'0000000101010011': 78 
+'0000000101010100': 65 
+47 
+'0000000101010110': 75 
+'0000000101010111': 86 
+'0000000101011000': 81 
+'0000000101011011': 64 
+48 
+'0000000101011100': 91 
+'0000000101011110': 79 
+'0000000101100011': 83 
+'0000000101100100': 77 
+49 
+'0000000101100111': 66 
+'0000000101101000': 82 
+'0000000101101011': 64 
+'0000000101101100': 79 
+50 
+'0000000101110010': 83 
+'0000000101110100': 77 
+'0000000101110110': 92 
+'0000000101111000': 83 
+51 
+'0000000101111100': 98 
+'0000000101111110': 86 
+'0000000110000001': 88 
+'0000000110000011': 88 
+52 
+'0000000110000111': 76 
+'0000000110001001': 78 
+'0000000110001011': 73 
+'0000000110001101': 87 
+
+10 
+ 
+53 
+'0000000110010011': 65 
+'0000000110010100': 80 
+'0000000110010111': 95 
+'0000000110011000': 84 
+54 
+'0000000110011011': 90 
+'0000000110011100': 68 
+'0000000110100001': 67 
+'0000000110100011': 71 
+55 
+'0000000110100100': 77 
+'0000000110100111': 78 
+'0000000110101000': 82 
+'0000000110101001': 87 
+56 
+'0000000110101011': 84 
+'0000000110101100': 71 
+'0000000110101101': 67 
+'0000000110110001': 86 
+57 
+'0000000110110100': 78 
+'0000000110111000': 76 
+'0000000110111001': 92 
+'0000000110111100': 82 
+58 
+'0000000110111101': 67 
+'0000000111000001': 70 
+'0000000111000010': 81 
+'0000000111000110': 100 
+59 
+'0000000111001001': 73 
+'0000000111001101': 69 
+'0000000111001110': 80 
+'0000000111010010': 75 
+60 
+'0000000111010100': 82 
+'0000000111010110': 72 
+'0000000111011000': 75 
+'0000000111011100': 76 
+61 
+'0000000111011110': 91 
+'0000000111100001': 72 
+'0000000111100100': 69 
+'0000000111101000': 74 
+62 
+'0000000111101001': 83 
+'0000000111101100': 69 
+'0000000111101101': 74 
+'0000000111110001': 78 
+63 
+'0000000111110010': 95 
+'0000000111110100': 80 
+'0000000111110110': 85 
+'0000000111111000': 66 
+64 
+'0000000111111001': 75 
+'0000000111111100': 59 
+'0000000111111101': 78 
+'0000000111111110': 78 
+ 
+Acknowledgements 
+This research was partially supported by the NSF ERI program, under award number 2138702. 
+The author acknowledges the use of IBM Quantum services for this work. The views expressed 
+are those of the author, and do not reflect the official policy or position of IBM or the IBM 
+Quantum team. 
+ 
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+
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf,len=1603
+page_content='1  Quantum Feasibility Labeling for NP-complete Vertex Coloring Problem  Junpeng Zhan1  Abstract: Many important science and engineering problems can be converted into NP-complete  problems which are of significant importance in computer science and mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Currently,  neither existing classical nor quantum algorithms can solve these problems in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' To  overcome this difficulty, this paper proposes a quantum feasibility labeling (QFL) algorithm to  label all possible solutions to the vertex coloring problem, which is a well-known NP-complete  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The variational quantum search (VQS) algorithm proposed in my previous work has been  demonstrated, up to 26 qubits, to achieve an exponential speedup in finding good element(s) from  an unstructured database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Using the labels and the associated possible solutions as input, the VQS  can find all feasible solutions to the vertex coloring problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The number of qubits and the circuit  depth required by the QFL each is a polynomial function of the number of vertices, the number of  edges, and the number of colors of a vertex coloring problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The QFL together with the VQS  could be the first algorithm to solve an NP-complete problem in polynomial time, provided that  the VQS is proved to be efficient for any number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Introduction  NP-complete problems [1], [2] are a type of problem for which it is believed that there is no  efficient algorithm for solving them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Many science and engineering problems can be converted  into NP-complete problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' These problems are typically characterized by the need to find the  optimal solution among a large number of possible solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Here are a few examples of the many science and engineering problems that can be converted  into NP-complete problems: Traveling Salesman Problem [3]–[5] (of interest in transportation and  logistics),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Knapsack Problem [6]–[9] (of interest in resource allocation and inventory  management),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Satisfiability Problem [10]–[14] (SAT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' of interest in computer science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' artificial  intelligence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' and logic),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Subset Sum Problem [1] (of interest in cryptography and computer  security),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Graph Coloring Problem [15],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' [16] (of interest in scheduling and resource allocation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Hamiltonian Cycle Problem [15]–[17] (of interest in network design),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' and the Steiner Tree  Problem [17] (of interest in telecommunications and computer networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Despite the many efforts [3]–[17] to develop classical algorithms for solving NP-complete  problems, it is generally believed that there is no classical algorithm that can solve NP-complete  problems in polynomial time, which means that the time and resources required to solve these  problems increase faster than a polynomial function of the size of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Whether NP-  complete problems can be solved efficiently is an extremely important but unsolved question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The  Clay Mathematics Institute has offered a prize of $1 million for a solution to the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Quantum computing has been the subject of intense research and development in recent years  due to its potential to solve certain types of problems much faster than classical computers [18]–  [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The progress in quantum computer hardware development has been particularly impressive  in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Researchers have made significant advances in the design and construction of  quantum computer hardware [28]–[33], and several companies have released commercial quantum  computers that are available for use by researchers and businesses [34]–[36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' These quantum  computers have demonstrated impressive performance on a variety of tasks [18], [28], [29], and  their capabilities are expected to continue to improve in the coming years [37], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  1 Department of Renewable Energy Engineering, Alfred University, Alfred, NY, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' E-mail: zhanj@alfred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='edu  2  In addition to the progress in quantum computer hardware development, there has also been  significant progress in the development of a wide range of quantum algorithms for a variety of  different types of problems [20], [21], [25], [39]–[49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Some of these algorithms have  demonstrated impressive performance on tasks that are difficult or impossible to solve using  classical algorithms [18], [28], and it is expected that the development of quantum algorithms will  continue to be an active area of research in the coming years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content="  Depending on whether classical computers are used, we can divide quantum algorithms into  1) pure quantum algorithms such as Shor's algorithm for factorization [24] and Grover's algorithm  for searching unordered lists [22], [23], and 2) variational quantum algorithms (VQAs) [50]–[52]  which involve both classical and quantum computers." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' VQAs have been successfully used in areas  such as optimization problems [53]–[55], quantum chemistry [32], [50], [52], [56], [57], and  machine learning [42], [58]–[63], and are expected to help achieve systematic quantum supremacy  over classical algorithms using hundreds of qubits which are already available [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Given the promising potential and recent success of quantum computing, it is natural to ask  whether it can provide a quantum exponential speedup in solving NP-complete problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Unfortunately, it is widely believed (though not proven) that the answer is no [64], [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' However,  a recent paper [25] proposed a variational quantum search (VQS) algorithm, which has been shown  to have an exponential advantage over Grover’s search algorithm in searching an unstructured  database, up to a limit of 26 qubits (the limit is due to the memory constraints of the 48-GB GPU  used in the demonstration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' According to that paper, a depth-10 quantum circuit can amplify the  total probability of k (k≥1) good elements, out of 2n elements represented by n+1 qubits, from  k/2n to nearly 1 for n up to 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Additionally, the maximum depth of quantum circuits in the VQS  increases linearly with the number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Given that Grover’s algorithm can provide quadratic  speedup in solving NP-complete problems, the VQS could solve them in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  This paper focuses on the vertex coloring problem [15], [16] which is a well-known NP-  complete problem in graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' It involves assigning colors to the vertices of a graph (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 1  shows a graph with 7 vertices) such that no two adjacent vertices have the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' It has many  practical applications, including scheduling, resource allocation, and network design, and has been  the subject of much research in both classical and quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Note that solving one NP-  complete is equivalent to solving all hundreds of NP-complete problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Paper [66] proposed a quantum method that has a complexity of O(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='9140n) in solving a k-  colorable n-vertex graph coloring problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Paper [67] presented a simple method based on  Grover’s algorithm to obtain a quadratic speedup in solving NP-complete problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  To efficiently solve the vertex coloring problem, this paper proposes a quantum feasibility  labeling (QFL) algorithm, which assigns a feasibility label to every possible feasible/infeasible  solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' By using the labels and the associated solutions as input, the VQS can then be directly  used to find the feasible solutions in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' If the VQS is proven to be efficient for more  than 26 qubits [25], the combination of the QFL and VQS will be the first quantum algorithm to  solve an NP-complete problem in polynomial time and will be a significant milestone and a major  achievement in the field of quantum computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Section 2 describes the vertex coloring problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Section 3 details the QFL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Results  are presented in Section 4 followed by conclusions given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  1  2  3  4  6  5  7  3  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' A graph with 7 vertices and 7 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' A line between two vertices indicates the two vertices  are directly connected, and we also say that the two vertices are adjacent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Vertex Coloring Problem  The constraints for a vertex coloring problem are: any two directly connected vertices should  not have the same color.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Here a solution in a ������������-vertex graph is represented as a vector of k elements,  where each element is a color and is associated with a vertex of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The solution is feasible  if all the constraints are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The solution is infeasible if at least one constraint is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Quantum Feasibility Labeling Algorithm  The QFL generates a feasibility label for each solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The feasibility label is represented by  a single qubit which is in either |0⟩ or |1⟩ state, indicating that the solution is infeasible or feasible,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The number of data qubits is set to the product of the number of vertices and the  number of qubits per vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  The number of possible colors for a vertex is ������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Then, the total number of possible solutions  to the vertex coloring with ������������ vertices is equal to ������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Let ������������ represent the minimal integer number  such that 2������������ ≥ ������������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Then ������������ × ������������  qubits represent 2������������������������  states which can represent all the ������������������������  solutions as 2������������������������ ≥ ������������������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Therefore, a circuit of ������������ × ������������ qubits is used to represent all the possible  solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The QFL uses a single qubit to represent the feasibility label of all possible solutions,  which is possible because 2������������������������+1 ≥ ������������������������ + ������������������������ , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=', adding one more qubit can represent the  feasibility label for every possible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Note that each color is uniquely represented by a unique value of a state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Take m=2 for example,  a 2-qubit state has four possible measurement values (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=', |00〉, |01〉, |10〉, and |11〉) and we can let  them represent four different colors, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='1 Quantum SO/SOR Module  Before detailing the QFL, we detail its core module, SO/SOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' First, a quantum subtraction  circuit compares two groups of m-qubit data (the blue part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 2 shows an example with m=2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  The quantum subtraction results are stored in m+1 Ancilla qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Then, a quantum OR circuit  (the pink part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 2 shows an example with three input qubits and one output qubit) obtains a  one-qubit feasibility label from the subtraction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' At last, a reset circuit (an example for the  2-qubit subtraction [68] and 3-input OR module is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 3) resets the state of each Ancilla  qubit to |0⟩ such that every subtraction can use the same Ancilla qubits without requesting new  qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  For the convenience of expression, the quantum subtraction and quantum OR circuits are  collectively referred to as an SO module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The SO module and the reset circuit are collectively  referred to as an SOR module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 2 shows an SOR module with data input on the far left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  To better understand the SO module, Table 1 provides its truth table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Note that the subtraction  circuit does not change the states of input data, which can be easily verified from Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  From Table 1, we can conclude that output o7=0 when q3q2 is equal to q1q0 and o7=1 when q3q2 is  different from q1q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' This is what we need, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=', the feasibility label is 1 when the colors of two  directly connected vertices are different and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  4  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Data input (left-hand side of the red dashed line) and an SOR module (right-hand side of  the red dashed line) consisting of a 2-qubit subtraction, a 3-input OR, and a reset circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The reset circuit used in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The quantum circuit for the Quantum Feasibility Labeling algorithm, where FLQ is short  for feasibility label qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  2-qubit Subtraction  0)  H  q1  H  0  data  qubits  q2  0)  H  q3  <0  reset  H  3-input OR  q4  04  <0  Ancilla  q5  05  10)  qubits  q6  06  <0l  q7  07  feasibility label qubit l0)ob  q1  q2  q3  04  05  06  (T  Hon  Hon  SO/SOR  n+1  n+1  Hon  Hon  AND  SO/SOR  SO/SOR  n+1  n+1  n+1  tu  FLQ 10)  L2  AND  FLQ 10)  L1  FLQ 10)  L4  L3  0)  <0  L55  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The truth table for the circuit given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 2, where ������������7 = ������������6|������������5|������������4 and ������������6������������5������������4 = ������������1������������0 −  ������������3������������2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Note: ������������6 = ������������6, ������������5 = ������������5  ���, ������������4 = ������������4  ���, where the quantum states given in the top row of the table  are indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 2 (on the right-hand sides of the blue and pink blocks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  row o7 o6 o5 o4 q6 q5 q4 q3 q2 q1 q0  1 0 0 1 1 0 0 0 0 0 0 0  2 0 0 1 1 0 0 0 0 1 0 1  3 0 0 1 1 0 0 0 1 0 1 0  4 0 0 1 1 0 0 0 1 1 1 1  5 1 0 0 0 0 1 1 0 0 1 1  6 1 0 0 1 0 1 0 0 0 1 0  7 1 0 0 1 0 1 0 0 1 1 1  8 1 0 1 0 0 0 1 0 0 0 1  9 1 0 1 0 0 0 1 0 1 1 0  10 1 0 1 0 0 0 1 1 0 1 1  11 1 1 0 0 1 1 1 0 1 0 0  12 1 1 0 0 1 1 1 1 0 0 1  13 1 1 0 0 1 1 1 1 1 1 0  14 1 1 0 1 1 1 0 1 0 0 0  15 1 1 0 1 1 1 0 1 1 0 1  16 1 1 1 0 1 0 1 1 1 0 0  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='2 Quantum Reset Circuit  Each m-qubit SO/SOR module needs m+1 Ancilla qubits that have initial states of |0〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Here  we use a reset circuit to return the state of each Ancilla qubit to |0〉 after passing through the  SO/SOR module such that the Ancilla qubits can be reused for the next SO/SOR module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The  basic idea is to use multi-controlled multi-target CNOT to change the state of each Ancilla qubit  to |0〉 according to the relationship between the input data and the output of the OR circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' For  example, for the circuit given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 2, the relationship (or truth table) between o4-o6 and q0-q3 is  given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' We can then use the values of q0-q3 as controls and let the qubits associated with  o4-o6 be the target(s) if they are in state |1〉, which can reset the outputs of o4-o6 to |0〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 3  provides an example reset circuit for the reset block (rightmost, in red) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 3 has 15  layers, each associated with a row in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Note that row 5 of Table 1 does not need a layer in  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 3 because its o4-o6 are already in state |0〉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='3 Construction of the Entire Quantum Circuit  Each SO/SOR module corresponds to a constraint of the vertex coloring problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Each  SO/SOR module has a one-qubit feasibility label, denoted as Li (shown in the lower-right corner  of each SO/SOR block in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 4), indicating whether the constraint is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' All constraints need  to be satisfied for a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Therefore, we use an AND circuit to combine two SO/SOR  together in a sequential way: starting from the third SO/SOR block, the output of the previous  AND circuit is used as the second input of the current AND circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  For the data input, each vertex has m qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' If a vertex is involved in multiple SO/SOR blocks,  all blocks should use the same m qubits for that vertex, which is possible because the states of the  data qubits do not change before and after the SO/SOR block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Also, the reset circuit does not  change the state of any data qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  The construction of the entire circuit of QFL is described in the following 4 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  6  Step 1 - Feasibility label: Use two SO/SOR modules (see the left and middle parts of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 4)  to obtain two feasibility labels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=', L1 and L2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 4) for the first two constraints, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Step 2 - Quantum AND: The two feasibility labels are converted into one feasibility label  using a quantum AND circuit (see the red part in the middle of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The output of this circuit  is a one-qubit feasibility label (L3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The logical relationship for this AND circuit can be  represented as L3=L1&L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Step 3 - Incremental step: Then, a new SOR circuit (see the right side of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 4) is added for  a new constraint of the vertex coloring problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The output at the feasibility label qubit of this  circuit (L4) and the feasibility label (L3) obtained from the previous AND circuit are used as inputs  of a new quantum AND circuit (see the rightmost red part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The output of the new  quantum AND circuit is a one-qubit feasibility label (L5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The  logical relationship for this circuit  can be expressed as L5=L3&L4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Step 4 - Repeat the incremental step for each remaining constraint of the vertex coloring  problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Now, the feasibility label qubit from the last quantum AND circuit and the data qubits together  represent all solutions and their feasibility labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='4 Quantum Subtraction with More Qubits  This section presents the m-qubit quantum subtraction circuit, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 2 (blue block) and  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 5 for m=2 and m=3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 5, the logical relationship of the 3-qubit quantum  subtraction can be expressed as ������������9������������8������������7������������6 = ������������2������������1������������0 − ������������5������������4������������3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Then, an m-qubit quantum  subtraction circuit can be derived, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=', m SUBT blocks are connected in series such that the indices  of the four qubits connected to the four wires of a block differ from those of its adjacent block by  exactly 1, respectively, where m≥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Quantum subtraction circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The top left panel in blue is a 1-qubit quantum subtraction  circuit having four qubits indicated by I, II, III, and IV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Its abstract form (denoted as  SUBT  II  II  TT  SUBT  III  III  III  IV  IV  IV  IVq0  1  1  q1  I  q2  1  I  SUBT  q3  II  II  q4  II  II  SUBT  q5  II  II  q6  III  III  SUBT  q7  IV  IV  III  I  q8  IV  IV  IHI  III  6b  IV  IV7  SUBT) is shown in the top right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The bottom panel shows a 3-qubit quantum subtraction  circuit consisting of three 1-qubit quantum subtraction circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The I, II, III, and IV indicate where  the four qubits of a SUBT block are connected to q0~q9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' For example, the four qubits of the  leftmost SUBT block are connected to q0, q3, q6, and q7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='5 Quantum OR Circuits with More Qubits  This section presents the m-qubit quantum OR circuit, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 2 (pink block) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 6  for m=3, and m=4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 6, the logical relationship of the 4-qubit quantum OR can  be expressed as ������������4 = ������������0|������������1|������������2|������������3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Then, an m-input quantum OR circuit can be derived and it can  be constructed using the pseudo code given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' A 4-input quantum OR circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Pseudo code for generating the m-input quantum OR circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='6 Complexity Analysis  This section calculates the number of qubits and the depth of the circuit required by the QFL  for solving a ������������-colorable n-vertex g-edge coloring problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  In the calculation given in the following two paragraphs, the reset circuit (the red block on the  far right in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 2) is used, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=', each SO/SOR module in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 4 uses SOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The total depth of the  entire circuit is ������������ �5������������ + 1 + �∑  �������������  ������������ �  ������������  ������������=1  + 2� + (22������������ − 1)� where the four terms in the square  brackets are the depths of the subtraction,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' AND,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' OR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' and reset circuits,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' and �������������  ������������ �  Input: m+1 qubits  Output: m-input quantum OR circuit  Add m−1 CNOT gates with the control qubits  located at qubits q0~qm−2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' and  the target qubit for each gate is located at  qubit qm+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Let u=2  while u≤m−1  Add a ������������������������(������������) gate with u control qubits  located at set ℒ and a target qubit located  at qubit qm+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Repeat the previous step for ℒ being any  combinations of u qubits from q0 to qm−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  u ← u+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Add a Pauli X gate to each of qubits q0~qm-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Add a ������������������������(������������) gate with a target qubit located  at qubit qm+1 and m control qubits located at  qubits q0~qm−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  qo  q1  q2  q3  948  denotes the number of i-combinations from m elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Note that ∑  �������������  ������������ �  ������������  ������������=1  = 2������������ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Then, the  total depth can be simplified to ������������(22������������ + 2������������ + 5������������ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Note that m is related to the number of  possible colors k, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=', ������������ = ⌈log2 ������������⌉ where the symbol ⌈⋅⌉ means rounding up to the nearest integer  number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Then the total depth can be written as ������������(������������2������������).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  The total number of qubits is ������������������������ + (������������ + 1) + ������������ + (������������ − 1) where the four terms correspond  to the numbers of data qubits, Ancilla qubits, feasibility label qubits, output qubits of quantum  AND circuits, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' It can be simplified as ������������������������ + ������������ + 2������������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  In the calculation given in this paragraph, the reset circuit is not used, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=', each SO/SOR  module in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 4 uses SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The total depth of the entire circuit is ������������ �5������������ + 1 + �∑  �������������  ������������ �  ������������  ������������=1  + 2��  which can be simplified to ������������[5������������ + 2������������ + 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Then the total depth can be written as ������������(������������������������).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' As the  reset circuit is not used, each SO module requires new Ancilla qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Therefore, the total number  of qubits is ������������������������ + (������������ + 1)������������ + ������������ + (������������ − 1) which can be rewritten as ������������������������ + (������������ + 3)������������ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Table 3 lists the circuit depths and the numbers of qubits in both situations, showing that using  the reset circuit reduces the required number of qubits at the expense of deeper circuit depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' When  k (the number of colors) is large, we recommend not using the reset circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' On the other hand,  when ������������ (the number of edges) is large and k is small, it is better to use the reset circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The circuit depths and the numbers of qubits required by the QFL with and without  using the reset circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  With Reset Circuit Circuit Depth  # of qubits  Yes  ������������(������������2������������)  ������������������������ + ������������ + 2������������  No  ������������(������������������������)  ������������������������ + (������������ + 3)������������ − 1  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Result  The QFL has been implemented on an IBM Qiskit [34] simulator to solve a 4-colorable 4-  vertex 3-edge coloring problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The three edges are 1-2, 1-3, and 1-4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=', vertex 1 connects to  vertices 2, 3, and 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Based on the principles of permutation and combination, vertex  1 can be colored with any of the four available colors, and then vertices 2, 3, and 4 can each be  colored with any of the remaining three colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' As a result, the number of possible feasible  solutions is 4 × 3 × 3 × 3 = 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The circuit used by the QFL is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' 4, where the three  SOR modules are for the three constraints, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' These modules use the same qubits for  vertex 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=', the output of one SOR block at these qubits is used as input for the next SOR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  The circuit is run and measured 20,000 times with the results shown in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The table  shows that 256 quantum states are obtained, which exactly corresponds to all possible  combinations of 8 qubits (counted from the far right of each state).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The feasibility label qubit is  the 9th qubit counted from the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The labels of the states shown in the first 37 rows are 0, while  those in rows 38-64 are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' A label of 0 represents an infeasible solution, while a label of 1 indicates  a feasible solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Considering that each row has 4 states, there are 148 infeasible states and 108  feasible states, which matches the number calculated in the previous paragraph and validates the  QFL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Conclusion  We have developed a QFL algorithm that provides a feasibility label for every possible solution  to the vertex coloring problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The output of this algorithm can be directly input into the VQS  algorithm, which can then find all feasible solutions in polynomial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The next step is to prove  that the VQS is efficient for any number of qubits, which would be a significant milestone, as it  would mean that NP-complete problems can be efficiently solved using quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content='  Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' Measurement results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
+page_content=' The strings in single quotes represent measured quantum states,  and the number after a colon is the number of times that the state shown in the same cell is  obtained from measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
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+page_content='Acknowledgements  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
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+page_content=' Arora and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
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+page_content=' Madigan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtAzT4oBgHgl3EQfn_1r/content/2301.01589v1.pdf'}
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+Mimicking non-ideal instrument behavior for hologram processing
+using neural style translation
+John S. Schreck1, Matthew Hayman2, Gabrielle Gantos1, Aaron Bansemer3, and David John Gagne1
+1National Center for Atmospheric Research (NCAR), Computational and Information Systems Laboratory, Boulder, CO,
+USA
+2National Center for Atmospheric Research (NCAR), Earth Observing Laboratory, Boulder, CO, USA
+3National Center for Atmospheric Research (NCAR), Mesoscale and Microscale Meteorology Laboratory, Boulder, CO, USA
+Correspondence: schreck@ucar.edu; mhayman@ucar.edu
+Abstract.
+Holographic cloud probes provide unprecedented information on cloud particle density, size and position. Each laser shot
+captures particles within a large volume, where images can be computationally refocused to determine particle size and shape.
+However, processing these holograms, either with standard methods or with machine learning (ML) models, requires consider-
+able computational resources, time and occasional human intervention. Current ML models are trained on simulated holograms
+obtained from the physical model of the probe since real holograms have no absolute truth labels. Any attempt to use another
+processing method to produce labels would be subject to errors that the ML model would subsequently inherit. Trained models
+perform well on real holograms only when image corruption is performed on the simulated images during training, thereby
+mimicking non-ideal conditions in the actual probe (Schreck et. al, 2022). Optimizing image corruption requires a cumbersome
+manual labeling effort.
+Here we demonstrate the feasibility of applying the neural style translation approach (Gatys et. al, 2016) to the simulated
+holograms. With a standard, pre-trained convolutional neural network (VGG-19), the simulated holograms are “stylized” to
+resemble the real ones obtained from the probe, while at the same time preserving the simulated image “content” (e.g. the
+particle locations and sizes). Several image similarity metrics concur that the stylized images are more like real holograms than
+the synthetic ones. With an ML model trained to predict particle locations and shapes on the stylized data sets, we observed
+comparable performance on both simulated and real holograms, obviating the need to perform manual labeling. The approach
+described here is not specific to hologram images and could be applied in other domains as a means for capturing noise and
+imperfections in observational instruments to make simulated data more like real world observations.
+1
+Introduction
+Holographic imaging enables 3D reconstruction of particle positions and sizes within a large sample volume. Through this tech-
+nique, ensembles of thousands of particles can be captured and characterized in a single instantaneous volume. This technique
+has been successfully applied to characterizing the microphysical properties of clouds with airborne in situ instruments such
+1
+arXiv:2301.02757v1  [physics.ins-det]  7 Jan 2023
+
+as the HOLODEC probe (Fugal et al., 2004; Spuler and Fugal, 2011) which is operated by NCAR and has been successfully
+deployed on several field campaigns.
+The HOLODEC instrument captures inline holograms of a 15 cm3 sample volume. A laser operating at 355 nm transmits
+an expanded beam between two probe arms. After the light passes through the sample volume, the light is imaged onto a CCD
+with an effective pixel resolution of 3 µm. The laser fires at a 3 Hz repetition rate, allowing for point-like captures of particle
+size distributions as the aircraft passes through a cloud with spatial separations of those points dictated by aircraft speed.
+While holographic imaging captures a significant amount of information in a single image, processing these holograms tends
+to be computationally intensive, often requiring human intervention to tune the processing approach. As a result, there has been
+substantial research into the use of machine learning to accelerate or improve processing of holographic particle data (Shimob-
+aba et al., 2019; Shao et al., 2020; Zhang et al., 2022). However, many of these efforts are focused on images with relatively
+small depth components which is distinct from HOLODEC requirements. Another key distinction is in the idyllic nature of
+the holograms that are processed. Processing solutions that work well on controlled laboratory or simulated holograms tend
+to perform suboptimally on operationally captured data from HOLODEC. One cannot expect the same consistent clarity and
+quality from an instrument on the outside of an aircraft, where a variety of operating and environmental conditions can cause
+the captured images to contain a significant amount of imperfections. The challenge of processing HOLODEC data necessitates
+tuning filters to minimize both false positives and missing true particles in holograms with non-negligible imperfections.
+These imperfections are difficult to model in simulation because they often depend on combinations of component imperfec-
+tions within the instrument. For example, small amounts of reflections from transmission optics can create modulation patterns
+and focus spots in reconstruction, laser modes deviate in amplitude and phase from a true Gaussian, vignetting can cause
+non-uniform field response and depend on beam pointing that varies with vibrations and temperature. Ultimately, developing
+processing techniques that can account for these and other (sometimes unknown) effects is essential to delivering high quality
+scientific observations.
+In a previous work, we outlined a processing pipeline concept to accelerate and improve processing of HOLODEC images
+through a combination of hardware acceleration and machine learning (Schreck et al., 2022). As is common in atmospheric
+sensing, actual hologram data lacked truth labels. In order to achieve an improvement in processing capability over the exist-
+ing solution, we developed an approach for corrupting simulated holograms in order to emulate the imperfections of actual
+HOLODEC data. A machine learning solution to process actual HOLODEC data was trained using simulated holograms
+(where labels are are known to near absolute accuracy) by adding random noise, applying blurring and skewing, and adjusting
+the image contrast. While the simulated training data did not visually appear similar to actual HOLODEC data, these image
+corruptions were sufficient to train a model that still out performed the particle recognition approaches used in the standard
+processing solution (Schreck et al., 2022). However, to succeed, each of these transforms needed to be tuned in hyperparameter
+optimization, which required manual labeling of thousands of reconstructed holograms. This is problematic because manual
+labeling requires considerable person-hours from well trained personnel, and (as we discovered in the process) human deter-
+mined labels have some non-zero but unquantified error. What is more, during the optimization, we simply guessed which
+types of corruptions to perform on the images. Finally, there may be many sensor situations where these simple corruption
+2
+
+Figure 1. (a) NCAR’s Mesa Lab taken to be the content image (Commons, 2007). (b) An image obtained from the James Webb telescope
+showing star formation is the style image (NASA, 2022). (c) The synthesized image is the product of the neural style transfer method that
+leverages the content and style images.
+operations do not suffice in developing a higher performance processing solution. In order to generalize to the broader domain
+of sensor signal processing, a solution should be able to address the specialized behavior of the sensor type and this mandates
+a more general ability to represent the difficult to model, non-ideal behaviors of the instrument.
+Here we introduce a new approach to bridging the gap between optimizing signal processing/machine learning solutions
+and the realities of operational sensor data. We show that the original neural style transfer method introduced by (Gatys et al.,
+2016), which utilizes a convolutional neural network (CNN) to perform image augmentation, can be used to produce realistic-
+looking, labeled hologram examples without the need for manually labeling as a means for tuning noise parameters. These
+stylized synethetic training data result in processing performance similar to the previous method, but without the requirement
+of human labeling and therefore represent a significant step toward producing an operation solution for processing HOLODEC
+data.
+2
+Methodology
+2.1
+Neural Style Transfer
+The original style transfer method leverages CNN representations of a “content” image and a “style” image to produce a third
+image where the style features are automatically synthesized with content features. For example, Figure 1 illustrates how the
+method is used to create an image of NCAR’s Mesa Lab shown in (a), stylized using a recent image from the James Webb
+telescope shown in (b) to create a synthesized image shown in (c). The synthesized image still clearly still shows the Mesa
+3
+
+(a)
+Content image
+(b)
+Style image
+(c)
+Generated imagePooling layer
+Layer output
+Backpropagation
+Input image
+Content loss
+Style Loss
+p
+Content image
+x
+Image parameters
+a
+Style image
+conv5
+conv4
+conv3
+conv2
+conv1
+p=
+a=
+conv5
+conv4
+conv3
+conv2
+conv1
+conv5
+conv4
+conv3
+conv2
+conv1
+x=
+Symbol Legend
+VGG-19
+Figure 2. Schematic illustration of the neural style transfer approach by Gatys et al. (2016) using the VGG-19 neural model, where the
+model’s 18 convolution layers are stacked into five blocks (orange squares). In this example, the output from the first convolution layer in
+each of the five blocks were used to compute style and content losses. The blocks represent the (same) model for inputs p, x, and a.
+Lab, however, overall the image colors, contrast, and line smoothnesses have been transformed relative to the original to also
+resemble the style image.
+The synthetic holograms come with labels and are free of noise, so they are identified as possessing the “content” we wish
+to preserve — that is the information about the particles within. The HOLODEC examples all contain varying degrees of
+noise/imperfections that we wish to effectively impart onto the synthetic examples, hence they are identified as the “style”
+examples. In this way, synthetic images are transformed to create a realistic training dataset with known labels.
+Figure 2 schematically illustrates the style transfer method using the VGG-19 network as the chosen CNN (Simonyan and
+Zisserman, 2015). The VGG model contains 19 layers and was originally trained to perform object recognition and localisation
+on the ImageNet dataset (Russakovsky et al., 2014). The model’s 18 convolution layers utilize the ReLU activation and are
+sequentially connected together into five blocks separated by 2D average Pooling layers, as is illustrated schematically in Fig-
+ure 2. The first two blocks contain two convolution layers, while the remaining three contain four. The position of a convolution
+4
+
+layer in VGG is characterized by integer l. All convolution layers in VGG-19 use a kernel size of 3x3 and stride and padding
+sizes both set to unity. Three fully-connected layers complete VGG-19. Thus, each column in Figure 2 is a single (the same)
+VGG-19 model (with the fully-connected layers discarded). Currently, a single color channel is used for the hologram images,
+so the very first convolution layer was replaced with one having the color channel equal to one.
+The style transfer approach begins by sampling random values for x, and freezing the weights of the all of the layers in
+the CNN. For the input content image p and generated image x, the output from convolution layer l are the feature matrix
+representations P l and F l, where P l
+ij and F l
+ij are the responses to the activation of the ith filter at position j in convolution
+layer at position l. A central question with style-transfer is the choice of loss function, and which layer outputs should be used
+to compute the loss. The choice will determine to what extent we need the full VGG-19 model, and will greatly affect run-time
+performance. We follow the original approach introduced by Gatys et al. (2016), but other effective loss functions have been
+developed (Li and Wand, 2016b; Li et al., 2017; Wilmot et al., 2017). A new image that matches the responses of the content
+image p may be found by updating the values of x via gradient descent using a squared-loss of the two feature representations
+Lcontent(p,x,l) = 1
+2
+�
+i,j
+(F l
+ij − P l
+ij)2
+(1)
+for convolution layer at position l, as is illustrated in Figure 2 by a blue hexagon. The “style” representation of an input image x
+is defined using a feature space for capturing texture information. This is taken to be the correlation between the outputs across
+the spatial domain for layer l, so that the expected correlation for style feature maps i and j are given by the Gram matrix
+Gl
+ij =
+�
+k
+Gl
+ikGl
+jk.
+(2)
+Now the image x may be updated via gradient descent so that it’s style features Gl match the style features Al of image a at
+layer l. The contribution coming from layer l is taken to be
+El =
+1
+4N 2
+l M 2
+l
+�
+i,j
+(Gl
+ij − Al
+ij)2
+(3)
+and the total style loss, illustrated by green circles in Figure 2, is
+Lstyle(a,x) =
+�
+l
+wlEl
+(4)
+where wl represents a weight for each contribution coming from layer l. Although not used in Gatys et al. (2016), the predicted
+images may sometimes contain higher-frequency noise which we would prefer to smooth out. This can be accomplished via a
+total variation term given by
+LT V (x) =
+�
+i,j
+|xi,j − xi+1,j| + |xi,j − xi,j+1|
+(5)
+for pixels i and j. For content image p, style image a, and generated image x, the total style transfer loss is then
+Ltotal(p,a,x) = αLcontent(p,x) + βLstyle(a,x) + γLT V (x)
+(6)
+5
+
+where the parameters α, β, and γ are constant weights, and only the output from a single layer is used to compute the content
+loss (see Figure 2). In general, if more than one layer output is selected for computing the content loss, the terms would be
+summed in Equation 6. We set α = γ = 1 and focus on the relationship between β and the predicted images below.
+In addition to β, the choice of which layer(s) to use in content and style representations are left as hyperparameters. For the
+content feature layer choice, the output from convolution layers farther from the input image should capture global features
+relative to those blocks nearer to the input image, which should capture more details of the input. Similar arguments apply to the
+style features, with the more distant convolution layers capturing more global styles and vice versa. Gatys et al. (2016) selected
+the output from the second convolution layer from the fourth block to compute the content loss (e.g. the 12th convolution layer
+in VGG-19), and the output from the first convolution layer in all five blocks to compute the style loss components (convolution
+layers at positions 1, 3, 5, 9 and 13). These choices result in a large CNN that greatly affects training performance when used
+with our relatively large holograms. We found that a much smaller model was just as capable as that used by Gatys et al. (2016)
+and was thus much faster to use. Specifically, we selected the fourth convolution layer outputs in VGG-19 for the content
+contributions and the first five convolution layer outputs in VGG-19 for the style contributions (with all weights equal to unity
+in Eq. 4), therefore only requiring layers in three of the five blocks in VGG-19. There may be a more optimal, faster model that
+achieves similar results, but we did not explore other combinations any further.
+Figure 2 shows that content image p and style image x are passed through the CNN and the outputs from selected feature
+layers are stored in each case. Initially white noise image x is then passed through the CNN and the outputs from the content
+and feature layers are used to compute the total loss according to Equation 6. The image pixels in x can then be updated via
+back-propagation and with successive iterations, x can learn to match the p’s content and a’s style features.
+2.2
+Metrics for comparing images
+The choice of α and β will determine how similar the generated image is to the content and style images. There are numerous
+metrics available for estimating the similarity between two images. For images x0 and x1, we select the structural similarity
+index measure (SSIM) (Wang et al., 2004) (Equation A1), a standard pixel-based metric that extracts information on structure,
+contrast, and luminance of an image that serves as the basis for comparison of two images. Additionally, a similarity metric
+based on the AlexNet convolutional neural network model (Krizhevsky et al., 2017), referred to as dAlex ≡ dAlex(x0,x1), is used
+to capture the distance between two images, x0 and x1, in deep feature space (Zhang et al., 2018). The SSIM ranges from -1
+to 1, with larger values indicating more similarity between images. The AlexNet metric ranges from 0 to 1 with smaller value
+indicating greater similarity; we use one minus the value so that a higher value indicates greater similarity in both metrics. For
+images x0 and x1, both metrics are computed by averaging over local differences between “patches” of size 11×11 for SSIM,
+and size 3 × 3 (the filter size used) for the AlexNet metric.
+2.3
+Image segmentation with a U-Net
+Figure 3 shows a neural segmentation model, for example a “U-Net” (Ronneberger et al., 2015; Shelhamer et al., 2017), that is
+used in the current study to process holograms and has been used in other studies (Shao et al., 2020). The model contains two
+6
+
+(ii) Convolutional neural 
+network 
+Encoder Decoder
+(iv) Mask prediction
+(iii) Noise added during training and 
+optimized with manual labeling
+No particles in 
+focus
+Particle is in 
+focus
+(i)
+(ii)
+HOLODEC 
+holograms
+(i) Input synthetic 
+holograms
+(i) Synthetic 
+holograms
+(ii) Stylize
+(b) Training with style transfer
+(c) Operation
+(a) Training and optimization with noise
+Figure 3. (a) U-Net segmentation model. (i) Example synthetic holograms (solid red lines) are passed into the encoder-head of the U-Net
+shown in (ii). (iii) During training, noise may optionally be added to the holograms. (iv) Predicted masks for the input examples, with a dark
+circle enclosing the in-focus particle. (b) (i) Synthetic images are (ii) stylized using HOLODEC holograms (dashed blue lines), which are
+then (iii) used to train a U-Net model. (c) In operation, HOLODEC holograms are passed through the U-Net trained on style-transferred
+images to obtain a mask prediction. In the example no particles were identified.
+CNN “heads” as illustrated in the figure. An encoder head takes in the input image and compresses it into a smaller dimensional
+representation of a fixed size. The decoder head up-samples the latent representation of the image and outputs a prediction the
+same size as the input image. Here, we use a sigmoid activation function on the decoder’s last layer that rescales the predicted
+outputs for all the pixels to lie within (0, 1). In other words, the model is trained to predict a probability that an in-focus particle
+occupies a pixel. If the probability exceeds a certain value, often 0.5 for binary tasks, the pixel is labeled 1 and 0 otherwise. The
+convolutional layers in the U-Net encoder head may also leverage pre-trained weights from the layers in other convolutional
+models, such as ResNet (He et al., 2016) trained on the ImageNet dataset, which frequently helps to speed training and also
+boost overall performance.
+7
+
+O2.4
+Datasets
+There are essentially four datasets we use in this work. The first two originate from full size images captured by HOLODEC or
+simulations of HOLODEC. These initial images consist of 4872-by-3248 pixel images which are processed and segmented to
+create images that can be readily processed by the U-Net. These images are also sampled to ensure a balance between in-focus
+particles and out-of-focus particles in the U-Net training process. This training data from the two sources are then used by
+style transfer to generate a realistic generated dataset (with content from the synthetic images and style from the HOLODEC
+images) which is used to train the U-Net. Here, style transfer is performed using reconstructed synthetic and HOLODEC
+images at different positions along z (see below). Finally a set of manually labeled HOLODEC holograms are used to evaluate
+the performance of the U-Net, and therefore the effectiveness of the style transfer solution.
+2.4.1
+HOLODEC and synthetic data
+A physical model of the instrument was used to generate 120 synthetic holograms which had optical settings identical to the
+HOLODEC instrument (Fugal et al., 2009). One hundred holograms were randomly selected for training, ten for validation,
+and the remaining ten for testing. Each hologram contained 500 spherically shaped particles randomly positioned along the x
+and y directions, positioned between minimum and maximum values of 14.072 and 158.928 millimeters along the z-direction,
+respectively. The diameter sizes were sampled with a gamma distribution.
+The HOLODEC data were selected from a set of several hundred holograms obtained from the RF07 subset from the Cloud
+Systems Evolution in the Trades (CSET) project that was originally obtained from June 1 to Aug 15, 2015 (Albrecht et al.,
+2019). They primarily contain liquid water particles of varying numbers and shape, were largely free of ice but contain other
+unidentified “artifacts” and image perturbations not captured by the physical model of the instrument. Overall, the known
+particle numbers range from none up to hundreds, with the average particle diameter less than 50 m.
+In accordance with the approach described in (Schreck et al., 2022), each hologram is initially processed by reconstructing
+1,000 planes along the z-axis (axis of laser propagation). This produces 1,000, 12 megapixel size images that must scanned
+for in-focus particles. These images are too large to be processed efficiently with a CNN. Therefore, following our previous
+approach (Schreck et al., 2022), each hologram reconstruction is broken up into 38-by-28 tiles each of 512 by 512 pixels. These
+tiles have an overlap stride of 128 pixels leaving 828 total tiles for each full image. This means that any one pixel will show up
+in more than 1 tile.
+For training using synthetic holograms, each was selected and then refocused along z for all of the 500 particles, from which
+a random grid tile containing the in-focus particle was selected and saved, resulting in 50,000 images (for details on refocusing,
+see Schreck et al. (2022)). Additionally, 25,000 tiles were selected where a particle was just out of focus by no more than 120
+micrometers. Finally, 25,000 tiles were randomly selected which did not contain in-focus particles. A comparable procedure
+was repeated to produce validation and test sets of tile images where half the images contained in-focus particles.
+For the RF07 subset used here, a hologram was drawn at random from several hundred, excluding 20 that were set aside previ-
+ously for manual examination (and described below). This raw data can be accessed at https://doi.org/10.5281/zenodo.6347222.
+8
+
+As the true particle positions are not known (and they are not needed as they will be used only for style transfer), the hologram
+was refocused at a random position between 14 and 160 millimeters along z, from which 10 tiles were randomly selected
+and saved from the refocused hologram. This process was repeated until about 100,000 image tiles were selected for training,
+10,000 for validation, and 10,000 for testing the mask prediction performance. Most of the tiles in each split did not have
+particles directly in-focus, but occasionally were pretty close.
+2.4.2
+Generated data sets with style transfer
+The labeled synthetic training splits are used as the content images while RF07 holograms, which come without labels, are the
+style images. Training, validation, and testing splits for 20 different values of β sampled evenly from approximately 10−5 to
+1015, with α = 1 and γ = 1 were created to find the optimal value of β. Once obtained, a data set containing training, validation,
+and testing splits was created with β ≡ β0 = 109, which we identify below as an optimal choice, and α and γ both equal to
+one. This is created to compare against the model trained on synthetic training split only and the model described in Schreck
+et al. (2022) where optimized noise (based on hand labeled data) was added during training.
+For each selection of β including β0, a randomly selected content (synthetic) image tile was selected and paired off with a
+randomly selected style (HOLODEC) image tile. Then, both content and style images were used to generate a training sample
+with the style-transfer method, with the mask labels for the generated example are inherited from the content image. For the
+β0 data set, 100,000, 20,000, and 20,000 style-generated images were created for training, validation, and testing the ability
+of U-Net model’s to reproduce the mask labels. For the other 20 values of β, a total of 20,000 training, 2,000 validation, and
+2,000 testing samples were produced in each case for the same purpose. In the latter data sets, we also used the same random
+seed so that all the pairs selected in each split for the different β choices (excluding β0) were the same.
+2.4.3
+Manually labeled HOLODEC data set
+The 20 manually labeled HOLODEC examples resulted in 2,356 total images of size 512 pixels by 512 pixels, where prospec-
+tive in-focus particles were positioned at the image center. The examples were manually labeled as containing an in-focus
+particle or not, along with an average reviewer confidence score ranging from zero to five (higher means more confident). The
+binary output of the U-Net is simply taken to be unity if any predicted value is larger than 0.5 and 0 otherwise. Many of the
+examples were subjective, and the total of number of particles we could identify was limited by previous modeling attempts,
+hence there could be more particles in these holograms still undiscovered.
+The total number of labeled examples was split into validation and testing data sets. The validation set contained 1,204 total
+images with 367 containing at least one in-focus particle. The testing set contained 1,154 images, where 874 contained at least
+one in-focus particle. The validation set was used to help guide the optimization of noise in our previous study that we compare
+with here (Schreck et al., 2022), while the test set represents the hold-out set of manually labeled examples.
+9
+
+3
+Results
+3.1
+Style transfer with holograms
+Figure 4 shows a synthetic hologram (content image) that contains a single in-focus particle, and different HOLODEC holo-
+gram examples (style images) that were used to generate the style transferred image, for α = 1, and β = 109. The HOLODEC
+example shown in Figure 4(a) was selected because it contains an in-focus particle. Note that while there appears to be a trans-
+fer of fine scale texture from the style images, particles from these style images are not transferred generated images. Thus
+there is no need to locate empty holograms for the style transfer images or perform careful QC in selecting the appropriate
+images for this process.
+The images in Figure 4(b-c) are typical examples where a U-Net trained only on synthetic images (left column) performs
+poorly on actual HOLODEC images due to a high false-positive rate. The interference fringes seen in the HOLODEC image
+in Figure 4(b) often produce patterns that resemble in-focus particles, while (c) shows an example where many small artefacts
+are present and have a round shape. In all three examples the in-focus particle in the synthetic example is still present in the
+generated examples, and the overall generated image contrast is much more comparable to the HOLODEC images.
+Figure 5(a) shows the training curve for the generated image shown in Figure 4(a). Figure 5(b) illustrates the progression of
+the predicted image at selected epochs. Starting initially with a random noise image for x, a very steep drop in the style loss
+is observed over the first few epochs. The content and TV losses also drop quickly after the first few epochs, then the curves
+mostly flatten after about 10 epochs, and the content contribution comes to dominate the total loss thereafter. In these examples,
+the choice of α/β determined the similarity between the generated image and content and style images used to create it. Here,
+according to the AlexNet metric, all the generated images are more similar to the HOLODEC images than to the synthetic
+image.
+Figure 6(a) and (b) show the SSIM and dAlex metrics versus β with α = 1. These metrics are computed for test data
+of HOLODEC-synthetic, HOLODEC-generated, and synthetic-generated pairs. Note that these metrics are constant for the
+HOLODEC-synthetic pairs because these images are unaffected by β.
+In both figures, four regions of interest are approximately identified with vertical lines. Region A corresponds with values of
+approximately β ≤ 10 where the content loss dominates the total loss. Accordingly, both metrics show that the style-transferred
+images strongly resemble the original synthetic images (see also Figure 4(a,b)). The main difference between the two metrics
+is that dAlex scores HOLODEC-generated pairs higher relative to HOLODEC-synthetic pairs than does SSIM. However, the
+curves are all qualitatively similar between the two metrics for all values of β.
+Examples of images from each region in Figure 6 are shown in Figure 7 using the content and style images from Figure 4(a)
+with α = 1 and varying β.
+Region B is identified approximately when 102 < β ≤ 107, which is when both metrics for synthetic-generated and HOLODEC-
+generated pairs drop significantly. The examples in Figure 4(c,d) show that the generated image starts to look significantly
+different compared with the original synthetic image as the style term begins to contribute. Although these examples may give
+the impression that the generated images still look like the synthetic images, they are in reality a mixed representation which
+10
+
+0
+740
+1480
+Y ( m)
+(a)
+Synthethic (content)
+HOLODEC (style)
+Synthesized
+0
+740
+1480
+Y ( m)
+(b)
+0
+740
+1480
+X ( m)
+0
+740
+1480
+Y ( m)
+(c)
+0
+740
+1480
+X ( m)
+0
+740
+1480
+X ( m)
+50
+100
+150
+Pixel value
+50
+55
+60
+65
+70
+75
+80
+85
+90
+Pixel value
+Figure 4. In (a-c), the same synthetic hologram is paired with a different HOLODEC hologram to produce a generated hologram using
+style transfer. The HOLODEC examples illustrate (a) an in-focus particle, (b) interference patterns and successive dark spots, and (c) small
+artefacts. U-Nets are observed to perform poorly on (b) and (c) when trained with synthetic images only. The content weight α = 1, while
+the style weight β = β0 = 109.
+11
+
+Figure 5. (a) The style, content, TV, and total loss are plotted versus the number of training epochs (the number of times the image was
+updated). (b) The initial random image and selected images at different training updates are shown. The pixel values shown ranged from 50
+to 90, as in Figure 4 for the HOLODEC and generated images.
+10
+4
+10
+1
+102
+105
+108
+1011
+1014
+Style weight,  
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+SSIM
+(a)
+A
+B
+C
+D
+H-Gen
+Syn-Gen
+H-Syn
+10
+4
+10
+1
+102
+105
+108
+1011
+1014
+Style weight,  
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+Similarity, dAlex
+(b)
+H-Gen
+Syn-Gen
+H-Syn
+Figure 6. (a) The average SSIM and (b) average dAlex scores computed using the test data sets, is shown versus the style weight β in
+Equation 6 with α and γ = 1.
+does not on average resemble the synthetic or HOLODEC hologram (for example, the average contrast difference for examples
+like those in Figure 4(c,d) is less than that for the synthetic images but still greater than what is observed in the HOLODEC
+examples). Around β = 104 both metrics reach a minimum before beginning to rise with increasing β.
+12
+
+(b)
+Random noise
+Update 10
+Update 50
+1480
+(ur)
+740
+Y
+0.
+Update 100
+Update 200
+Update 500
+1480
+(ur)
+740
+Y
+0
+740
+1480 0
+740
+1480 0
+740
+1480
+X (μm)
+X (μum)
+X (μm)(a)
+Style
+104
+Content
+TV
+103
+Total
+102
+101
+Loss 
+100
+10-1
+10-2
+10-3
+100
+101
+102
+Image updatesRegion C is identified approximately when 107 < β ≤ 1012. Both SSIM and dAlex continue to increase for synthetic-
+generated pairs and peak near β = 109, though the peak is more pronounced for SSIM. Both scores for HOLODEC-generated
+pairs also increase and reach plateau values, which are higher compared against that in regions A and B, indicating the images
+have been stylized to some degree while the the Synthetic-Generated scores indicate the image still retains content features.
+Clearly, the neural dAlex determines the generated holograms are more similar to HOLODEC holograms compared to SSIM,
+but overall both metrics indicate the predicted images have struck a particular balance between retained content and learned
+style features. Figure 4(e-g) show that the in-focus particle remains in place and roughly the same shape, while, for example,
+the image contrast is adjusted to be more like that of the HOLODEC images.
+Finally, region D is identified by β > 1012, which is when the style term dominates the loss. The dAlex score for style-
+generated pairs remains high while that for content-generated pairs converges to the same value for content-style pairs, which
+is to say the metric cannot differentiate the generated image from style images. However, Figure 4(h,i) show that the in-focus
+particle and other content from the synthetic image has largely been removed. Furthermore, what appear to be fake particles
+are now being introduced into the generated images. In other words, the original image was over-stylized.
+Region A represents the instances where output has very little style imposed on the content images and therefore most closely
+resembles idealized synthetic data, which we previously established in Schreck et al. (2022) makes a poor U-Net training data
+set when the objective is to process actual HOLODEC data. Images from region D represent instances where content is largely
+absent from output. Clearly we cannot expect to train a U-Net to recognize particles that are not in the input images. Without
+having to resort to manual labeled HOLODEC images, the obvious choice would be to select the value of β around β0 = 109
+in region C, when both metrics for synthetic-generated pairs peak, and the generated examples appear to be more similar to
+HOLODEC examples according to dAlex.
+The performance of the U-Net trained on the generated images is then expected to peak at mask reproduction when β is
+around β0 = 109, as the predicted images are the most similar to the content images in region C. Also, because the generated
+images are similar to the HOLODEC images in region C relative to A and B, we might expect U-Net models trained on them
+to perform the best on HOLODEC images. If this is the case, one can select the value of β by finding the value where the
+generated images are more similar to the HOLODEC images according to dAlex and SSIM. This selection should result in the
+best generated dataset for training the U-Net.
+3.2
+Performance on synthetic and HOLODEC images
+In order to test the hypothesis that the optimal β can be found analytically from dAlex and SSIM, the generated images (and
+their corresponding labels) at the different values of β are used to train U-Net models, as is shown schematically in Figure 3(b).
+The model architecture and training hyperparameters were all identical for each different value of β (see Appendix A1 for more
+details). Post-training, each U-Net model is run in operation mode on the test set of generated images and the manually labeled
+test set of actual HOLODEC tiles. Figure 8(a) shows several binary performance metrics for the U-Net at reproducing the mask
+labels accompanying the generated images, while Figure 8(b) shows similar quantities for particle detection in true HOLODEC
+tiles. Since none of the manually labeled tile examples were used in the creation of the style-generated images, all 2,356 were
+13
+
+0
+740
+1480
+Y ( m)
+A
+(a)
+10
+3.95
+A
+(b)
+101.32
+B
+(c)
+103.42
+0
+740
+1480
+Y ( m)
+B
+(d)
+104.47
+C
+(e)
+107.63
+C
+(f)
+108.68
+0
+740
+1480
+X ( m)
+0
+740
+1480
+Y ( m)
+C
+(g)
+109.74
+0
+740
+1480
+X ( m)
+D
+(h)
+1011.84
+0
+740
+1480
+X ( m)
+D
+(i)
+1013.95
+50
+100
+150
+(a-c) Pixel value
+40
+60
+80
+100
+(d-f) Pixel value
+50
+60
+70
+80
+90
+(g-i) Pixel value
+Figure 7. (a-i) Examples of generated holograms using the content and style images from Figure 4(a) and different style weights. The region
+is listed in the bottom right.
+used in the figure. The metrics are the the F1 score, the area-under-the-ROC-curve (AUC), the probability of detection (POD),
+and the false alarm rate (FAR).
+14
+
+Syn 0
+10
+4
+10
+1
+102
+105
+108
+1011
+1014
+Style weight, 
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Mask metric
+(a)
+Synthetic holograms
+F1
+AUC
+POD
+FAR
+Syn 0
+10
+4
+10
+1
+102
+105
+108
+1011
+1014
+Style weight, 
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+In-focus metric
+(b)
+Labeled HOLODEC holograms
+Figure 8. (a) Several binary metrics for the U-Net’s mask prediction performance are plotted versus style weight β for synthetic or style-
+generated images. (b) The same binary metrics are shown for the particle detection task in manually labeled HOLODEC images versus β.
+As the models did not rely on the manual examples during training, all 2,356 examples were used in the figure. The x-axis labels "Syn" and
+"0" refer to U-Nets trained on synthetic images and generated images with β = 0, respectively.
+Figure 8(a) shows overall gradually diminishing training performance with increased β. This is expected as the pristine
+simulations are increasingly stylized with non-ideal instrument noise and artifacts, making particle detection more difficult. In
+region A, when the generated holograms resemble the synthetic examples, the mask prediction performance is very high and
+generally flat. But then in region B, when the generated holograms are modified more by the style image, there is a clear drop
+in the AUC and POD while the FAR rises. As β continues to grow and the generated images start to resemble both content
+(synthetic) and style (HOLODEC) examples, the performance rises and reaches a peak around β = β0 before dropping, first
+slowly and then very suddenly once in region D. We expect this peak in region C to be less than that in A where the images
+lack the characteristic noise of the instrument. In region D, the AUC is about 0.5 which should be expected as the over-stylized
+images and the mask labels retain very little content from the actual particles. The trained U-Net outputs all, or too many 1s
+when trained on over-stylized images, so POD quickly approaches 1. We also computed diameters for the predicted masks and
+found no significant deviation from the mean true value for the test set, for all choices of β except those in region D.
+Overall, Figure 8(b) shows that the relevant range of β may be selected according to our hypothesis. Relative to the U-Net
+trained only on synthetic images (labeled ‘Syn’ on the x-axis in Figure 8(b)), the performance on the HOLODEC examples
+in region A is signified by lower PODs and higher FARs, although the AUC values are all approximately around 0.6. Even
+the model trained on generated images with β = 0 showed a higher FAR relative to ’Syn’, indicating that the U-Net trained
+on idealized synthetic holograms tends to under predict particles, to the benefit of a low FAR. We should note here that
+the difference in performance between the synthetic trained and β = 0 cases indicates that setting β = 0 does not perfectly
+15
+
+Metric
+F1
+AUC
+POD
+FAR
+CSI
+Syn/Gen
+HOLO
+Syn/Gen
+HOLO
+Syn/Gen
+HOLO
+Syn/Gen
+HOLO
+Syn/Gen
+HOLO
+Synthetic
+0.977
+0.220
+0.992
+0.448
+0.979
+0.106
+0.025
+0.387
+0.956
+0.100
+Optimized noise
+0.968
+0.866
+0.985
+0.808
+0.967
+0.922
+0.031
+0.096
+0.937
+0.840
+Stylized
+0.966
+0.864
+0.988
+0.805
+0.968
+0.922
+0.035
+0.097
+0.935
+0.839
+Table 1. Several metrics are listed for each hyperparameter optimized model and were computed on the synthetic or generated (Syn/Gen),
+and HOLODEC (HOLO) test data sets, for mask and particle detection (binary) predictions, respectively. The validation set of labeled
+HOLODEC images was used to optimize the noise parameters in the Optimized noise model.
+reproduce the content image, resulting in slightly lower U-Net performance. However, neither case is relevant to the objective
+of this work since both cases fail to perform well on HOLODEC data. In region B, the particle detection performance generally
+rises even as the mask performance drops. However, the particle detection performance zig-zags quite significantly making a
+selection of β unreliable without manual labeled examples. In Region C, the detection performance reaches an approximate
+plateau, with high AUC values and consistently low FAR values. The performance seems to slightly decrease as the style
+weight approaches values in region D. In region D, the model is unable to train effectively, resulting in an output of all 1s. This
+results in a high POD and FAR with AUC at 0.5. Region C models also performed the best overall on the HOLODEC images
+relative to A, B and D. As expected, models in Region D do not perform at all on the manual examples, as signified by AUCs
+close to 0.5.
+3.3
+Comparison of optimized models
+Now that the style weight may be selected without relying on manual examples for optimization, we selected β = β0 = 109 and
+trained a model on the style-transferred images. Table 1 compares this model (‘Stylized’) versus a model trained on idealized
+synthetic images only (‘Synthetic’), and a model trained on synthetic images with noise introduced during training (‘Optimized
+noise’, where the optimization required manual labels). In all three cases we performed extensive hyperparameter optimization
+to find optimal architectures and training parameters using the mask performance on the validation set of generated images as
+the optimization metric. The model which leveraged noise during training additionally used the manually labeled validation
+HOLODEC examples to guide optimization of the added noise. The same binary metrics in Figure 8 are listed as well as the
+critical success index (CSI) for mask performance on test images (either synthetic or generated) and detection performance on
+test HOLODEC images. As the validation set of manually labeled examples were used to select the noise added which resulted
+in the Optimized noise, only the testing split is used in Table 1.
+Table 1 shows that the Stylized model and the Optimized noise model gave nearly the same performance on all metrics on
+both data sets even though the Stylized approach does not require manual labeling. The Optimized noise model outperforms
+Stylized by hundredths of a percent in each metric category. The Synthetic model showed better mask performance on synthetic
+data. However, the Synthetic demonstrated much worse performance at particle detection on real HOLODEC data compared
+16
+
+0
+20
+40
+60
+80
+0
+50
+100
+150
+True positive count
+d  = 17.9 m
+(a)(i)
+Synthetic
+0
+20
+40
+60
+80
+d  = 15.94 m
+(b)(i)
+Optimized noise
+0
+20
+40
+60
+80
+d  = 17.72 m
+(c)(i)
+Stylized
+0
+20
+40
+60
+80
+Predicted diamater ( m)
+0
+50
+100
+150
+False positive count
+d  = 22.84 m
+(a)(ii)
+0
+20
+40
+60
+80
+Predicted diamater ( m)
+d  = 14.8 m
+(b)(ii)
+0
+20
+40
+60
+80
+Predicted diamater ( m)
+d  = 22.6 m
+(c)(ii)
+Figure 9. (a-c) Histograms of computed particle diameters for the three models for the test set of manually labeled HOLODEC images. The
+dashed vertical line in each panel denotes the predicted average particular diameter ⟨d⟩.
+to the other two models. For example Table 1 shows that it had a much higher FAR, and the CSI was more than 70% lower
+compared to the other models.
+Lastly, as the Optimized noise and Stylized models produced nearly identical performance, the diameters for the predicted
+particles in the test set of manually labeled HOLODEC examples are computed to probe how the models describe the particles
+shapes. Figure 9 shows histograms of the predicted diameters for the three models. A particle diameter was computed for a
+predicted mask if at least one in-focus particle was present. From the predicted mask, any pixel labeled 1 was grouped with
+a neighboring pixel along x or y (but not along the diagonal in the plane) if it was also labeled 1. Therefore, breaks between
+groups defines multiple particles. The diameter of a group was taken to be it’s maximum extent in either x or y.
+Figure 9(a)(i) shows the Synthetic model expectantly under-predicting the true particles compared with the other two mod-
+els, and nearly as many false positive predictions were made relative to true positive, as seen in Figure 9(a)(ii). Meanwhile, the
+Optimized noise and Stylized models predicted similar distributions, but which differed from the Synthetic distribution. Both
+models predicted two peaks centered approximately around 15 and 22 µm, respectively, with the Stylized model having the
+more pronounced second peak. The Stylized model also predicts slightly greater numbers of larger particles relative to Opti-
+mized noise that are also sometimes observed by the Synthetic model. As a result, the overall average diameter is predicted
+to be about 17.8 µm versus 15.9 µm for Stylized and Optimized noise models, respectively. Figure 9(b)(ii) and (c)(ii) further
+show that both models mostly predict false diameters at 15 µm or smaller.
+17
+
+4
+Discussion
+Overall, the performances observed on both synthetic and HOLODEC images with the Optimized noise and the Style model
+were comparable, with both models performing well against models trained on the synthetic holograms only. This shows the
+advantage of using the style-transfer method for hologram image augmentation, which obviated the need to perform man-
+ual labeling as a means for optimizing noise added during the training of U-Net models. Additionally, we did not have to
+ambiguously choose the types of augmentations performed on the images, as was required for the Optimized model. Further-
+more, the application of style transfer to images that were used for training, rather than applied during operation, means that
+computational performance is also comparable with the Optimized noise model.
+The main computational bottleneck involving the style-transfer method occurs during the creation of style-transferred train-
+ing data sets. Many generated images needed to be created, one at a time, from a unique content-style image pair, that all
+required the iterative optimization described above. The image sizes used were also quite large. That further slowed training
+times and made it cumbersome to create large data sets used for exploring the different weight combinations. Improved ap-
+proaches could be focused on replacing the optimization with a trained neural network that predicts the stylized image. This
+can lead to several orders of magnitude speed-up over the original method when feed-forward architectures are utilized (Li
+and Wand, 2016a; Ulyanov et al., 2016; Huang and Belongie, 2017). These may include the StyleGAN method (Karras et al.,
+2021), which integrates the concepts of style transfer with a generative adversarial network (Goodfellow et al., 2020) and al-
+lows for fine-tune control over the relative strength of image features at different scales. For holograms, the StyleGAN could
+potentially be used to more selectively control specific content or style features on demand (for example, adding more features
+pertaining to artefacts).
+We also only worked with CSET holograms, in particular the RF07 data split, which had limited particle densities as well
+as sizes. How well a trained model, which leveraged style images from one field campaign, works on other campaigns remains
+to be tested in future work. Style-generated hologram data sets, which do not resemble operational inputs, plausibly would
+not be feasible as training targets, as we learned in Schreck et al. (2022). In such a scenario, the holograms from the latest
+field campaign could be used as style examples to generate the relevant training data set. The StyleGAN approach again could
+potentially take advantage of all the different types of holograms obtained from field campaigns (and all of the unique objects
+contained in each one), since it may be capable of leveraging multiple styles and content features arising in the data sets.
+Finally, as the primary objective of this study was to remove the manual-labeling step without sacrificing performance, we
+did not try to assess the physical realness of the generated holograms since during operation only real HOLODEC holograms
+are used. However, the question remains as to whether they are physically reasonable. One way to test this assumption could
+be to perform style-transfer using full-size holograms, which would be computationally challenging but not impossible with
+current GPUs, and then refocus them via wave-propagation along the focal plane. This would require faithfully reconstructing
+the real and imaginary components of the electric field, rather than just the intensity, as was done here since style-transfer was
+applied to tiles selected from the full-sized holograms after they had been propagated.
+18
+
+Parameter
+Synthetic
+Optimized noise
+Stylized (β0)
+10−5 < β ≤ 1015
+Learning rate
+3.86 × 10−4
+2.46 × 10−4
+8.5 × 10−5
+1 × 10−3
+Training loss
+Focal-Tyversky
+Focal-Tyversky
+IoU
+Focal-Tyversky
+Segmentation model
+U-Net
+LinkNet
+U-Net
+U-Net
+Encoder model
+EfficientNet-b0
+Xception
+DenseNet121
+VGG11
+Tile transform
+255
+Normalized
+Standard
+Symmetric
+L2 regularization
+0.0
+0.0
+2.1 × 10−7
+6.0 × 10−6
+Gaussian blur σ
+-
+2.125
+-
+-
+Gaussian noise
+-
+0.326
+-
+-
+Brightness factor
+-
+1.270
+-
+-
+Table A1. The values of the best hyperparameters in the optimization studies for the neural segmentation models for the three species. The
+batch size was fixed at 16.
+5
+Conclusions
+In summary, we have shown that the style-transfer algorithm is an effective approach for translating synthetic holograms,
+which were created using an idealized physical model of the instrument, into holograms that resemble those actually observed
+by the instrument. In principle, the application of style-transfer to perform the image augmentations should not be limited to
+holograms since non-ideal instrument behavior is a problem across many domains. When the synthetic holograms are used in
+machine learning models to predict masks around in-focus particles as was done here, noise had to be injected onto the images
+during training so that the model performed well on both synthetic and raw data. However, the choice of noise transformations
+and parameters can only be selected after expensive manual labeling of raw images and hyperparameter optimization. The style-
+transfer approach, which transformed the clean hologram data set into one more resembling observed holograms, delivered the
+same mask prediction performance without the cumbersome requirement of manual labeling. Furthermore, both models had
+comparable performance on a small set of manually labeled HOLODEC examples, and predicted similar distributions for the
+particle diameters.
+A1
+Model and training hyperparameters
+Table A1 lists the best parameters found for the segmentation models investigated. See reference Schreck et al. (2022) for
+more details on the hyperparameter optimization of segmentation models with hologram data sets. The segmentation models
+found to be optimal in Table A1 were the U-Net (Ronneberger et al., 2015) and LinkNet (Chaurasia and Culurciello, 2017),
+while the pre-trained encoder model weights considered were DenseNet-121 (Huang et al., 2017), Xception (Chollet, 2017),
+EfficientNet-b0 (Tan and Le, 2019), and VGG-11 (Simonyan and Zisserman, 2014). See the package segmentation-models-
+pytorch located at https://github.com/qubvel/segmentation_models.pytorch for more details on the segmentation and encoder
+models. The optimal training losses found were the intersection over union (IOU) and Focal-Tyversky losses (Lin et al.,
+19
+
+2017; Salehi et al., 2017). For additional definitions of each loss function, see the Holodec-ML software package located at
+https://github.com/NCAR/holodec-ml. We also utilized pre-trained weights obtained from the ImageNet data set in all trained
+models. The tile transforms were applied to a tile just before being passed into a segmentation model include dividing all pixels
+by 255, re-scaling all pixels to lie between 0 and 1 (Normalized) or -1 and 1 (Symmetric), or re-scaling each tile by subtracting
+the pixel mean and dividing by the square of the variance (Standard).
+A2
+Structural similarity index metric (SSIM)
+The global structure similarity definition for comparing images x and y is given by
+SSIM(x,y) =
+(2µxµy + C1)(2σxy + C2)
+(µ2x + µ2y + C1)(σ2x + σ2y + C2)
+(A1)
+where C1 and C2 are fixed constants, and µ and σ are defined as
+µx = 1
+N
+N
+�
+i=1
+xi
+σ2
+x =
+1
+N − 1
+N
+�
+i=1
+(xi − µx)2
+σxy =
+1
+N − 1
+N
+�
+i=1
+(xi − µx)(yi − µy).
+(A2)
+In practice, we use the mean structural similarity index which averages over an 11 by 11 circular-symmetric Gaussian Weight-
+ing function (Wang et al., 2004).
+A3
+Data sets
+The HOLODEC and synthetic data sets can be accessed at https://doi.org/10.5281/zenodo.6347222. The labeled HOLODEC
+examples were a subset of the RF07 data set (validation set ID 0-9, testing set ID 10-19), while the synthetic holograms were
+generated with simulations. All holograms in the RF07 data set were used for creating the style-transferred data sets. See
+Section 2.4 for more details. The style generated data sets can be built using the Holodec-ML software package located at
+https://github.com/NCAR/holodec-ml.
+Acknowledgements. This material is based upon work supported by the National Center for Atmospheric Research, which is a major facility
+sponsored by the National Science Foundation under Cooperative Agreement No. 1852977. We would like to acknowledge high-performance
+computing support from Cheyenne and Casper Computational and Information Systems Laboratory, CISL (2020) provided by NCAR’s
+Computational and Information Systems Laboratory, sponsored by the National Science Foundation. The neural networks described here
+and simulation code used to train and test the models are archived at https://github.com/NCAR/holodec-ml. All HOLODEC and synthetic
+hologram data sets created for this study are available at https://doi.org/10.5281/zenodo.6347222
+20
+
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+
diff --git a/fdE0T4oBgHgl3EQf6AIY/content/tmp_files/load_file.txt b/fdE0T4oBgHgl3EQf6AIY/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf,len=811
+page_content='Mimicking non-ideal instrument behavior for hologram processing  using neural style translation  John S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Schreck1, Matthew Hayman2, Gabrielle Gantos1, Aaron Bansemer3, and David John Gagne1  1National Center for Atmospheric Research (NCAR), Computational and Information Systems Laboratory, Boulder, CO,  USA  2National Center for Atmospheric Research (NCAR), Earth Observing Laboratory, Boulder, CO, USA  3National Center for Atmospheric Research (NCAR), Mesoscale and Microscale Meteorology Laboratory, Boulder, CO, USA  Correspondence: schreck@ucar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' mhayman@ucar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='edu  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Holographic cloud probes provide unprecedented information on cloud particle density, size and position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Each laser shot  captures particles within a large volume, where images can be computationally refocused to determine particle size and shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  However, processing these holograms, either with standard methods or with machine learning (ML) models, requires consider-  able computational resources, time and occasional human intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Current ML models are trained on simulated holograms  obtained from the physical model of the probe since real holograms have no absolute truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Any attempt to use another  processing method to produce labels would be subject to errors that the ML model would subsequently inherit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Trained models  perform well on real holograms only when image corruption is performed on the simulated images during training, thereby  mimicking non-ideal conditions in the actual probe (Schreck et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' al, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Optimizing image corruption requires a cumbersome  manual labeling effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Here we demonstrate the feasibility of applying the neural style translation approach (Gatys et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' al, 2016) to the simulated  holograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' With a standard, pre-trained convolutional neural network (VGG-19), the simulated holograms are “stylized” to  resemble the real ones obtained from the probe, while at the same time preserving the simulated image “content” (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' the  particle locations and sizes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Several image similarity metrics concur that the stylized images are more like real holograms than  the synthetic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' With an ML model trained to predict particle locations and shapes on the stylized data sets, we observed  comparable performance on both simulated and real holograms, obviating the need to perform manual labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The approach  described here is not specific to hologram images and could be applied in other domains as a means for capturing noise and  imperfections in observational instruments to make simulated data more like real world observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  1  Introduction  Holographic imaging enables 3D reconstruction of particle positions and sizes within a large sample volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Through this tech-  nique, ensembles of thousands of particles can be captured and characterized in a single instantaneous volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' This technique  has been successfully applied to characterizing the microphysical properties of clouds with airborne in situ instruments such  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='02757v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='ins-det]  7 Jan 2023  as the HOLODEC probe (Fugal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Spuler and Fugal, 2011) which is operated by NCAR and has been successfully  deployed on several field campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  The HOLODEC instrument captures inline holograms of a 15 cm3 sample volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' A laser operating at 355 nm transmits  an expanded beam between two probe arms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' After the light passes through the sample volume, the light is imaged onto a CCD  with an effective pixel resolution of 3 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The laser fires at a 3 Hz repetition rate, allowing for point-like captures of particle  size distributions as the aircraft passes through a cloud with spatial separations of those points dictated by aircraft speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  While holographic imaging captures a significant amount of information in a single image, processing these holograms tends  to be computationally intensive, often requiring human intervention to tune the processing approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' As a result, there has been  substantial research into the use of machine learning to accelerate or improve processing of holographic particle data (Shimob-  aba et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Shao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' However, many of these efforts are focused on images with relatively  small depth components which is distinct from HOLODEC requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Another key distinction is in the idyllic nature of  the holograms that are processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Processing solutions that work well on controlled laboratory or simulated holograms tend  to perform suboptimally on operationally captured data from HOLODEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' One cannot expect the same consistent clarity and  quality from an instrument on the outside of an aircraft, where a variety of operating and environmental conditions can cause  the captured images to contain a significant amount of imperfections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The challenge of processing HOLODEC data necessitates  tuning filters to minimize both false positives and missing true particles in holograms with non-negligible imperfections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  These imperfections are difficult to model in simulation because they often depend on combinations of component imperfec-  tions within the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' For example, small amounts of reflections from transmission optics can create modulation patterns  and focus spots in reconstruction, laser modes deviate in amplitude and phase from a true Gaussian, vignetting can cause  non-uniform field response and depend on beam pointing that varies with vibrations and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Ultimately, developing  processing techniques that can account for these and other (sometimes unknown) effects is essential to delivering high quality  scientific observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  In a previous work, we outlined a processing pipeline concept to accelerate and improve processing of HOLODEC images  through a combination of hardware acceleration and machine learning (Schreck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' As is common in atmospheric  sensing, actual hologram data lacked truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' In order to achieve an improvement in processing capability over the exist-  ing solution, we developed an approach for corrupting simulated holograms in order to emulate the imperfections of actual  HOLODEC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' A machine learning solution to process actual HOLODEC data was trained using simulated holograms  (where labels are are known to near absolute accuracy) by adding random noise, applying blurring and skewing, and adjusting  the image contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' While the simulated training data did not visually appear similar to actual HOLODEC data, these image  corruptions were sufficient to train a model that still out performed the particle recognition approaches used in the standard  processing solution (Schreck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' However, to succeed, each of these transforms needed to be tuned in hyperparameter  optimization, which required manual labeling of thousands of reconstructed holograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' This is problematic because manual  labeling requires considerable person-hours from well trained personnel, and (as we discovered in the process) human deter-  mined labels have some non-zero but unquantified error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' What is more, during the optimization, we simply guessed which  types of corruptions to perform on the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Finally, there may be many sensor situations where these simple corruption  2  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (a) NCAR’s Mesa Lab taken to be the content image (Commons, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (b) An image obtained from the James Webb telescope  showing star formation is the style image (NASA, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (c) The synthesized image is the product of the neural style transfer method that  leverages the content and style images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  operations do not suffice in developing a higher performance processing solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' In order to generalize to the broader domain  of sensor signal processing, a solution should be able to address the specialized behavior of the sensor type and this mandates  a more general ability to represent the difficult to model, non-ideal behaviors of the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Here we introduce a new approach to bridging the gap between optimizing signal processing/machine learning solutions  and the realities of operational sensor data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' We show that the original neural style transfer method introduced by (Gatys et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=',  2016), which utilizes a convolutional neural network (CNN) to perform image augmentation, can be used to produce realistic-  looking, labeled hologram examples without the need for manually labeling as a means for tuning noise parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' These  stylized synethetic training data result in processing performance similar to the previous method, but without the requirement  of human labeling and therefore represent a significant step toward producing an operation solution for processing HOLODEC  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  2  Methodology  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='1  Neural Style Transfer  The original style transfer method leverages CNN representations of a “content” image and a “style” image to produce a third  image where the style features are automatically synthesized with content features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' For example, Figure 1 illustrates how the  method is used to create an image of NCAR’s Mesa Lab shown in (a), stylized using a recent image from the James Webb  telescope shown in (b) to create a synthesized image shown in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The synthesized image still clearly still shows the Mesa  3  (a)  Content image  (b)  Style image  (c)  Generated imagePooling layer  Layer output  Backpropagation  Input image  Content loss  Style Loss  p  Content image  x  Image parameters  a  Style image  conv5  conv4  conv3  conv2  conv1  p=  a=  conv5  conv4  conv3  conv2  conv1  conv5  conv4  conv3  conv2  conv1  x=  Symbol Legend  VGG-19  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Schematic illustration of the neural style transfer approach by Gatys et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (2016) using the VGG-19 neural model, where the  model’s 18 convolution layers are stacked into five blocks (orange squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' In this example, the output from the first convolution layer in  each of the five blocks were used to compute style and content losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The blocks represent the (same) model for inputs p, x, and a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Lab, however, overall the image colors, contrast, and line smoothnesses have been transformed relative to the original to also  resemble the style image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  The synthetic holograms come with labels and are free of noise, so they are identified as possessing the “content” we wish  to preserve — that is the information about the particles within.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The HOLODEC examples all contain varying degrees of  noise/imperfections that we wish to effectively impart onto the synthetic examples, hence they are identified as the “style”  examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' In this way, synthetic images are transformed to create a realistic training dataset with known labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Figure 2 schematically illustrates the style transfer method using the VGG-19 network as the chosen CNN (Simonyan and  Zisserman, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The VGG model contains 19 layers and was originally trained to perform object recognition and localisation  on the ImageNet dataset (Russakovsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The model’s 18 convolution layers utilize the ReLU activation and are  sequentially connected together into five blocks separated by 2D average Pooling layers, as is illustrated schematically in Fig-  ure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The first two blocks contain two convolution layers, while the remaining three contain four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The position of a convolution  4  layer in VGG is characterized by integer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' All convolution layers in VGG-19 use a kernel size of 3x3 and stride and padding  sizes both set to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Three fully-connected layers complete VGG-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Thus, each column in Figure 2 is a single (the same)  VGG-19 model (with the fully-connected layers discarded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Currently, a single color channel is used for the hologram images,  so the very first convolution layer was replaced with one having the color channel equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  The style transfer approach begins by sampling random values for x, and freezing the weights of the all of the layers in  the CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' For the input content image p and generated image x, the output from convolution layer l are the feature matrix  representations P l and F l, where P l  ij and F l  ij are the responses to the activation of the ith filter at position j in convolution  layer at position l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' A central question with style-transfer is the choice of loss function, and which layer outputs should be used  to compute the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The choice will determine to what extent we need the full VGG-19 model, and will greatly affect run-time  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' We follow the original approach introduced by Gatys et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (2016), but other effective loss functions have been  developed (Li and Wand, 2016b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Wilmot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' A new image that matches the responses of the content  image p may be found by updating the values of x via gradient descent using a squared-loss of the two feature representations  Lcontent(p,x,l) = 1  2  �  i,j  (F l  ij − P l  ij)2  (1)  for convolution layer at position l, as is illustrated in Figure 2 by a blue hexagon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The “style” representation of an input image x  is defined using a feature space for capturing texture information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' This is taken to be the correlation between the outputs across  the spatial domain for layer l, so that the expected correlation for style feature maps i and j are given by the Gram matrix  Gl  ij =  �  k  Gl  ikGl  jk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  (2)  Now the image x may be updated via gradient descent so that it’s style features Gl match the style features Al of image a at  layer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The contribution coming from layer l is taken to be  El =  1  4N 2  l M 2  l  �  i,j  (Gl  ij − Al  ij)2  (3)  and the total style loss, illustrated by green circles in Figure 2, is  Lstyle(a,x) =  �  l  wlEl  (4)  where wl represents a weight for each contribution coming from layer l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Although not used in Gatys et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (2016), the predicted  images may sometimes contain higher-frequency noise which we would prefer to smooth out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' This can be accomplished via a  total variation term given by  LT V (x) =  �  i,j  |xi,j − xi+1,j| + |xi,j − xi,j+1|  (5)  for pixels i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' For content image p, style image a, and generated image x, the total style transfer loss is then  Ltotal(p,a,x) = αLcontent(p,x) + βLstyle(a,x) + γLT V (x)  (6)  5  where the parameters α, β, and γ are constant weights, and only the output from a single layer is used to compute the content  loss (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' In general, if more than one layer output is selected for computing the content loss, the terms would be  summed in Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' We set α = γ = 1 and focus on the relationship between β and the predicted images below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  In addition to β, the choice of which layer(s) to use in content and style representations are left as hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' For the  content feature layer choice, the output from convolution layers farther from the input image should capture global features  relative to those blocks nearer to the input image, which should capture more details of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Similar arguments apply to the  style features, with the more distant convolution layers capturing more global styles and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Gatys et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (2016) selected  the output from the second convolution layer from the fourth block to compute the content loss (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' the 12th convolution layer  in VGG-19), and the output from the first convolution layer in all five blocks to compute the style loss components (convolution  layers at positions 1, 3, 5, 9 and 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' These choices result in a large CNN that greatly affects training performance when used  with our relatively large holograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' We found that a much smaller model was just as capable as that used by Gatys et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (2016)  and was thus much faster to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Specifically, we selected the fourth convolution layer outputs in VGG-19 for the content  contributions and the first five convolution layer outputs in VGG-19 for the style contributions (with all weights equal to unity  in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' 4), therefore only requiring layers in three of the five blocks in VGG-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' There may be a more optimal, faster model that  achieves similar results, but we did not explore other combinations any further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Figure 2 shows that content image p and style image x are passed through the CNN and the outputs from selected feature  layers are stored in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Initially white noise image x is then passed through the CNN and the outputs from the content  and feature layers are used to compute the total loss according to Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The image pixels in x can then be updated via  back-propagation and with successive iterations, x can learn to match the p’s content and a’s style features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='2  Metrics for comparing images  The choice of α and β will determine how similar the generated image is to the content and style images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' There are numerous  metrics available for estimating the similarity between two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' For images x0 and x1, we select the structural similarity  index measure (SSIM) (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2004) (Equation A1), a standard pixel-based metric that extracts information on structure,  contrast, and luminance of an image that serves as the basis for comparison of two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Additionally, a similarity metric  based on the AlexNet convolutional neural network model (Krizhevsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2017), referred to as dAlex ≡ dAlex(x0,x1), is used  to capture the distance between two images, x0 and x1, in deep feature space (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The SSIM ranges from -1  to 1, with larger values indicating more similarity between images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The AlexNet metric ranges from 0 to 1 with smaller value  indicating greater similarity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' we use one minus the value so that a higher value indicates greater similarity in both metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' For  images x0 and x1, both metrics are computed by averaging over local differences between “patches” of size 11×11 for SSIM,  and size 3 × 3 (the filter size used) for the AlexNet metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='3  Image segmentation with a U-Net  Figure 3 shows a neural segmentation model, for example a “U-Net” (Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Shelhamer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2017), that is  used in the current study to process holograms and has been used in other studies (Shao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The model contains two  6  (ii) Convolutional neural  network  Encoder Decoder  (iv) Mask prediction  (iii) Noise added during training and  optimized with manual labeling  No particles in  focus  Particle is in  focus  (i)  (ii)  HOLODEC  holograms  (i) Input synthetic  holograms  (i) Synthetic  holograms  (ii) Stylize  (b) Training with style transfer  (c) Operation  (a) Training and optimization with noise  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (a) U-Net segmentation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (i) Example synthetic holograms (solid red lines) are passed into the encoder-head of the U-Net  shown in (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (iii) During training, noise may optionally be added to the holograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (iv) Predicted masks for the input examples, with a dark  circle enclosing the in-focus particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (b) (i) Synthetic images are (ii) stylized using HOLODEC holograms (dashed blue lines), which are  then (iii) used to train a U-Net model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (c) In operation, HOLODEC holograms are passed through the U-Net trained on style-transferred  images to obtain a mask prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' In the example no particles were identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  CNN “heads” as illustrated in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' An encoder head takes in the input image and compresses it into a smaller dimensional  representation of a fixed size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The decoder head up-samples the latent representation of the image and outputs a prediction the  same size as the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Here, we use a sigmoid activation function on the decoder’s last layer that rescales the predicted  outputs for all the pixels to lie within (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' In other words, the model is trained to predict a probability that an in-focus particle  occupies a pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' If the probability exceeds a certain value, often 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='5 for binary tasks, the pixel is labeled 1 and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The  convolutional layers in the U-Net encoder head may also leverage pre-trained weights from the layers in other convolutional  models, such as ResNet (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2016) trained on the ImageNet dataset, which frequently helps to speed training and also  boost overall performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  7  O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='4  Datasets  There are essentially four datasets we use in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The first two originate from full size images captured by HOLODEC or  simulations of HOLODEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' These initial images consist of 4872-by-3248 pixel images which are processed and segmented to  create images that can be readily processed by the U-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' These images are also sampled to ensure a balance between in-focus  particles and out-of-focus particles in the U-Net training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' This training data from the two sources are then used by  style transfer to generate a realistic generated dataset (with content from the synthetic images and style from the HOLODEC  images) which is used to train the U-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Here, style transfer is performed using reconstructed synthetic and HOLODEC  images at different positions along z (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Finally a set of manually labeled HOLODEC holograms are used to evaluate  the performance of the U-Net, and therefore the effectiveness of the style transfer solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='1  HOLODEC and synthetic data  A physical model of the instrument was used to generate 120 synthetic holograms which had optical settings identical to the  HOLODEC instrument (Fugal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' One hundred holograms were randomly selected for training, ten for validation,  and the remaining ten for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Each hologram contained 500 spherically shaped particles randomly positioned along the x  and y directions, positioned between minimum and maximum values of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='072 and 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='928 millimeters along the z-direction,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The diameter sizes were sampled with a gamma distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  The HOLODEC data were selected from a set of several hundred holograms obtained from the RF07 subset from the Cloud  Systems Evolution in the Trades (CSET) project that was originally obtained from June 1 to Aug 15, 2015 (Albrecht et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=',  2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' They primarily contain liquid water particles of varying numbers and shape, were largely free of ice but contain other  unidentified “artifacts” and image perturbations not captured by the physical model of the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Overall, the known  particle numbers range from none up to hundreds, with the average particle diameter less than 50 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  In accordance with the approach described in (Schreck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2022), each hologram is initially processed by reconstructing  1,000 planes along the z-axis (axis of laser propagation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' This produces 1,000, 12 megapixel size images that must scanned  for in-focus particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' These images are too large to be processed efficiently with a CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Therefore, following our previous  approach (Schreck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2022), each hologram reconstruction is broken up into 38-by-28 tiles each of 512 by 512 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' These  tiles have an overlap stride of 128 pixels leaving 828 total tiles for each full image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' This means that any one pixel will show up  in more than 1 tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  For training using synthetic holograms, each was selected and then refocused along z for all of the 500 particles, from which  a random grid tile containing the in-focus particle was selected and saved, resulting in 50,000 images (for details on refocusing,  see Schreck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (2022)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Additionally, 25,000 tiles were selected where a particle was just out of focus by no more than 120  micrometers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Finally, 25,000 tiles were randomly selected which did not contain in-focus particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' A comparable procedure  was repeated to produce validation and test sets of tile images where half the images contained in-focus particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  For the RF07 subset used here, a hologram was drawn at random from several hundred, excluding 20 that were set aside previ-  ously for manual examination (and described below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' This raw data can be accessed at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='6347222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  8  As the true particle positions are not known (and they are not needed as they will be used only for style transfer), the hologram  was refocused at a random position between 14 and 160 millimeters along z, from which 10 tiles were randomly selected  and saved from the refocused hologram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' This process was repeated until about 100,000 image tiles were selected for training,  10,000 for validation, and 10,000 for testing the mask prediction performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Most of the tiles in each split did not have  particles directly in-focus, but occasionally were pretty close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='2  Generated data sets with style transfer  The labeled synthetic training splits are used as the content images while RF07 holograms, which come without labels, are the  style images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Training, validation, and testing splits for 20 different values of β sampled evenly from approximately 10−5 to  1015, with α = 1 and γ = 1 were created to find the optimal value of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Once obtained, a data set containing training, validation,  and testing splits was created with β ≡ β0 = 109, which we identify below as an optimal choice, and α and γ both equal to  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' This is created to compare against the model trained on synthetic training split only and the model described in Schreck  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (2022) where optimized noise (based on hand labeled data) was added during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  For each selection of β including β0, a randomly selected content (synthetic) image tile was selected and paired off with a  randomly selected style (HOLODEC) image tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Then, both content and style images were used to generate a training sample  with the style-transfer method, with the mask labels for the generated example are inherited from the content image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' For the  β0 data set, 100,000, 20,000, and 20,000 style-generated images were created for training, validation, and testing the ability  of U-Net model’s to reproduce the mask labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' For the other 20 values of β, a total of 20,000 training, 2,000 validation, and  2,000 testing samples were produced in each case for the same purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' In the latter data sets, we also used the same random  seed so that all the pairs selected in each split for the different β choices (excluding β0) were the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='3  Manually labeled HOLODEC data set  The 20 manually labeled HOLODEC examples resulted in 2,356 total images of size 512 pixels by 512 pixels, where prospec-  tive in-focus particles were positioned at the image center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The examples were manually labeled as containing an in-focus  particle or not, along with an average reviewer confidence score ranging from zero to five (higher means more confident).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The  binary output of the U-Net is simply taken to be unity if any predicted value is larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='5 and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Many of the  examples were subjective, and the total of number of particles we could identify was limited by previous modeling attempts,  hence there could be more particles in these holograms still undiscovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  The total number of labeled examples was split into validation and testing data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The validation set contained 1,204 total  images with 367 containing at least one in-focus particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The testing set contained 1,154 images, where 874 contained at least  one in-focus particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The validation set was used to help guide the optimization of noise in our previous study that we compare  with here (Schreck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2022), while the test set represents the hold-out set of manually labeled examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  9  3  Results  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='1  Style transfer with holograms  Figure 4 shows a synthetic hologram (content image) that contains a single in-focus particle, and different HOLODEC holo-  gram examples (style images) that were used to generate the style transferred image, for α = 1, and β = 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The HOLODEC  example shown in Figure 4(a) was selected because it contains an in-focus particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Note that while there appears to be a trans-  fer of fine scale texture from the style images, particles from these style images are not transferred generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Thus  there is no need to locate empty holograms for the style transfer images or perform careful QC in selecting the appropriate  images for this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  The images in Figure 4(b-c) are typical examples where a U-Net trained only on synthetic images (left column) performs  poorly on actual HOLODEC images due to a high false-positive rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The interference fringes seen in the HOLODEC image  in Figure 4(b) often produce patterns that resemble in-focus particles, while (c) shows an example where many small artefacts  are present and have a round shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' In all three examples the in-focus particle in the synthetic example is still present in the  generated examples, and the overall generated image contrast is much more comparable to the HOLODEC images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Figure 5(a) shows the training curve for the generated image shown in Figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Figure 5(b) illustrates the progression of  the predicted image at selected epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Starting initially with a random noise image for x, a very steep drop in the style loss  is observed over the first few epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The content and TV losses also drop quickly after the first few epochs, then the curves  mostly flatten after about 10 epochs, and the content contribution comes to dominate the total loss thereafter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' In these examples,  the choice of α/β determined the similarity between the generated image and content and style images used to create it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Here,  according to the AlexNet metric, all the generated images are more similar to the HOLODEC images than to the synthetic  image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Figure 6(a) and (b) show the SSIM and dAlex metrics versus β with α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' These metrics are computed for test data  of HOLODEC-synthetic, HOLODEC-generated, and synthetic-generated pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Note that these metrics are constant for the  HOLODEC-synthetic pairs because these images are unaffected by β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  In both figures, four regions of interest are approximately identified with vertical lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Region A corresponds with values of  approximately β ≤ 10 where the content loss dominates the total loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Accordingly, both metrics show that the style-transferred  images strongly resemble the original synthetic images (see also Figure 4(a,b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The main difference between the two metrics  is that dAlex scores HOLODEC-generated pairs higher relative to HOLODEC-synthetic pairs than does SSIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' However, the  curves are all qualitatively similar between the two metrics for all values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Examples of images from each region in Figure 6 are shown in Figure 7 using the content and style images from Figure 4(a)  with α = 1 and varying β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Region B is identified approximately when 102 < β ≤ 107, which is when both metrics for synthetic-generated and HOLODEC-  generated pairs drop significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The examples in Figure 4(c,d) show that the generated image starts to look significantly  different compared with the original synthetic image as the style term begins to contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Although these examples may give  the impression that the generated images still look like the synthetic images, they are in reality a mixed representation which  10  0  740  1480  Y ( m)  (a)  Synthethic (content)  HOLODEC (style)  Synthesized  0  740  1480  Y ( m)  (b)  0  740  1480  X ( m)  0  740  1480  Y ( m)  (c)  0  740  1480  X ( m)  0  740  1480  X ( m)  50  100  150  Pixel value  50  55  60  65  70  75  80  85  90  Pixel value  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' In (a-c), the same synthetic hologram is paired with a different HOLODEC hologram to produce a generated hologram using  style transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The HOLODEC examples illustrate (a) an in-focus particle, (b) interference patterns and successive dark spots, and (c) small  artefacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' U-Nets are observed to perform poorly on (b) and (c) when trained with synthetic images only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The content weight α = 1, while  the style weight β = β0 = 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  11  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (a) The style, content, TV, and total loss are plotted versus the number of training epochs (the number of times the image was  updated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (b) The initial random image and selected images at different training updates are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The pixel values shown ranged from 50  to 90, as in Figure 4 for the HOLODEC and generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  10  4  10  1  102  105  108  1011  1014  Style weight,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='6  SSIM  (a)  A  B  C  D  H-Gen  Syn-Gen  H-Syn  10  4  10  1  102  105  108  1011  1014  Style weight,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='9  Similarity, dAlex  (b)  H-Gen  Syn-Gen  H-Syn  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (a) The average SSIM and (b) average dAlex scores computed using the test data sets, is shown versus the style weight β in  Equation 6 with α and γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  does not on average resemble the synthetic or HOLODEC hologram (for example, the average contrast difference for examples  like those in Figure 4(c,d) is less than that for the synthetic images but still greater than what is observed in the HOLODEC  examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Around β = 104 both metrics reach a minimum before beginning to rise with increasing β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  12  (b)  Random noise  Update 10  Update 50  1480  (ur)  740  Y  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Update 100  Update 200  Update 500  1480  (ur)  740  Y  0  740  1480 0  740  1480 0  740  1480  X (μm)  X (μum)  X (μm)(a)  Style  104  Content  TV  103  Total  102  101  Loss  100  10-1  10-2  10-3  100  101  102  Image updatesRegion C is identified approximately when 107 < β ≤ 1012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Both SSIM and dAlex continue to increase for synthetic-  generated pairs and peak near β = 109, though the peak is more pronounced for SSIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Both scores for HOLODEC-generated  pairs also increase and reach plateau values, which are higher compared against that in regions A and B, indicating the images  have been stylized to some degree while the the Synthetic-Generated scores indicate the image still retains content features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Clearly, the neural dAlex determines the generated holograms are more similar to HOLODEC holograms compared to SSIM,  but overall both metrics indicate the predicted images have struck a particular balance between retained content and learned  style features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Figure 4(e-g) show that the in-focus particle remains in place and roughly the same shape, while, for example,  the image contrast is adjusted to be more like that of the HOLODEC images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Finally, region D is identified by β > 1012, which is when the style term dominates the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The dAlex score for style-  generated pairs remains high while that for content-generated pairs converges to the same value for content-style pairs, which  is to say the metric cannot differentiate the generated image from style images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' However, Figure 4(h,i) show that the in-focus  particle and other content from the synthetic image has largely been removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Furthermore, what appear to be fake particles  are now being introduced into the generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' In other words, the original image was over-stylized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Region A represents the instances where output has very little style imposed on the content images and therefore most closely  resembles idealized synthetic data, which we previously established in Schreck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (2022) makes a poor U-Net training data  set when the objective is to process actual HOLODEC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Images from region D represent instances where content is largely  absent from output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Clearly we cannot expect to train a U-Net to recognize particles that are not in the input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Without  having to resort to manual labeled HOLODEC images, the obvious choice would be to select the value of β around β0 = 109  in region C, when both metrics for synthetic-generated pairs peak, and the generated examples appear to be more similar to  HOLODEC examples according to dAlex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  The performance of the U-Net trained on the generated images is then expected to peak at mask reproduction when β is  around β0 = 109, as the predicted images are the most similar to the content images in region C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Also, because the generated  images are similar to the HOLODEC images in region C relative to A and B, we might expect U-Net models trained on them  to perform the best on HOLODEC images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' If this is the case, one can select the value of β by finding the value where the  generated images are more similar to the HOLODEC images according to dAlex and SSIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' This selection should result in the  best generated dataset for training the U-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='2  Performance on synthetic and HOLODEC images  In order to test the hypothesis that the optimal β can be found analytically from dAlex and SSIM, the generated images (and  their corresponding labels) at the different values of β are used to train U-Net models, as is shown schematically in Figure 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  The model architecture and training hyperparameters were all identical for each different value of β (see Appendix A1 for more  details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Post-training, each U-Net model is run in operation mode on the test set of generated images and the manually labeled  test set of actual HOLODEC tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Figure 8(a) shows several binary performance metrics for the U-Net at reproducing the mask  labels accompanying the generated images, while Figure 8(b) shows similar quantities for particle detection in true HOLODEC  tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Since none of the manually labeled tile examples were used in the creation of the style-generated images, all 2,356 were  13  0  740  1480  Y ( m)  A  (a)  10  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='95  A  (b)  101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
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+page_content='74  0  740  1480  X ( m)  D  (h)  1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='84  0  740  1480  X ( m)  D  (i)  1013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='95  50  100  150  (a-c) Pixel value  40  60  80  100  (d-f) Pixel value  50  60  70  80  90  (g-i) Pixel value  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (a-i) Examples of generated holograms using the content and style images from Figure 4(a) and different style weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The region  is listed in the bottom right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  used in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The metrics are the the F1 score, the area-under-the-ROC-curve (AUC), the probability of detection (POD),  and the false alarm rate (FAR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  14  Syn 0  10  4  10  1  102  105  108  1011  1014  Style weight,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
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+page_content='0  Mask metric  (a)  Synthetic holograms  F1  AUC  POD  FAR  Syn 0  10  4  10  1  102  105  108  1011  1014  Style weight,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
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+page_content='0  In-focus metric  (b)  Labeled HOLODEC holograms  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (a) Several binary metrics for the U-Net’s mask prediction performance are plotted versus style weight β for synthetic or style-  generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (b) The same binary metrics are shown for the particle detection task in manually labeled HOLODEC images versus β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  As the models did not rely on the manual examples during training, all 2,356 examples were used in the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The x-axis labels "Syn" and  "0" refer to U-Nets trained on synthetic images and generated images with β = 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Figure 8(a) shows overall gradually diminishing training performance with increased β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' This is expected as the pristine  simulations are increasingly stylized with non-ideal instrument noise and artifacts, making particle detection more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' In  region A, when the generated holograms resemble the synthetic examples, the mask prediction performance is very high and  generally flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' But then in region B, when the generated holograms are modified more by the style image, there is a clear drop  in the AUC and POD while the FAR rises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' As β continues to grow and the generated images start to resemble both content  (synthetic) and style (HOLODEC) examples, the performance rises and reaches a peak around β = β0 before dropping, first  slowly and then very suddenly once in region D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' We expect this peak in region C to be less than that in A where the images  lack the characteristic noise of the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' In region D, the AUC is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='5 which should be expected as the over-stylized  images and the mask labels retain very little content from the actual particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The trained U-Net outputs all, or too many 1s  when trained on over-stylized images, so POD quickly approaches 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' We also computed diameters for the predicted masks and  found no significant deviation from the mean true value for the test set, for all choices of β except those in region D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Overall, Figure 8(b) shows that the relevant range of β may be selected according to our hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Relative to the U-Net  trained only on synthetic images (labeled ‘Syn’ on the x-axis in Figure 8(b)), the performance on the HOLODEC examples  in region A is signified by lower PODs and higher FARs, although the AUC values are all approximately around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
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+page_content=' Even  the model trained on generated images with β = 0 showed a higher FAR relative to ’Syn’, indicating that the U-Net trained  on idealized synthetic holograms tends to under predict particles, to the benefit of a low FAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' We should note here that  the difference in performance between the synthetic trained and β = 0 cases indicates that setting β = 0 does not perfectly  15  Metric  F1  AUC  POD  FAR  CSI  Syn/Gen  HOLO  Syn/Gen  HOLO  Syn/Gen  HOLO  Syn/Gen  HOLO  Syn/Gen  HOLO  Synthetic  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
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+page_content='839  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Several metrics are listed for each hyperparameter optimized model and were computed on the synthetic or generated (Syn/Gen),  and HOLODEC (HOLO) test data sets, for mask and particle detection (binary) predictions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The validation set of labeled  HOLODEC images was used to optimize the noise parameters in the Optimized noise model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  reproduce the content image, resulting in slightly lower U-Net performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' However, neither case is relevant to the objective  of this work since both cases fail to perform well on HOLODEC data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' In region B, the particle detection performance generally  rises even as the mask performance drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' However, the particle detection performance zig-zags quite significantly making a  selection of β unreliable without manual labeled examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' In Region C, the detection performance reaches an approximate  plateau, with high AUC values and consistently low FAR values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The performance seems to slightly decrease as the style  weight approaches values in region D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' In region D, the model is unable to train effectively, resulting in an output of all 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' This  results in a high POD and FAR with AUC at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Region C models also performed the best overall on the HOLODEC images  relative to A, B and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' As expected, models in Region D do not perform at all on the manual examples, as signified by AUCs  close to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='3  Comparison of optimized models  Now that the style weight may be selected without relying on manual examples for optimization, we selected β = β0 = 109 and  trained a model on the style-transferred images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Table 1 compares this model (‘Stylized’) versus a model trained on idealized  synthetic images only (‘Synthetic’), and a model trained on synthetic images with noise introduced during training (‘Optimized  noise’, where the optimization required manual labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' In all three cases we performed extensive hyperparameter optimization  to find optimal architectures and training parameters using the mask performance on the validation set of generated images as  the optimization metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The model which leveraged noise during training additionally used the manually labeled validation  HOLODEC examples to guide optimization of the added noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The same binary metrics in Figure 8 are listed as well as the  critical success index (CSI) for mask performance on test images (either synthetic or generated) and detection performance on  test HOLODEC images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' As the validation set of manually labeled examples were used to select the noise added which resulted  in the Optimized noise, only the testing split is used in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Table 1 shows that the Stylized model and the Optimized noise model gave nearly the same performance on all metrics on  both data sets even though the Stylized approach does not require manual labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The Optimized noise model outperforms  Stylized by hundredths of a percent in each metric category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The Synthetic model showed better mask performance on synthetic  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' However, the Synthetic demonstrated much worse performance at particle detection on real HOLODEC data compared  16  0  20  40  60  80  0  50  100  150  True positive count  d  = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='9 m  (a)(i)  Synthetic  0  20  40  60  80  d  = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='94 m  (b)(i)  Optimized noise  0  20  40  60  80  d  = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='72 m  (c)(i)  Stylized  0  20  40  60  80  Predicted diamater ( m)  0  50  100  150  False positive count  d  = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='84 m  (a)(ii)  0  20  40  60  80  Predicted diamater ( m)  d  = 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='8 m  (b)(ii)  0  20  40  60  80  Predicted diamater ( m)  d  = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='6 m  (c)(ii)  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (a-c) Histograms of computed particle diameters for the three models for the test set of manually labeled HOLODEC images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The  dashed vertical line in each panel denotes the predicted average particular diameter ⟨d⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  to the other two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' For example Table 1 shows that it had a much higher FAR, and the CSI was more than 70% lower  compared to the other models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Lastly, as the Optimized noise and Stylized models produced nearly identical performance, the diameters for the predicted  particles in the test set of manually labeled HOLODEC examples are computed to probe how the models describe the particles  shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Figure 9 shows histograms of the predicted diameters for the three models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' A particle diameter was computed for a  predicted mask if at least one in-focus particle was present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' From the predicted mask, any pixel labeled 1 was grouped with  a neighboring pixel along x or y (but not along the diagonal in the plane) if it was also labeled 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Therefore, breaks between  groups defines multiple particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The diameter of a group was taken to be it’s maximum extent in either x or y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Figure 9(a)(i) shows the Synthetic model expectantly under-predicting the true particles compared with the other two mod-  els, and nearly as many false positive predictions were made relative to true positive, as seen in Figure 9(a)(ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Meanwhile, the  Optimized noise and Stylized models predicted similar distributions, but which differed from the Synthetic distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Both  models predicted two peaks centered approximately around 15 and 22 µm, respectively, with the Stylized model having the  more pronounced second peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The Stylized model also predicts slightly greater numbers of larger particles relative to Opti-  mized noise that are also sometimes observed by the Synthetic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' As a result, the overall average diameter is predicted  to be about 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='8 µm versus 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='9 µm for Stylized and Optimized noise models, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Figure 9(b)(ii) and (c)(ii) further  show that both models mostly predict false diameters at 15 µm or smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  17  4  Discussion  Overall, the performances observed on both synthetic and HOLODEC images with the Optimized noise and the Style model  were comparable, with both models performing well against models trained on the synthetic holograms only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' This shows the  advantage of using the style-transfer method for hologram image augmentation, which obviated the need to perform man-  ual labeling as a means for optimizing noise added during the training of U-Net models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Additionally, we did not have to  ambiguously choose the types of augmentations performed on the images, as was required for the Optimized model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Further-  more, the application of style transfer to images that were used for training, rather than applied during operation, means that  computational performance is also comparable with the Optimized noise model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  The main computational bottleneck involving the style-transfer method occurs during the creation of style-transferred train-  ing data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Many generated images needed to be created, one at a time, from a unique content-style image pair, that all  required the iterative optimization described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The image sizes used were also quite large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' That further slowed training  times and made it cumbersome to create large data sets used for exploring the different weight combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Improved ap-  proaches could be focused on replacing the optimization with a trained neural network that predicts the stylized image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' This  can lead to several orders of magnitude speed-up over the original method when feed-forward architectures are utilized (Li  and Wand, 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Ulyanov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Huang and Belongie, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' These may include the StyleGAN method (Karras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=',  2021), which integrates the concepts of style transfer with a generative adversarial network (Goodfellow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2020) and al-  lows for fine-tune control over the relative strength of image features at different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' For holograms, the StyleGAN could  potentially be used to more selectively control specific content or style features on demand (for example, adding more features  pertaining to artefacts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  We also only worked with CSET holograms, in particular the RF07 data split, which had limited particle densities as well  as sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' How well a trained model, which leveraged style images from one field campaign, works on other campaigns remains  to be tested in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Style-generated hologram data sets, which do not resemble operational inputs, plausibly would  not be feasible as training targets, as we learned in Schreck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' In such a scenario, the holograms from the latest  field campaign could be used as style examples to generate the relevant training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The StyleGAN approach again could  potentially take advantage of all the different types of holograms obtained from field campaigns (and all of the unique objects  contained in each one), since it may be capable of leveraging multiple styles and content features arising in the data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Finally, as the primary objective of this study was to remove the manual-labeling step without sacrificing performance, we  did not try to assess the physical realness of the generated holograms since during operation only real HOLODEC holograms  are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' However, the question remains as to whether they are physically reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' One way to test this assumption could  be to perform style-transfer using full-size holograms, which would be computationally challenging but not impossible with  current GPUs, and then refocus them via wave-propagation along the focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' This would require faithfully reconstructing  the real and imaginary components of the electric field, rather than just the intensity, as was done here since style-transfer was  applied to tiles selected from the full-sized holograms after they had been propagated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  18  Parameter  Synthetic  Optimized noise  Stylized (β0)  10−5 < β ≤ 1015  Learning rate  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='86 × 10−4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='46 × 10−4  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='5 × 10−5  1 × 10−3  Training loss  Focal-Tyversky  Focal-Tyversky  IoU  Focal-Tyversky  Segmentation model  U-Net  LinkNet  U-Net  U-Net  Encoder model  EfficientNet-b0  Xception  DenseNet121  VGG11  Tile transform  255  Normalized  Standard  Symmetric  L2 regularization  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='1 × 10−7  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='0 × 10−6  Gaussian blur σ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='125 Gaussian noise 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='326 Brightness factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='270 Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The values of the best hyperparameters in the optimization studies for the neural segmentation models for the three species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The  batch size was fixed at 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  5  Conclusions  In summary, we have shown that the style-transfer algorithm is an effective approach for translating synthetic holograms,  which were created using an idealized physical model of the instrument, into holograms that resemble those actually observed  by the instrument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' In principle, the application of style-transfer to perform the image augmentations should not be limited to  holograms since non-ideal instrument behavior is a problem across many domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' When the synthetic holograms are used in  machine learning models to predict masks around in-focus particles as was done here, noise had to be injected onto the images  during training so that the model performed well on both synthetic and raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' However, the choice of noise transformations  and parameters can only be selected after expensive manual labeling of raw images and hyperparameter optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The style-  transfer approach, which transformed the clean hologram data set into one more resembling observed holograms, delivered the  same mask prediction performance without the cumbersome requirement of manual labeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Furthermore, both models had  comparable performance on a small set of manually labeled HOLODEC examples, and predicted similar distributions for the  particle diameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  A1  Model and training hyperparameters  Table A1 lists the best parameters found for the segmentation models investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' See reference Schreck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' (2022) for  more details on the hyperparameter optimization of segmentation models with hologram data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The segmentation models  found to be optimal in Table A1 were the U-Net (Ronneberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2015) and LinkNet (Chaurasia and Culurciello, 2017),  while the pre-trained encoder model weights considered were DenseNet-121 (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2017), Xception (Chollet, 2017),  EfficientNet-b0 (Tan and Le, 2019), and VGG-11 (Simonyan and Zisserman, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' See the package segmentation-models-  pytorch located at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='com/qubvel/segmentation_models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='pytorch for more details on the segmentation and encoder  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The optimal training losses found were the intersection over union (IOU) and Focal-Tyversky losses (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=',  19  2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' Salehi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' For additional definitions of each loss function, see the Holodec-ML software package located at  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='com/NCAR/holodec-ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' We also utilized pre-trained weights obtained from the ImageNet data set in all trained  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The tile transforms were applied to a tile just before being passed into a segmentation model include dividing all pixels  by 255, re-scaling all pixels to lie between 0 and 1 (Normalized) or -1 and 1 (Symmetric), or re-scaling each tile by subtracting  the pixel mean and dividing by the square of the variance (Standard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  A2  Structural similarity index metric (SSIM)  The global structure similarity definition for comparing images x and y is given by  SSIM(x,y) =  (2µxµy + C1)(2σxy + C2)  (µ2x + µ2y + C1)(σ2x + σ2y + C2)  (A1)  where C1 and C2 are fixed constants, and µ and σ are defined as  µx = 1  N  N  �  i=1  xi  σ2  x =  1  N − 1  N  �  i=1  (xi − µx)2  σxy =  1  N − 1  N  �  i=1  (xi − µx)(yi − µy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  (A2)  In practice, we use the mean structural similarity index which averages over an 11 by 11 circular-symmetric Gaussian Weight-  ing function (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  A3  Data sets  The HOLODEC and synthetic data sets can be accessed at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='6347222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The labeled HOLODEC  examples were a subset of the RF07 data set (validation set ID 0-9, testing set ID 10-19), while the synthetic holograms were  generated with simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' All holograms in the RF07 data set were used for creating the style-transferred data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' See  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='4 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The style generated data sets can be built using the Holodec-ML software package located at  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='com/NCAR/holodec-ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' This material is based upon work supported by the National Center for Atmospheric Research, which is a major facility  sponsored by the National Science Foundation under Cooperative Agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' 1852977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' We would like to acknowledge high-performance  computing support from Cheyenne and Casper Computational and Information Systems Laboratory, CISL (2020) provided by NCAR’s  Computational and Information Systems Laboratory, sponsored by the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' The neural networks described here  and simulation code used to train and test the models are archived at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='com/NCAR/holodec-ml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content=' All HOLODEC and synthetic  hologram data sets created for this study are available at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/fdE0T4oBgHgl3EQf6AIY/content/2301.02757v1.pdf'}
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diff --git a/gtE2T4oBgHgl3EQfcAfW/content/tmp_files/2301.03892v1.pdf.txt b/gtE2T4oBgHgl3EQfcAfW/content/tmp_files/2301.03892v1.pdf.txt
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+Frascati Physics Series Vol. 74 (2022)
+ISBN: 978-88-86409-76-6
+Frontier Objects in Astrophysics and Particle Physics
+September 25- October 1, 2022
+CONSTRAINTS ON DARK ENERGY FROM THE ABUNDANCE OF
+MASSIVE GALAXIES
+Paola Santini
+INAF - Osservatorio Astronomico di Roma, via di Frascati 33, 00078 Monte Porzio Catone (RM), Italy
+Nicola Menci
+INAF - Osservatorio Astronomico di Roma, via di Frascati 33, 00078 Monte Porzio Catone (RM), Italy
+Marco Castellano
+INAF - Osservatorio Astronomico di Roma, via di Frascati 33, 00078 Monte Porzio Catone (RM), Italy
+Abstract
+This conference proceedings paper provides a short summary of the constraints pre-
+sented by Menci et al. (2020) and Menci et al. (2022) to dynamical dark energy
+models.
+Dynamical dark energy (DDE) models have been proposed to address several
+observational tensions arising within the standard Λ cold dark matter (ΛCDM) sce-
+nario. Different DDE models, parameterized by different combinations of the local
+value of the equation-of-state parameter w0 and its time derivative wa, predict different
+maximal abundance of massive galaxies in the early Universe. We use the observed
+abundance of massive galaxies already in place at z ≳ 4.5 to constrain DDE models.
+To this aim, we consider four independent probes: (i) the observed stellar mass func-
+tion at z ∼ 6 from the CANDELS survey; (ii) the estimated volume density of massive
+haloes derived from the observation of massive, star-forming galaxies detected in the
+submillimeter range at z ∼ 5; (iii) the rareness of the most massive system detected at
+z ∼ 7 by the SPT survey; (iv) the abundance of massive (M > 1010.5M⊙) galaxies at
+arXiv:2301.03892v1  [astro-ph.CO]  10 Jan 2023
+
+z ∼ 10 as inferred from early JWST observations. Our probes exclude a major frac-
+tion of the DDE parameter space that is allowed by other existing probes. In particular,
+early JWST results, if confirmed, are in tension with the standard ΛCDM scenario at
+a 2σ confidence level.
+1
+Introduction
+The current theory of structure formation envisages all cosmic structures to form
+from the collapse and the growth of initially tiny density perturbations of the dark
+matter (DM) density field in a universe characterized by an accelerated expansion.
+Such an acceleration indicates that the dominant component (with density parameter
+ΩΛ ≃ 0.7) of the cosmic fluid must be composed of some form of dark energy (DE),
+with equation-of-state parameter w = p/ρ ≤ −1/3. Although the nature of such a
+component remains unknown, the simplest model assumes DE to be connected with
+the vacuum energy, the so-called cosmological constant, with w = −1. When coupled
+with the assumption that DM is composed of nonrelativistic particles at decoupling,
+such a scenario leads to the Λ cold DM (ΛCDM) standard cosmological model [3].
+While measurements of the Cosmic Microwave Background (CMB) have pro-
+vided a first, strong confirmation of such a scenario, tensions have recently emerged
+([1] and references therein) and have stimulated an extended effort toward the inves-
+tigation of more complex cosmological models. One of the simplest physical alter-
+natives is a DE with a time-dependent equation of state, also called dynamical dark
+energy (DDE) (see [4] for a review).
+The abundance of massive galaxies at high redshift constitutes a powerful probe
+for cosmological models. In fact, in the standard CDM scenario, large-mass DM
+haloes become progressively rarer with increasing redshift. The exponential high-
+mass tail of the mass function of DM haloes is expected to shift toward progressively
+smaller masses for increasing redshift (see, e.g., [5] for a review) at a rate that depends
+on the assumed cosmology. Hence, the comparison of the predicted abundance of
+massive DM haloes at increasingly larger redshift with the observed abundance of
+galaxies with corresponding stellar mass M∗ provides increasingly strong constraints
+on the assumed cosmological framework. Indeed, viable cosmological models must
+allow for an evolution of the initial density perturbations that is fast enough to match
+the abundance of massive galaxies observed to be in place at early epochs.
+Under (extremely) conservative assumptions and considering different observ-
+
+ables, we compare the maximal abundance of massive galaxies predicted in different
+DDE models at high redshifts with the measured abundance of the most massive sys-
+tems observed to be already in place at the same redshifts. This conference proceed-
+ings paper summarizes the results that are presented and discussed in [1, 2].
+2
+Method
+We compute the expected abundance of DM haloes, as a function of redshift z and
+DM mass Mh, in different DDE models adopting the most conservative assumptions.
+We adopt the Sheth & Tormen [6] mass function, within the Press and Schechter
+formalism. Besides being physically motivated and tested against N-body simulations
+for a variety of cosmologies, it has the most extended high-mass tail among the differ-
+ent proposed forms, hence representing the most conservative choice.
+The high-mass exponential cutoff in the Sheth & Tormen mass function is criti-
+cally determined by the cosmic expansion rate and by the growth factor, which depend
+on the equation of state of DE. For the latter, we use the CPL parameterization [7, 8]
+in terms of the scale factor a:
+w(a) = w0 + wa(1 − a)
+(1)
+where the parameter w0 represents the value of w at the present epoch, while wa is
+its look-back time variation wa = −dw/da. In the above parameterization, the stan-
+dard ΛCDM cosmology corresponds to w0 = −1 and wa = 0. For each combination
+(w0, wa) , we can compute the expected number of DM haloes of mass Mh as a func-
+tion of redshift. We refer the reader to [1] for a full description of the methodology
+and assumptions on the various involved cosmological parameters.
+To compare these cosmological predictions on abundance of DM haloes with
+the measured abundance of galaxies it is necessary to take into account the baryon
+physics. However, baryonic effects are degenerate with cosmology in determining the
+expected galaxy abundance. This can be bypassed by noticing that the ratio of galaxy
+baryonic components (stellar mass or gas mass) to DM halo mass has an absolute
+maximum at the cosmic baryon fraction fb ( fb ≃ 0.16, [9]). In fact, the observed
+abundance of galaxies with large mass in the baryonic component Mb places a lower
+limit on the abundance of DM haloes with masses Mh ≥ Mb/ fb. Such a constraint
+can be used to rule out cosmological models that do not allow for a sufficiently rapid
+growth of galactic DM haloes. In other words, since galaxies cannot outnumber their
+
+DM haloes, any (w0, wa) combinations for which φw0,wa(Mh ≥ Mb/ fb, z) ≤ φobs(Mb, z)
+can be excluded.
+Due to the exponential cutoff of the DM halo mass function and to its rapid
+redshift evolution, the highest masses at the highest redshifts put the most stringent
+limits. To adopt the most conservative assumptions, all our choices aim at maximizing
+φw0,wa(Mh, z) and minimizing φobs(Mb, z).
+3
+Results
+We describe here the various observables considered to constrain the w0−wa parameter
+space and the relevant regions excluded.
+3.1
+Stellar mass function at z = 6 from the CANDELS survey
+We first compare with the observed stellar mass distribution of massive, distant galax-
+ies. Since stellar mass is a time-integrated quantity, it is not much sensitive to the de-
+tails of the star formation history [10] and can be easily related to the DM mass of the
+host halo. An extended wavelength coverage is essential for estimating stellar masses
+from spectral energy distribution (SED) fitting, while a combination of survey volume
+and depth is required to measure the abundance of distante, massive and rare galaxies.
+The CANDELS project [11, 12], taking advantage of the optical/near-infrared/mid-
+infrared imaging provided by HST, Spitzer, and VLT on almost 1000 arcmin2 down
+to faint fluxes, provides an ideal data set to base such a measurement. Here, we use
+the mass function derived by Grazian et al. [13], who used a spectral-fitting technique
+to derive stellar masses for a galaxy sample with high-quality photometric redshifts
+based on the CANDELS GOODS-South and UDS fields.
+We focus on their largest stellar mass bin (centered on M∗ = 8 × 1010M⊙, as-
+suming a Salpeter initial mass function [14]) at z = 5.5−6.5. These high redshifts and
+large masses ensure that the mass functions predicted by the different DDE models
+are in the full exponential regime, and are steep enough to make the comparison with
+the observed number density discriminant for the different DDE models.
+We associate the stellar mass M∗ to the host halo DM mass Mh using the relation
+M∗ = F fbMh, where F describes the efficiency of baryon conversion into stars. We
+consider three cases: (a) the extremely unrealistic case F = 1, corresponding to a
+complete conversion; (b) the more realistic case F = 0.25, as obtained from abundance
+matching techniques (e.g., [16]) - this value however is derived assuming a ΛCDM
+
+Figure 1: Exclusion regions at a 2σ confidence level in the w0 − wa plane derived
+from the observed CANDELS stellar mass function at z ∼ 6 [13]. The brown, red,
+and orange regions correspond to the assumption of F = 1, F = 0.5 and F = 0.25,
+respectively (see text). Our exclusion region is compared with the 2σ and 3σ contours
+allowed by CMB+weak lensing (green regions) and by the combination of the same
+data with the Hubble diagram of supernovae and quasars (blue region), derived from
+Figure 4 of [15]. The black dot corresponds to the ΛCDM case (w0 = −1, wa = 0).
+halo mass function; and (c) the conservative value F = 0.5, derived as a conservative
+upper limit on the star formation efficiency from hydrodynamical N-body simulations.
+Before comparing the predicted number density of DM haloes with observa-
+tions, we rescale the observed volumes and luminosities from a ΛCDM assumption to
+a generic cosmology through the factors fVol = VΛ/Vw0,wa and flum = D2
+L,w0,wa/D2
+L,Λ,
+respectively, where V is the cosmological volume and D2
+L is the square luminosity
+distance used to convert observed fluxed into luminosities, hence stellar masses. In
+summary, for each combination (w0, wa), we compare the volume-corrected, observed
+abundance of galaxies ˜φ = φobs fVol with stellar mass M∗ = 8 flum1010M⊙ at z ∼ 6 with
+the predicted number density of DM haloes with DM masses larger than M∗/(F fb),
+i.e. φw0,wa(Mh ≥ M∗/(F fb)). The confidence for the exclusion Pexcl of each considered
+DDE model is obtained from the probability distribution function p(˜φ) as the proba-
+bility that the measured abundance ˜φ is larger than number density predicted by the
+
+2
+ExcludedbyCANDELS(2fields)
+stellarmassfunctionatz-6byGrazianetal.(2015)
+1
+0
+SNae+QSo
+-1
+CMB+WeakLensing
+-2
+-2.0
+-1.5
+-1.0
+-0.5
+w
+0model, i.e., Pexcl(w0, wa) =
+� ∞
+φw0,wa p(˜φ)d ˜φ.
+We show in Figure 1 the region of the w0 −wa parameter space excluded at a 2σ
+confidence level (i.e., Pexcl ≥ 0.95) for F = 1, F = 0.5 and F = 0.25. The exclusion
+region is overplotted on the regions allowed by CMB and weak lensing observations,
+and on the one derived by the combination of the same data with the Hubble diagram
+of supernovae and distant quasars [15]. Our probe significantly restricts the region in
+the DDE parameter space allowed by other methods. In particular, we exclude an ap-
+preciable part of the region favored by the distant quasar method. Very similar results
+are obtained by comparing DDE predictions to the [17] mass function of CANDELS
+galaxies.
+3.2
+Submillimeter detected massive galaxies at z ∼ 5
+The population of galaxies identified in rest-frame optical and ultraviolet is known
+to under represent the most massive galaxies, which have rich dust content and/or
+old stellar populations. These are, however, detectable at submillimeter (sub-mm)
+wavelengths. Wang et al. [18] performed detailed sub-mm (870 µm) observations
+with ALMA of a sample of IRAC–bright galaxies. They detected 39 star-forming
+objects at z > 3, which are unseen in even the deepest near-infrared (H-band) imaging
+with the HST (“H-dropouts”) and proved to be massive galaxies with stellar mass
+extending up to M∗ ≃ 3 × 1011M⊙, with a median mass M∗ ≃ 4 × 1010M⊙.
+For such objects, we follow a procedure similar to that explained in the previous
+section. We compute the number density of galaxies with stellar masses in the bin
+10.25 ≤ log(M∗/M⊙) ≤ 10.75 (dominating the statistics of observed objects) at red-
+shifts z = 4.5–5.5, and derive the corresponding 2σ lower limit φlow(M∗) = 1.8 × 10−5
+Mpc−3. To relate the observed stellar mass M∗ to the DM mass Mh, we first adopt
+the highly conservative assumption Mh = M∗/ fb (i.e., F = 1). The comparison al-
+lows us to exclude (at a 2σ confidence level) the combinations (w0, wa) for which
+φw0,wa < φlow. The result is shown as a brown exclusion region in Figure 2.
+Of course, the above approach is very conservative, since we assume that the
+whole baryonic mass is in stars, and that the baryon mass of DM haloes is related to
+the DM mass through the universal baryon fraction (no loss of baryons). However, the
+very fact that the objects are characterized by high star formation rates (≳ 200M⊙yr−1,
+[18]) indicates that a sizable fraction of baryons is in the form of gas. Properly ac-
+counting for such a gas fraction would yield larger values of Mh associated with the
+observed M∗ and, hence, tighter constraints. However, gas mass estimates for these
+
+Figure 2: Exclusion regions at a 2σ confidence level in the w0 − wa plane derived
+from the observed abundance of luminous sub-mm galaxies at z = 4.5 − 5.5 [18].
+The brown region corresponds to the assumption that the observed stellar masses are
+related to the DM mass through the baryon fraction fb (M∗ = fbMh). The red region
+corresponds to adoption of the DM mass derived from the measured cross-correlation
+function of H-dropouts (see text).
+objects are affected by extremely large uncertainties (∼ a factor of 10) related to the
+uncertainties affecting the photometric redshifts and to all the assumptions required
+to convert the sub-mm continuum flux into a gas mass. To bypass this and to derive
+more realistic constraints for DDE models, we analyze the clustering properties of the
+H-dropouts. We base our analysis on the procedure adopted by [18], who estimated
+the halo mass function from the measured correlation function, modified as described
+in [1] to be adapted to a generic cosmology. We find that Mh = 1013M⊙ constitutes
+a 2σ lower limit for the value of the DM mass for any DDE model. The resulting
+exclusion region in the w0 − wa plane is shown in red in Figure 2. While the ΛCDM
+model remains marginally consistent with the observations, a much larger fraction
+of the w0 − wa parameter space is excluded by the abundance of optically invisible,
+sub-mm galaxies at z ∼ 5.
+
+2
+Excludedby
+the abundance of submm galaxies
+at z=4.5-5.5
+1
+0
+SNae+QSo
+-1
+-2
+-2.0
+-1.5
+-1.0
+-0.5
+w
+03.3
+SPT0311-58 at z = 6.9
+The most massive system detected at z ≥ 6 is a far-infrared luminous object at red-
+shift z = 6.9 identified in the 2500 deg2 South Pole Telescope (SPT) survey [19].
+High-resolution imaging revealed this source (denoted SPT0311-58) to be a pair of ex-
+tremely massive star-forming galaxies, with the larger galaxy (SPT0311-58W) form-
+ing stars at a rate of 2900 M⊙/yr and largely dominating over the companion. An
+elongated faint object seen at optical and near-infrared wavelengths is consistent with
+a nearly edge-on spiral galaxy at z ≃ 1.4 acting as a gravitational lens for the source,
+with an estimated magnification µ = 2.
+The molecular gas mass was estimated from ALMA observations, both based
+on the CO luminosity and from a radiative transfer model [20]. We adopt the latter
+value of MH2 ≃ (3.1 ± 1.9) × 1011M⊙ as a baseline, since it is based on a detailed fit
+with a model built ad-hoc to study the interstellar medium properties of this object
+and does not require assumptions on the conversion factor from the CO line to the H2
+mass [19]. To estimate the DM mass of the host halo of this galaxy, we assume that
+Mh = (M∗ + Mgas)/ fb, where Mgas is the total gas mass. Since the stellar mass cannot
+be directly measured due to the extremely faint optical emission of the galaxy (likely
+due to the large dust extinction), we can infer a lower limit on the stellar content
+from existing measurements of the molecular gas fraction fH2 = MH2/(M∗ + MH2).
+We consider the most conservative value fH2 = 0.8 measured on high-z star-forming
+galaxies (see references in [1]) and a more realistic value fH2 = 0.4 suggested by both
+theoretical models [21] and observations [22]. Assuming that H2 constitutes 80% of
+the gas mass at high redshifts (an upper limit according to [23, 24]) leads to a DM
+mass ranging from ≃ 2 × 1012M⊙ to ≃ 6 × 1012M⊙. An even larger DM mass would
+be consistent with the observations if the object lost the majority of its molecular gas
+content.
+To estimate the rareness of such a system in all the considered DDE cosmolo-
+gies, we compute the Poisson probability of finding such a massive object within the
+volume probed by the SPT survey, for different combinations (w0, wa). We follow the
+method in [25] adapted to a generic cosmology and take into account the uncertainties
+in the measured value of MH2 as described in [1]. We consider the total area of the
+SPT survey, although the effective area is potentially much smaller. In fact, most of
+the objects in the survey are strongly lensed, indicating that a source must be grav-
+itationally lensed to exceed the flux threshold for inclusion in the observations. For
+this reason, in the following we also consider the case of an effective area reduced
+
+by 1/10. From the rareness, we compute the associated 2σ exclusion regions in the
+w0 − wa plane. The result is shown in Figure 3.
+Figure 3: Exclusion regions at a 2σ confidence level in the w0 − wa plane for two
+different inferred DM masses of SPT0311–58: 2 × 1012M⊙ (red area) and 2 × 1012M⊙
+(yellow area). The left panel assumes the full SPT survey area of 2500 deg2 while the
+right one assumes 250 deg2.
+In the case Mh = 6 × 1011M⊙, corresponding to the assumption of fH2 = 0.4
+for the H2 gas fraction, a major portion of the w0 − wa plane is excluded, although
+the ΛCDM case (w0 = −1, wa = 0) remains allowed. The excluded region includes
+both the larger wa cases allowed by the quasar method (blue region) and the cases
+w0 ≥ −0.6 allowed by the CMB + weak lensing results, which shows the potential
+impact of our results. Tighter constraints are obtained assuming an area of 250 deg2,
+shown in the right panel of Figure 3.
+3.4
+High-z galaxies from early JWST results
+We finally exploit the very recent, early JWST results to derive even tighter constraints
+on DDE models. We compare the maximum stellar mass density ρmax,w0,wa(> M∗)
+allowed by different cosmologies with the unexpected large stellar mass density mea-
+sured by Labb´e et al. [26], who observed seven galaxies with M∗ ≥ 1010M⊙ at
+7 < z < 11. To derive the most stringent limits on cosmological models, we fo-
+cus on their most massive bin, i.e. M∗ ≥ 1010.5M⊙ (derived assuming a conservative
+Chabrier initial mass function [27]), in the redshift range 9 < z < 11, yielding a stellar
+
+2
+2
+ExcludedbySPT031158atz=6.9
+ExcludedbySPT031158atz=6.9
+(22
+=2500deg)
+1
+1
+0
+0
+SNaet
+SNae+QSO
++QSO
+-1
+-1
+-2
+-2
+-2.0
+-1.5
+-1.0
+-0.5
+-2.0
+-1.5
+-1.0
+-0.5
+M
+W
+0mass density of ρobs ≃ 106M⊙/Mpc3.
+We compute the predicted maximal stellar mass density for different (w0, wa)
+combinations. We assume Mh = M∗/ fb and adopt an even more conservative value for
+the baryon fraction of fb = 0.18 [28]. We rescale the observed stellar mass density for
+the volume and luminosity correction factors to convert from ΛCDM to a generic DDE
+model as explained in Sect. 3.1. We derive the proper confidence level for exclusion
+for each considered cosmology by calculating the probability that ρmax,w0,wa(> ¯M∗) <
+ρobs(>
+¯M∗), where ¯M∗ = 1010.5M⊙. We run a Monte Carlo simulation to account for
+the observational uncertainties, assigning an errobar of 0.5 dex to the stellar mass to
+account for systematics related with the SED fitting procedure [10].
+The resulting exclusion regions in the w0 − wa parameter space is shown in Fig-
+ure 4 for different confidence levels, and compared with regions allowed by existing
+probes. The ΛCDM case is excluded at almost 2σ level, while a major fraction of the
+parameter space is excluded with high confidence level. Our probe severely restricts
+the region in the DDE parameter space allowed by other methods, and exclude almost
+all the region favored by the distant quasar method.
+4
+Discussion and Conclusions
+We have determined exclusion regions in the w0 −wa parameter space of DDE models
+from the abundance of massive galaxies at early (z > 4.5) epochs. Adopting the most
+conservative assumptions for the ratio between the observed baryonic component and
+the DM mass, as well as conservative choices for the cosmological parameters, we
+have derived robust constraints that do not depend on the details of the baryon physics
+involved in galaxy formation. In addition, our results do not depend on the nature of
+the DM component [2].
+All our probes exclude a major fraction of the parameter space favored by the
+quasar distances [15], including their best-fit combination w0 ≃ −0.8 and wa ≃ −1.5.
+If confirmed, recent JWST observations are in tension with a ΛCDM scenario at a
+2σ confidence level. Our results leave open the possibility that the present tension
+in the value of H0 between the values derived from Planck and those obtained from
+local luminosity distance measurements [29] may be solved in DDE models, since the
+combinations (w0, wa) that allow for the reconciliation of the different observations
+[30, 31] include values outside our exclusion region.
+Our constraints will be greatly tightened when improved, reliable measurements
+of the actual baryon fraction in galaxies, and of the relative weight of each baryonic
+
+Figure 4: Exclusion regions in the w0 − wa plane derived from the observed stellar
+mass density at z = 10 [26]. The excluded regions above each coloured line corre-
+spond to different confidence levels shown in the upper bar. Our exclusion regions
+are compared with the 2σ and 3σ contours allowed by CMB+weak lensing (grey and
+dark-grey regions) and by the combination of the same data with the Hubble diagram
+of supernovae and quasars (blue regions), derived from Fig. 4 of [15].
+component, will be available. Increasing the statistics of high-redshift massive objects
+will also greatly tighten present constraints by reducing the uncertainties associated
+with the low abundance of these galaxies.
+A critical issue is associated with the systematics dominating the error budget
+in the mass estimates, especially at high redshift. The advent of JWST has for the first
+time opened the possibility to measure the rest-frame optical emission at high redshift
+[32], which was previously possible, despite being subject to a high noise level, only
+for very few bright and isolated sources detected with the Spitzer telescope. Early
+JWST observations have revealed their potential impressive impact in constraining
+cosmological models, as also shown by independent analyses [33, 34]. However,
+JWST observations are extremely new and may still be subject to revision. In particu-
+lar, the results of [26] were derived on the basis of the first set of calibrations released
+by STScI. A 10-20% level revision in the NIRCam calibrations [35] is not expected to
+yield to revisions of the stellar mass-to-light ratios of the targets large enough to affect
+
+confidence level for exclusion
+0.90
+0.92
+0.95
+0.99
+1.00
+2
+1
+0
+r
+W
+-1
+SNae+QSo
+-2
+-2.0
+-1.8
+-1.6
+-1.4
+-1.2
+-1.0
+-0.8
+-0.6
+Wosignificantly mass estimates and our conclusions. Nevertheless, we caution that the
+effect on the overall shape of the galaxy SED (as well as the assumptions on the star
+formation histories adopted in the SED-fitting procedure [36]) may reflect in a non-
+linear way on the estimated physical parameters of some objects. Finally, we have just
+started studying in detail the physics of z ∼ 10 galaxies, and cannot exclude that the
+star formation process can be significantly different from the lower redshift Universe,
+where our models and estimate procedures are calibrated. In particular, as discussed
+by [37], the increase of gas temperatures in star-forming, high-redshift galaxies could
+result in an increasing contribution of massive stars to the galactic light, which would
+yield significantly lower values for the stellar masses compared to those measured by
+[26].
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+page_content=' Italy  Marco Castellano  INAF - Osservatorio Astronomico di Roma,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' via di Frascati 33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' 00078 Monte Porzio Catone (RM),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Italy  Abstract  This conference proceedings paper provides a short summary of the constraints pre-  sented by Menci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' (2020) and Menci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' (2022) to dynamical dark energy  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Dynamical dark energy (DDE) models have been proposed to address several  observational tensions arising within the standard Λ cold dark matter (ΛCDM) sce-  nario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Different DDE models, parameterized by different combinations of the local  value of the equation-of-state parameter w0 and its time derivative wa, predict different  maximal abundance of massive galaxies in the early Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' We use the observed  abundance of massive galaxies already in place at z ≳ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5 to constrain DDE models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  To this aim, we consider four independent probes: (i) the observed stellar mass func-  tion at z ∼ 6 from the CANDELS survey;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' (ii) the estimated volume density of massive  haloes derived from the observation of massive, star-forming galaxies detected in the  submillimeter range at z ∼ 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' (iii) the rareness of the most massive system detected at  z ∼ 7 by the SPT survey;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' (iv) the abundance of massive (M > 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5M⊙) galaxies at  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='03892v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='CO]  10 Jan 2023  z ∼ 10 as inferred from early JWST observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Our probes exclude a major frac-  tion of the DDE parameter space that is allowed by other existing probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' In particular,  early JWST results, if confirmed, are in tension with the standard ΛCDM scenario at  a 2σ confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  1  Introduction  The current theory of structure formation envisages all cosmic structures to form  from the collapse and the growth of initially tiny density perturbations of the dark  matter (DM) density field in a universe characterized by an accelerated expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Such an acceleration indicates that the dominant component (with density parameter  ΩΛ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='7) of the cosmic fluid must be composed of some form of dark energy (DE),  with equation-of-state parameter w = p/ρ ≤ −1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Although the nature of such a  component remains unknown, the simplest model assumes DE to be connected with  the vacuum energy, the so-called cosmological constant, with w = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' When coupled  with the assumption that DM is composed of nonrelativistic particles at decoupling,  such a scenario leads to the Λ cold DM (ΛCDM) standard cosmological model [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  While measurements of the Cosmic Microwave Background (CMB) have pro-  vided a first, strong confirmation of such a scenario, tensions have recently emerged  ([1] and references therein) and have stimulated an extended effort toward the inves-  tigation of more complex cosmological models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' One of the simplest physical alter-  natives is a DE with a time-dependent equation of state, also called dynamical dark  energy (DDE) (see [4] for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  The abundance of massive galaxies at high redshift constitutes a powerful probe  for cosmological models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' In fact, in the standard CDM scenario, large-mass DM  haloes become progressively rarer with increasing redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The exponential high-  mass tail of the mass function of DM haloes is expected to shift toward progressively  smaller masses for increasing redshift (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=', [5] for a review) at a rate that depends  on the assumed cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Hence, the comparison of the predicted abundance of  massive DM haloes at increasingly larger redshift with the observed abundance of  galaxies with corresponding stellar mass M∗ provides increasingly strong constraints  on the assumed cosmological framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Indeed, viable cosmological models must  allow for an evolution of the initial density perturbations that is fast enough to match  the abundance of massive galaxies observed to be in place at early epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Under (extremely) conservative assumptions and considering different observ-  ables, we compare the maximal abundance of massive galaxies predicted in different  DDE models at high redshifts with the measured abundance of the most massive sys-  tems observed to be already in place at the same redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' This conference proceed-  ings paper summarizes the results that are presented and discussed in [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  2  Method  We compute the expected abundance of DM haloes, as a function of redshift z and  DM mass Mh, in different DDE models adopting the most conservative assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  We adopt the Sheth & Tormen [6] mass function, within the Press and Schechter  formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Besides being physically motivated and tested against N-body simulations  for a variety of cosmologies, it has the most extended high-mass tail among the differ-  ent proposed forms, hence representing the most conservative choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  The high-mass exponential cutoff in the Sheth & Tormen mass function is criti-  cally determined by the cosmic expansion rate and by the growth factor, which depend  on the equation of state of DE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' For the latter, we use the CPL parameterization [7, 8]  in terms of the scale factor a:  w(a) = w0 + wa(1 − a)  (1)  where the parameter w0 represents the value of w at the present epoch, while wa is  its look-back time variation wa = −dw/da.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' In the above parameterization, the stan-  dard ΛCDM cosmology corresponds to w0 = −1 and wa = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' For each combination  (w0, wa) , we can compute the expected number of DM haloes of mass Mh as a func-  tion of redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' We refer the reader to [1] for a full description of the methodology  and assumptions on the various involved cosmological parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  To compare these cosmological predictions on abundance of DM haloes with  the measured abundance of galaxies it is necessary to take into account the baryon  physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' However, baryonic effects are degenerate with cosmology in determining the  expected galaxy abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' This can be bypassed by noticing that the ratio of galaxy  baryonic components (stellar mass or gas mass) to DM halo mass has an absolute  maximum at the cosmic baryon fraction fb ( fb ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='16, [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' In fact, the observed  abundance of galaxies with large mass in the baryonic component Mb places a lower  limit on the abundance of DM haloes with masses Mh ≥ Mb/ fb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Such a constraint  can be used to rule out cosmological models that do not allow for a sufficiently rapid  growth of galactic DM haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' In other words, since galaxies cannot outnumber their  DM haloes, any (w0, wa) combinations for which φw0,wa(Mh ≥ Mb/ fb, z) ≤ φobs(Mb, z)  can be excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Due to the exponential cutoff of the DM halo mass function and to its rapid  redshift evolution, the highest masses at the highest redshifts put the most stringent  limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' To adopt the most conservative assumptions, all our choices aim at maximizing  φw0,wa(Mh, z) and minimizing φobs(Mb, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  3  Results  We describe here the various observables considered to constrain the w0−wa parameter  space and the relevant regions excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='1  Stellar mass function at z = 6 from the CANDELS survey  We first compare with the observed stellar mass distribution of massive, distant galax-  ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Since stellar mass is a time-integrated quantity, it is not much sensitive to the de-  tails of the star formation history [10] and can be easily related to the DM mass of the  host halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' An extended wavelength coverage is essential for estimating stellar masses  from spectral energy distribution (SED) fitting, while a combination of survey volume  and depth is required to measure the abundance of distante, massive and rare galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  The CANDELS project [11, 12], taking advantage of the optical/near-infrared/mid-  infrared imaging provided by HST, Spitzer, and VLT on almost 1000 arcmin2 down  to faint fluxes, provides an ideal data set to base such a measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Here, we use  the mass function derived by Grazian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' [13], who used a spectral-fitting technique  to derive stellar masses for a galaxy sample with high-quality photometric redshifts  based on the CANDELS GOODS-South and UDS fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  We focus on their largest stellar mass bin (centered on M∗ = 8 × 1010M⊙, as-  suming a Salpeter initial mass function [14]) at z = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' These high redshifts and  large masses ensure that the mass functions predicted by the different DDE models  are in the full exponential regime, and are steep enough to make the comparison with  the observed number density discriminant for the different DDE models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  We associate the stellar mass M∗ to the host halo DM mass Mh using the relation  M∗ = F fbMh, where F describes the efficiency of baryon conversion into stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' We  consider three cases: (a) the extremely unrealistic case F = 1, corresponding to a  complete conversion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' (b) the more realistic case F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='25, as obtained from abundance  matching techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=', [16]) - this value however is derived assuming a ΛCDM  Figure 1: Exclusion regions at a 2σ confidence level in the w0 − wa plane derived  from the observed CANDELS stellar mass function at z ∼ 6 [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The brown, red,  and orange regions correspond to the assumption of F = 1, F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5 and F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='25,  respectively (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Our exclusion region is compared with the 2σ and 3σ contours  allowed by CMB+weak lensing (green regions) and by the combination of the same  data with the Hubble diagram of supernovae and quasars (blue region), derived from  Figure 4 of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The black dot corresponds to the ΛCDM case (w0 = −1, wa = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  halo mass function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' and (c) the conservative value F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5, derived as a conservative  upper limit on the star formation efficiency from hydrodynamical N-body simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Before comparing the predicted number density of DM haloes with observa-  tions, we rescale the observed volumes and luminosities from a ΛCDM assumption to  a generic cosmology through the factors fVol = VΛ/Vw0,wa and flum = D2  L,w0,wa/D2  L,Λ,  respectively, where V is the cosmological volume and D2  L is the square luminosity  distance used to convert observed fluxed into luminosities, hence stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' In  summary, for each combination (w0, wa), we compare the volume-corrected, observed  abundance of galaxies ˜φ = φobs fVol with stellar mass M∗ = 8 flum1010M⊙ at z ∼ 6 with  the predicted number density of DM haloes with DM masses larger than M∗/(F fb),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' φw0,wa(Mh ≥ M∗/(F fb)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The confidence for the exclusion Pexcl of each considered  DDE model is obtained from the probability distribution function p(˜φ) as the proba-  bility that the measured abundance ˜φ is larger than number density predicted by the  2  ExcludedbyCANDELS(2fields)  stellarmassfunctionatz-6byGrazianetal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' (2015)  1  0  SNae+QSo  1  CMB+WeakLensing  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5  w  0model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=', Pexcl(w0, wa) =  � ∞  φw0,wa p(˜φ)d ˜φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  We show in Figure 1 the region of the w0 −wa parameter space excluded at a 2σ  confidence level (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=', Pexcl ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='95) for F = 1, F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5 and F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The exclusion  region is overplotted on the regions allowed by CMB and weak lensing observations,  and on the one derived by the combination of the same data with the Hubble diagram  of supernovae and distant quasars [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Our probe significantly restricts the region in  the DDE parameter space allowed by other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' In particular, we exclude an ap-  preciable part of the region favored by the distant quasar method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Very similar results  are obtained by comparing DDE predictions to the [17] mass function of CANDELS  galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='2  Submillimeter detected massive galaxies at z ∼ 5  The population of galaxies identified in rest-frame optical and ultraviolet is known  to under represent the most massive galaxies, which have rich dust content and/or  old stellar populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' These are, however, detectable at submillimeter (sub-mm)  wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' [18] performed detailed sub-mm (870 µm) observations  with ALMA of a sample of IRAC–bright galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' They detected 39 star-forming  objects at z > 3, which are unseen in even the deepest near-infrared (H-band) imaging  with the HST (“H-dropouts”) and proved to be massive galaxies with stellar mass  extending up to M∗ ≃ 3 × 1011M⊙, with a median mass M∗ ≃ 4 × 1010M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  For such objects, we follow a procedure similar to that explained in the previous  section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' We compute the number density of galaxies with stellar masses in the bin  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='25 ≤ log(M∗/M⊙) ≤ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='75 (dominating the statistics of observed objects) at red-  shifts z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5, and derive the corresponding 2σ lower limit φlow(M∗) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='8 × 10−5  Mpc−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' To relate the observed stellar mass M∗ to the DM mass Mh, we first adopt  the highly conservative assumption Mh = M∗/ fb (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=', F = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The comparison al-  lows us to exclude (at a 2σ confidence level) the combinations (w0, wa) for which  φw0,wa < φlow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The result is shown as a brown exclusion region in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Of course, the above approach is very conservative, since we assume that the  whole baryonic mass is in stars, and that the baryon mass of DM haloes is related to  the DM mass through the universal baryon fraction (no loss of baryons).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' However, the  very fact that the objects are characterized by high star formation rates (≳ 200M⊙yr−1,  [18]) indicates that a sizable fraction of baryons is in the form of gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Properly ac-  counting for such a gas fraction would yield larger values of Mh associated with the  observed M∗ and, hence, tighter constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' However, gas mass estimates for these  Figure 2: Exclusion regions at a 2σ confidence level in the w0 − wa plane derived  from the observed abundance of luminous sub-mm galaxies at z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5 − 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5 [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  The brown region corresponds to the assumption that the observed stellar masses are  related to the DM mass through the baryon fraction fb (M∗ = fbMh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The red region  corresponds to adoption of the DM mass derived from the measured cross-correlation  function of H-dropouts (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  objects are affected by extremely large uncertainties (∼ a factor of 10) related to the  uncertainties affecting the photometric redshifts and to all the assumptions required  to convert the sub-mm continuum flux into a gas mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' To bypass this and to derive  more realistic constraints for DDE models, we analyze the clustering properties of the  H-dropouts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' We base our analysis on the procedure adopted by [18], who estimated  the halo mass function from the measured correlation function, modified as described  in [1] to be adapted to a generic cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' We find that Mh = 1013M⊙ constitutes  a 2σ lower limit for the value of the DM mass for any DDE model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The resulting  exclusion region in the w0 − wa plane is shown in red in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' While the ΛCDM  model remains marginally consistent with the observations, a much larger fraction  of the w0 − wa parameter space is excluded by the abundance of optically invisible,  sub-mm galaxies at z ∼ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  2  Excludedby  the abundance of submm galaxies  at z=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5  1  0  SNae+QSo  1  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5  w  03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='3  SPT0311-58 at z = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='9  The most massive system detected at z ≥ 6 is a far-infrared luminous object at red-  shift z = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='9 identified in the 2500 deg2 South Pole Telescope (SPT) survey [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  High-resolution imaging revealed this source (denoted SPT0311-58) to be a pair of ex-  tremely massive star-forming galaxies, with the larger galaxy (SPT0311-58W) form-  ing stars at a rate of 2900 M⊙/yr and largely dominating over the companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' An  elongated faint object seen at optical and near-infrared wavelengths is consistent with  a nearly edge-on spiral galaxy at z ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='4 acting as a gravitational lens for the source,  with an estimated magnification µ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  The molecular gas mass was estimated from ALMA observations, both based  on the CO luminosity and from a radiative transfer model [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' We adopt the latter  value of MH2 ≃ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='9) × 1011M⊙ as a baseline, since it is based on a detailed fit  with a model built ad-hoc to study the interstellar medium properties of this object  and does not require assumptions on the conversion factor from the CO line to the H2  mass [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' To estimate the DM mass of the host halo of this galaxy, we assume that  Mh = (M∗ + Mgas)/ fb, where Mgas is the total gas mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Since the stellar mass cannot  be directly measured due to the extremely faint optical emission of the galaxy (likely  due to the large dust extinction), we can infer a lower limit on the stellar content  from existing measurements of the molecular gas fraction fH2 = MH2/(M∗ + MH2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  We consider the most conservative value fH2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='8 measured on high-z star-forming  galaxies (see references in [1]) and a more realistic value fH2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='4 suggested by both  theoretical models [21] and observations [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Assuming that H2 constitutes 80% of  the gas mass at high redshifts (an upper limit according to [23, 24]) leads to a DM  mass ranging from ≃ 2 × 1012M⊙ to ≃ 6 × 1012M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' An even larger DM mass would  be consistent with the observations if the object lost the majority of its molecular gas  content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  To estimate the rareness of such a system in all the considered DDE cosmolo-  gies, we compute the Poisson probability of finding such a massive object within the  volume probed by the SPT survey, for different combinations (w0, wa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' We follow the  method in [25] adapted to a generic cosmology and take into account the uncertainties  in the measured value of MH2 as described in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' We consider the total area of the  SPT survey, although the effective area is potentially much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' In fact, most of  the objects in the survey are strongly lensed, indicating that a source must be grav-  itationally lensed to exceed the flux threshold for inclusion in the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' For  this reason, in the following we also consider the case of an effective area reduced  by 1/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' From the rareness, we compute the associated 2σ exclusion regions in the  w0 − wa plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The result is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Figure 3: Exclusion regions at a 2σ confidence level in the w0 − wa plane for two  different inferred DM masses of SPT0311–58: 2 × 1012M⊙ (red area) and 2 × 1012M⊙  (yellow area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The left panel assumes the full SPT survey area of 2500 deg2 while the  right one assumes 250 deg2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  In the case Mh = 6 × 1011M⊙, corresponding to the assumption of fH2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='4  for the H2 gas fraction, a major portion of the w0 − wa plane is excluded, although  the ΛCDM case (w0 = −1, wa = 0) remains allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The excluded region includes  both the larger wa cases allowed by the quasar method (blue region) and the cases  w0 ≥ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='6 allowed by the CMB + weak lensing results, which shows the potential  impact of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Tighter constraints are obtained assuming an area of 250 deg2,  shown in the right panel of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='4  High-z galaxies from early JWST results  We finally exploit the very recent, early JWST results to derive even tighter constraints  on DDE models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' We compare the maximum stellar mass density ρmax,w0,wa(> M∗)  allowed by different cosmologies with the unexpected large stellar mass density mea-  sured by Labb´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' [26], who observed seven galaxies with M∗ ≥ 1010M⊙ at  7 < z < 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' To derive the most stringent limits on cosmological models, we fo-  cus on their most massive bin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' M∗ ≥ 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5M⊙ (derived assuming a conservative  Chabrier initial mass function [27]), in the redshift range 9 < z < 11, yielding a stellar  2  2  ExcludedbySPT031158atz=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='9  ExcludedbySPT031158atz=6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='9  (22  =2500deg)  1  1  0  0  SNaet  SNae+QSO  +QSO  1  1  2  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5  M  W  0mass density of ρobs ≃ 106M⊙/Mpc3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  We compute the predicted maximal stellar mass density for different (w0, wa)  combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' We assume Mh = M∗/ fb and adopt an even more conservative value for  the baryon fraction of fb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='18 [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' We rescale the observed stellar mass density for  the volume and luminosity correction factors to convert from ΛCDM to a generic DDE  model as explained in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' We derive the proper confidence level for exclusion  for each considered cosmology by calculating the probability that ρmax,w0,wa(> ¯M∗) <  ρobs(>  ¯M∗), where ¯M∗ = 1010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' We run a Monte Carlo simulation to account for  the observational uncertainties, assigning an errobar of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5 dex to the stellar mass to  account for systematics related with the SED fitting procedure [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  The resulting exclusion regions in the w0 − wa parameter space is shown in Fig-  ure 4 for different confidence levels, and compared with regions allowed by existing  probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The ΛCDM case is excluded at almost 2σ level, while a major fraction of the  parameter space is excluded with high confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Our probe severely restricts  the region in the DDE parameter space allowed by other methods, and exclude almost  all the region favored by the distant quasar method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  4  Discussion and Conclusions  We have determined exclusion regions in the w0 −wa parameter space of DDE models  from the abundance of massive galaxies at early (z > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5) epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Adopting the most  conservative assumptions for the ratio between the observed baryonic component and  the DM mass, as well as conservative choices for the cosmological parameters, we  have derived robust constraints that do not depend on the details of the baryon physics  involved in galaxy formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' In addition, our results do not depend on the nature of  the DM component [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  All our probes exclude a major fraction of the parameter space favored by the  quasar distances [15], including their best-fit combination w0 ≃ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='8 and wa ≃ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  If confirmed, recent JWST observations are in tension with a ΛCDM scenario at a  2σ confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Our results leave open the possibility that the present tension  in the value of H0 between the values derived from Planck and those obtained from  local luminosity distance measurements [29] may be solved in DDE models, since the  combinations (w0, wa) that allow for the reconciliation of the different observations  [30, 31] include values outside our exclusion region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Our constraints will be greatly tightened when improved, reliable measurements  of the actual baryon fraction in galaxies, and of the relative weight of each baryonic  Figure 4: Exclusion regions in the w0 − wa plane derived from the observed stellar  mass density at z = 10 [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The excluded regions above each coloured line corre-  spond to different confidence levels shown in the upper bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Our exclusion regions  are compared with the 2σ and 3σ contours allowed by CMB+weak lensing (grey and  dark-grey regions) and by the combination of the same data with the Hubble diagram  of supernovae and quasars (blue regions), derived from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' 4 of [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  component, will be available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Increasing the statistics of high-redshift massive objects  will also greatly tighten present constraints by reducing the uncertainties associated  with the low abundance of these galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  A critical issue is associated with the systematics dominating the error budget  in the mass estimates, especially at high redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The advent of JWST has for the first  time opened the possibility to measure the rest-frame optical emission at high redshift  [32], which was previously possible, despite being subject to a high noise level, only  for very few bright and isolated sources detected with the Spitzer telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Early  JWST observations have revealed their potential impressive impact in constraining  cosmological models, as also shown by independent analyses [33, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' However,  JWST observations are extremely new and may still be subject to revision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' In particu-  lar, the results of [26] were derived on the basis of the first set of calibrations released  by STScI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' A 10-20% level revision in the NIRCam calibrations [35] is not expected to  yield to revisions of the stellar mass-to-light ratios of the targets large enough to affect  confidence level for exclusion  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='90  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='92  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='95  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='99  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='00  2  1  0  r  W  1  SNae+QSo  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='6  Wosignificantly mass estimates and our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Nevertheless, we caution that the  effect on the overall shape of the galaxy SED (as well as the assumptions on the star  formation histories adopted in the SED-fitting procedure [36]) may reflect in a non-  linear way on the estimated physical parameters of some objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Finally, we have just  started studying in detail the physics of z ∼ 10 galaxies, and cannot exclude that the  star formation process can be significantly different from the lower redshift Universe,  where our models and estimate procedures are calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' In particular, as discussed  by [37], the increase of gas temperatures in star-forming, high-redshift galaxies could  result in an increasing contribution of massive stars to the galactic light, which would  yield significantly lower values for the stellar masses compared to those measured by  [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  References  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
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+page_content='  Huterer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
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+page_content=' Dark energy two decades after: observables, probes,  consistency tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
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+page_content='  Del Popolo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' & Yesilyurt, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The cosmological mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Astronomy  Reports 51, 709–734 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Sheth, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=', Mo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' & Tormen, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Ellipsoidal collapse and an improved  model for the number and spatial distribution of dark matter haloes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Royal As-  tronomical Society, Monthly Notices 323, 1–12 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Chevallier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' & Polarski, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Accelerating Universes with Scaling Dark Matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  International Journal of Modern Physics D 10, 213–223 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Linder, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Exploring the Expansion History of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Physical Review  Letters 90, 091301 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Planck 2018 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' CMB power spectra and like-  lihoods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Astronomy & Astrophysics 641, A5 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Santini, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Stellar Masses from the CANDELS Survey: The GOODS-South  and UDS Fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The Astrophysical Journal 801, 97 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Grogin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' CANDELS: The Cosmic Assembly Near-infrared Deep Ex-  tragalactic Legacy Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The Astrophysical Journal Supplement Series 197,  35–+ (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Koekemoer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' CANDELS: The Cosmic Assembly Near-infrared Deep  Extragalactic Legacy Survey - The Hubble Space Telescope Observations, Imag-  ing Data Products and Mosaics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The Astrophysical Journal Supplement Series  197, 36–+ (May 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Grazian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The galaxy stellar mass function at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5≤z≤7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='5 in the CAN-  DELS/UDS, GOODS-South, and HUDF fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Astronomy & Astrophysics 575,  A96 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Salpeter, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The Luminosity Function and Stellar Evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The Astrophysi-  cal Journal 121, 161–+ (1955).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Risaliti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' & Lusso, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Cosmological Constraints from the Hubble Diagram of  Quasars at High Redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Nature Astronomy 3, 272–277 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Behroozi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' & Silk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The most massive galaxies and black holes allowed by  ΛCDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Royal Astronomical Society, Monthly Notices 477, 5382–5387 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Duncan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The mass evolution of the first galaxies: stellar mass functions  and star formation rates at 4 < z < 7 in the CANDELS GOODS-South field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Royal Astronomical Society, Monthly Notices 444, 2960–2984 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Wang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' A dominant population of optically invisible massive galaxies in  the early Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Nature 572, 211–214 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Marrone, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Galaxy growth in a massive halo in the first billion years of  cosmic history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Nature 553, 51–54 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Strandet, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' ISM Properties of a Massive Dusty Star-forming Galaxy  Discovered at z ∼ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The Astrophysical Journal Letters 842, L15 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
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+page_content='  Narayanan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
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+page_content=' & Dav´e, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Galaxy gas fractions at high redshift:  the tension between observations and cosmological simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Royal Astro-  nomical Society, Monthly Notices 426, 1178–1184 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Tacconi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Phibss: Molecular Gas Content and Scaling Relations in z ˜  1-3 Massive, Main-sequence Star-forming Galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The Astrophysical Journal  768, 74 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Lagos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Cosmic evolution of the atomic and molecular gas con-  tents of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Royal Astronomical Society, Monthly Notices 418, 1649–1667  (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Lagos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Which galaxies dominate the neutral gas content of the  Universe?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Royal Astronomical Society, Monthly Notices 440, 920–941 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Harrison, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' & Hotchkiss, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' A consistent approach to falsifying ΛCDM with  rare galaxy clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Journal of Cosmology and Astroparticle Physics 2013, 022  (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Labbe, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
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+page_content='  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Chabrier, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The Galactic Disk Mass Function: Reconciliation of the Hubble  Space Telescope and Nearby Determinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The Astrophysical Journal Letters  586, L133–L136 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Planck Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Planck 2015 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' XIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Dark energy and modified  gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Astronomy & Astrophysics 594, A14 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
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+page_content='  Kamionkowski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' & Riess, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' The Hubble Tension and Early Dark Energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Invited Review for Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' arXiv: 2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='04492 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content='  Di Valentino, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=', Melchiorri, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=', Linder, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
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+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
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+page_content=' A JWST/NIRCam Study of Key Contributors to Reionization:  The Star-forming and Ionizing Properties of UV-faint z ∼ 7 − 8 Galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gtE2T4oBgHgl3EQfcAfW/content/2301.03892v1.pdf'}
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+This work has been submitted to the IEEE for possible publication. Copyright may be transferred
+without notice, after which this version may no longer be accessible.
+arXiv:2301.02297v1  [cs.RO]  5 Jan 2023
+
+IEEE TRANSACTIONS ON ROBOTICS. PREPRINT VERSION. ACCEPTED NOVEMBER, 2022
+1
+Improving Self-Consistency in Underwater Mapping
+through Laser-Based Loop Closure (Extended)
+Thomas Hitchcox, Student Member, IEEE, and James Richard Forbes, Member, IEEE
+Abstract—Accurate,
+self-consistent
+bathymetric
+maps
+are
+needed to monitor changes in subsea environments and infras-
+tructure. These maps are increasingly collected by underwater
+vehicles, and mapping requires an accurate vehicle navigation
+solution. Commercial off-the-shelf (COTS) navigation solutions
+for underwater vehicles often rely on external acoustic sensors for
+localization, however survey-grade acoustic sensors are expensive
+to deploy and limit the range of the vehicle. Techniques from
+the field of simultaneous localization and mapping, particularly
+loop closures, can improve the quality of the navigation solution
+over dead-reckoning, but are difficult to integrate into COTS
+navigation systems. This work presents a method to improve
+the self-consistency of bathymetric maps by smoothly integrating
+loop-closure measurements into the state estimate produced by a
+commercial subsea navigation system. Integration is done using
+a white-noise-on-acceleration motion prior, without access to
+raw sensor measurements or proprietary models. Improvements
+in map self-consistency are shown for both simulated and
+experimental datasets, including a 3D scan of an underwater
+shipwreck in Wiarton, Ontario, Canada.
+Index Terms—Marine robotics, sensor fusion, SLAM, commer-
+cial off-the-shelf (COTS) systems.
+I. INTRODUCTION
+A
+CCURATE, self-consistent bathymetric maps are crit-
+ical for assessing the health of subsea environments
+and infrastructure. Increasingly, these maps are collected by
+autonomous underwater vehicles (AUVs) using a variety of
+on-board sensors, including cameras [1–3], sonar [4–6], and
+laser scanners [7, 8]. Since the map is constructed using the
+estimated AUV trajectory, long-term navigation accuracy is a
+prerequisite for building accurate maps.
+The standard navigation solution for commercial AUVs is
+a commercial off-the-shelf (COTS) inertial navigation sys-
+tem (INS), with acoustic aiding from a Doppler velocity
+log (DVL). The dead-reckoned precision of these systems is
+measured by drift rate as a percent of distance traveled, with
+high-quality DVL-INS systems achieving a drift rate of as
+low as 0.01 %. However, without localizing measurements the
+precision of the state estimate will deteriorate without bound,
+impacting long-term accuracy.
+Manuscript received 30 May 2022; revised 21 September 2022; accepted
+22 November, 2022. This work was supported in part by the Natural Sciences
+and Engineering Research Council of Canada and in part by Voyis Imaging
+Inc. through the Collaborative Research and Development program. The work
+of Thomas Hitchcox was supported by the McGill Engineering Doctoral
+Award program. This paper was recommended for publication by Associate
+Editor Maurice Fallon and Editor Francois Chaumette upon evaluation of the
+reviewers’ comments.
+T.
+Hitchcox
+(corresponding
+author)
+and
+J.
+R.
+Forbes
+are
+with
+the Department of Mechanical Engineering, McGill University, Mon-
+treal, QC H3A 0C3, Canada. thomas.hitchcox@mail.mcgill.ca,
+james.richard.forbes@mcgill.ca.
+Fig. 1: A point cloud map of an actual shipwreck collected
+in Wiarton, Ontario, Canada, where colour represents relative
+depth. The top map is generated using the state estimate pro-
+duced by a commercial off-the-shelf (COTS) Doppler velocity
+log-aided inertial measurement system (DVL-INS). The bot-
+tom map is generated using the proposed method, which con-
+ditions the DVL-INS estimate on loop-closure measurements
+without access to the raw DVL-INS sensor measurements. Note
+the improvement in map self-consistency when loop-closure
+measurements are included.
+Since GPS signals attenuate rapidly in water, AUV localiza-
+tion is primarily done using acoustics [9]. For example, long
+baseline (LBL) arrays are acoustic beacons installed on the
+seafloor that trilaterate the position of an AUV, much like an
+“acoustic GPS.” Short baseline (SBL) and ultrashort baseline
+(USBL) systems are affixed to a surface vessel, and measure
+the acoustic range and bearing of an underwater vehicle. These
+sensors are frequently deployed in a commercial setting, and
+have been used to aid AUV navigation in the literature, for
+example [10].
+Acoustic positioning systems enable accurate and precise
+AUV trajectory estimates, however they are expensive to
+deploy and limit the mission domain of the vehicle. For
+example, LBL systems are time-consuming to install and
+calibrate, while SBL and USBL systems require the presence
+of a large surface vessel. In addition, acoustic positioning
+
+3m2
+IEEE TRANSACTIONS ON ROBOTICS. PREPRINT VERSION. ACCEPTED NOVEMBER, 2022
+systems produce measurements with limited precision, which
+may lead to small irregularities in a composite map built from
+several overlapping measurements of the same area. This in
+turn may make it difficult to assess relative distances and
+deformation, or other measurements critical to subsea safety.
+A. Motivation
+Loop closures play a central role in many simultaneous lo-
+calization and mapping (SLAM) algorithms, whereby a vehicle
+returns to and is able to recognize a previously explored region
+of the map. Loop-closure measurements effectively “reset”
+any navigation drift accumulated throughout the loop [11],
+resulting in navigation solutions that are both more accurate
+and more precise than dead-reckoning, without the need
+for external localizing measurements. Multiple loop closures
+over time lead to bounded navigation drift and a more self-
+consistent map estimate, whereby the resulting map is free of
+irregularities and “double vision” effects produced by poorly
+aligned measurements, an example of which is shown in Fig. 1.
+This is not to be confused with the term consistent, which in
+the context of state estimation describes a solution for which
+the covariance bounds accurately reflect the error in the mean
+state estimate [12, Sec. 5.4.2].
+Previous applications of SLAM for underwater mapping
+leverage loop-closure measurements to improve map self-
+consistency. However, these applications have largely been
+implemented on research platforms with access to raw sensor
+measurements and full knowledge of the state estimation algo-
+rithm. In contrast, commercial “strapdown” DVL-INS systems
+for subsea navigation produce a state estimate, and due to their
+proprietary nature rarely provide access to
+1) raw sensor measurements, including interoceptive mea-
+surements uk, such as from an IMU, and exteroceptive
+measurements yℓ, such as from a DVL;
+2) a process model of the form
+˜xk = fk−1(xk−1, uk−1),
+(1)
+which describes how the vehicle moves throughout time;
+3) sensor models of the form
+˜yℓ = gℓ(xℓ, vℓ),
+(2)
+which allow for predicted measurements; and
+4) sensor noise and bias specifications, for example
+u(t) = ¯u(t) + β(t) + w(t),
+(3a)
+˙β(t) ∼ N(0, Q ˙βδ(t − t′)),
+(3b)
+w(t) ∼ N(0, Qwδ(t − t′)),
+(3c)
+where u is known to be corrupted by time-varying random
+walk bias β and Gaussian white noise w, characterized
+by power spectral densities Q ˙β and Qw, respectively.
+Commercial DVL-INS systems, for example the Sonardyne
+SPRINT-Nav 500 [13], are effectively “black boxes,” and
+their lack of transparency makes it difficult to incorporate
+loop-closure measurements using conventional state estimation
+tools [14, 15], as illustrated by the factor graph [16] in Fig. 2.
+T0
+T1
+T2
+TK
+· · ·
+e0(ˇT0, ˇP0)
+e1
+e2
+eK
+Fig. 2: A factor graph depicting the output from a commercial
+DVL-INS. Only the estimated trajectory ˇT0:K and incomplete
+marginal covariance matrices ˇP0:K are available, leading to
+the formation of prior factors e0:K. Without factors linking
+adjacent nodes, loop-closure corrections cannot propagate
+throughout the graph, and the trajectory cannot be updated.
+B. Prior Work
+The field of simultaneous localization and mapping has
+found ample application in the domain of subsea robotics.
+For example, [4] produced a self-consistent bathymetric map
+of two hydrothermal vents by aligning point cloud submaps
+generated using multibeam sonar. A distributed particle map-
+ping algorithm was described in [5], where particle weighting
+was determined based on the innovation between multibeam
+sonar measurements and the existing map. However, the map
+resolution was limited by the selection of a grid cell size.
+This limitation was later addressed in [17], which adopted
+Gaussian processes as a map representation. The submap
+alignment approach was followed by [18], which demonstrated
+improvements in submap simplification and point cloud align-
+ment in a harbour scanning application. Harbour scanning
+and surveillance was also the subject of [19], which used a
+feature-based approach to align point clouds collected from
+imaging sonar. Recent research has focused on more structured
+environments, for example ship hull inspection [1, 6, 20–22]
+and subsea infrastructure [7].
+These studies generally had access to the information enu-
+merated in Section I-A, and as a result were able to in-
+corporate loop-closure measurements using conventional state
+estimation techniques. For example, [4] applied loop-closure
+measurements within an extended Kalman filtering framework,
+and enjoyed access to raw navigation sensor measurements as
+well as a vehicle process model. Individual particles in [5]
+and [17] were propagated forward using DVL measurements
+and a constant-velocity motion model. The research platform
+used in the related ship hull inspection studies [1, 6, 20–22]
+produced raw DVL, IMU, and depth sensor measurements,
+while the platform in [3] had access to a variety of raw
+sensor measurements including stereo vision and profiling
+sonar. These applications used conventional pose-graph SLAM
+to incorporate loop-closure measurements produced by various
+exteroceptive sensors.
+C. Contribution
+This work describes a novel approach to underwater map-
+ping using a high-resolution laser line scanner and the output
+of a commercial DVL-INS navigation system. First, this work
+develops a robust laser-based front-end algorithm to produce
+high-precision loop-closure measurements by aligning point
+cloud scans collected in challenging underwater environments.
+
+HITCHCOX AND FORBES: IMPROVING SELF-CONSISTENCY IN UNDERWATER MAPPING THROUGH LASER-BASED LOOP CLOSURE (EXTENDED)
+3
+Next, this work shows how to cleanly fuse loop-closure mea-
+surements into the output of a survey-grade COTS DVL-INS
+system via factor graph optimization. As these commercial
+systems are typically “black boxes” which only provide a
+navigation estimate, the proposed approach shows how to
+systematically incorporate loop-closure measurements without
+access to raw sensor measurements or other information
+typically required in state estimation tasks. In contrast to
+previous approaches, the proposed methodology also enforces
+a smoothness requirement on the posterior trajectory estimate.
+This eliminates discontinuities often encountered in dead-
+reckoned trajectory estimates, and is critical for accurate
+feature detection in laser submaps. In summary, the proposed
+methodology describes a robust and comprehensive system
+for high-precision, self-consistent underwater mapping using
+COTS navigation systems. Improvements to both map self-
+consistency and the relative accuracy of the trajectory esti-
+mate are rigorously evaluated in simulation and on an actual
+underwater mapping dataset.
+D. Paper Organization
+This paper is organized as follows. Section II contains
+preliminary information on conventions used, state estimation
+on matrix Lie groups, and batch state estimation. Section III
+introduces the methodology, including the formulation of loop-
+closure measurements from laser scan data and the construc-
+tion of the batch optimization problem. Section IV contains
+results on simulated and field datasets. The paper concludes
+in Section V with a review of the findings and opportunities
+for future work.
+II. PRELIMINARIES
+A. Reference Frames and Navigation Conventions
+This section discusses the conventions for attitude and
+displacement used in this paper. A three-dimensional dextral
+reference frame Fa is composed of three orthonormal physical
+basis vectors. The position of physical point z relative to point
+w, denoted by r−→
+zw, is resolved in reference frame Fa as rzw
+a
+and in reference frame Fb as rzw
+b . These these quantities are
+related via rzw
+a
+= Cabrzw
+b , where Cab is a direction cosine ma-
+trix, C ∈ SO(3) = {C ∈ R3×3 | CCT = 1, det C = +1} [23].
+Time-varying quantities are indicated by the subscript (·)k,
+for example rzkw
+a
+describes the position of moving point z at
+time tk. In this work, point z is affixed to the vehicle, while
+w denotes the stationary point in the world. Body frame Fb
+rotates with the vehicle, while local geodetic frame Fa remains
+stationary. Both Fb and Fa are north-east-down (NED), in
+agreement with maritime convention.
+B. Matrix Lie Groups
+The attitude and position of a vehicle at time tk, collectively
+referred to as the vehicle’s “pose,” may be conveniently
+represented in 3D space as an element of matrix Lie group
+SE(3) [23, Sec. 7.1.1],
+Tzkw
+abk =
+�Cabk
+rzkw
+a
+0
+1
+�
+∈ SE(3),
+(4)
+with SE(3) = {T ∈ R4×4 | C ∈ SO(3), r ∈ R3}. Associated
+with every matrix Lie group is a matrix Lie algebra, defined as
+the tangent space at the group identify [24]. For SE(3), this is
+se(3) ≜ T1SE(3). For estimation problems involving matrix
+Lie groups, the matrix Lie algebra is a convenient space to
+represent perturbations and uncertainty. An element of se(3)
+is given by [25, Sec. 2.3]
+ξ∧ =
+�φ
+ρ
+�∧
+=
+�
+���
+0
+−φ3
+φ2
+ρ1
+φ3
+0
+−φ1
+ρ2
+−φ2
+φ1
+0
+ρ3
+0
+0
+0
+0
+�
+��� ∈ se(3),
+(5)
+where (·)∧ : R6 → se(3) is an isometric operator. The inverse
+of this operator is (·)∨ : se(3) → R6, such that (ξ∧)∨ = ξ. A
+Lie group and Lie algebra are related through the exponential
+map, which for matrix Lie groups is the matrix exponential,
+T = exp(ξ∧).
+(6)
+The matrix logarithm is used to return to the Lie algebra via
+ξ∧ = log(T).
+(7)
+Elements of the matrix Lie algebra are combined according to
+the Baker-Campbell-Hausdorff (BCH) equation,
+γ∧ = log
+�
+exp(ξ∧) exp(η∧)
+�
+.
+(8)
+An approximation to the BCH equation for ξ ≫ η is
+γ∧ ≈ (ξ + Jr(ξ)−1η)∧,
+(9)
+where Jr is the right Jacobian of SE(3) [23, Sec. 7.1.5].
+Errors on matrix Lie groups are defined multiplicatively.
+This work uses the left-invariant error definition,
+δT = T−1˜T,
+(10)
+where T is the current state estimate and ˜T is a state estimate
+generated from sensor measurements or prior information. The
+corresponding perturbation scheme is
+T = ¯T exp(−δξ∧),
+(11)
+with perturbation δξ ∼ N(0, P), P = E[δξ δξT] ∈ R6×6. Note
+the negative sign in (11) ensures consistency with the left-
+invariant error definition (10). The state estimate is therefore
+defined by mean estimate ¯T and covariance P.
+This work makes frequent use of the adjoint matrix Ad(T),
+which maps perturbations about the group identity to other
+group elements [24]. Formally,
+Ad(T)δξ ≜
+�
+Tδξ∧T−1�∨
+.
+(12)
+For SE(3), the adjoint matrix is [25]
+Ad(T) =
+� C
+0
+r×C
+C
+�
+,
+(13)
+where (·)× is the skew-symmetric operator [23, Sec. 7.1.2].
+The adjoint matrix is represented in the matrix Lie algebra as
+�
+ad(ξ∧
+1 )ξ2
+�∧ ≜
+�
+ξ∧
+1 , ξ∧
+2
+�
+= ξ∧
+1 ξ∧
+2 − ξ∧
+2 ξ∧
+1 ,
+(14)
+where [·, ·] is the Lie bracket [26, Sec. 10.2.6]. For se(3),
+ad(ξ∧) =
+�φ×
+0
+ρ×
+φ×
+�
+.
+(15)
+
+4
+IEEE TRANSACTIONS ON ROBOTICS. PREPRINT VERSION. ACCEPTED NOVEMBER, 2022
+C. Gaussian Processes
+A continuous-time Gaussian process (GP) may be viewed
+as a distribution over functions,
+f(t) ∼ GP
+�
+µ(t), Σ(t, t′)
+�
+,
+(16)
+where µ(t) is the mean function, and Σ(t, t′) is the covariance
+function [27, Sec. 2.2][23, Sec. 2.3]. For any finite collection
+of time steps t0:K, f(t0:K) follows a joint Gaussian distri-
+bution. The covariance function determines how individual
+function samples fi(t) covary over time. For example, a GP for
+which the covariance over time is large will be smoother than
+a GP for which the covariance over time is small. This work
+uses the zero-mean white noise GP, given by [23, Sec. 2.3]
+w(t) ∼ GP(0, Qδ(t − t′)),
+(17)
+where Q is a power spectral density matrix and δ(·) is the
+Dirac delta function.
+D. The White-Noise-On-Acceleration Motion Prior
+The white-noise-on-acceleration (WNOA) motion prior may
+be summarized by the following set of stochastic differential
+equations [28],
+˙T(t) = T(t)ϖb(t)∧,
+(18a)
+˙ϖb(t) ∼ GP(0, Qδ(t − t′)).
+(18b)
+Equation (18a) describes the continuous-time state kinemat-
+ics for SE(3), with ϖb the generalized velocity, such that
+Tϖ∧
+b ∈ se(3), with T a time increment. The subscript (·)b has
+been included to emphasize that ϖ is a body-frame quantity.
+The time rate of change of ϖ is distributed according to the
+zero-mean white noise Gaussian process in (18b), with power
+spectral density Q. Note that Q is a hyperparameter that needs
+to be tuned. This motion prior helps to enforce smoothness
+is the posterior state estimate, as E[ ˙ϖ] = 0. In discrete time,
+this implies E[ϖk] = ϖk−1. The white noise assumption also
+preserves sparsity in the upcoming batch problem [29].
+The WNOA assumption is reasonable in the context of
+subsea navigation, as AUV kinematics evolve slowly over
+time. With the inclusion of ϖ, the augmented navigation state
+becomes the ordered pair
+X = (T, ϖ) ∈ SE(3) × R6.
+(19)
+E. Batch State Estimation
+Given
+a
+set
+of
+exteroceptive
+measurements
+{yℓ}L
+ℓ=1,
+interoceptive measurements {uk}K−1
+k=0 , and prior estimate
+Y0 = ¯Y0 exp(−δη∧
+0 ), S0 =E[δη0 δηT
+0 ], the standard approach
+to batch estimation is to produce a maximum a posteriori
+(MAP) solution, given by
+ˆX = arg max
+X
+p
+�
+X | y1:L, u0:K−1, Y0
+�
+.
+(20)
+Under the Markov assumption, the joint probability in (20)
+may be factored as
+ˆX = arg max
+X
+L
+�
+ℓ=1
+p
+�
+yℓ
+��Xℓ
+� K
+�
+k=1
+p
+�
+Xk
+��Xk−1, uk−1
+�
+p
+�
+X0
+��Y0
+�
+.
+(21)
+Taking the negative log likelihood of (21) results in a nonlinear
+least-squares problem of the form
+ˆX = arg min
+X
+J(X),
+(22)
+where the objective function J(X) is given by
+J(X) = 1
+2
+L
+�
+ℓ=1
+��eℓ(¯yℓ, gℓ(¯Xℓ, 0))
+��2
+R−1
+ℓ
++ 1
+2
+��e0(¯Y0, ¯X0)
+��2
+S−1
+0
++ 1
+2
+K
+�
+k=1
+��ek(fk−1(¯Xk−1, ¯uk−1), ¯Xk)
+��2
+Q−1
+k .
+(23)
+In (23), ek, eℓ, and e0 are the interoceptive, exteroceptive,
+and prior errors, respectively, while fk−1 and gℓ represent the
+nonlinear process and measurement models, respectively. The
+notation ∥e∥2
+Σ−1 = eTΣ−1e denotes the squared Mahalanobis
+distance, and Qk and Rℓ represent the discrete-time covariance
+on the interoceptive and exteroceptive errors, respectively. To
+minimize (22), (23) is repeatedly linearized about the current
+state estimate ¯X, and the local minimizing solution found
+using, for example, Gauss-Newton or Levenberg-Marquardt.
+III. METHODOLOGY
+This section describes the primary contributions of this
+paper, namely the formulation of laser-based loop-closure
+measurements and the smooth incorporation of these measure-
+ments into a COTS DVL-INS trajectory estimate. An overview
+of the upcoming methodology is shown in Fig. 3.
+A. Loop Closures from Subsea Point Cloud Scans
+To correct for drift in the DVL-INS trajectory estimate,
+loop-closure measurements are obtained by aligning sections
+of the point cloud scan collected using a Voyis Insight Pro
+underwater laser scanner. The raw laser profiles are first
+filtered and registered to the trajectory estimate to produce
+a 3D point cloud. Loop-closure opportunities are identified at
+path crossings, and alignment is performed using a multi-step
+point cloud alignment algorithm.
+Prior DVL-INS
+trajectory
+estimate
+WNOA motion prior
+(Sec. II-D)
+Laser point
+measurements
+Other terms
+(Sec. III-B1)
+Loop-closure
+measurements
+(Sec. III-A)
+Batch state
+estimation
+(Sec. III-B)
+eobs.
+k
+erel.
+k
+eℓ
+rpzk
+bk
+ˇTzkw
+abk
+ek
+Posterior
+estimate ˆT
+zkw
+abk
+Fig. 3: A visual overview of Section III. Note the colour of
+the error terms is consistent with the factor graph of Fig. 6.
+
+HITCHCOX AND FORBES: IMPROVING SELF-CONSISTENCY IN UNDERWATER MAPPING THROUGH LASER-BASED LOOP CLOSURE (EXTENDED)
+5
+1) Point Cloud Generation
+The Voyis Insight Pro underwater laser scanner, pictured in
+Fig. 4, records 2D profile measurements of the seabed at a
+frequency of 20 Hz. To construct a 3D point cloud, individual
+laser profiles are registered to the prior DVL-INS trajectory
+estimate ˇTzkw
+abk via�rpw
+a
+1
+�
+= ˇTzkw
+abk Tsz
+bℓ
+�rpsk
+ℓk
+1
+�
+,
+(24)
+where rpsk
+ℓk
+∈ R3 is a laser measurement of point p at time
+tk resolved in the sensor frame, and Tsz
+bℓ ∈ SE(3) is a static
+extrinsics matrix. Where necessary, the DVL-INS trajectory is
+interpolated according to [30, Sec. 2.4]
+ˇTj = ˇTi exp
+�
+α log
+�
+ˇT−1
+i
+ˇTk
+��
+,
+(25a)
+α = tj − ti
+tk − ti
+,
+(25b)
+where ˇTj = ˇTzjw
+abj , and ti < tj < tk. The result of these oper-
+ations is a filtered point cloud P resolved in the local geodetic
+frame, P = {rpiw
+a
+}N
+i=1.
+2) Point Cloud Alignment
+The objective of point cloud alignment is to combine
+partially overlapping scans of the same 3D object or scene.
+In the context of SLAM, point cloud alignment is often
+performed to estimate the relative pose between two or more
+observations, for example to reduce odometry drift [31] or
+to bound navigation drift over time by closing large loops in
+the trajectory [32]. More formally, the problem of point cloud
+alignment may be expressed as
+T⋆
+12 = arg min
+T∈SE(3)
+1
+2
+N
+�
+i=1
+M
+�
+j=1
+bij·wij·
+��eij
+�
+T12, rpiz2
+b2
+, rpjz1
+b1
+���2
+Σ−1
+ij ,
+(26)
+where the pose T⋆
+12 =
+�
+Tz2z1
+b1b2
+�⋆ optimally aligns source cloud
+S = {rpiz2
+b2
+}N
+i=1 to target cloud T = {rpjz1
+b1
+}M
+j=1. The Boolean
+value bij = {0, 1} assumes a value of 1 if (pi, pj) represents an
+inlier correspondence, while wij ∈ [0, 1] is a correspondence
+weight, often computed using a robust cost function [33].
+T⋆
+12 is optimal in the sense that it minimizes the sum of
+squared weighted errors, often a combination of point-to-
+point and point-to-plane errors [34, 35], with associated error
+covariance Σij(Ri, Rj). Ri and Rj represent the covariance
+on the point measurements rpiz2
+b2
+and rpjz1
+b1
+, respectively, with
+Ri = E[δri δrT
+i ], δri = rpiz2
+b2
+− ¯rpiz2
+b2
+. A depiction of the point
+cloud alignment problem is shown in Fig. 5.
+Fig. 4: An Insight Pro underwater laser scanner developed by
+Voyis Imaging Inc. The beam emitter is on the left, while
+the camera is on the right. 3D point clouds are produced by
+triangulating the laser beam. The baseline between the emitter
+and the camera is approximately 1 m.
+Fig. 5: Generating a loop-closure measurement by aligning
+source submap S to target submap T . The vehicle trajec-
+tory appears as a dashed line. Submaps T = {rpjz1
+b1
+}M
+j=1 and
+S = {rpiz2
+b2
+}N
+i=1 are constructed from the point measurements
+at vehicle poses Tz1w
+ab1
+and Tz2w
+ab2 , respectively. Point cloud
+alignment produces the loop-closure measurement Tz2z1
+b1b2 .
+In this work, loop-closure locations are identified at simple
+path crossings on the (x, y) plane, at time stamps tℓ1 and tℓ2,
+with tℓ1 < tℓ2. Ordinarily, cross-covariance information would
+be used to determine the search region, and thus the required
+size of the submaps to construct, using a squared Mahalanobis
+distance test [36], however this information is absent from the
+DVL-INS trajectory estimate. Instead, given the inherently low
+drift rate of the DVL-INS [13], the source and target clouds
+are constructed using a simple (x, y) distance threshold, e.g.
+rpw
+a
+∈ T
+�� ���
+1
+0
+�
+(rpw
+a
+− ˇra
+zℓ1w)
+��
+2 ≤ δr⋆.
+(27)
+In this work a constant value of δr⋆ = 5 m appears to work
+well, however a gradually increasing threshold related to the
+length of the trajectory could also be used.
+To provide a body-frame relative pose measurement (26),
+the point measurements are first resolved in the body frames,
+�rpzℓ
+bℓ
+1
+�
+=
+�ˇTzℓw
+abℓ
+�−1 �rpw
+a
+1
+�
+.
+(28)
+The point clouds are preprocessed by downsampling to a 5 cm
+grid, which reduces the amount of point data by approximately
+a factor of 10 while still preserving high-frequency features of
+the scanned object. Normal vectors are then estimated using
+the 40 nearest Euclidean neighbours. To account for cases of
+large navigation drift between observations, the TEASER++
+coarse alignment algorithm [37] is used to initialize an iterative
+closest point (ICP)-based fine alignment algorithm. To run
+TEASER++, FPFH feature descriptors [38] are computed at
+3D SIFT keypoints [39], and a set of putative correspon-
+dences is formed from the 10 nearest neighbour matches in
+33-dimensional FPFH space. Keypoints and descriptors are
+computed using the Point Cloud Library (PCL) v1.9 [40].
+Default values are used for all TEASER++ parameters.
+The combination of SIFT keypoints and FPFH descriptors
+was selected for this application following an alignment study
+on the shipwreck field dataset introduced in Section IV-D.
+
+2Z1
+616
+IEEE TRANSACTIONS ON ROBOTICS. PREPRINT VERSION. ACCEPTED NOVEMBER, 2022
+In this dataset, a vehicle makes eight passes over a small
+shipwreck, producing eight point cloud submaps and 28 unique
+submap pairs. 27 of the 28 pairs were then aligned using
+TEASER++ and different detector/descriptor combinations,
+with one submap pair excluded due to insufficient overlap. The
+study includes three keypoint detectors and two 3D feature
+descriptors. The keypoint detectors are SIFT, ISS [41], and
+Harris 3D keypoints [42], while the feature descriptors are
+FPFH and SHOT [43]. These detectors and descriptors were
+included in the study both due to their prevalence in the
+point cloud alignment literature and the availability of an
+open-source implementation in PCL v1.9. SIFT and Harris
+3D keypoint parameters were tuned slightly to obtain several
+hundred keypoints in each submap, while default values were
+used for ISS keypoints. For fairness, both FPFH and SHOT
+descriptors used the same search radius value of 0.25 m.
+Each submap pair was then aligned by TEASER++ using
+each of the six detector/descriptor combinations. The results
+are given in Table I, which lists summary statistics on at-
+titude errors ∥δφ∥ and position errors ∥δρ∥ in the format
+50 % · 90 % · MAX. Pose errors were computed between
+each TEASER++ relative pose estimate ˜Ti and the ground-
+truth relative pose Ti, computed from a well-initialized ICP
+alignment, as
+δξi = log(T−1
+i
+˜Ti)∨.
+(29)
+Examining Table I, the combination of SIFT keypoints and
+SHOT descriptors (second row) delivers the lowest median
+attitude error (0.42 deg), as well as the lowest 50 % and 90 %
+position errors. However, this combination produced at least
+one outlier measurement from the 27 submap pairs, while the
+SIFT+FPFH combination (first row) produced zero outliers. In
+addition, the SIFT+FPFH combination produced reasonable
+median and 90 % errors. Note the extremely large position
+errors in Table I are due to failed alignments producing a
+180 deg “flip” of the (relatively flat) point cloud submaps. The
+submaps are measured at a range of approximately 7 m, thus
+“flipped” alignments produce a relative body-frame position
+error of more than twice this value.
+As the objective of a coarse alignment algorithm is to
+robustly initialize ICP as close to ground-truth as possible,
+TABLE I: Summary statistics from the keypoint detec-
+tor and descriptor alignment study, reported in the format
+50 % · 90 % · MAX. Attitude errors ∥δφ∥ and position errors
+∥δρ∥ are computed according to (29). The lowest value in
+each column is indicated in bold font. Si = SIFT, I = ISS,
+H = Harris3D, F = FPFH, and So = SHOT.
+KP
+D
+∥δφ∥ [deg]
+∥δρ∥ [m]
+Si
+F
+0.44 ·
+1.05
+·
+1.54
+0.04 · 0.09 · 0.16
+So
+0.42 ·
+0.86
+· 92.59
+0.03 · 0.07 · 3.33
+I
+F
+0.49 ·
+0.84
+· 52.34
+0.04 · 0.09 · 2.31
+So
+0.61 · 176.28 · 179.72
+0.05 · 20.12 · 22.56
+H
+F
+0.83 · 21.14 · 178.44
+0.08 · 3.05 · 22.48
+So
+0.83 · 52.32 · 164.84
+0.06 · 4.22 · 18.28
+TABLE II: ICP preprocessing and alignment parameters
+Stage
+Configuration
+Description
+Preprocessing
+VoxelGrid
+Downsample to 5 cm grid
+Normals
+40 nearest neighbours
+Keypoints
+3D SIFT
+PCL v1.9 implementation
+Descriptors
+FPFH
+PCL v1.9 implementation
+Coarse align.
+TEASER++
+10 matches, default params.
+ICP data assn.
+KDTree
+Single nearest neighbour
+ICP error min. Mixed
+Pt-Pl if v(pj) < 3e−2
+Outlier reject.
+FRMSD
+Default params. from [49]
+Termination
+Diff.
+∥δφi∥2 < 1e−2 rad,
+and
+∥δρi∥2 < 1e−3 m
+Counter
+20 iterations max
+the SIFT+FPFH combination was selected for this application.
+Note that TEASER++ was chosen for the coarse alignment
+algorithm as it has been shown in extensive point cloud
+alignment studies [37] to outperform other coarse alignment
+methods, for example FGR [44] and RANSAC [45].
+For the fine alignment step, this work uses the Weighted
+Optimal Linear Attitude and Translation Estimator (WOLATE)
+algorithm [46] within an ICP-based alignment scheme. Align-
+ment errors are formulated between each point in the source
+cloud and their single nearest neighbour in the target cloud.
+A combination of point-to-point and point-to-plane errors
+are used, with the surface variation v(pj) [47] of the target
+points determining the type of error used for each asso-
+ciation. Following the study in [48], the Fractional Root
+Mean Squared Distance (FRMSD) robust cost function [49]
+is used for outlier rejection when aligning structured scans,
+such as shipwrecks. The algorithm terminates when the pose
+differential δξi between two successive iterations falls below a
+threshold, or when a maximum number of iterations is reached.
+Following the recommendations in [50] for best practices when
+reporting ICP algorithms, the preprocessing steps and relevant
+parameters are summarized in Table II.
+3) Loop-Closure Measurement Model
+Point cloud alignment yields the loop-closure measurement
+Ξℓ1ℓ2 ≜ T
+zℓ2zℓ1
+bℓ1bℓ2 =
+�
+T
+zℓ1w
+abℓ1
+�−1
+T
+zℓ2w
+abℓ2 ∈ SE(3),
+(30)
+and, given the perturbation scheme (11), the noise model is
+Ξℓ1ℓ2 = gℓ(ˇTℓ1, ˇTℓ2, δξΞ)
+(31a)
+= ¯Ξℓ1ℓ2 exp(−δξ∧
+Ξ),
+(31b)
+δξΞ ∼ N(0, RΞ),
+(31c)
+where the shorthand ˇTℓi = ˇT
+zℓiw
+abℓi , i = 1, 2 is used for read-
+ability. The covariance RΞ on the loop-closure measurement
+may be obtained from the point cloud alignment algorithm in a
+number of ways, for example the linearization-based approach
+in [51].
+
+HITCHCOX AND FORBES: IMPROVING SELF-CONSISTENCY IN UNDERWATER MAPPING THROUGH LASER-BASED LOOP CLOSURE (EXTENDED)
+7
+B. Updating the Trajectory
+1) Formulating the Objective Function
+The objective is now to condition the prior DVL-INS
+trajectory estimate on the newly available loop-closure mea-
+surements. This is accomplished through nonlinear batch state
+estimation, described in Section II-E. This section describes
+how the error terms in the batch problem are formulated, and
+Fig. 6 shows the resulting factor graph.
+First, the prior, process, and measurement errors must be
+defined. Given the augmented navigation state (19), errors
+must be defined for the SE(3) pose and for the generalized
+velocity. Using both the left-invariant error definition (10) and
+the constant velocity WNOA motion prior, the prior error is
+e0 =
+�
+eξ
+0
+eϖ
+0
+�
+=
+�
+log
+�
+T−1
+0 Y0
+�∨
+ϖ0 − ψ0
+�
+,
+(32)
+where (Y0, ψ0) is the prior estimate on the first navigation
+state. The process errors take the form
+ek =
+�
+eξ
+k
+eϖ
+k
+�
+=
+�
+log
+�
+T−1
+k ˜Tk
+�∨
+ϖk − ϖk−1
+�
+,
+(33)
+where the predicted pose at time tk,
+˜Tk = fk−1(Tk−1, ϖk−1) = Tk−1 exp(Tϖ∧
+k−1),
+(34)
+arises from a forward Euler discretization of the continuous-
+time SE(3) kinematics (18a) over an integration period of
+T = tk − tk−1. The loop-closure errors are
+eℓ = T−1
+ℓ2 ˜Tℓ2 = T−1
+ℓ2 Tℓ1Ξℓ1ℓ2,
+(35)
+where Tℓ1 and Tℓ2 are the two poses involved in loop-closure
+measurement ℓ. Additionally, it was discovered in testing that
+including a relative pose constraint between each subsequent
+pair of poses helped the loop-closure correction to propagate
+throughout the trajectory. The relative pose errors take the
+same form as the loop-closure errors,
+erel.
+k
+= T−1
+k ˜Tk = T−1
+k Tk−1Ξk−1,k,
+(36)
+where the relative pose measurements are taken directly from
+the initializing solution,
+Ξk−1,k = ˇT−1
+k−1ˇTk.
+(37)
+Finally, since roll, pitch, and depth are directly observable
+AUV states [5], errors are included of the form
+eobs.
+k
+=
+�
+�
+eφ1
+k
+eφ2
+k
+ez
+k
+�
+� = DEk log
+�
+T−1
+k ˇTk
+�∨
+,
+(38)
+where
+D =
+�1
+0
+0
+0
+0
+1
+�
+∈ R3×6,
+Ek =
+�1
+CabkJℓ(eφ
+k)
+�
+,
+(39)
+where Jℓ is the left Jacobian of SO(3) [23, Sec. 7.1.3], and
+eξ
+k =
+�
+(eφ
+k)T
+(eρ
+k)T�T .
+(40)
+The least-squares objective function (23) is then augmented as
+Jaug.(X) = J(X) + 1
+2
+K
+�
+k=1
+���erel.
+k
+��2
+R−1
+rel. +
+��eobs.
+k
+��2
+R−1
+obs.
+�
+,
+(41)
+X0
+X1
+X2
+XK
+· · ·
+e0
+eobs.
+1
+eobs.
+2
+eobs.
+K
+eℓ
+e1
+eK
+erel.
+1
+erel.
+2
+erel.
+3
+erel.
+K
+e2
+e3
+Fig. 6: The factor graph corresponding to the batch state
+estimation problem. The formation of WNOA factors ek and
+relative pose factors erel.
+k
+allow corrections from loop-closure
+factor eℓ to propagate throughout the graph.
+where Rrel. and Robs. are considered to be additional hyperpa-
+rameters. Together, the relative pose errors (36) promote loop-
+closure propagation, while the WNOA errors (33) promote
+smoothing. The hyperparameters Q and Rrel. may be tuned
+to control the smoothness of the posterior solution, while
+Robs. is tuned to ensure the posterior does not stray too far
+in observable dimensions.
+The batch estimation problem is visualized in the factor
+graph of Fig. 6. Note that, in contrast to the initial factor graph
+in Fig. 2, there are now factors linking adjacent nodes. This
+will allow corrections from the loop-closure measurements to
+propagate throughout the pose graph, as required.
+2) Minimizing the Objective Function
+To minimize (41), the estimation errors are repeatedly
+linearized about the current navigation state estimate ¯X. Per-
+turbing the navigation state as
+T = ¯T exp(−δξ∧),
+(42a)
+ϖ = ¯ϖ + δϖ,
+(42b)
+δx =
+�
+δξT
+δϖT�T ,
+(42c)
+the prior error (32) is linearized as
+e0 = ¯e0 + F0
+0δx0 + M0δy0,
+(43)
+where δy0 =
+�
+δηT
+0
+δψT
+0
+�T, S0 = E[δy0 δyT
+0 ], and where the
+prior Jacobians are
+F0
+0 = blkdiag(Jℓ(¯eξ
+0)−1, 1),
+(44a)
+M0 = blkdiag(−Jr(¯eξ
+0)−1, −1),
+(44b)
+with Jℓ being the left Jacobian of SE(3). Note that detailed
+derivations of the work appearing in this section are available
+in the supplementary material in Appendix A. The discrete-
+time process errors (33) are linearized as
+ek = ¯ek + Fk
+k−1δxk−1 + Fk
+kδxk,
+(45)
+where the process error Jacobians are given by
+Fk
+k−1 =
+�Uk−1
+Vk−1
+0
+−1
+�
+,
+(46a)
+Uk−1 = − Jr(¯eξ
+k)−1 Ad(exp(−T ¯ϖ∧
+k−1)),
+Vk−1 = TJr(¯eξ
+k)−1Jr(T ¯ϖk−1),
+Fk
+k =
+�
+Jℓ(¯eξ
+k)−1
+0
+0
+1
+�
+.
+(46b)
+
+8
+IEEE TRANSACTIONS ON ROBOTICS. PREPRINT VERSION. ACCEPTED NOVEMBER, 2022
+The loop-closure errors (35) are linearized as
+eℓ = ¯eℓ + Hℓ
+ℓ1δξℓ1 + Hℓ
+ℓ2δξℓ2 + MℓδξΞ,
+(47)
+with corresponding Jacobians
+Hℓ
+ℓ1 = − Jr(¯eℓ)−1 Ad(¯Ξ−1
+ℓ1ℓ2),
+(48a)
+Hℓ
+ℓ2 = Jℓ(¯eℓ)−1,
+(48b)
+Mℓ = − Jr(¯eℓ)−1.
+(48c)
+The relative pose errors (36) are linearized in the same manner.
+Finally, errors on the observable states (38) are linearized by
+approximating
+Cabk = ¯Cabk exp
+�
+−δφ×
+k
+�
+≈ ¯Cabk
+�
+1 − δφ×
+k
+�
+.
+(49)
+Assuming eρ
+k → 0 as the optimization proceeds, this yields
+eobs.
+k
+= ¯eobs.
+k
++ Hobs.
+k δξk,
+(50)
+Hobs.
+k
+= DEk(¯Cabk,¯eφ
+k)Jℓ(¯eξ
+k)−1.
+(51)
+The final step is to determine the covariance on the discrete-
+time WNOA process errors. This is done by discretizing the
+power spectral density Q via [52, (4.110)]
+Qk =
+� tk
+tk−1
+A(tk, s)L(s)Q(s) (A(tk, s)L(s))T ds,
+(52)
+where A, L characterize the continuous-time error kinematics,
+which for the WNOA motion prior take the form
+δ˙x(t) =
+�− ad( ¯ϖ∧)
+−1
+0
+0
+�
+�
+��
+�
+A
+δx(t) +
+�0
+1
+�
+����
+L
+δw(t),
+(53)
+with δw(t) ∼ GP(0, Q(t − t′)). The exact solution to (52)
+may be obtained via the matrix exponential [53], however to
+avoid this expensive operation this work makes use of a third-
+order approximation in A [52, (4.119)],
+Qk ≈ TΥ + T 2
+2
+�
+AΥ + ΥAT�
++ T 3
+6
+�
+A2Υ + 2AΥAT + Υ
+�
+AT�2�
+(54)
++ T 4
+24
+�
+A3Υ + 3A2ΥAT + 3AΥ
+�
+AT�2
++ Υ
+�
+AT�3�
+,
+where Υ = LQLT. Finally, the minimizing solution for a
+single iteration of Gauss-Newton is
+δx⋆ =
+� δξ⋆
+δϖ⋆
+�
+= −
+�
+ΓTWΓ
+�−1
+ΓTWe.
+(55)
+Jacobian Γ is given by
+Γ =
+�
+FT
+HT
+Hrel.T
+Hobs.T�T
+,
+(56)
+F =
+�
+����
+F0
+0
+F1
+0
+F1
+1
+...
+...
+FK
+K−1
+FK
+K
+�
+���� ,
+(57)
+H =
+�
+��
+H1
+ℓ1
+H1
+ℓ2
+...
+HL
+ℓ1
+HL
+ℓ2
+�
+�� ,
+(58)
+Hrel. =
+�
+��
+Hrel.,1
+0
+Hrel.,1
+1
+...
+...
+Hrel.,K
+K−1
+Hrel.,K
+K
+�
+�� ,
+(59)
+Hobs. =
+�
+��
+0
+Hobs.
+1
+...
+Hobs.
+K
+�
+�� ,
+(60)
+and weighting matrix W = Σ−1 is described by
+Σ = blkdiag
+�
+M0S0MT
+0 , Q1:K, R1:L, Rrel.
+1:K, Robs.
+1:K
+�
+,
+(61)
+where, for the loop-closure errors,
+Rℓ = MℓRΞMT
+ℓ .
+(62)
+The column matrix of errors is simply
+e =
+�
+eT
+0
+eT
+1:K
+eT
+1:L
+�
+erel.
+1:K
+�T
+�
+eobs.
+1:K
+�T�T
+.
+(63)
+Finally, in accordance with the perturbation scheme (42a, 42b),
+the state update is given by
+T ← T exp(−δξ∧
+⋆ ),
+(64a)
+ϖ ← ϖ + δϖ⋆.
+(64b)
+3) Rejecting False Loop-Closure Measurements
+Measurement outliers are inevitable in real-world robotics
+problems, and a robust implementation of the proposed
+methodology requires a method to identify and reject false
+loop-closure measurements. Many approaches exist in the lit-
+erature for rejecting loop-closure measurement outliers, for ex-
+ample switchable constraints [54], expectation-maximization
+[55], and graduated non-convexity [56].
+This application uses a recently developed adaptive robust
+cost function (RCF) to reject false loop-closure measure-
+ments, owing to its ability to handle multivariate, mixed-
+unit error definitions, such as loop-closure errors (35), in a
+statistically sound manner [57]. The RCF assigns a weight
+wℓ(ϵℓ(eℓ)) ∈ (0, 1] to loop-closure error eℓ according to the
+Mahalanobis distance associated with the error,
+ϵℓ(eℓ) = ∥eℓ∥Σ−1
+ℓ
+∈ R≥0,
+(65)
+where, ordinarily, the covariance Σℓ on the (relative) loop-
+closure measurement would be the relative uncertainty com-
+puted between the two vehicle poses involved in the mea-
+surement [58]. Since the DVL-INS output does not contain
+the joint covariance information required to properly compute
+Σℓ, a constant value is used here,
+Σℓ = blkdiag(σ2
+φout1, σ2
+ρout1),
+(66)
+with σφout = 1 deg and σρout = 1 m. These 1σ values reflect the
+low heading uncertainty of the survey-grade DVL-INS used in
+the field experiments, as well as the static search bound used
+for the loop-closure detection method (27).
+
+HITCHCOX AND FORBES: IMPROVING SELF-CONSISTENCY IN UNDERWATER MAPPING THROUGH LASER-BASED LOOP CLOSURE (EXTENDED)
+9
+IV. RESULTS
+A. Assessing the Quality of the State Estimate
+The methodology described in Section III conditions an
+existing state estimate on newly available loop-closure mea-
+surements. Since loop closures provide relative constraints
+between poses, as shown in Fig. 6, it is expected that this
+approach will
+1) reduce relative pose errors throughout the trajectory; and
+2) produce a more self-consistent point cloud map, as mea-
+sured by a reduction in the point disparity error in
+overlapping regions.
+1) Measuring Errors in the Estimated Trajectory
+A pose-based relative error metric based on [59] is used
+to measure the accuracy of the estimated trajectory. Let
+ˆTk ∈ SE(3) represent the estimated pose at time tk. The pose
+at time tk relative to the pose at time tℓ is
+δˆTℓk = ˆT−1
+ℓ
+ˆTk,
+(67)
+where ˆTℓ is taken to be the earliest pose involved in any
+loop-closure measurement. The relative pose error may then
+be expressed as
+Erel.
+k
+= δT−1
+ℓk δˆTℓk =
+�δCk
+δrk
+0
+1
+�
+,
+(68)
+where δTℓk is (67) evaluated using the ground-truth trajectory.
+Relative pose errors on SE(3) may then be expressed in the
+Lie algebra as
+δerel.
+k
+= log
+�
+Erel.
+k
+�∨
+=
+�
+(δφrel.
+k )T (δρrel.
+k )T�T .
+(69)
+However, since the AUV trajectories studied in this work
+are largely planar, the accuracy of the trajectory estimate is
+reported as the norm of the relative displacement error δrk
+projected on the (x, y) plane,
+erel.
+k
+=
+���1
+0�
+δrk
+��
+2 .
+(70)
+Assessing performance using a relative metric such as (68)
+avoids the problem of aligning the estimated and ground-truth
+trajectories, which would be required if attempting to provide
+an absolute performance metric [59]. In addition to being
+intuitive and easy to visualize, the relative planar displacement
+metric (70) allows for direct comparison to other navigation
+solutions within the subsea industry, where position drift is
+often reported on the (x, y) plane as a percentage of distance
+traveled. For estimates incorporating multiple loop closures
+along the length of the trajectory, the relative displacement
+error (70) is expected to remain bounded over time.
+2) Measuring Self-Consistency in the Point Cloud Map
+Performance is also assessed by evaluating the self-
+consistency in overlapping regions of the point cloud map. A
+point cloud map generated from an accurate trajectory estimate
+is expected to be well-aligned, or “crisp.” In contrast, a map
+produced using a drifting trajectory estimate will see “double
+vision” effects in overlapping regions due to poorly aligned
+scans. To assess self-consistency in the point cloud map, this
+work uses the point disparity metric from [60]. For point
+clouds S = {rpiw
+a
+}N
+i=1 and T = {rpjw
+a
+}M
+j=1, this metric is
+erel.
+j
+= ∥rpjw
+a
+− rpiw
+a
+∥2 ,
+(71)
+where point pj in T is the nearest Euclidean neighbour to point
+pi in S. Note (71) is only computed within the intersection
+of S and T to prevent cloud size from biasing the metric.
+The point disparity metric is relative, and may be computed
+without access to ground-truth information [60], making it
+especially important for field trials where a ground-truth map
+is not available. Note the point disparity metric is susceptible
+to map-to-map error, whereby an erroneous group of two
+or more well-aligned submaps would produce low disparity
+errors, despite separation from the true submap group. The
+results in the following studies were visually checked to ensure
+the absence of this error, though extending (71) to account for
+map-to-map error is an interesting avenue for future research.
+B. Hyperparameter Values
+The methodology described in Section III involves three sets
+of hyperparameters. These are
+1) Q, the PSD on the white noise Gaussian process;
+2) Rrel.
+k , the covariance on the relative pose errors; and
+3) Robs.
+k , the covariance on the roll, pitch, and depth errors,
+all of which are assumed to be observable.
+The hyperparameter sets take the form
+Q = diag(Q ˙ω1, Q ˙ν1),
+(72a)
+Rrel.
+k
+= diag(σ2
+φ1, σ2
+ρ1),
+(72b)
+Robs.
+k
+= diag(σ2
+rp1, σ2
+z ),
+(72c)
+where Q ˙ω, Q ˙ν are the power spectral densities on the body-
+centric angular and linear acceleration, respectively, and σ2
+φ,
+σ2
+ρ are the variances on the body-centric angular and linear
+displacement, respectively. σ2
+rp is the variance on vehicle roll
+and pitch, and σ2
+z is the variance on vehicle depth. While
+this hyperparameter structure is simple, it was found to work
+well for both simulated and field experiments, and generally
+makes physical sense. For example, [61] scales the values of
+a diagonal Q matrix to reflect the nonholonomic constraints
+of an automobile. In contrast, AUVs are highly maneuverable,
+leading to the selection of isotropic hyperparameters in (72).
+This section contains results from both simulated and field
+experiments. The hyperparameter values used to obtain each
+set of results are summarized in Table III. These values
+were hand-selected to produce good results without extensive
+tuning, and were modified based on the quality of the DVL-
+INS state estimate and the frequency at which DVL-INS data
+were available.
+TABLE III: Hyperparameter values used in experiments
+Hyperparameter
+Unit
+Simulated
+Field
+Q ˙ω
+rad2 s−3
+1e−2
+1e−2
+Q ˙ν
+m2 s−3
+9e−4
+1e−4
+σφ
+rad
+1e−3
+1e−6
+σρ
+m
+1e−4
+1e−6
+σrp
+deg
+5
+5
+σz
+m
+0.25
+0.25
+
+10
+IEEE TRANSACTIONS ON ROBOTICS. PREPRINT VERSION. ACCEPTED NOVEMBER, 2022
+C. Simulation Results: AUV Area Inspection
+Simulated output from a DVL-INS and a corresponding
+ground-truth trajectory were provided by industry collaborator
+Sonardyne. The DVL-INS output contains latitude and longi-
+tude, depth, and roll-pitch-yaw estimates for a simulated AUV
+deployment. At each time step, marginal variance estimates
+are available for the depth and heading states, and a joint
+Fig. 7: The simulated dataset. A single tie-line is intersected
+on the left by multiple “lawnmower” inspection passes.
+covariance estimate is available for planar position in the local
+geodetic frame. 30 min of DVL-INS output data is available at
+a frequency of 5 Hz. Note that the DVL-INS output has been
+heavily degraded by Sonardyne to better assess the ability of
+loop closures to mitigate navigation drift and does not reflect
+the performance of Sonardyne commercial products.
+The ground-truth trajectory is shown in Fig. 7, where the
+vehicle starts with a single tie-line followed by multiple
+planar “lawnmower” passes over a large inspection area.
+Loop closures occur at the eight intersections between the
+lawnmower passes and the tie-line. The prior “INS” estimate
+is then conditioned on the loop-closure measurements using
+the methodology from Section III to produce a posterior
+“INS+LC” trajectory estimate. Both estimates are then com-
+pared to the ground-truth solution (“GT”) using the metrics
+from Section IV-A. The application and propagation of loop-
+closure measurements within the WNOA framework is ex-
+pected to produce a more accurate navigation solution with a
+correspondingly more self-consistent point cloud map.
+(a) INS top
+(b) INS isometric. Heatmap scale is identical to Fig. 8a.
+(c) INS+LC top
+(d) INS+LC isometric. Heatmap scale is identical to Fig. 8c.
+Fig. 8: Heatmaps representing the relative displacement error for the INS and INS+LC trajectories. The left column shows a
+top view, while the right column shows an isometric view to illustrate the smoothing effects of the WNOA terms. Note the
+series of spikes at the far ends of Fig. 8d, where the vehicle completes low-radius turns to initiate the next survey pass.
+
+m
+.0
+error
+0.8
+: displacement
+0.6
+Relative 
+300
+.0
+1150
+350
+1100
+Northing [m]
+1050
+Easting m
+400
+10001200
+0.50+
+0.40
+1150
+error
+0.30
+Relative displacement
+1100
+0.20
+1050
+0.10
+1000
+0.00
+300
+350
+400
+Easting
+mm
+.0
+error
+0.8
+Relative displacement
+0.6
+0.4
+0.2
+300
+0.0
+1150
+350
+1100
+Northing [m]
+1050
+Easting m
+400
+100025
+min]
+20
+1600
+time
+Depth
+1610
+15
+300
+10
+1150
+5
+350
+1100
+1050
+Easting [m]
+Northing [m]
+0
+400
+10001200
+0.50+
+0.40
+1150
+error
+0.30
+Relative displacement
+1100
+0.20
+1050
+0.10
+1000
+0.00
+300
+350
+400
+Easting
+mHITCHCOX AND FORBES: IMPROVING SELF-CONSISTENCY IN UNDERWATER MAPPING THROUGH LASER-BASED LOOP CLOSURE (EXTENDED)
+11
+To illustrate the improvements in accuracy and the smooth-
+ing effect from the WNOA terms, Fig. 8 displays the relative
+displacement error erel.
+k
+as a “heatmap” for the prior INS and
+posterior INS+LC trajectories. From Fig. 8a it is clear that the
+prior estimate is not accurate, with relative displacement errors
+exceeding 0.5 m for much of the trajectory. Incorporating loop-
+closure measurements improves the accuracy of the trajectory
+estimate, as demonstrated by the cooler colours throughout the
+heatmap in Fig. 8c. The improvement is particularly noticeable
+around the tie-line at the bottom of Fig. 8c, however for many
+of the passes the effects extend hundreds of meters beyond
+the loop-closure location, increasing the accuracy across the
+entire inspection area. Relative displacement errors of 0.5 m
+may seem inconsequential at this scale, but may prove critical
+for certain subsea activities such as jumper pipe installation.
+The smoothing effects from the WNOA motion prior are
+(a) An underwater scene generated in the Stonefish AUV simulator [62]. From left to right: boat hull, dragon, propeller, and armadillo.
+The hull and propellor are available in Stonefish, while the dragon and armadillo are from the Stanford 3D Scanning Repository [63].
+(b) Prior elevation (INS)
+(c) Prior disparity (INS)
+(d) Posterior elevation (INS+LC)
+(e) Posterior disparity (INS+LC)
+(f) Ground-truth elevation (INS+GPS)
+(g) Ground-truth disparity (INS+GPS)
+Fig. 9: Scanning a 3D scene in the Stonefish AUV simulator [62] to evaluate the quality of the point cloud map. The four
+models in Fig. 9a are positioned along the tie-line near the loop-closure locations, and are scanned multiple times as the AUV
+completes the trajectory. Errors in the trajectory estimate are easily seen in the point disparity maps in the right column. Note
+the areas of high point disparity in the ground-truth map Fig. 9g are due to occlusion of the range-bearing scanner. Scanning
+the 3D models at different orientations produces different occlusion patterns, leading to non-overlapping areas of the map. This
+in turn leads to large nearest-neighbour distances to points in other scans, and thus a high point disparity error.
+
+A1607
+1606
+UU
+1605
+340
+1607
+335
+330
+1010
+325
+Easting[m]
+320
+Northing「ml
+315
+1604
+10000.30→+
+0.24
+0.18
+1605
+340
+1607
+0.12
+335
+330
+1010
+325
+0.06
+Easting[m]
+320
+Northing「m
+3151607
+1606
+U
+1605
+epth
+340
+1607
+335
+1605 
+330
+1010
+325
+Easting[m]
+320
+Northing「m
+315
+1604
+10000.30→+
+0.24
+0.18
+1605
+340
+1607
+0.12
+335
+330
+1010
+325
+0.06
+Easting[m]
+320
+Northing「m
+315
+10001607
+1606
+1605
+epth
+340
+1607
+335
+1605 
+330
+1010
+325
+Easting[m]
+320
+Northing「ml
+1604
+315
+10000.30→+
+0.24
+0.18
+1605
+340
+1607
+0.12
+335
+330
+1010
+325
+0.06
+Easting[m]
+320
+Northing「m
+315
+100012
+IEEE TRANSACTIONS ON ROBOTICS. PREPRINT VERSION. ACCEPTED NOVEMBER, 2022
+(a) Empirical probability density functions (EPDF)
+(b) Empirical cumulative distribution functions (ECDF)
+Fig. 10: Distributions on the point disparity error for each
+of the three navigation solutions for the simulated AUV area
+inspection dataset. The INS+LC solution greatly improves on
+the prior INS solution, virtually eliminating errors beyond
+6 cm. The relatively uniform error distribution for the INS
+solution is visible in the heatmap of Fig. 9c.
+visible in Fig. 8b and 8d, which are, respectively, isometric
+views of Fig. 8a and 8c. Here, the relative displacement error
+is represented both by the heatmap and the plot elevation.
+The prior INS estimate shows visible step changes in the
+relative displacement error, which are characteristic of the
+correction step of a filter and likely represent the effects of
+DVL measurements within the DVL-INS estimation algorithm.
+In contrast, the posterior INS+LC estimate has been visibly
+smoothed due to the presence of the WNOA error terms.
+Trajectory smoothing is important in this context, as the point
+cloud map is generated by registering individual laser profiles
+to the trajectory estimate. A trajectory with step changes will
+produce a map with step changes, which will surely impact
+TABLE IV: Critical values from the ECDF of Fig. 10b. For
+example, for the INS solution, 95.45 % (2σ) of point disparity
+errors are below 23.39 cm. Occluded areas in the simulated
+point cloud scan are responsible for the large errors at the
+upper ends of the ECDF distributions.
+Solution
+50 %
+1σ
+2σ
+3σ
+INS
+6.23 cm
+8.32 cm
+23.39 cm
+56.15 cm
+INS+LC
+1.61 cm
+2.27 cm
+11.60 cm
+44.70 cm
+INS+GPS
+0.81 cm
+1.13 cm
+11.48 cm
+45.64 cm
+front-end activities such as feature detection and point cloud
+alignment. A smooth, self-consistent map is also visually
+appealing, and will improve the accuracy of subsea metrology.
+Note the series of “spikes” in the relative displacement error
+of Fig. 8d, corresponding to the posterior trajectory estimate.
+These spikes occur at the beginning and end of each low-
+radius turn, suggesting that the smoothing effect of the WNOA
+terms may be erasing trajectory information in high-curvature
+regions. A geometry-based trajectory upsampling approach,
+for example based on scale-invariant density [64], is expected
+to resolve this, and will be explored as part of future work.
+To evaluate the effects of the optimization on map quality,
+an underwater scene was constructed and scanned using the
+open-source AUV simulator Stonefish [62]. Raw laser profiles
+were collected along the ground-truth trajectory, and were then
+registered to the INS and INS+LC trajectories to produce,
+respectively, the prior and posterior point cloud maps. The
+3D scene in Stonefish, as well as the resulting elevation and
+point disparity maps, are shown throughout Fig. 9.
+Compared to the ground-truth disparity map Fig. 9g, the
+prior map Fig. 9c shows high point disparity errors throughout,
+indicating a self-inconsistent map. In contrast, the disparity
+errors are largely resolved by the INS+LC solution, which
+incorporates both loop-closure measurements and smoothing
+into the INS estimate. The improvements are quantified in
+Fig. 10 by plotting an empirical probability density function
+(EPDF) and an empirical cumulative distribution function
+(ECDF) of the disparity error for each of the three solutions.
+For a quantitative comparison, Table IV lists critical values
+drawn from the ECDF curves. The INS+LC solution improves
+on the INS solution by producing a larger fraction of points
+with a lower point disparity error. This is especially evident
+in Fig. 10b, where the INS+LC solution converges to the GT
+solution around 6 cm. Assuming the remaining 5 % of errors
+lie in occluded regions, as explained in the caption of Fig. 9,
+the INS+LC solution has effectively eliminated point disparity
+errors beyond 6 cm. In contrast, 20 % of disparity errors from
+the prior INS solution exceed 10 cm. “Double-vision” effects
+of this magnitude arising from poor scan alignment are sure
+to complicate inspection and metrology tasks, even within the
+small domain of this simulation. Following the methodology
+from Section III, the INS+LC solution has produced a smooth,
+crisp, self-consistent point cloud map from which relative
+distance measurements may accurately be drawn.
+
+HITCHCOX AND FORBES: IMPROVING SELF-CONSISTENCY IN UNDERWATER MAPPING THROUGH LASER-BASED LOOP CLOSURE (EXTENDED)
+13
+(a) The test trajectory, with shipwreck
+section in green in northeast corner.
+(b) Shipwreck section, with different
+trajectory estimates and shipwreck area.
+(c) The boxed region from Fig. 11b, showing the outline
+of the main shipwreck structure.
+Fig. 11: Field deployment in Colpoy’s Bay, Wiarton, Ontario, Canada. The full trajectory is shown in Fig. 11a. The shipwreck
+section, shown in green in northeast corner of Fig. 11a, is 0.58 km long and took approximately 10.5 min to complete. The
+trajectory makes eight passes over the shipwreck area. The different navigation solutions are summarized in Table V.
+D. Field Results: Wiarton Shipwreck
+A field trial was conducted with Voyis Imaging Inc. in
+Colpoy’s Bay, Wiarton, Ontario, Canada. The bay is shallow
+and contains multiple shipwrecks and other manmade struc-
+tures, making it an ideal test location for Voyis’s surface vessel.
+The full test trajectory is shown in blue in Fig. 11a. A section
+of the trajectory, highlighted in green in the northeast corner of
+Fig. 11a, makes eight passes over a small shipwreck. Fig. 11b
+shows this section in detail, and Fig. 11c shows the main
+shipwreck structure segmented from the lakebed. This section
+of the trajectory, which is approximately 580 m long and took
+10.5 min to complete, is the focus of the field results.
+The surface vessel was equipped with a Sonardyne SPRINT-
+Nav 500 DVL-aided INS, a u-blox ZED-F9P high precision
+GNSS module equipped with a u-blox ANN-MB series high
+precision multi-band antenna, and a Voyis Insight Pro under-
+water laser scanner. GNSS data were post-processed using the
+Canadian Spacial Reference System Precise Point Positioning
+(CSRS-PPP) application [65], which in a recent study was
+found to be capable of measuring 2D position with a precision
+of 2 cm (1σ) [66].
+Three navigation solutions were generated from these data.
+The first solution is a dead-reckoned DVL-INS trajectory
+(“INS”), where the positioning precision of the SPRINT-
+Nav 500 has been manually degraded by Sonardyne from
+the nominal value of 0.02 % of distance traveled (CEP50)
+[52, Sec. 4.9.1.2][13]. The DVL-INS output is available at
+10 Hz. Note that this solution is representative of the state-
+of-the-art for high-grade commercial systems, and will be
+used to benchmark the proposed methodology. The second
+solution, referred to as “INS+LC,” applies the methodology
+of Section III to incorporate loop-closure measurements into
+TABLE V: Understanding the different navigation solutions
+for the Wiarton shipwreck field dataset.
+Solution
+Description
+INS
+Dead-reckoned DVL-INS solution, with position
+precision manually degraded by Sonardyne.
+INS+LC
+The dead-reckoned DVL-INS trajectory estimate
+conditioned on loop-closure measurements. Raw
+sensor measurements from the DVL-INS are
+inaccessible, and GNSS data is not used as part
+of this solution.
+INS+GPS
+DVL-aided INS solution with GNSS correction.
+the dead-reckoned DVL-INS estimate. Batch processing was
+performed offline in MATLAB, taking approximately 90 s to
+converge on a laptop with an E3-1505M v5 CPU and 16 GB
+of RAM. It is important to note that this solution is produced
+using the DVL-INS state estimate, without access to the raw
+DVL-INS sensor measurements. The third solution fuses the
+DVL-INS output with the GNSS data to form a ground-
+truth estimate (“INS+GPS”). The three navigation solutions
+are summarized in Table V, and the trajectory estimates are
+overlaid on the shipwreck area in Fig. 11b and 11c.
+Incorporating loop-closure measurements produces a more
+accurate trajectory estimate, as measured by the relative dis-
+placement error (70). Fig. 12 shows the relative displacement
+error, measured against the ground-truth INS+GPS estimate,
+for the dead-reckoned INS trajectory and the INS+LC trajec-
+tory with an increasing number of loop closures. The loop-
+
+N
+200 m
+500 ftINS
+-INS+LC
+680
+-INS+GPS
+670
+660
+Im
+650
+640
+630
+620
+610
+50
+60
+70
+80
+90
+Easting
+ImINS
+654
+-INS+LC
+INS+GPS
+652
+B
+650
+Northing 
+648
+646
+644
+64
+66
+68
+70
+72
+Easting|m14
+IEEE TRANSACTIONS ON ROBOTICS. PREPRINT VERSION. ACCEPTED NOVEMBER, 2022
+Fig. 12: Relative displacement errors for trajectory estimates
+incorporating an increasing number of loop closures. Loop-
+closure locations are marked as vertical dashed lines, and
+“INS+XLC” indicates the first X loop closures were used in
+generating the estimate. Incorporating loop closures bounds
+navigation drift over time. For numerical results, see Table VI.
+closure locations are marked with vertical dashed lines, with
+the first observation of the shipwreck occurring approximately
+40 s in to the trajectory. The relative displacement drift in the
+INS trajectory estimate increases without bound, while the
+maximum displacement error decreases monotonically as more
+loop-closure measurements are applied. Even a single loop-
+closure measurement at the end of the trajectory is effective in
+bounding the relative displacement drift over time, as demon-
+strated by the cyan line in Fig. 12. From Table VI, which
+summarizes the maximum error and final error as a percent of
+distance traveled for the different solutions, the final drift error
+for the “INS+1LC (last)” solution is 6.82e−3 % of distance
+traveled. This particular solution suggests an order of magni-
+tude improvement over state-of-the-art DVL-INS systems [13].
+Importantly, the dashed yellow “INS+0LC” curve in Fig. 12
+indicates that the posterior solution does not deviate far from
+the prior DVL-INS solution when loop-closure measurements
+are absent. This suggests that the proposed methodology may
+still be used to smooth the DVL-INS solution in the absence
+of loop-closure measurements, without sacrificing solution
+accuracy. For example, the maximum position drift error for
+the “INS-0LC” solution tabulated in Table VI is only 9 mm
+TABLE VI: Summary of drift errors from Fig. 12, with values
+drawn after the first shipwreck observation at 40 s.
+Solution
+Max drift [m]
+Final %DT
+INS
+0.658
+10.98e−2
+INS+0LC
+0.667
+10.82e−2
+INS+1LC
+0.378
+5.99e−2
+INS+3LC
+0.255
+4.12e−2
+INS+5LC
+0.150
+2.25e−2
+INS+7LC
+0.084
+6.83e−3
+INS+1LC (last)
+0.224
+6.82e−3
+greater than the maximum drift observed in the “INS” solution.
+However, note the proposed methodology is intended to be
+used in a targeted fashion in situations where at least one
+loop-closure measurement is available.
+In addition to the relative displacement errors summarized
+in Fig. 12, relative pose errors (69) are computed across
+the trajectory for the prior “INS” and posterior “INS+LC”
+solutions, with summary statistics given in Table VII. Relative
+attitude errors in Table VII are decomposed into body-centric
+roll, pitch, and yaw, while the rightmost column gives statistics
+on the Euclidean norm of the body-centric relative position er-
+rors. Interestingly, the proposed methodology has produced an
+increase in the relative body-centric pitch and roll errors, from
+median values of 1.1e−3 deg and 1.5e−5 deg, respectively,
+to 1.1e−1 deg and 1.2e−1 deg, respectively. Relative body-
+centric yaw errors remain largely unchanged by the proposed
+methodology, while trends in the relative body-centric position
+error generally follow the trend of the relative displacement
+error plotted in Fig. 12. For example, 90 % of body-centric
+position errors for the prior “INS” solution fall below 0.630 m,
+while the corresponding value for the posterior “INS+LC”
+solution is 0.108 m.
+An increase in roll and pitch errors may seem concerning,
+however the posterior errors remain low and bounded. Such
+errors were likely introduced in this field trial through a
+combination of small angular errors in the INS-laser extrinsics
+estimate and the relatively weak pitch and roll prior used in
+the optimization (see (38) and the value of hyperparameter σrp
+in Table III). The more important result is that relative body-
+centric position errors remain low and bounded when multiple
+loop-closure measurements are present.
+Barring measurement outliers, finding that loop closures
+improve trajectory accuracy is not particularly surprising in a
+conventional state estimation context. However, these results
+have been achieved following the methodology of Section III,
+TABLE VII: Summary statistics on relative pose errors (69) computed for the Wiarton field trial. Relative attitude errors |δφrel.
+i |
+have been broken down by component, while the rightmost column gives statistics on the norm of the relative body-centric
+position error. Cumulative statistics for each error are reported in the format 50 % · 75 % · 90 %.
+Solution
+|δφrel.
+1 | [deg]
+|δφrel.
+2 | [deg]
+|δφrel.
+3 | [deg]
+∥δρrel.∥ [m]
+INS
+1.1e−3 · 1.9e−3 · 2.5e−3
+1.5e−3 · 2.4e−3 · 3.5e−3
+1.5e−2 · 1.9e−2 · 2.2e−2
+0.482 · 0.562 · 0.630
+INS+LC
+1.1e−1 · 2.7e−1 · 5.3e−1
+1.2e−1 · 3.8e−1 · 5.7e−1
+1.5e−2 · 2.0e−2 · 2.2e−2
+0.077 · 0.096 · 0.108
+
+HITCHCOX AND FORBES: IMPROVING SELF-CONSISTENCY IN UNDERWATER MAPPING THROUGH LASER-BASED LOOP CLOSURE (EXTENDED)
+15
+without access to raw sensor measurements, a vehicle process
+model, exteroceptive sensor models, or sensor noise and bias
+specifications. The loop-closure corrections have instead been
+smoothly integrated into the DVL-INS estimate using the
+factor graph illustrated in Fig. 6, improving the accuracy of
+the trajectory estimate.
+Incorporating loop-closure measurements produces a more
+self-consistent point cloud map, as measured by the point
+disparity error (71). Fig. 13 shows the point disparity in the
+shipwreck area as a heatmap, for each of the three navigation
+solutions. The disparity is computed for each of the eight
+passes over the wreck, and is the Euclidean distance from
+each point in one pass to its nearest neighbour in the remaining
+seven passes. A highly accurate trajectory estimate is expected
+to produce a tightly overlapping, crisp point cloud map from
+the composite scans, with a low point disparity error.
+From a qualitative evaluation of Fig. 13b, the dead-reckoned
+INS trajectory estimate has clearly produced a self-inconsistent
+point cloud map. Areas around the ribs of the shipwreck have
+point disparity errors of around 20 cm, while one of the passes
+shows relatively large errors on the seabed owing to drift in
+the depth dimension. In contrast, both the posterior and the
+ground-truth estimates have produced highly self-consistent
+maps, with low point disparity errors throughout.
+Interestingly, the INS+LC solution produces a point cloud
+map that is more self-consistent than the ground-truth estimate.
+This is difficult to judge qualitatively from Fig. 13, however
+Fig. 14 shows the EPDF and ECDF of the disparity error
+for each of the three navigation solutions. Critical values
+from the ECDF are tabulated in Table VIII. In Fig. 14a,
+(a) Prior elevation (INS)
+(b) Prior disparity (INS)
+(c) Posterior elevation (INS+LC)
+(d) Posterior disparity (INS+LC)
+(e) Ground-truth elevation (INS+GPS)
+(f) Ground-truth disparity (INS+GPS)
+Fig. 13: Visualizing the point disparity error in the shipwreck area for the three navigation solutions. Left column: colour map
+indicates depth, and has been included for context. Right column: colour map indicates point disparity.
+
+5
+6.50
+Depth「m
+655
+6.25
+650
+6.00
+5.75
+645
+70
+Northingm
+65
+Easting|m0.20+
+5
+0.16
+655
+0.12
+650
+0.08
+0.04
+645
+70
+Northing「m
+65
+EastingIm5
+6.50
+6.25
+m
+655
+Depth [1
+6.00
+650
+5.75
+645
+70
+Northingm
+65
+Easting|m0.20+
+5
+m
+0.16
+7
+655
+0.12
+650
+0.08
+0.04
+645
+70
+Northingm
+65
+Easting「m5
+6.50
+6.25
+Depth「m
+655
+650
+6.00
+5.75
+645
+70
+Northingm
+65
+Easting|m0.20+
+5
+0.16
+655
+0.12
+650
+0.08
+0.04
+645
+70
+Northing「m
+65
+Easting「m16
+IEEE TRANSACTIONS ON ROBOTICS. PREPRINT VERSION. ACCEPTED NOVEMBER, 2022
+(a) Empirical probability density functions (EPDF)
+(b) Empirical cumulative distribution functions (ECDF)
+Fig. 14: Distributions on the point disparity error for each of
+the three navigation solutions for the Wiarton shipwreck field
+dataset. The posterior INS+LC solution produces a more self-
+consistent point cloud map than the ground-truth INS+GPS
+solution, likely owing to a combination of residual navigation
+and extrinsics errors. Critical values from the ECDF are
+summarized in Table VIII.
+the INS+LC curve peaks to the left of the INS+GPS curve,
+indicating a lower overall point disparity error and thus a more
+self-consistent point cloud map [5]. This is likely due to a
+combination of small estimation errors in the ground-truth
+solution and small errors in the scanner extrinsics estimate
+Tsz
+bℓ from (24). It should therefore come as no surprise that
+the INS+LC solution delivers a more self-consistent map,
+as the point disparity error is precisely what is minimized
+during point cloud alignment (26). For additional images of
+the shipwreck area generated using the prior and posterior
+navigation solutions, see Appendix B.
+TABLE VIII: Critical values from the ECDF of Fig. 14b.
+For example, for the INS solution, 95.45 % (2σ) of point
+disparity errors are below 8.51 cm. Note the improvement in
+the INS+LC solution over the INS+GPS solution.
+Solution
+50 %
+1σ
+2σ
+3σ
+INS
+1.51 cm
+2.24 cm
+8.51 cm
+18.89 cm
+INS+LC
+0.75 cm
+1.00 cm
+2.35 cm
+7.23 cm
+INS+GPS
+1.08 cm
+1.45 cm
+3.49 cm
+8.63 cm
+Again, this improvement in map self-consistency has been
+achieved without access to the standard ingredients available in
+typical state estimation problems. Visualizing the point cloud
+map and the resulting disparity errors is a straightforward
+way to verify that the loop-closure measurements have been
+successfully applied, and that the updates have been smoothly
+propagated throughout the trajectory.
+Compared to the GPS-aided solution, the improvement in
+map self-consistency that comes from leveraging loop-closure
+measurements may appear modest. However, an improvement
+on the order of centimeters may be consequential for certain
+subsea inspection tasks, such as measuring deformation in
+manmade structures. In this respect, the methodology of Sec-
+tion III offers a valuable addition to inspection and metrology
+work. This is especially true for dead-reckoned solutions, but
+remains true even when localizing measurements are available,
+for example LBL, USBL, or GPS measurements.
+A Monte Carlo experiment was conducted on the Wiarton
+field dataset to test the effectiveness of the loop-closure mea-
+surement outlier rejection method discussed in Section III-B3.
+To run the experiment, between one and five of the seven loop-
+closure measurements were randomly replaced by randomly
+generated measurements. Thirty Monte Carlo trials were con-
+ducted for each outlier corruption level, for a total of 150 trials.
+The number of trials at each corruption level was chosen to
+provide a representative statistical sample. Additionally, the
+experimental results obtained using 30 trials per corruption
+level were very similar to results obtained when using 20 and
+25 trials per level.
+The
+outlier
+measurements
+were
+generated
+to
+mimic
+the
+outliers
+experimentally
+observed
+in
+the
+detector/descriptor
+study
+in
+Section
+III-A2.
+Outlier
+position measurements were uniformly sampled so that
+∥rxy∥ ≤ 5 m
+and
+rz ∈ [[−0.5 m, 0.5 m] ∪ [13.5 m, 14.5 m]],
+with rout =
+�
+(rxy)T
+rz�T. This reflects both the planar search
+bound used to detect loop-closure candidates (27) as well as
+the range “flipping” effect discussed in Section III-A2. Outlier
+attitude measurements were uniformly sampled according to
+φout
+j
+∈ (−π, π] rad, j = 1, 2, 3. An outlier measurement Ξout
+ℓ1ℓ2
+is then generated according
+Ξout
+ℓ1ℓ2 =
+�Cout(φout)
+rout
+0
+1
+�
+.
+(73)
+Results from this experiment are summarized throughout
+Fig. 15. All 150 Monte Carlo trials are plotted in Fig. 15a,
+along with the ground-truth “INS+GPS” trajectory and the
+
+HITCHCOX AND FORBES: IMPROVING SELF-CONSISTENCY IN UNDERWATER MAPPING THROUGH LASER-BASED LOOP CLOSURE (EXTENDED)
+17
+(a) 150 Monte Carlo trajectories
+(b) Shipwreck section
+(c) Mean navigation drift by outlier corruption level
+Fig. 15: Trajectory estimates and associated relative drift errors (70) for 150 Monte Carlo trials. For each trial, between 1 and 5 of
+the seven loop-closure measurements are replaced by outlier measurements (73), with 30 trials conducted per outlier corruption
+level. The outlier rejection method discussed in Section III-B3 is effective in rejecting false loop-closure measurements, with
+no failures visible in Fig. 15a. The zoom in Fig. 15b show a graceful decay from the ground-truth “INS+GPS trajectory” to
+the prior “INS” estimate as more outlier measurements are added. This is confirmed by the relative displacement error plot in
+Fig. 15c, which simply follows the trend of measurement removal first seen in the ablation study of Fig. 12. The worst-case
+position drift at each time step measured across all 150 trials is shown as a grey patch in Fig. 15c. When compared against the
+relative displacement error from the prior “INS” estimate (blue line), the proposed methodology is seen to produce estimates
+that are, at any given time, no worse than the prior estimate, even in instances with extreme outlier rates.
+prior “INS” trajectory estimate. No visible navigation failures
+are seen in Fig. 15a, implying the adaptive robust cost func-
+tion is effective in rejecting false loop-closure measurements.
+A zoom of the shipwreck region in Fig. 15b shows the
+Monte Carlo trajectory samples gracefully decaying from the
+ground-truth solution to the prior estimate as more outliers
+are included. This behaviour is also seen in the relative
+displacement error plot of Fig. 15c, where the mean relative
+(x, y) navigation drift (70) is plotted for each outlier corruption
+level. The trend of higher outlier rates producing larger relative
+drift values is reminiscent of the ablation study summarized
+in Fig. 12, in which loop-closure measurements are simply
+removed from the solution. This provides sound evidence
+that the proposed outlier rejection algorithm is successful in
+identifying and removing false loop-closure measurements.
+Finally, the grey patch in Fig. 15c shows the worst-case
+relative displacement error at each time step across all 150
+Monte Carlo trials. Compared to the prior “INS” estimate
+(blue curve), it is clear that, for this dataset, the proposed
+methodology delivers worst-case posterior estimates that are,
+at any given time, no worse than the prior estimate, even in
+instances with extreme outlier rates.
+V. CONCLUSION
+This works presents a novel and comprehensive method
+for systematically conditioning the output of a COTS DVL-
+INS navigation system on loop-closure measurements for the
+purpose of improving the self-consistency [4, 59] of the re-
+sulting bathymetric map. The method relies on a combination
+of relative pose and white-noise-on-acceleration [28] error
+terms to smoothly integrate the measurements in a batch state
+estimation framework.
+The first contribution of this work is the development of
+a robust front-end algorithm for computing high-precision
+loop-closure measurements from 3D scans of challenging
+underwater environments. Second, loop-closure measurements
+are cleanly incorporated into an existing state estimate via a
+factor graph optimization framework, without access to raw
+sensor measurements, sensor models, or other information
+typically required in conventional state estimation problems.
+The effectiveness of the proposed method was demon-
+strated for both simulated and field datasets using loop-closure
+measurements from an underwater laser scanner. The same
+simple hyperparameter structure was used for both studies,
+with good results. For the field results, conditioning the dead-
+reckoned DVL-INS estimate on loop-closure measurements
+produced a markedly more self-consistent point cloud map of
+an underwater shipwreck. Incorporating all seven loop-closure
+measurements resulted in a maximum relative position drift
+of 8.4 cm over a 576 m trajectory, with a final position error
+of 6.83e−3 % of distance traveled. This represents an order
+of magnitude improvement over unaided commercial DVL-
+INS systems. Additionally, the proposed methodolgy was
+demonstrated to be robust to false loop-closure measurements.
+Future work will primarily focus on hyperparameter training
+through an expectation-maximization framework, for example
+[67] and [68]. The algorithm will be tested over longer tra-
+jectories with more varied terrain, including open seabed [8].
+Finally, future work may also incorporate image information,
+in the form of conventional image descriptors and textured
+point cloud maps.
+
+680
+670
+660
+Northing
+650
+640
+-INS
+630
+INS+7LC. 1 outlier
+INS+7LC. 2 outlier
+-INS+7LC.3 outlier
+620
+INS+7LC. 4 outlier
+INS+7LC,5 outlier
+-INS+GPS
+610
+50
+60
+70
+80
+90
+Easting[m654
+652
+650
+Northing
+648
+INS
+INS+7LC.1 outlier
+646
+suu
+INS+7LC. 2 outlier
+-INS+7LC.3 outlier
+-INS+7LC.4 outlier
+644
+-INS+7LC.5 outlier
+INS+GPS
+64
+66
+68
+70
+72
+Easting
+mO.
+INS
+INS+7LC. 1 outlier
+0.6
+INS+7LC. 2 outlier
+INS+7LC.3 outlier
+INS+7LC. 4 outlier
+e
+0.5
+INS+7LC,5 outlier
+error
+Worst case (150 trials)
+0.4
+0.3
+0.2
+0.1
+0
+100
+200
+300
+400
+500
+600
+0
+Time18
+IEEE TRANSACTIONS ON ROBOTICS. PREPRINT VERSION. ACCEPTED NOVEMBER, 2022
+ACKNOWLEDGMENT
+The authors would like to thank Ryan Wicks of Voyis
+for providing experimental data and guidance, and Martin
+Jørgensen of Sonardyne International for providing simulation
+data and helpful feedback.
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+2020.
+Thomas Hitchcox received his B.Eng. and M.Eng.
+degrees in mechanical engineering in 2015 and 2018,
+respectively, from McGill University, Montreal, QC,
+Canada. He is currently a Ph.D. Candidate with the
+Department of Mechanical Engineering at McGill.
+His research interests include state estimation, com-
+puter vision, and robust algorithms for point cloud
+filtering and alignment.
+James Richard Forbes James Richard Forbes re-
+ceived the B.A.Sc. degree in Mechanical Engineer-
+ing (Honours, Co-op) from the University of Water-
+loo, Waterloo, ON, Canada in 2006, and the M.A.Sc.
+and Ph.D. degrees in Aerospace Science and Engi-
+neering from the University of Toronto Institute for
+Aerospace Studies (UTIAS), Toronto, ON, Canada
+in 2008 and 2011, respectively. James is currently
+an Associate Professor and William Dawson Scholar
+in the Department of Mechanical Engineering at
+McGill University, Montreal, QC, Canada. James is
+a Member of the Centre for Intelligent Machines (CIM), and a Member of
+the Group for Research in Decision Analysis (GERAD). James was awarded
+the McGill Association of Mechanical Engineers (MAME) Professor of the
+Year Award in 2016, the Engineering Class of 1944 Outstanding Teaching
+Award in 2018, and the Carrie M. Derick Award for Graduate Supervision
+and Teaching in 2020. The focus of James’ research is navigation, guidance,
+and control of robotic systems.
+
+20
+IEEE TRANSACTIONS ON ROBOTICS. PREPRINT VERSION. ACCEPTED NOVEMBER, 2022
+APPENDIX A
+SUPPORTING DERIVATIONS
+A. Introduction
+The purpose of this appendix is to derive in detail the prior, process, and loop closure Jacobians appearing in Section III-B2
+of “Improving Self-Consistency in Underwater Mapping through Laser-Based Loop Closure.” Key identities from matrix Lie
+group theory are reviewed in Section A-B, and the white-noise-on-acceleration (WNOA) motion prior [A1] is reviewed in
+Section A-C. Section A-D examines the WNOA error kinematics. Finally, the necessary Jacobians are derived in Section A-E.
+The intent of this appendix is to make these derivations accessible, with key steps and identities indicated throughout. For a
+more detailed treatment of matrix Lie group theory, please consult the references cited throughout, particularly [A2] and [A3].
+B. Preliminaries
+1) Matrix Lie groups
+A matrix Lie group G is a set of real, invertible n × n matrices that is closed under matrix multiplication. Associated with
+every matrix Lie group is a matrix Lie algebra g, defined as the tangent space at the group identity, g ≜ T1G. The matrix Lie
+algebra is a vector space closed under the operation of the Lie bracket [A4, Sec. 10.2.6]. It is often more convenient to work
+with isometric representations of matrix Lie algebra elements, namely ξ ∈ Rd, where ξ∧ ∈ g.
+A Lie group and its corresponding Lie algebra are related through the exponential map. For matrix Lie groups this is simply
+the matrix exponential [A2, Sec. 7.1.3]. For X ∈ G, this leads to expressions of the form
+X = exp(ξ∧),
+(A.1)
+where ξ∧ is the representation of X in g, and ξ is the representation of X in Rd. Finally, the matrix logarithm is used to move
+from the matrix Lie group to the matrix Lie algebra, as in
+ξ∧ = log (X) .
+(A.2)
+The relationship between a matrix Lie group, its associated matrix Lie algebra, and the isometric space Rd, as well as several
+other quantities discussed throughout this document, is illustrated in Fig. A.1, which is inspired by, but modified from, [A5].
+G ⊂ Rn×n
+g ≜ T1G ⊂ Rn×n
+1
+0
+X
+TXG
+log (X)
+exp(A)
+A
+A∨
+ξ
+ξ∧
+Rd
+Ad(X)A
+A′
+ξ′
+Ad(X)ξ
+Fig. A.1: Matrix Lie groups, modified from [A5]. Mappings are shown between the matrix Lie group element X, its matrix Lie
+algebra representation A = ξ∧, and its Rd representation ξ. Note the difference between the adjoint operator Ad(·) and the
+adjoint matrix, Ad(·), discussed in Section A-B4. All perturbations are modeled in the matrix Lie algebra g (orange), defined
+as the tangent space at the group identity.
+
+HITCHCOX AND FORBES: IMPROVING SELF-CONSISTENCY IN UNDERWATER MAPPING THROUGH LASER-BASED LOOP CLOSURE (EXTENDED)
+21
+2) Errors and perturbations on matrix Lie groups
+There are four ways to define a matrix Lie group error [A6]. These are summarized in Table A.1, along with their
+corresponding perturbation schemes. The error definition and perturbation scheme are linked. For example the selection of a
+left-invariant error definition necessitates the use of a left-invariant perturbation scheme. To see this, consider
+δX = X−1 ¯X,
+(A.3a)
+exp(δξ∧) = X−1 ¯X,
+X exp(δξ∧) = XX−1 ¯X,
+X = ¯X exp(−δξ∧),
+(A.3b)
+where (A.3a) is the left-invariant matrix Lie group error and (A.3b) is the left-invariant perturbation scheme. This work uses
+a left-invariant error definition.
+3) The Baker-Campbell-Hausdorff (BCH) equation
+The BCH equation describes how to combine elements of the matrix Lie algebra [A4, Sec. 10.2.7],
+c∧ = log
+�
+exp(a∧) exp(b∧)
+�
+,
+(A.4)
+where a∧, b∧, c∧ ∈ g. Elements of g are therefore correctly combined on the group G, through application of the exponential
+map. However, the following approximation,
+c ≈ a + Jr(a)−1b,
+(A.5)
+is valid if b is small [A2, Sec. 7.1.5], where Jr(ξ) is the right Jacobian of G. The following approximation,
+c ≈ a + b,
+(A.6)
+is valid if both a and b are small. This leads to the following three useful identities related to the BCH equation,
+exp(ξ∧) exp(δξ∧) ≈ exp
+�
+(ξ + Jr(ξ)−1δξ)∧�
+,
+(A.7a)
+exp((ξ + δξ)∧) ≈ exp(ξ∧) exp
+�
+(Jr(ξ)δξ)∧�
+,
+(A.7b)
+exp(δξ∧
+1 ) exp(δξ∧
+2 ) ≈ exp
+�
+(δξ1 + δξ2)∧�
+.
+(A.7c)
+4) The adjoint operator and the adjoint matrix
+The adjoint operator maps the effects of perturbations about the group identity to other group elements. For X ∈ G and
+ξ∧ ∈ g, the adjoint operator Ad : g → g is defined as [A7, Sec. 2.5]
+Ad(X)ξ∧ ≜ Xξ∧X−1.
+(A.8)
+The adjoint matrix Ad : Rd → Rd encodes the effects of the adjoint operator directly on Rd [A3],
+Ad(X)ξ ≜
+�
+Xξ∧X−1�∨
+.
+(A.9)
+The adjoint matrix may also be defined in terms of the left and right group Jacobians [A2, Sec. 7.1.5],
+Ad(X) ≜ Jℓ(ξ)Jr(ξ)−1,
+(A.10)
+where Jℓ(ξ) = Jr(−ξ). Finally, the adjoint matrix exists in the matrix Lie algebra as
+�
+ad(ξ∧
+1 )ξ2
+�∧ ≜
+�
+ξ∧
+1 , ξ∧
+2
+�
+= ξ∧
+1 ξ∧
+2 − ξ∧
+2 ξ∧
+1 ,
+(A.11)
+where ξ∧
+1 , ξ∧
+2 ∈ g and [·, ·] is the Lie bracket [A4, Sec. 10.2.6]. Ad and ad are related through the exponential map,
+Ad(X) = exp
+�
+ad(ξ∧)
+�
+,
+(A.12)
+where X = exp(ξ∧).
+TABLE A.1: Matrix Lie group error definitions and corresponding perturbation schemes.
+Error definition
+Matrix Lie group error
+Perturbation scheme
+Right invariant
+δX = ¯XX−1
+X = exp(−δξ∧)¯X
+Right perturbation
+δX = X¯X−1
+X = exp(δξ∧)¯X
+Left invariant
+δX = X−1 ¯X
+X = ¯X exp(−δξ∧)
+Left perturbation
+δX = ¯X−1X
+X = ¯X exp(δξ∧)
+
+22
+IEEE TRANSACTIONS ON ROBOTICS. PREPRINT VERSION. ACCEPTED NOVEMBER, 2022
+C. The white-noise-on-acceleration motion prior
+The white-noise-on-acceleration (WNOA) motion prior, modified slightly from [A1], may be summarized by the following
+set of nonlinear stochastic differential equations (SDEs),
+˙T(t) = T(t)ϖb(t)∧,
+(A.13a)
+˙ϖb(t) = wb(t),
+(A.13b)
+wb(t) ∼ GP(0, Qδ(t − t′)),
+(A.13c)
+where the time argument is included to emphasize that (A.13) evolves in continuous time, and the subscript (·)b is included
+to emphasize that the generalized velocity ϖb is a body-frame quantity. The WNOA prior promotes constant body-centric
+velocity (smoothing) throughout in the trajectory. The navigation state is defined as the ordered pair
+X = (T, ϖ) ∈ SE(3) × R6,
+(A.14)
+with T ∈ SE(3) and Tϖ∧ ∈ g, where T is a time increment. Following a left-invariant perturbation scheme for the pose, the
+navigation state is perturbed as
+T = ¯T exp(−δξ∧),
+(A.15a)
+ϖ = ¯ϖ + δϖ.
+(A.15b)
+Equation (A.13) may be divided into a set of deterministic mean equations,
+˙¯T = ¯T ¯ϖ∧,
+(A.16a)
+˙¯ϖ = 0,
+(A.16b)
+and a separate SDE describing the perturbations [A1],
+�
+δ ˙ξ(t)
+δ ˙ϖ(t)
+�
+= A
+� δξ(t)
+δϖ(t)
+�
++ L δw(t),
+(A.17)
+with L =
+�0
+1�T, and where
+δw(t) ∼ GP(0, Qδ(t − t′)).
+(A.18)
+To formulate a batch estimation problem, the continuous-time error kinematics A must be derived and discretized.
+D. Deriving the WNOA state error kinematics on SE(3) × R6
+The WNOA state error kinematics are derived in this section. The continuous-time state error kinematics are first obtained
+by linearizing the navigation state kinematics, and are then discretized exactly via the matrix exponential.
+Following the perturbation scheme (A.15), approximating exp(−δξ∧) ≈ (1 − δξ∧), and ignoring higher-order terms, the
+continuous-time pose kinematics (A.13a) are perturbed as
+˙T = Tϖ∧,
+d
+dt
+�¯T exp(−δξ∧)
+�
+= ¯T exp(−δξ∧)( ¯ϖ + δϖ)∧,
+˙¯T − ˙¯Tδξ∧ − ¯Tδ ˙ξ∧ ≈ ¯T ¯ϖ∧ + ¯Tδϖ∧ − ¯Tδξ∧ ¯ϖ∧,
+¯Tδ ˙ξ∧ = − ¯Tδϖ∧ + ¯Tδξ∧ ¯ϖ∧ − ¯T ¯ϖ∧δξ∧,
+δ ˙ξ∧ = − δϖ∧ + δξ∧ ¯ϖ∧ − ¯ϖ∧δξ∧,
+δ ˙ξ = − ad( ¯ϖ∧)δξ − δϖ.
+(A.19)
+Equation (A.19) describes the continuous-time pose error kinematics. Inserting (A.19) into (A.17) yields
+�
+δ ˙ξ(t)
+δ ˙ϖ(t)
+�
+�
+��
+�
+δ˙x(t)
+=
+�− ad( ¯ϖ∧)
+−1
+0
+0
+�
+�
+��
+�
+A(t)
+� δξ(t)
+δϖ(t)
+�
+�
+��
+�
+δx(t)
++
+�0
+1
+�
+����
+L(t)
+δw(t),
+(A.20)
+which describes the continuous-time state error kinematics.
+The continuous-time state error kinematics will now be discretized, to provide a check on the solution when deriving the
+discrete-time batch Jacobians in Section A-E2. The matrix A(t) is discretized exactly via the matrix exponential [A8, Sec. 3.5.4].
+
+HITCHCOX AND FORBES: IMPROVING SELF-CONSISTENCY IN UNDERWATER MAPPING THROUGH LASER-BASED LOOP CLOSURE (EXTENDED)
+23
+Considering Ak−1 = exp(TA), where T = tk − tk−1, the first few powers of An are
+A2 =
+�ad( ¯ϖ∧)2
+ad( ¯ϖ∧)
+0
+0
+�
+,
+(A.21a)
+A3 =
+�− ad( ¯ϖ∧)3
+− ad( ¯ϖ∧)2
+0
+0
+�
+.
+(A.21b)
+The matrix A(t) is unfortunately not nilpotent, but may be written in closed form by noting
+− ad(ξ∧) = ad(−ξ∧),
+(A.22a)
+exp(ad(ξ∧)) = Ad(exp(ξ∧)).
+(A.22b)
+Writing out the first few terms of Ak−1 = exp(TA) component-wise,
+exp(TA) =
+�A11
+k−1
+A12
+k−1
+0
+1
+�
+,
+A11
+k−1 = 1 + ad(−T ¯ϖ∧) + 1
+2 ad(−T ¯ϖ∧)2 + 1
+6 ad(−T ¯ϖ∧)3 + · · · ,
+A12
+k−1 = − T1 − T
+2 ad(−T ¯ϖ∧) − T
+6 ad(−T ¯ϖ∧)2 + · · · ,
+exp(TA) =
+��∞
+n=0
+1
+n! ad(−T ¯ϖ∧)n
+−T �∞
+n=0
+1
+(1+n)! ad(−T ¯ϖ∧)n
+0
+1
+�
+,
+Ak−1 =
+�Ad(exp(−T ¯ϖ∧
+k−1))
+−TJr(T ¯ϖk−1)
+0
+1
+�
+.
+(A.23)
+Matrix Ak−1 describes the discrete-time state error kinematics for the WNOA motion prior.
+E. Deriving the batch Jacobians
+The prior, process, and loop closure Jacobians are derived in this section. Each derivation starts with the respective discrete-
+time error definitions, and uses the identities given throughout Section A-B to arrive at the final result. Note throughout that
+¯(·) is used to denote a mean state estimate, and ˜(·) is used to denote a state estimate generated from sensor measurements or
+prior information.
+1) Deriving the Jacobians on the prior error
+With a left-invariant pose error definition, the prior navigation state error is
+e0 =
+�
+eξ
+0
+eϖ
+0
+�
+=
+�
+log
+�
+T−1
+0 Y0
+�∨
+ϖ0 − ψ0
+�
+,
+(A.24)
+where (Y0, ψ0) is the prior estimate on the first navigation state. The objective is to perturb (A.24) to first order with respect to
+the design variables T0 and ϖ0 to recover the Jacobian matrices. In the case of the prior pose error, the BCH identities (A.7)
+given in Section A-B3 will be used to manipulate the resulting expression into a form that matches the left-invariant error
+definition introduced in Table A.1. With δT0 = exp((eξ
+0)∧), ˜T0 = Y0, and following the left-invariant perturbation scheme
+(A.15a), the prior pose error is linearized as
+δT0 = T−1
+0 ˜T0
+= T−1
+0 Y0
+= exp(δξ∧
+0 )¯T−1
+0 ¯Y0 exp(−δη∧
+0 )
+= exp(δξ∧
+0 )δ¯T0 exp(−δη∧
+0 )
+= δ¯T0 δ¯T−1
+0
+exp(δξ∧
+0 )δ¯T0 exp(−δη∧
+0 )
+= δ¯T0 exp((Ad(δ¯T−1
+0 )δξ0)∧) exp(−δη∧
+0 ).
+(A.25a)
+Note that the mean prior pose error δ¯T0 in (A.25a) has been relocated to the far left through use of the adjoint matrix (A.9),
+matching the form of the left-invariant error. However, (A.25a) contains two exp(·) terms, which must be combined to match
+the left-invariant error definition. Using BCH identity (A.7c) to combine the perturbations in (A.25a) and continuing,
+δT0 ≈ δ¯T0 exp((Ad(δ¯T−1
+0 )δξ0 − δη0)∧),
+exp((eξ
+0)∧) = exp((¯eξ
+0)∧) exp((Ad(δ¯T−1
+0 )δξ0 − δη0)∧).
+(A.25b)
+
+24
+IEEE TRANSACTIONS ON ROBOTICS. PREPRINT VERSION. ACCEPTED NOVEMBER, 2022
+Finally, to obtain a linear expression, use BCH identity (A.7a) to combine all terms on the matrix Lie algebra in (A.25b),
+exp((eξ
+0)∧) ≈ exp((¯eξ
+0 + Jr(¯eξ
+0)−1(Ad(δ¯T−1
+0 )δξ0 − δη0))∧),
+eξ
+0 = ¯eξ
+0 + Jr(¯eξ
+0)−1(Ad(δ¯T−1
+0 )δξ0 − δη0).
+(A.25c)
+To simplify the Jacobian associated with prior pose perturbation δξ0 in (A.25c), identity (A.10) is used to produce
+Jr(ξ)−1 Ad(X−1) = Jr(ξ)−1 Ad((exp(ξ∧))−1)
+= Jr(ξ)−1 Ad(exp(−ξ∧))
+= Jℓ(−ξ)−1 Ad(exp(−ξ∧))
+= Jr(−ξ)−1
+= Jℓ(ξ)−1.
+(A.26)
+Applying this to (A.25c), the linearized prior pose error becomes
+eξ
+0 = ¯eξ
+0 + Jℓ(¯eξ
+0)−1δξ0 − Jr(¯eξ
+0)−1δη0.
+(A.27)
+The prior generalized velocity error is linearized as
+eϖ
+0 = ( ¯ϖ0 + δϖ0) − (¯ψ0 + δψ0)
+= ¯eϖ
+0 + δϖ0 − δψ0.
+(A.28)
+Combining (A.27) and (A.28) yields the prior Jacobian,
+�
+δeξ
+0
+δeϖ
+0
+�
+� �� �
+δe0
+=
+�
+Jℓ(¯eξ
+0)−1
+0
+0
+1
+�
+�
+��
+�
+F0
+0
+� δξ0
+δϖ0
+�
+� �� �
+δx0
++
+�
+−Jr(¯eξ
+0)−1
+0
+0
+−1
+�
+�
+��
+�
+M0
+�δη0
+δψ0
+�
+� �� �
+δy0
+.
+(A.29)
+2) Deriving the Jacobians on the WNOA error
+With a left-invariant pose error definition, the WNOA process (constant body velocity) navigation state errors are
+ek =
+�
+eξ
+k
+eϖ
+k
+�
+=
+�
+log
+�
+T−1
+k ˜Tk
+�∨
+ϖk − ϖk−1
+�
+.
+(A.30)
+The objective is to linearize (A.30), using the BCH identities (A.7) to produce an expression that looks like a left-invariant
+error. With δTk = exp((eξ
+k)∧) and ˜Tk = Tk−1 exp(Tϖ∧
+k−1), T = tk − tk−1, the WNOA pose error is linearized as
+δTk = T−1
+k ˜Tk
+= T−1
+k Tk−1 exp(Tϖ∧
+k−1)
+= exp(δξ∧
+k )¯T−1
+k ¯Tk−1 exp(−δξ∧
+k−1) exp(T( ¯ϖk−1 + δϖk−1)∧).
+(A.31a)
+Using BCH identity (A.7b) to separate the terms in the last exponential and continuing from (A.31a),
+δTk ≈ exp(δξ∧
+k )¯T−1
+k ¯Tk−1 exp(−δξ∧
+k−1) exp(T ¯ϖ∧
+k−1) exp(T(Jr(T ¯ϖk−1)δϖk−1)∧)
+= exp(δξ∧
+k )¯T−1
+k ¯Tk−1 exp(T ¯ϖ∧
+k−1) exp(−T ¯ϖ∧
+k−1) exp(−δξ∧
+k−1) exp(T ¯ϖ∧
+k−1)
+× exp(T(Jr(T ¯ϖk−1)δϖk−1)∧)
+= exp(δξ∧
+k )¯T−1
+k ¯Tk−1 exp(T ¯ϖ∧
+k−1) exp(−(Ad(exp(−T ¯ϖ∧
+k−1))δξk−1)∧)
+× exp(T(Jr(T ¯ϖk−1)δϖk−1)∧)
+= exp(δξ∧
+k )δ¯Tk exp(−(Ad(exp(−T ¯ϖ∧
+k−1))δξk−1)∧) exp(T(Jr(T ¯ϖk−1)δϖk−1)∧)
+= δ¯Tkδ¯T−1
+k
+exp(δξ∧
+k )δ¯T exp(−(Ad(exp(−T ¯ϖ∧
+k−1))δξk−1)∧) exp(T(Jr(T ¯ϖk−1)δϖk−1)∧)
+= δ¯Tk exp((Ad(δ¯T−1
+k )δξk)∧) exp(−(Ad(exp(−T ¯ϖ∧
+k−1))δξk−1)∧) exp(T(Jr(T ¯ϖk−1)δϖk−1)∧),
+exp((eξ
+k)∧) = exp((¯eξ
+k)∧) exp((Ad(δ¯T−1
+k )δξk)∧) exp(−(Ad(exp(−T ¯ϖ∧
+k−1))δξk−1)∧)
+× exp(T(Jr(T ¯ϖk−1)δϖk−1)∧).
+(A.31b)
+Using BCH identity (A.7c) to combine all perturbation terms in (A.31b) and continuing,
+exp((eξ
+k)∧) ≈ exp((¯eξ
+k)∧) exp((Ad(δ¯T−1
+k )δξk − Ad(exp(−T ¯ϖ∧
+k−1))δξk−1 + TJr(T ¯ϖk−1)δϖk−1)∧).
+(A.31c)
+
+HITCHCOX AND FORBES: IMPROVING SELF-CONSISTENCY IN UNDERWATER MAPPING THROUGH LASER-BASED LOOP CLOSURE (EXTENDED)
+25
+Equation (A.31c) now resembles a left-invariant error, as required. Combining all terms on the matrix Lie algebra via BCH
+identity (A.7a) in order to produce a linear expression, and continuing from (A.31c),
+exp((eξ
+k)∧) ≈ exp((¯eξ
+k + Jr(¯eξ
+k)−1(Ad(δ¯T−1
+k )δξk − Ad(exp(−T ¯ϖ∧
+k−1))δξk−1+TJr(T ¯ϖk−1)δϖk−1))∧),
+(A.31d)
+eξ
+k = ¯eξ
+k + Jr(¯eξ
+k)−1(Ad(δ¯T−1
+k )δξk − Ad(exp(−T ¯ϖ∧
+k−1))δξk−1 + TJr(T ¯ϖk−1)δϖk−1)
+= ¯eξ
+k − Jr(¯eξ
+k)−1 Ad(exp(−T ¯ϖ∧
+k−1))δξk−1 + TJr(¯eξ
+k)−1Jr(T ¯ϖk−1)δϖk−1 + Jℓ(¯eξ
+k)−1δξk.
+(A.31e)
+Note BCH identity (A.10) was used to simplify (A.31e), for details see (A.26). The generalized velocity error is linearized as
+eϖ
+k = (ϖk + δϖk) − (ϖk−1 + δϖk−1)
+= ¯eϖ
+k + δϖk − δϖk−1.
+(A.32)
+Collecting (A.31e) and (A.32), the WNOA error Jacobians are given by
+�
+δeξ
+k
+δeϖ
+k
+�
+� �� �
+δek
+=
+�
+−Jr(¯eξ
+k)−1 Ad(exp(−T ¯ϖ∧
+k−1))
+TJr(¯eξ
+k)−1Jr(T ¯ϖk−1)
+0
+−1
+�
+�
+��
+�
+Fk
+k−1
+� δξk−1
+δϖk−1
+�
+�
+��
+�
+δxk−1
++
+�
+Jℓ(¯eξ
+k)−1
+0
+0
+1
+�
+�
+��
+�
+Fk
+k
+� δξk
+δϖk
+�
+� �� �
+δxk
+.
+(A.33)
+Note that Fk
+k−1 is the negative of Ak−1 from (A.23), with the exception of the Jr(¯eξ
+k)−1 terms in the top row owing to the
+application of BCH identity (A.7a) when moving from (A.31c) to (A.31d). The discrete-time state error kinematics Ak−1, also
+known as the transition matrix [A2, Sec. 3.1.1], are expected to appear at this location of the batch problem [A2, Sec. 3.1.2],
+and therefore (A.23) provides a useful check on the solution.
+3) Deriving the Jacobians on the loop closure error
+Again using a left-invariant pose error definition, the loop closure error is
+eℓ = log
+�
+T−1
+ℓ2 ˜Tℓ2
+�∨
+.
+(A.34a)
+With δTℓ = exp(e∧
+ℓ ) and ˜Tℓ2 = Tℓ1Ξℓ1ℓ2, the loop close error is linearized as
+δTℓ =T−1
+ℓ2 ˜Tℓ2
+= T−1
+ℓ2 Tℓ1Ξℓ1ℓ2
+= exp(δξ∧
+ℓ2)¯T−1
+ℓ2 ¯Tℓ1 exp(−δξ∧
+ℓ1)¯Ξℓ1ℓ2 exp(−δξ∧
+Ξ)
+= exp(δξ∧
+ℓ2)¯T−1
+ℓ2 ¯Tℓ1 ¯Ξℓ1ℓ2 ¯Ξ−1
+ℓ1ℓ2 exp(−δξ∧
+ℓ1)¯Ξℓ1ℓ2 exp(−δξ∧
+Ξ)
+= exp(δξ∧
+ℓ2)δ¯Tℓ exp(−(Ad(¯Ξ−1
+ℓ1ℓ2)δξℓ1)∧) exp(−δξ∧
+Ξ)
+= δ¯Tℓ δ¯T−1
+ℓ
+exp(δξ∧
+ℓ2)δ¯Tℓ exp(−(Ad(¯Ξ−1
+ℓ1ℓ2)δξℓ1)∧) exp(−δξ∧
+Ξ)
+= δ¯Tℓ exp((Ad(δ¯T−1
+ℓ )δξℓ2)∧) exp(−(Ad(¯Ξ−1
+ℓ1ℓ2)δξℓ1)∧) exp(−δξ∧
+Ξ),
+exp(e∧
+ℓ ) = exp(¯e∧
+ℓ ) exp((Ad(δ¯T−1
+ℓ )δξℓ2)∧) exp(−(Ad(¯Ξ−1
+ℓ1ℓ2)δξℓ1)∧) exp(−δξ∧
+Ξ)
+(A.34b)
+≈ exp(¯e∧
+ℓ ) exp((Ad(δ¯T−1
+ℓ )δξℓ2 − Ad(¯Ξ−1
+ℓ1ℓ2)δξℓ1 − δξΞ)∧).
+(A.34c)
+BCH identity (A.7c) was applied to combine all perturbation terms in (A.34b). Continuing from (A.34c) and applying BCH
+identify (A.7a) to combine all terms on the matrix Lie algebra,
+exp(e∧
+ℓ ) ≈ exp((¯eℓ + Jr(¯eℓ)−1(Ad(δ¯T−1
+ℓ )δξℓ2 − Ad(¯Ξ−1
+ℓ1ℓ2)δξℓ1 − δξΞ))∧),
+eℓ = ¯eℓ + Jr(¯eℓ)−1(Ad(δ¯T−1
+ℓ )δξℓ2 − Ad(¯Ξ−1
+ℓ1ℓ2)δξℓ1 − δξΞ)
+= ¯eℓ − Jr(¯eℓ)−1 Ad(¯Ξ−1
+ℓ1ℓ2)δξℓ1 + Jℓ(¯eℓ)−1δξℓ2 − Jr(¯eℓ)−1δξΞ.
+(A.34d)
+Again, identity (A.10) was used to simplify (A.34d), for details see (A.26). The loop closure Jacobians are therefore
+δeℓ = −Jr(¯eℓ)−1 Ad(¯Ξ−1
+ℓ1ℓ2)
+�
+��
+�
+Hℓ
+ℓ1
+δξℓ1 + Jℓ(¯eℓ)−1
+�
+��
+�
+Hℓ
+ℓ2
+δξℓ2 −Jr(¯eℓ)−1
+�
+��
+�
+Mℓ
+δξΞ.
+(A.35)
+
+26
+IEEE TRANSACTIONS ON ROBOTICS. PREPRINT VERSION. ACCEPTED NOVEMBER, 2022
+Appendix A References
+[A1]
+S. Anderson and T. D. Barfoot, “Full STEAM ahead: Exactly sparse Gaussian process regression for batch continuous-time trajectory estimation on
+SE(3),” in Proc. IEEE/RSJ Int. Conf. Intell. Robots Syst. (IROS), IEEE, 2015, pp. 157–164.
+[A2]
+T. D. Barfoot, State Estimation for Robotics. Cambridge University Press, 2017.
+[A3]
+J. Sola, J. Deray, and D. Atchuthan, “A micro Lie theory for state estimation in robotics,” arXiv preprint arXiv:1812.01537, 2018.
+[A4]
+G. S. Chirikjian, Stochastic Models, Information Theory, and Lie Groups, Volume 2: Analytic Methods and Modern Applications. Springer Science &
+Business Media, 2011, vol. 2.
+[A5]
+G. Bourmaud, R. M´egret, M. Arnaudon, and A. Giremus, “Continuous-discrete extended Kalman filter on matrix Lie groups using concentrated
+Gaussian distributions,” Journal of Mathematical Imaging and Vision, vol. 51, no. 1, pp. 209–228, 2015.
+[A6]
+J. Arsenault, “Practical considerations and extensions of the invariant extended Kalman filtering framework,” M.S. thesis, Department of Mechanical
+Engineering, McGill University, 2019.
+[A7]
+E. Eade, “Lie groups for computer vision,” Tech. Rep., 2014.
+[A8]
+J. Farrell, Aided Navigation: GPS with High Rate Sensors. McGraw-Hill, Inc., 2008.
+APPENDIX B
+ADDITIONAL ALIGNMENT IMAGES
+(a) Prior elevation map (INS)
+(b) Prior time stamp map (INS)
+(c) Posterior elevation map (INS+LC)
+(d) Posterior time stamp map (INS+LC)
+Fig. B.1: Images of the Wiarton shipwreck area, comparing the prior INS point cloud map in the top row to the posterior
+INS+LC point cloud map in the bottom row. In the right column, colour denotes relative trajectory time, from low (blue) to
+high (red). The time stamp plots are included to highlight the many passes involved in generating the final point cloud map.
+(a) Prior time stamp map (INS), zoom
+(b) Posterior time stamp map (INS+LC), zoom
+Fig. B.2: A zoom of the shipwreck area, corresponding to the images in Fig. 1. The INS+LC solution on the right delivers a
+markedly more crisp, self-consistent point cloud map, suitable for inspection and metrology work.
+
+hhh 
\ No newline at end of file
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+arXiv:2301.04775v1  [eess.SY]  12 Jan 2023
+On Phase Change Rate Maximization with
+Practical Applications ⋆
+C.-Y. Kao ∗ S. Hara ∗∗ Y. Hori ∗∗∗ T. Iwasaki ∗∗∗∗ S. Z. Khong †
+∗ Dept. of Electrical Engineering, National Sun Yat-Sen University, Taiwan.
+(e-mail: cykao@mail.ee.nsysu.edu.tw)
+∗∗ Global Scientific Information and Computing Center, Tokyo Institute of
+Technology, Japan. (e-mail: shinji hara@ipc.i.u-tokyo.ac.jp)
+∗∗∗ Applied Physics and Physico-Informatics, Keio University, Japan.
+(e-mail: yhori@appi.keio.ac.jp)
+∗∗∗∗ Dept. of Mechanical and Aerospace Engineering, University of
+California at Los Angeles, USA. (e-mail: tiwasaki@ucla.edu)
+† Independent Researcher. (email: szkhongwork@gmail.com)
+Abstract: We recapitulate the notion of phase change rate maximization and demonstrate the
+usefulness of its solution on analyzing the robust instability of a cyclic network of multi-
+agent systems subject to a homogenous multiplicative perturbation. Subsequently, we apply
+the phase change rate maximization result to two practical applications. The first is a magnetic
+levitation system, while the second is a repressilator with time-delay in synthetic biology. We
+also state results on robust instability analysis of digital control systems by making use of the
+bilinear transformation.
+Keywords: phase change rate maximization, instability analysis, strong stabilization
+1. INTRODUCTION
+Robustness against model uncertainties for feedback sys-
+tems has been recognized as one of the important issues
+in control theory from the practical application viewpoint
+over forty years since the 1980s. The most typical and
+successful theory is the H∞ control which includes robust
+stability and robust stabilization against norm-bounded
+dynamic uncertainties. See e.g., (Zhou, 1996) and the ref-
+erences therein.
+A counterpart of the robust stability analysis is the so-
+called “robust instability analysis” for nominally unstable
+feedback systems, and the problem is to find a stable per-
+turbation with the smallest H∞-norm which stabilizes the
+system. A practical motivation of the analysis is maintain-
+ing nonlinear oscillations caused by instability of an equi-
+librium point for dynamical systems arising in neuro-
+science and synthetic biology. See (Hara, 2020) and (Hara,
+2021) for applications to the FitzHugh-Nagumo neuron
+model and repressilator model, respectively.
+The instability analysis problem is closely related to the
+strong stabilization, i.e., stabilization by a stable con-
+troller (Youla, 1974; Zeren, 2000; Ohta, 2001). Actually,
+it is equivalent to strong stabilization by a minimum-
+norm controller. The problem is extremely difficult due
+to the following two reasons: (i) non-convexity nature
+of minimum-norm controller synthesis and (ii) no upper
+bound on the order of stable stabilizing controllers. In
+other words, the robust instability analysis is similar to
+⋆ This work was supported in part by the National Science and Tech-
+nology Council of Taiwan, under grant MOST 110-2221-E-110-047-MY3.
+This work has been submitted to IFAC for possible publication.
+the robust stability analysis in terms of the problem for-
+mulation, but it is quite different technically and much
+more challenging as optimization problems.
+Recently, the authors proposed a new optimization prob-
+lem, which we call the “Phase Change Rate Maximization
+Problem” in order to provide an almost complete solu-
+tion to the robust instability analysis for some classes
+of systems with one or two unstable poles (Hara, 2022).
+The problem is to find a real-rational transfer function
+such that its peak gain occurs at a given frequency ωp
+with a prescribed phase value, and the phase change rate
+(PCR) at ωp is the maximum among those satisfying the
+constraints. The essential idea behind is the following.
+One of the key factors for the difficulty of robust in-
+stability analysis is that we cannot detect the transition
+from instability to stability by the presence of a pole on
+the imaginary axis (which successfully characterizes the
+transition in the opposite direction, making the robust
+stability analysis tractable). Hence we need an additional
+criterion for the transition. It turned out, roughly speak-
+ing, that the positivity of the PCR of the loop transfer
+function at the peak gain frequency is an indication of the
+instability-to-stability transition for certain systems. The
+aforementioned paper showed that the maximum PCR
+is attained by a first-order all-pass function and derived
+conditions under which the exact robust instability anal-
+ysis is possible in terms of the PCR.
+The purpose of this paper is twofold. The first purpose is
+to supplement the theoretical results in (Hara, 2022) by a
+more comprehensive example than those in the reference
+and illustrate how the PCR plays an important role for the
+
+exact robust instability analysis. The class of systems is
+given as cyclic networks of homogeneous agents, where
+by changing the number of agents we can treat a variety
+of situations with respect to the location of stable and
+unstable complex poles with relatively small dampings.
+We focus especially on the relationship between the sign
+of the PCR and the stable/unstable poles which are fairly
+close to the imaginary axis and represent under what
+situation we can get the exact result. The second purpose
+is to show that the PCR condition derived in (Hara, 2022)
+works well for two practical applications, namely (i) a
+minimum-norm strong stabilization for magnetic levita-
+tion systems and (ii) an exact robust instability analysis
+for the repressilator with time delay. The target systems of
+the former and the latter cases are in G0
+1 (one unstable pole
+with the peak gain attained at zero frequency) and G#
+2
+(two unstable poles with the peak gain attained at non-
+zero frequency) , respectively, for which we can get the
+exact results. This means that the theoretical foundation
+in (Hara, 2022) can be practically useful although the class
+of applicable systems may appear restricted. An applica-
+tion of strong stabilization in digital control setting is also
+presented to show the effectiveness in practice.
+The remainder of this paper is organized as follows. Sec-
+tion 2 is devoted to a brief summary of the PCR maxi-
+mization problem presented in (Hara, 2022) and an illus-
+trative example. Section 3 provides two practical applica-
+tions. An extension to strong stabilization in the digital
+control setting with application to the magnetic levitation
+system is discussed in Section 4. Section 5 summarizes
+the contributions of this paper and addresses some future
+research directions.
+Notation and Terminology: The set of real numbers is
+denoted by R. ℜ(s) and ℑ(s) denote the real and imag-
+inary parts of a complex number s, respectively. The set
+of proper real rational functions of one complex variable
+s is denoted by Rp. Let L∞ denote the set of functions that
+are bounded on the imaginary axis jR. The subset of L∞
+which consists of real rational functions bounded on jR is
+denoted by RL∞. The stable subsets of L∞ and RL∞ are
+denoted by H∞ and RH∞, respectively. The norms in L∞
+and H∞ are denoted by ∥ · ∥L∞ and ∥ · ∥H∞, respectively.
+The open (closed) left and right half complex planes are
+abbreviated as OLHP (CLHP) and ORHP (CRHP), re-
+spectively.
+The following terminology will be used for a rational
+function h ∈ Rp throughout the paper: h is called “stable”
+(or “exponentially stable”) if all the poles of h are in the
+OLHP; “marginally stable” if all the poles of h are in the
+CLHP and any pole of h on the imaginary axis is simple;
+“unstable” (or “exponentially unstable”) if at least one of
+the poles of h is in the ORHP.
+2. PHASE CHANGE RATE MAXIMIZATION
+In this section, we introduce the PCR maximization prob-
+lem, and motivate the problem by instability analysis and
+strong stabilization.
+2.1 Problem Formulation
+Given ωp > 0 and θp ∈ [0, 2π), we consider the following
+“phase change rate” maximization problem
+sup
+f∈RH∞
+θ′
+f(ωp) s.t. ∥f∥H∞ = |f(jωp)|, θf(ωp) = θp,
+(1)
+where θf(ω) denotes the phase angle of f(jω), and θ′
+f(ω)
+is its derivative. In other words, we seek a function f from
+RH∞, whose H∞-norm occurs at frequency ωp and phase
+at ωp is constrained to be θp, and has the maximal “phase
+change rate” among all functions which satisfy the same
+constraints. Such problem arises from robust instability
+analysis and minimum-norm strong stabilization as ex-
+plained below.
+Consider a class of unstable systems defined by
+G := {g ∈ RL∞ | g is strictly proper and unstable}. (2)
+The robust instability radius (RIR) for g ∈ G, denoted by
+ρ∗(g) ∈ R, with respect to real rational dynamic perturba-
+tion δ ∈ RH∞, is defined as the smallest magnitude of the
+perturbation that internally stabilizes the system:
+ρ∗(g) :=
+inf
+δ∈S(g) ∥δ∥H∞,
+(3)
+where S(g) is the set of real-rational, proper, stable trans-
+fer functions internally stabilizing g, i.e.,
+S(g) := {δ ∈ RH∞ : δ(s)g(s) = 1 ⇒ ℜ(s) < 0,
+δ(s) = 0, ℜ(s) > 0 ⇒ |g(s)| < ∞ }.
+(4)
+The optimization problem stated in (3) is identical to the
+so-called “minimum-norm strong stabilization” problem
+for a given (unstable) plant g, where the minimum-norm
+controller sought is required to be stable itself. It is no-
+ticed from the well known result on strong stabilizability
+in (Youla, 1974) that ρ∗(g) is finite if and only if the Parity
+Interlacing Property (PIP) is satisfied, i.e., the number of
+unstable real poles of g between any pair of real zeros in
+the closed right half complex plane (including zero at ∞)
+is even. Consequently, the class of systems of our interest
+is defined as
+Gn := {g ∈ G | g has n unstable poles and
+satisfies the PIP condition},
+(5)
+where n is a natural number. Let g ∈ G be given. We have
+the following lower bound for ρ∗(g) (see (Hara, 2021))
+ρ∗(g) ≥ 1/∥g∥L∞,
+∥g∥L∞ := sup
+ω∈R
+|g(jω)|.
+(6)
+When ρ∗(g) is exactly equal to its lower bound 1/∥g∥L∞,
+we say g has the exact RIR. It has been shown in (Hara,
+2021) that, if f with ∥f∥H∞ = 1/∥g∥L∞ marginally stabi-
+lizes g with a single pair of poles on the imaginary axis,
+then g has the exact RIR. Moreover, based on an extended
+version of the Nyquist criteria, necessary and sufficient
+conditions were derived in (Hara, 2022) for marginal sta-
+bilization of g, which in turn are sufficient conditions for
+obtaining the exact RIR of g. As a part of the necessary
+and sufficient condition for f with ∥f∥H∞ = 1/∥g∥L∞ to
+marginally stabilize g, the open-loop transfer function gf
+must satisfy the following loop-gain and PCR conditions:
+g(jωp)f(jωp) = 1, θ′
+gf(ωp) = θ′
+g(ωp) + θ′
+f(ωp) > 0,
+
+where ωp is the frequency where the L∞-gain of g oc-
+curs. Searching for such an f boils down to solving a
+PCR optimization problem of the form described in (1),
+where the phase θf(ωp) is constrained to −θg(ωp) (and
+the magnitude of f at ωp is irrelevant to PCR optimiza-
+tion, as positive scaling of f will not change its phase or
+phase change rate). The solution of the problem provides
+a tight condition for g to be marginally stabilizable. In
+the next subsection, we summarize the theoretical foun-
+dation in (Hara, 2022).
+2.2 The Solution and its Application to Instability Analysis
+The PCR optimization in (1) can be solved by first nar-
+rowing down the feasible set using the following sets of
+functions:
+RFωp,θp := {f ∈ RH∞ : 1 = ∥f∥H∞ = |f(ωp)|,
+(7)
+θf(ωp) = θp}.
+Oωp,θp := {f ∈ RH∞ : f is minimum phase,
+(8)
+|f(jωp)| = ∥f∥H∞, and θf(ωp) = θp}.
+AP ωp,θp := {f ∈ RH∞ : |f(jω)| = 1, ∀ω,
+(9)
+|f(jωp)| = ∥f∥H∞, and θf(ωp) = θp}.
+Note that the constraint on the magnitude of the H∞-
+norm of functions in RF•,• and AP •,• bears no signifi-
+cance as explained previously. The constraint is placed for
+convenience only. The first result gives an upper bound
+on the PCR for functions in Oωp,θp.
+Proposition 1. Let θp ∈ (−π, π] and f ∈ Oωp,θp be given. If
+ωp ̸= 0, then θ′
+f(ωp) ≤ − |θp/ωp|. Moreover, if ωp = 0, then
+θ′
+f(ωp) ≤ 0.
+Proposition 1 establishes that, for a stable minimum-
+phase system, its PCR at the peak-frequency (i.e., where
+the H∞-norm occurs) is always non-positive. Since any
+RH∞ function can be factorized as multiplication of an
+all-pass function and a minimum-phase function, Propo-
+sition 1 suggests that the PCR maximization problem
+over the set RF•,• boils down to the problem over the
+set AP •,•. This is indeed the case, as the following propo-
+sition states.
+Proposition 2. Given ωp ̸= 0 and θp ∈ (−π, π] (mod 2π),
+we have
+sup
+f∈RFωp,θp
+θ′
+f(ωp) =
+sup
+f∈AP ωp,θp
+θ′
+f(ωp) = −
+����
+sin(θp)
+ωp
+���� .(10)
+Moreover, when θp ̸∈ {0, π}, the supremum is attained by
+the first-order all-pass function of the form f(s) = a−s
+a+s or
+f(s) = s−a
+a+s. When θp ∈ {0, π}, the supremum is attained
+by a zeroth-order all-pass functions; i.e., f(s) = 1 or
+f(s) = −1. For ωp = 0, the only feasible phase angles
+are θp ∈ {0, π} (mod 2π). In this case,
+sup
+f∈RF0,θp
+θ′
+f(0) =
+sup
+f∈AP 0,θp
+θ′
+f(0) = 0.
+The supremum is attained by f(s) = 1 or f(s) = −1.
+Using the solutions stated in Proposition 2, the following
+results were derived for two subclasses of Gn defined by
+G0
+n := {g ∈ Gn | ∥g∥L∞ = |g(0)| > |g(jω)| ∀ω ̸= 0},
+(11)
+G#
+n := {g ∈ Gn | ∃ ωp > 0 such that
+∥g∥L∞ = |g(jωp)| > |g(jω)| ∀ω ̸= ±ωp}
+(12)
+based on an extended Nyquist criterion (Hara, 2022).
+Theorem 1.
+(I) Given g ∈ G0
+n, g can be marginally stabilized by a
+stable system f with ∥f∥H∞ = 1/∥g∥L∞ = 1/|g(0)| if
+and only if n = 1 and
+θ′
+g(0) > 0.
+(13)
+(II) Given g ∈ G#
+n for which the peak gain occurs at ωp,
+g can be marginally stabilized by a stable system f
+with ∥f∥H∞ = 1/∥g∥L∞ = 1/|g(jωp)| if and only if
+n = 2 and
+θ′
+g(ωp) >
+����
+sin(θg(ωp))
+ωp
+���� .
+(14)
+Note that the marginally stabilizing controllers for cases
+(I) and (II) can be taken as the zeroth-order and the first-
+order all-pass functions, respectively, as suggested by
+Proposition 2.
+As marginal stabilization of a system guarantees the exact
+RIR for the system, Theorem 1 immediately leads to
+sufficient conditions for attaining the exact RIR of systems
+in G1 and G2. Furthermore, necessary conditions can also
+be derived based on the following result, which gives a
+PCR condition on the loop-transfer function at the peak
+frequency when the closed-loop system has all its pole in
+the closed left half plane.
+Lemma 1. (Hara, 2022, Lemma 5) Given ωc
+≥
+0, an
+integer n ≥ 1, and a transfer function L ∈ Gn, consider
+the positive feedback system with loop transfer function
+L satisfying the following condition
+1 = |L(jωp)| = ∥L∥L∞,
+|L(jω)| < |L(jωp)|, ∀ω ̸= ±ωp.
+(15)
+If the feedback system has all its poles in the CLHP, then
+θ′
+L(ωp) ≥ 0.
+Based on Theorem 1 and Lemma 1, we have necessary
+conditions and sufficient conditions for the exact RIR as
+follows.
+Theorem 2. Let g ∈ G be given. Suppose g(jω) takes the
+peak gain at ωp and consider the exact RIR condition
+ρ∗(g) = 1/∥g∥L∞ = 1/|g(jωp)|.
+(16)
+(I) Suppose g ∈ G0
+1 and ωp = 0. Then
+θ′
+g(ωp) > 0 ⇒ (16) ⇒ θ′
+g(ωp) ≥ 0.
+(II) Suppose g ∈ G#
+2 and ωp > 0. Then
+θ′
+g(ωp) > ̺(ωp) ⇒ (16) ⇒ θ′
+g(ωp) ≥ ̺(ωp),
+where
+̺(ω) :=
+����
+sin(θg(ω))
+ω
+���� .
+(III) For any g ∈ G#
+1 , we have ρ∗(g) > 1/∥g∥L∞.
+For the proofs of these results, readers are referred to
+Section 4 of (Hara, 2022). Also note that, the necessary
+conditions in statements (I) and (II) hold in fact for sys-
+tems in G0
+n and G#
+n , respectively, for any n.
+
+2.3 An Illustrative Example
+In this subsection we illustrate, by a numerical example,
+how the PCR condition effectively works for the robust
+instability analysis. Consider a class of positive feedback
+systems of which the loop transfer functions are repre-
+sented by
+h(s) =
+−k
+(s + 1)2m+1 , m = 1, 2, . . .,
+(17)
+where we assume that the loop-gain k > 0 is large enough
+so that the closed-loop system is exponentially unstable.
+Our interest here is to assess robust instability against
+a ball type multiplicative stable perturbation; in other
+words, the perturbed system ˜h has the form
+˜h(s) = (1 + δ(s))h(s), δ(s) ∈ RH∞.
+(18)
+Such a setting may arise when one considers a cyclic
+network with 2m + 1 identical agents with a multiplica-
+tive uncertainty present for the loop. The corresponding
+characteristic equation of the closed-loop system is given
+by 1 − gm(s)δ(s) = 0, where
+gm(s) :=
+h(s)
+1 − h(s) =
+−k
+(s + 1)2m+1 + k .
+(19)
+For k = 20, we observe that gm ∈ G#
+2 for 1 ≤ m ≤ 7,
+and gm ∈ G#
+4 when 8 ≤ m ≤ 20. The unstable poles
+of gm increases further when m becomes bigger. Table 1
+summarizes the findings for m = 1 to 20.
+For 1 ≤ m ≤ 4, gm has one peak gain, while g5 has
+two peak gains. In all these cases, the PCR condition
+stated in Theorem 1 holds at the global peak frequencies.
+See Fig. 1(a) and 1(b) for an illustration of the magni-
+tude profiles of g4 and g5. For g5, applying Proposition 2
+we obtain the first-order all-pass function of the form
+δgl,5(s) =
+1
+1.0896
+�
+s−24.426
+s+24.426
+�
+, which marginally stabilizes
+g5 and the closed-loop system has a pair of poles at
+±jωp = ±j(0.322). In this case, we conclude that g5 has
+the exact RIR equal to 1/|g5(j(0.322))| = 1/1.0896.
+For m = 6, 7, the PCR condition fails at the global peak
+frequencies for gm. However for each case, there is a local
+peak frequency where the PCR holds. See Fig. 1(c) for
+an illustration of the magnitude profile of g6. Further
+examination reveals that the global peak-gain is due to a
+pair of dominating stable poles, while the local peak-gain
+is the result of a pair of unstable poles which is further
+away from the imaginary axis compared to the dominat-
+ing stable poles. Take g6 for example. Applying Proposi-
+tion 2 at the global and local peak frequencies, we obtain
+first-order all-pass functions δgl,6(s) =
+1
+1.3976
+�
+−s+1.2522
+s+1.2522
+�
+and δlc,6(s) =
+1
+1.0811
+�
+s−18.02
+s+18.02
+�
+, respectively. The closed-
+loop system with δgl,6 is exponentially unstable, which
+has two unstable poles and two imaginary-axis poles. It
+appears that δgl,6 pushes the dominating stable poles to
+the imaginary axis while leaving the unstable poles in
+the ORHP. On the other hand, the closed-loop system
+with δlc,6 is marginally stable with a pair of poles at
+±jωp = ±j(0.276). In this case, g6 does not have exact
+RIR, and ρ∗(g6) ∈ (1/1.3976, 1/1.0811]. Note that ρ∗(g6) is
+strictly larger than 1/∥g6∥L∞ = 1/1.3976, as the necessary
+condition stated in statement (II) of Theorem 2 is violated.
+For 8 ≤ m ≤ 13, gm has two peak-gains and both are
+caused by unstable poles. The PCR condition holds at
+both peak frequencies. For 14 ≤ m ≤ 16, a third peak
+is formed, which is caused by a pair of stable poles.
+The PCR of gm is negative at this peak (let’s call it a
+“stable peak”). For 17 ≤ m ≤ 20, the stable peak over-
+takes the other two peaks and becomes the global peak.
+See Fig. 1(d) to 1(f) for an illustration of the magnitude
+profiles of g8, g16 and g17. Now consider g8. The first-
+order all-pass functions obtained by the global and lo-
+cal peak frequencies are δgl,8(s) =
+1
+5.4116
+�
+s−2.749
+s+2.749
+�
+and
+δlc,8(s) =
+1
+1.073
+�
+s−29.498
+s+29.498
+�
+, respectively. The closed-loop
+system with δgl,8 is exponentially unstable; apparently
+δgl,8 pushes a pair of unstable poles to the imaginary axis
+while leaving the other pair in the ORHP. Similar to g6,
+δlc,8 is able to marginally stabilize g8, and therefore we
+have ρ∗(g8) ∈ [1/5.4116, 1/1.073]. Note that we cannot
+yet exclude the possibility that ρ∗(g8) = 1/5.4116 since
+no necessary condition is violated. For g9 to g16, we have
+similar results, where the inverse of the L∞-gain of gm
+gives a lower bound and the second peak-gain of gm gives
+an upper bound. For g17 to g20, the situation is slightly
+different. For those systems, their PCRs at the global peak
+frequencies violate the necessary condition for having
+exact RIR’s. Therefore, we know that ρ∗(gm) is strictly
+larger than 1/∥gm∥L∞, for m = 17, · · · , 20. For each of
+these system, an upper bound for ρ∗ is obtained using
+their respective third peak-gains.
+3. PRACTICAL APPLICATIONS
+In this section, we apply our main results to analyze
+(in)stability properties of system models that are derived
+from real-world applications. In Section 3.1 we consider
+linearized models for magnetic levitation systems. These
+models belong to the class G0
+1. In Section 3.2 we consider
+linearized models for a certain gene regulatory network
+called “repressilator”. These models belong to the class
+G#
+2 . The goal is to illustrate that our results are applicable
+to real applications to provide useful information.
+3.1 Strong Stabilization for Magnetic Levitation Systems
+A typical linearized model for the magnetic levitation
+system (Namerikawa, 2001) at an equilibrium is a third-
+order system of the following form
+g(s) =
+k
+(−s2 + p2)(τs + 1),
+(20)
+where the pair of poles at ±p is due to the mechanical
+aspect of the system while the stable pole at −τ −1 comes
+from the electrical part. Typically, we have τ−1 ≫ p and
+therefore it is generally reasonable to neglect the factor
+(τs + 1) from the dynamical model for control design
+purpose. Here we will show that, for the purpose of
+minimum-norm strong stabilization, ignoring the (τs+1)
+factor will lead to a wrong conclusion. For the reduced
+second-order model:
+
+Table 1. Summary of the numbers of peak-gains, satisfaction of the PCR conditions, whether
+exact RIR occurs, etc. among different cases.
+m
+1 − 4
+5
+6 − 7
+8 − 13
+14 − 16
+17 − 20
+# of unstable poles
+2
+2
+2
+4
+4
+4
+# of peak-gains
+1
+2
+2
+2
+3
+3
+# of unstable peak-gains
+1
+1
+1
+2
+2
+2
+# of stable peak-gains
+0
+1
+1
+0
+1
+1
+global peak-gain is (s./us.)?
+us
+us
+s
+us
+us
+s
+PCR holds at global peak?
+y
+y
+n
+y
+y
+n
+PCR holds at a local peak?
+n/a
+n
+y
+y
+y
+y
+RIR = 1/∥gm∥L∞ ?
+y
+y
+n
+inc
+inc
+n
+RIR > 1/∥gm∥L∞ ?
+n
+n
+y
+inc
+inc
+y
+Abbreviation: ’s.’ – stable; ’us.’ – unstable; ’y’ – yes; ’n’ – no; ’n/a’ – not applicable; ’inc’ – inconclusive
+Fig. 1. Magnitude profile of gm for m = 4, 5, 6, 8, 16, 17. For m = 4 to 6, gm has one pair of unstable poles, while it has
+two pairs for the other three cases. The red color indicates the frequency ranges where the PCR condition holds.
+A gain-peak where the PCR condition does not hold appears to be caused by a pair of stable poles.
+gr(s) =
+k
+(−s2 + p2),
+(21)
+we have the following result.
+Proposition 3. Consider gr(s) in (21). Then
+inf
+c∈S(gr) ∥c∥H∞ =
+1
+|gr(0)| =
+1
+∥gr∥L∞
+= p2
+k ,
+(22)
+where the infimum is obtained by the sequence of stabi-
+lizing controllers cǫ, where
+cǫ(s) = p2
+k + ǫs + z
+s + d,
+0 < z < d
+with ǫ → 0. Here z and d can be any positive real
+numbers, as long as z < d.
+Proof. It can be readily verified that the poles of the closed-
+loop system [gr, cǫ] is governed by the characteristic equa-
+tion s3 + ds2 + kǫs + kǫz = 0. Applying the Routh-
+Hurwitz stability criterion, we conclude that the closed-
+loop system is stable for any ǫ > 0, d > z > 0. Clearly,
+∥cǫ∥H∞ = p2/k + ǫ. Since 1/∥gr∥L∞ is a lower bound for
+the infimum, we have
+p2
+k =
+1
+∥gr∥L∞
+≤
+inf
+c∈S(gr) ∥c∥H∞ ≤ ∥cǫ∥H∞ = p2
+k + ǫ.
+Since ∥cǫ∥H∞ → p2/k as ǫ → 0, we conclude that the
+infimum is equal to p2/k.
+Relating to Theorem 2, note that gr ∈ G1
+0 with θ′
+gr(0) = 0.
+This is a critical case, as gr does not violate the necessary
+condition for acquiring the exact RIR, but does not satisfy
+the sufficient PCR condition, either. Nonetheless, here we
+are able to show that, the exact RIR of gr is achievable,
+and the infimum of the strong stabilization problem is
+obtained by a constant feedback equal to 1/∥gr∥L∞ =
+1/|gr(0)|.
+For the third-order model g, the outcome of minimum-
+norm strong stabilization is very different. Based on The-
+orem 2, we have the following result.
+Proposition 4. Consider g(s) in (20). Then
+inf
+c∈S(g) ∥c∥H∞ > p2
+k =
+1
+|g(0)| =
+1
+∥g∥L∞
+.
+(23)
+Proof. We note that the third-order model g also belongs
+to G0
+1. Furthermore, it can be readily verified that the PCR
+of g at the zero frequency is −τ. Since having a non-
+
+Figure (a)
+Figure (b)
+Figure (c)
+1.2
+1.2
+1.5
+m=4
+m=5
+9=w
+Magitude
+0.5
+0.6
+0.6
+0.4
+0.4
+0
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+Frequency (rad/sec)
+Frequency (rad/sec)
+Frequency (rad/sec)
+Figure (d)
+Figure (e)
+Figure (f)
+6
+1.5
+1.5
+m=8
+m=16
+m=17
+0.5
+0
+0
+0
+0
+0.2
+0.4
+0.6
+0.8
+0
+0.2
+0.4
+0.6
+0.8
+1
+0
+0.2
+0.4
+0.6
+0.8
+1
+Frequency (rad/sec)
+Frequency (rad/sec)
+Frequency (rad/sec)negative PCR is a necessary condition for systems in G0
+1
+to have the exact RIR by Theorem 2, the infimum of the
+minimum-norm strong stabilization for g is strictly larger
+than 1/∥g∥L∞ = p2/k.
+For obtaining an upper bound of the infimum, let us
+introduce a phase-lead compensator to raise the PCR of
+g. Consider f(s) =
+(τc+τ)s+1
+τcs+1
+and gc(s) = g(s)f(s). The
+compensated plant gc has 0 phase change rate at the zero
+frequency for any τc > 0. This can be readily verified by
+checking the imaginary part of
+d
+dω log(gc(jω)) at the zero
+frequency. Furthermore, one can also verify that gc ∈ G0
+1 if
+and only if τc ≤ 1/(p2τ). This can be shown by computing
+the real part of
+d
+dω log(gc(jω)), which reveals that
+• Real
+� d
+dω log(gc(jω))|ω=0
+�
+= 0;
+• when τc ≤ 1/(p2τ),
+d
+dω log |gc(jω)| < 0 for any ω > 0;
+• when τc > 1/(p2τ),
+d
+dω log |gc(jω)| > 0 for ω → 0+.
+and hence the claim. Setting τc = 1/(p2τ), we have the
+following result.
+Proposition 5. The compensated plant gc satisfies
+inf
+c∈S(gc) ∥c∥H∞ =
+1
+|gc(0)| =
+1
+∥gc∥L∞
+= p2
+k ,
+(24)
+where the infimum is obtained by the sequence of stabi-
+lizing controllers cǫ,
+cǫ(s) = p2
+k + ǫ
+s + ǫ2
+s + q/(τc + τ).
+Here ǫ is a sufficiently (in fact, arbitrarily) small positive
+number and q > 0 is chosen sufficiently large for given ǫ.
+The infimum in (24) in turn implies
+inf
+c∈S(g) ∥c∥H∞ ≤ p2(1 + p2τ 2)/k.
+Proof. One can verify that the characteristic equation of
+the closed-loop system [gc, cǫ] has the form s5 + [(q +
+1)d]s4 + [qd2]s3 + [kǫd]s2 + [kǫ( ˆd + ǫ2d)]s + [kǫ3 ˆd], where
+d := (τ+τc)/(ττc) = τ −1+p2τ, and ˆd := 1/(ττc) = p2. The
+goal here is to select parameters ǫ > 0 and q > 0 such that
+the roots of the polynomial are all in the open left-half
+plane. Applying the Routh-Hurwitz stability criterion,
+one concludes that it is so when ǫ is sufficiently small and,
+corresponding to an ǫ, q is chosen sufficiently large. The
+infimum in (24) is obtained by taking ǫ → 0. Furthermore,
+the analysis implies that fcǫ is a stabilizing controller for
+g. Since ∥fcǫ∥H∞ → p2(1 + p2τ 2)/k as ǫ → 0, it implies
+p2(1 + p2τ 2)/k is an upper bound for infc∈S(g) ∥c∥H∞.
+Remark 1. Since τ −1 ≫ p, we have 1 + p2τ 2 ≈ 1. That
+is, the upper bound on the norm of the minimum-norm
+strong stabilizing controller is very close to the lower
+bound p2/k.
+3.2 Robust Instability Analysis for Repressilator
+Consider a biological network oscillator called the repres-
+silator with three dynamical units in a cyclic loop (Elowitz,
+2000). Its linearized model is given by
+ξ = he(s)ξ, he(s) =
+−ke
+(s + α1)(s + α2)(s + α3),
+where ξ is a variable, and ke > 0 depends on the equilib-
+rium state of the original nonlinear system (Hara, 2021).
+The nominal system with the characteristic equation 1 =
+he(s) is exponentially unstable. The subscript •e is used to
+indicate that the quantity • depends on the equilibrium
+state, which in turn depends on the perturbation of the
+DC-gain of the system, denoted by e. For more details
+about the repressilator model, see (Hara, 2021).
+Here we are interested in assessing robust instability
+against a ball type multiplicative stable perturbation
+when the nominal dynamics are further complicated by
+time-delay. We use the fifth-order Pad´e approximation for
+the time-delay in order to keep the model rational. Let
+Dτ(s) denote the Pad´e approximation of the time-delay
+transfer function e−τs. The corresponding characteristic
+equation is 1 − δ(s)ge(s) = 0, where
+ge(s) =
+he(s)Dτ(s)
+1 − he(s)Dτ(s).
+Using the parameters α1 = 0.4621, α2 = 0.5545, α3 =
+0.3697, we investigate the case where e = 0; i.e., no
+perturbation on the DC-gain. In this case, we have k0 =
+2.216 and the exact RIR of g0 are confirmed when τ = 0,
+see Hara (2021). In what follows, we examine the effect of
+the time-delay on the exact RIR.
+Numerical computations show that g0 ∈ G#
+2
+for τ ∈
+[0, 4.771]. The PCR condition holds at the peak-gain fre-
+quency of g0 up to τ = 3.481, and ceases to hold when
+τ = 3.482. Thus, g0 has exact RIR for τ ∈ [0, 3.481].
+Furthermore, one can verify that when τ is large enough,
+a pair of stable poles of g0 creates a gain-peak. When τ =
+3.482, this “stable peak” becomes dominant and the PCR
+condition ceases to hold at the global peak frequency.
+However, the condition holds at the local (second) peak
+frequency. See Fig. 2 for an illustration. More specifically,
+when τ
+= 3.482, ∥g0∥L∞
+= |g(j1.5009)| = 1.10273,
+while a local peak occurs at ω = 0.396 rad/sec with
+|g(j0.396)| = 1.10268. The first-order all-pass function
+1
+1.10268
+�
+s−18.8246
+s+18.8246
+�
+, obtained by applying Proposition 2 to
+the local peak frequency, marginally stabilizes g0. Thus,
+we conclude that 1/1.10273 < ρ∗(g0) ≤ 1/1.10268 when
+τ = 3.482.
+For τ = 3.4, a marginally stabilizing perturbation with
+norm equal to 1/∥g0∥L∞ is
+1
+1.1044
+�
+s−18.4747
+s+18.4747
+�
+. This per-
+turbation is further multiplied by a high-pass filter to
+make the DC-gain of δ(s) equal to zero. Specifically, δ(s)
+is defined by
+δ(s) = s + 0.01γ
+s + ξ
+· (1 + ǫ)
+1
+1.1044
+�s − 18.4747
+s + 18.4747
+�
+,
+(25)
+where γ = −1/(1 + ǫ) with a small non-negative number
+ǫ. The closed-loop systems of g0 is marginally stabilized
+with ǫ = 0. The nonlinear repressilator models with
+ǫ = 0.95 and ǫ = 1.05 were simulated, and the results
+are shown in Fig. 3 (left and right figures, respectively).
+Clearly, δ(s) with ǫ = 0.95 is not able to stabilize g0 and
+the closed-loop system exhibits oscillatory behavior. On
+the other hand, δ(s) with ǫ = 1.05 stabilizes g0 and the
+oscillatory behavior ceases to exist.
+
+Remark 2. In E.coli cells, the delay factor mainly repre-
+sents the protein maturation time, which is usually 6 to 60
+minutes. For the repressilator model presented in this sec-
+tion, the unit of time is “hour”; therefore, the delay time τ
+of the range [0.1, 1] corresponds to realistic scenarios. Our
+analysis shows that the L∞-norm of g0 gives the exact
+RIR for τ ∈ [0, 3.481] , which indicates that it is a useful
+metric for determining the instability (i.e., oscillation) of
+practical repressilators.
+Fig. 2. Magnitude profile of g0. The red color indicates the
+frequency range where the PCR condition holds.
+Fig. 3. Time-course simulations of the closed-loop sys-
+tems. Left: g0 and δ(s) with ǫ = 0.95. Right: g0 and
+δ(s) with ǫ = 1.05.
+4. EXTENSIONS TO DIGITAL CONTROL SYSTEMS
+4.1 Robust Instability Analysis of Discrete-Time Systems via
+Bilinear Transformation to Continuous-Time Systems
+In this section, we propose a procedure for robust insta-
+bility analysis and finding minimum-norm strongly stabi-
+lizing controller for discrete-time LTI (unstable) systems.
+For a discrete-time LTI system with transfer function g(z),
+its stability can be assessed by the location of its poles
+inside, on, and/or outside the unit circle. Let the open
+unit disk be denoted by Ω, the unit circle by ∂Ω, and
+the outside of the closed unit disk by Ωc, respectively.
+For a discrete-time transfer function, the “stable” region
+is Ω, “unstable region” is Ωc, and “stability boundary” is
+∂Ω, which corresponds to the OLHP, the ORHP, and jR,
+respectively, for its continuous-time counterpart.
+It is well-known that the bilinear transformation
+z = 1 + s
+1 − s
+⇔
+s = z − 1
+z + 1
+is a one-to-one mapping between the regions in each
+aforementioned pairs; that is, s ∈ OLHP ⇔ z ∈ Ω; s ∈
+ORHP ⇔ z ∈ Ωc; s ∈ jR ⇔ z ∈ ∂Ω. Therefore, when
+we apply the bilinear transformation to a discrete-time
+transfer function, its stability property is preserved by the
+resulting continuous-time representative, and vice versa.
+Moreover, it is straightforward to see that the norm of the
+transfer function is also preserved. Observing these, we
+propose the following procedure for applying our results
+in Theorems 1 and 2 to discrete-time systems.
+Given a discrete-time transfer function gd(z), we verify
+whether infc∈S(gz) ∥c∥H∞ = 1/∥gz∥L∞ (exact RIR).
+• Step 1: Apply bilinear transformation z =
+1+s
+1−s to
+find the continuous-time representative gd,c of gd;
+i.e., gd,c(s) := gd
+�
+1+s
+1−s
+�
+.
+• Step 2(a): If gd,c ∈ G0
+1 and θ′
+gd,c(0) > 0, or gd,c ∈
+G#
+n and θ′
+gd,c(ωp) > | sin(θgd,c(ωp))|/|ωp|, then gd has
+exact RIR. Proceed to Step 3.
+• Step 2(b): If gd,c ∈ G0
+n and θ′
+gd,c(0) < 0, or gd,c ∈ G#
+n
+and θ′
+gd,c(ωp) < | sin(θgd,c(ωp))|/|ωp|, then gd does not
+have exact RIR. infc∈S(gz) ∥c∥H∞ is strictly larger than
+1/∥gz∥L∞. The analysis is completed and stop.
+• Step 2(c): If gd,c ∈ G#
+1 , then gd does not have exact
+RIR. infc∈S(gz) ∥c∥H∞ is strictly larger than 1/∥gz∥L∞.
+The analysis is completed and stop.
+• Step 2(d): If gd,c does not belong to one of the sce-
+narios described in 2(a) to 2(c), then the analysis is
+inconclusive. Stop.
+• Step 3: Obtain the all-pass function c∗(s) which
+marginally stabilizes gd,c. Then c∗,d(z) := c∗
+�
+z−1
+z+1
+�
+is the all-pass function that marginally stabilizes gd.
+The analysis is completed and stop.
+4.2 Application to the Magnetic Levitation System
+In this section, we apply the procedure outlined in the
+previous section to a strong stabilization problem of the
+magnetic levitation system presented in Section 3.1 in the
+digital control setting, and discuss the sampling affect.
+For simplicity, we consider the second-order reduced
+model gr(s) in (21) of the magnetic levitation system. For
+the digital control setting, assume an ideal sampler and a
+synchronized zeroth-order holder are placed around the
+continuous-time plant, which leads to the following time-
+discretized model
+gd(z) = κ
+z + 1
+(z − e−pT )(z − epT ),
+(26)
+where κ := k(1 − epT )(1 − e−pT )/(2p2) and T is the
+sampling period. Applying the bilinear transformation
+z ← (1 + s)/(1 − s), the continuous-time representative
+of gd(z) has the following form
+gd,c(s) = kc
+(1 − s)
+(s − q)(s + q),
+where kc = 2κ/((1 + epT )(1 + e−pT )) and q = (1 −
+e−pT )/(1 + e−pT ). Note that kc can also be expressed as
+kc = −kq2/p2. Therefore, we have qd,c(0) = k/p2 = gr(0),
+which is also equal to |gd(0)| = ∥gd∥L∞. We have the
+following result.
+Proposition 6. Consider the continuous-time representa-
+tion gd,c of the discrete-time plant gd. We have
+inf
+c∈S(gd,c) ∥c∥H∞ > p2
+k =
+1
+|gd,c(0)| =
+1
+∥gd,c∥L∞
+.
+(27)
+
+1.2
+T=3.482
+Megnitude
+0.8
+0.6
+0.4
+0.2
+0.2
+0.4
+0.6
+0.8
+1
+1.2
+1.4
+1.6
+1.8
+2
+Freguency(rad/sec)This in turn implies the discrete-time plant gd can not
+be stabilized by a stable controller with norm arbitrarily
+close to 1/|gd(1)| = 1/∥gd∥L∞.
+Proof. Proposition 6 follows from the facts that
+• gd,c ∈ G0
+1. This is verified by
+d
+dω log |gd,c(jω)| = ω(−ω2 − 2ω2 + q2(q2 − 2))
+(ω2 + 1)(ω2 + q2)2
+.
+Since q ∈ (0, 1), the derivative is 0 when ω = 0, and
+strictly negative when ω ̸= 0.
+• The PCR of gd,c at zero frequency is −1. This is
+verified by the imaginary part of
+d
+dω log(gd,c(jω)),
+which is equal to
+−j
+1 − jω −
+j
+jω − q −
+j
+jω + q .
+When ω = 0, the imaginary part is equal to −1.
+Thus, gd,c violates the necessary condition for having
+exact RIR stated in statement (I) of Theorem 2, and
+hence comes inequality (27). This in turn implies that the
+discrete-time model gd can not be stabilized by a stable
+controller with norm arbitrarily close to the inverse of the
+L∞-norm of gd.
+Proposition 6 is in stark contrast to Proposition 3, where
+the continuous-time plant gr is shown to have strongly
+stabilizing controller with its norm arbitrarily close to
+1/|gr(0)|. It appears that the very act of sampling and
+the introduction of the zeroth-order-hold discretization
+decimate this property. To obtain an upper bound of the
+minimum-norm strongly stabilizing controller of gd, we
+apply a lead compensator to gd,c, similar to that intro-
+duced for the third-order continuous-time plant. One can
+readily verify that, with a lead compensator, ˜gd,c(s) =
+gd,c(s) (τ+1)s+1
+τs+1
+belongs to G0
+1 with zero PCR at the zero
+frequency for any 0 < τ ≤ 1/q2 − 1 = 4 e−pT /(1 − e−pT )2.
+Setting τ to be 1/q2 − 1, we have the following result.
+Proposition 7. Let τ = 1/q2 − 1. The plant ˜gd,c satisfies
+inf
+c∈S(˜gd,c) ∥c∥H∞ =
+1
+|˜gd,c(0)| =
+1
+∥˜gd,c∥L∞
+= p2
+k ,
+(28)
+where the infimum is obtained by the sequence of stabi-
+lizing controllers cǫ,
+cǫ(s) = p2
+k + ǫp2τ
+kq2
+�
+s + ǫ2
+τ
+1+τ
+s + (α + ǫ)(1 + τ)
+�
+.
+Here ǫ is a sufficiently (in fact, arbitrarily) small positive
+number, and α > 0 is chosen sufficiently large for given ǫ.
+This in turn implies infc∈S(gd,c) ∥c∥H∞ ≤ p2/(k(1 − q2)).
+Proof. It can be verified that the characteristic equation of
+the closed-loop system with the controller cǫ has the form
+s4 + α(1 + τ)s3 + τǫ(1 − ǫ2)s2
++ ǫ
+�
+1 + ǫ2
+τ 2
+1 + τ
+�
+s + ǫ3
+τ
+1 + τ = 0.
+Let ai, i = 0, · · · , 3, denote the coefficients of si, i =
+0, · · · , 3, respectively. Applying the Routh-Hurwitz sta-
+bility criterion, the closed-loop system is stable if and
+only if a1a2a3 −a2
+1 −a0a2
+3 > 0. Now one can readily verify
+that, for every small enough ǫ, there is a corresponding
+large enough α such that the inequality holds. The infi-
+mum in (28) is obtained by taking ǫ to 0.
+Remark 3. By Propositions 6 and 7, we have p2/k <
+infc∈S(gd,c) ∥c∥H∞ ≤ p2/(k(1 − q2)). The upper bound
+has a factor 1/(1 − q2) = (1 + e−pT )2/(4 e−pT ). Note
+that this factor approaches 1 as e−pT approaches 1, or as
+pT approaches 0. Thus, when T approaches 0, the result
+matches that of the continuous-time 2nd-order reduced
+model gr.
+The closed-loop system of gd,c(s) with
+c∗(s) = p2
+k
+�(1 + τ)s + 1
+τs + 1
+�
+has three poles at the origin. Applying the bilinear trans-
+formation s ← z−1
+z+1, we obtain the discrete-time controller
+cd,∗(z) := c∗( z−1
+z+1). One can verify that the closed-loop
+system of gd(z) with cd,∗(z) has three poles at 1.
+5. CONCLUDING REMARKS
+We recalled the phase change rate maximization problem
+and solution from Hara (2022) and illustrated the latter’s
+utility in the robust instability analysis of a cyclic net-
+work of homogenous multi-agent systems subject to an
+identical multiplicative stable perturbation on each agent.
+We also applied the result to two practical applications
+— magnetic levitation systems and repressilators with
+time-delay. In addition, robust instability of digital con-
+trol systems was characterized via the use of the bilinear
+transformation. An interesting future research direction
+involves examining the robust instability of a cyclic net-
+work subject to heterogeneous multiplicative perturba-
+tions on the agents.
+REFERENCES
+K. Zhou and J. Doyle and K. Glover. Robust and Optimal
+Control, Prentice Hall, New Jersey, 1996.
+S. Hara, T. Iwasaki, and Y. Hori. Robust instability anal-
+ysis with neuronal dynamics. IEEE Conf. Dec. Contr.,
+2020 (arXiv 2003.01868).
+S. Hara, T. Iwasaki, and Y. Hori. Instability Margin Anal-
+ysis for Parametrized LTI Systems with Application to
+Repressilator. Automatica, 2021.
+D.C. Youla, J.J. Bongiorno, Jr., and C.N. Lu. Single-loop
+feedback-stabilization of linear multivariable dynami-
+cal plants. Automatica, vol.10, pp.159-173, 1974.
+M. Zeren and H. Ozbay.
+On the strong stabilization
+and stableH∞-controller design problems for MIMO
+systems. Automatica, vol.36, pp.1675-1684, 2000.
+Y. Ohta, H. Maeda, S. Kodama, and K. Yamamoto.
+A study on unit interpolation with rational analytic
+bounded functions. Trans. of the Society of Instrument
+and Control Engineers, no. 1, pp. 124-129, 2001.
+S. Hara, C.Y. Kao, S.Z. Khong, T. Iwasaki, and Y. Hori.
+Exact Instability Margin Analysis and Minimum Norm
+Strong Stabilization–phase change rate maximization–
+submitted to IEEE Trans. on Automatic Control, 2022
+(arXiv 2202.09500)
+M. B. Elowitz and S. Leibler.
+A synthetic oscillatory
+network of transcriptional regulators Nature, vol. 403,
+no. 6767, pp. 335–338, 2000.
+
+T. Namerikawa and M. Fujita Uncertainty structure and
+µ-synthesis of a magnetic suspension system
+IEEJ
+Transactions on Electronics, Information and Systems, vol.
+121, no. 6, pp. 1080–1087, 2001.
+
diff --git a/jNE5T4oBgHgl3EQfGA7Z/content/tmp_files/2301.05428v1.pdf.txt b/jNE5T4oBgHgl3EQfGA7Z/content/tmp_files/2301.05428v1.pdf.txt
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+Experimental Observation of Topological Quantum Criticality
+Sonja Barkhofen,1 Syamsundar De,1, 2 Jan Sperling,1 Christine Silberhorn,1
+Alexander Altland,3 Dmitry Bagrets,3 Kun Woo Kim,4 and Tobias Micklitz5
+1Integrated Quantum Optics Group, Institute for Photonic Quantum Systems (PhoQS),
+Paderborn University, Warburger Straße 100, 33098 Paderborn, Germany
+2Advanced Technology Development Centre, Indian Institue of Technology Kharagpur, Kharagpur 721302, India
+3Institut f¨ur Theoretische Physik, Universit¨at zu K¨oln, Z¨ulpicher Straße 77, 50937 K¨oln, Germany
+4Department of Physics, Chung-Ang University, 06974 Seoul, Republic of Korea
+5Centro Brasileiro de Pesquisas F´ısicas, Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro, Brazil
+(Dated: January 16, 2023)
+We report on the observation of quantum criticality forming at the transition point between
+topological Anderson insulator phases in a one-dimensional photonic quantum walk with spin. The
+walker’s probability distribution reveals a time-staggered profile of the dynamical spin-susceptibility,
+recently suggested as a smoking gun signature for topological Anderson criticality in the chiral
+symmetry class AIII. Controlled breaking of phase coherence removes the signal, revealing its origin
+in quantum coherence.
+Introduction:—The
+presence
+of
+disorder
+in
+one-
+dimensional (1d) systems generically causes Anderson lo-
+calization of single particle states at microscopically short
+length scales [1, 2]. The single known exception to this
+rule is quantum criticality between different symmetry
+protected topological phases [3, 4].
+At criticality, the
+number of topological boundary states changes, and the
+only way to do so is by hybridization through the bulk.
+This topologically enforced delocalization trumps Ander-
+son localization and leads to the transient formation of an
+extended quantum critical state whose exotic properties
+include the extremely (logarithmically) slow spreading of
+wave packages, or vanishing typical (but finite ensemble
+averaged) conductance [5]. In this paper, we report on
+the experimental observation of such criticality between
+topological phases in symmetry class AIII.
+The experimental realization of this setting is challeng-
+ing. It requires precision control over an internal degree
+of freedom, or ‘spin’, difficult to achieve in ultracold atom
+setups, otherwise tailored to the observation of Anderson
+localization [6–8]. Second, the identification of reluctant
+(logarithmically slow) delocalization appears to require
+signal observation over exponentially long time scales.
+However, the toolbox of quantum optics experimen-
+tation turns out sufficiently versatile to overcome these
+challenges. In a recent work, we proposed the blueprint of
+a photonic quantum simulator of the extended state, and
+a delicate time-staggered signature in the spin suscepti-
+bility as smoking gun evidence for criticality already on
+short time scales [9]. We here report on the experimen-
+tal realization of this proposal within a tunable optical
+linear network.
+Quantum walk protocol:—A schematic of the photonic
+quantum simulator is shown in Fig. 1. The optical linear
+network simulates a one-dimensional quantum walk of a
+spin-1/2 particle with single time-step Floquet evolution,
+ˆU( ˆϕ, ˆθ) = R3( ˆϕ
+2 )R1(
+ˆθ
+2) ˆT R1(
+ˆθ
+2)R3( ˆϕ
+2 ),
+(1)
+FIG. 1. Experimental setup based on an unbalanced Mach-
+Zehnder interferometer with dynamic coin and deterministic
+in/out coupling (see text for details).
+and step- and coin-operators,
+ˆT =
+�
+q
+(|q + 1, ↑⟩⟨↑, q| + |q − 1, ↓⟩⟨↓, q|) ,
+(2)
+Rj(ˆα) =
+�
+q,σσ′
+|q, σ⟩
+�
+e−iαq ˆσi�
+σσ′ ⟨q, σ′|,
+j = 1, 3.
+(3)
+Here the sums are over lattice sites (with unit spacing)
+q ∈ Z, and spin-orientations σ, σ′ ∈ {↑, ↓} parametriz-
+ing the walker’s internal degrees of freedom, with Pauli
+matrices ˆσi, i = 1, 2, 3 operating on the latter. Through-
+out the work we denote eigenstates of ˆσ3 and ˆσ2 by ↑, ↓
+and +, −, respectively. The latter will play a major role,
+and are also referred to as circular right (R)/left (L) po-
+larized states. Static disorder is introduced by drawing
+site-dependent angles ϕq, θq from some distribution (see
+also below), which leaves the average values ¯ϕ, ¯θ as tun-
+ing parameters.
+The operator
+ˆU
+possesses the (chiral) symmetry
+ˆσ2 ˆU ˆσ2 = ˆU †, putting it into class AIII of the classi-
+fication scheme.
+Eigenstates, |ψ⟩, of Floquet systems
+with chiral symmetries come in pairs with quasi-energies
+arXiv:2301.05428v1  [quant-ph]  13 Jan 2023
+
+EOM2
+laser
+H
+EOM3
+H
+EOM12
+FIG. 2. Experimental (left panel) and numerical (center panel) probability distributions for static (purple/red) and dephasing
+(orange/brown) disorder. The distributions are averaged over 500 disorder realizations detected in the circular R/L basis with
+circular R input state, γ = θEOM = π/8 for step 14. Right panel: Both, in experiment and numerical simulation, distributions
+are well described by exponential profiles for static, respectively, Gaussian profiles for dephasing disorder. Dotted lines with
+slope 1 (red) and 2 (blue) here are introduced for comparison.
+ϵ0 ± ω, mirror symmetric around center energies, ϵ0.
+Presently, we have four of those, ϵ0 = 0, π, ±π/2. The
+first emerges as a direct consequence of chirality: if |ψ⟩
+is a state with quasi-energy ϵ, then ˆσ2|ψ⟩ is the one
+with energy −ϵ.
+The remaining three originate in a
+second symmetry, our walk operator anticommutes with
+ˆS ≡ �
+q |q⟩(−1)q⟨q|, which is to say that it hops between
+neighboring sites, . . . q ←→ (q + 1) ←→ (q + 2) . . .. It is
+straightforward to check that ˆS|ψ⟩ is an eigenstate with
+energy ϵ+π[10]. Hence, the entire spectrum is π-shift in-
+variant, explaining the second mirror symmetry 0 → π.
+The remaining pair is best understood by defining an
+auxiliary operator V = iU. With (ˆσ2 ˆS)V (ˆσ2 ˆS) = V †, it
+follows that 0 and π are mirror energies of V . However,
+V and U have the same eigenstates, with quasi-energies
+shifted by a factor i = eiπ/2. This explains why the (0, π)
+pair of V becomes the (−π/2, π/2) pair of our operator U.
+At critical points separating topologically distinct An-
+derson insulating phases, gaps in the quasienergy spec-
+trum of the clean system close and delocalized states at
+the above quasienergies emerge.
+Considering the walk
+(1), we find that it contains a critical point for the av-
+erage disorder value ¯θ = 0 [9] at which all four energies
+simultaneously are critical [11]. Finite constant values
+¯θ gap out the pair (0, π) while a staggered configura-
+tion θs = const. × (−)q gaps out (−π/2, π/2). However,
+throughout we will keep the system at ¯θ = 0 and identify
+signatures of the esnuing delocalized states.
+Time-staggered spin-polarization:—Anderson critical
+states retain ‘memory’ of the anti-unitary symmetries
+defining them [5]: consider the average probability for
+a walker initially prepared on site q = 0 in spin-state σ
+to be found after t time-steps at a distance q in spin-state
+σ′,
+Pσ′σ(t, q) = ⟨|⟨q, σ′| ˆU t|0, σ⟩|2⟩θ.
+(4)
+Our linear optical network gives direct experimental ac-
+cess to this observable, where the average ⟨. . . ⟩θ is over
+multiple runs, each for a randomly drawn binary config-
+uration θq ∈ {±θ} at constant ϕq = 0. Preparing the
+walker in a ˆσ2-eigenstate σ ∈ {+, −}, the probabilities
+P+,σ and P−,σ differ [9], reflecting the origin of critical-
+ity in a symmetry involving ˆσ2. This asymmetry, absent
+in Anderson localized phases, motivates the introduction
+of the spin-polarization
+∆P(t) ≡
+�
+q
+(P−−(t, q) − P+−(t, q)) ,
+(5)
+sampled over all sites as a unique diagnostic of topological
+quantum criticality.
+What makes the measurement of ∆P challenging is
+that all states in the quasienergy spectrum contribute to
+P, while only the states of distinguished quasi-energies
+(0, π) and (−π/2, π/2) contribute to the asymmetry [12].
+More precisely, the states defined by ˆσ2 yield a signal with
+smooth time dependence, while those with associated to
+ˆσ2 ˆS yield a staggered signal ∆P(t) = (−1)t|∆P(t)|, cor-
+responding to the sign alternation of the ˆS-operator as
+introduce above [9]. The added contribution of all states
+is thus expected to show spectral peaks at ω = 0, π in the
+Fourier transform of ∆P(t). This is our principal exper-
+imental signature of AIII topological quantum criticality
+in the walk.
+Simulator and experimental realization:—Fig. 1 shows
+the experimental setup for our quantum simulator. At its
+core is a Mach-Zehnder interferometer with a feedback
+loop realising the chiral translation ˆT, Eq. (2), via time-
+multiplexing: A laser pulse is split into horizontal and
+
+experiment
+numerics
+experiment
+0.12
+0.12
+In(-In (q)/ 5(0)2)
+0
+ dephasing
+I dephasing
+ static
+I static
+0.1
+0.1
+-2
+dephasing
+0.08
+0.08
+static
+.4
+2
+0.5
+1
+1.5
+2
+2.5
+In q
+numerics
+0
+1(o)/(b)u-)ul
+0.04
+0.04
+2
+0.02
+0.02
+dephasing
+static
+0
+0
+.4
+-12
+-8
+-4 
+0
+4
+8
+12
+-12
+-8
+-4
+0
+4
+8
+12
+0.5
+1.5
+2
+2.5
+1
+q
+q
+In g3
+FIG. 3.
+Comparison of experimental results and numerical simulations for critical quantum walks with static (upper panels)
+and dephasing (left-bottom panel) disorder and up to t = 15 time steps. Results are averaged over 500 different realisations of
+the quantum walk and error bars indicate the statistical error of the 500 measurements. Top-left: Average spin-polarization
+⟨∆P(t)⟩. Top-right: Corresponding power spectra, S(ω), of the static binary disorder. Bottom-left: The power spectra of
+quantum walk with dephasing disorder are shown. Bottom-right: The distribution of power spectrum S(ω = 0) and S(ω = π)
+is plotted by sampling 1000 points where each data point is obtained from the spin polarization averaged over 500 random
+disorder realizations.
+vertical polarization orientations (the ‘spin’-components
+↑, ↓) and send through fiber lines of different lengths. In
+this way positions on the lattice are mapped onto time
+domain, with time delay between vertically/horizontally
+polarized pulses defining the lattice constant [13–15].
+The network architecture offers full control over the dy-
+namic coin operations ˆR, Eq. (3) via polarization rota-
+tions [14, 16], and allows to measure the spin resolved
+probability distributions Pσ′σ defined in Eq. (4) [17]. It
+thus provides the key ingredients of our proposal. In the
+following, we discuss the concrete realization of this pro-
+tocol in the quantum optical simulator. Readers primar-
+ily interested in results are invited to continue reading in
+the following section.
+We realize the quantum walker by a weak coherent
+laser pulse at telecom wavelength and its polarisation
+acts as the internal degree of freedom. The initial polar-
+isation is set at quarter wave plate QWP1 at -45◦, such
+that left circular light L enters the setup. The dynamic
+coin operation is accomplished by an EOM, implement-
+ing the voltage dependent polarisation rotation
+R1(θU) =
+� cos(θU)
+−i sin(θU)
+−i sin(θU)
+cos(θU)
+�
+.
+(6)
+As the EOM switches fast enough to address each pulse
+(i.e. each quantum walk position) individually it is ca-
+pable of realising both static and dynamic disorder. Its
+programmability makes the recording of hundreds of pat-
+terns in short time without manual setup of parameters
+possible. After the coin operation, the light pulses are
+split according to their polarisation at a polarising beam
+splitter (PBS1) and are routed into the two arms of the
+interferometer. The pulses propagating in the upper arm
+are retarded by a time delay of ∆τ ≈ 105 ns relative to
+those in the lower arm. The conditional routing of the
+photons through the long or short fibre realises the chi-
+ral translation operator, Eq. (2). The time delay ∆τ and
+roundtrip time τRT here define the lattice spacing and
+single time step duration, respectively. After PBS2 the
+
+static disorder
+0.06
+0.04
+0.02
+<()
+P
+0
+V
+-0.02
+-0.04
+numeric
+experiment
+-0.06
+5
+6
+7
+8
+9
+10
+11
+12
+13
+14
+qstatic disorder
+0.7
+numeric
+experiment
+0.6
+0.5
+(3)
+0.4
+S
+0.3
+0.2
+0.1
+0
+0.2
+0.4
+0.6
+0.8
+1
+w (2元)dephasing disorder
+0.7
+numerics
+0.6
+experiment
+0.5
+0.4
+(3)
+S
+0.3
+0.2
+0.1
+0
+0.2
+0.4
+0.6
+0.8
+1
+w (2元)Distribution
+static(num)
+0.8
+dephasing(num)
+0.7
+static(exp)
+dephasing(exp)
+0.6
+0.5
+3
+0.4
+S
+0.3
+0.2
+0.1
+0
+0
+0.2
+0.4
+0.6
+0.8
+S(w =0)4
+pulses are feedbacked in the loop and the dynamics con-
+tinues until the desired final step, in which the train of
+pulses is deterministically coupled out by the EOMs 2
+and 3. Given that these EOMs flip the polarization for
+the in and outcoupling the full dynamics of an N step
+evolution is described by
+ˆUfull = ˆσ1(T −1R1(θU))N ˆσ1T.
+The application of the operator T describes the first tran-
+sition through the fibre arms in which the role of the
+horizontally and the vertically polarized light is swapped
+with respect to the following roundtrips. The consecutive
+ˆσ1 matrices signify the polarization flip by the EOMs 2
+and 3 for in and outcoupling. As ˆσ1 commutes with R1
+and using T = ˆσ1T −1ˆσ1 we obtain for the full evolution
+ˆUfull = (TR1(θU))NT.
+which satisfies the chiral symmetries introduced above.
+The deterministic in- and outcoupling of the light, as
+already introduced in [15, 18], minimizes the roundtrip
+losses with respect to the probabilistic version used ear-
+lier [13] and thus enables the recording of longer dynam-
+ics. The detection units consists of two superconducting
+nanowire single photon detectors (SNSPDs) with dead
+times well below the pulse separation, which in combina-
+tion with PBS3 and a QWP at 45◦ enable the measure-
+ment of each pulse intensity in the circular R/L basis.
+Results:—We obtain the spin-resolved probability dis-
+tribution by recording measurements for 500 different re-
+alisations of polarization rotations simulating the static
+binary disorder discussed above. To define a reference
+frame without Anderson localization to compare to, we
+additionally simulate dynamic disorder for which local-
+ization is absent due to the destruction of phase coher-
+ence. The latter is realized via application of the polar-
+ization rotations randomly distributed in time, at other-
+wise identical system parameters.
+The comparison between the two recordings is pre-
+sented in Fig. 2, at time step t = 14, traced over polari-
+sation. At these values, the Anderson localized and the
+diffusively spreading wave package do not yet differ by
+much in absolute values. Instead, the difference shows in
+the shape of the distribution. That is, the convex profile
+of an exponential distribution in the Anderson localized
+case, compared to the concave form of a Gaussian diffu-
+sive profile, as verified in the right panel. (The boundary
+peaks in the diffusive case represent a small fraction of
+quasi-ballistically propagating contributions which dis-
+appear in the numerical simulation of longer runs, see
+Supplemental Material.)
+For each random realization of binary angles we extract
+the spatially integrated spin-polarization, Eq. (5), which
+we then average over different angle realizations.
+The
+upper left panel in Figure 3 shows experimental data,
+and corresponding numerical simulations, for ⟨∆P(t)⟩
+between t = 5 and 14 time steps, averaged over 500
+realizations of static binary disorder.
+The upper-right
+panel exhibits the resulting power spectrum, S(ω) =
+| ¯P(ω)|2/ �
+ω | ¯P(ω)|2, where ¯P(ω) = �14
+t=5 eiωt⟨∆P(t)⟩.
+For comparison, we show in the lower left panel the
+corresponding power spectrum for dynamic binary dis-
+order.
+The latter has a structureless random pattern
+with contributions of the same order from all frequen-
+cies, as expected for a noisy spin polarization. In con-
+trast, the spin polarization for static disorder shows the
+time-staggering predicted for the topological quantum
+critical state, which is also witnessed by the pronounced
+peak at ω = π in the power spectrum. The relatively
+small number of time-steps probed in the experiment im-
+plies sizeable fluctuations of random parameters, which
+need to be taken into account in the interpretation of
+data: for a given run, the average value of tmax ran-
+domly drawn angular parameters scales as ∼ O(1/
+√
+t),
+and a value of comparable magnitude for the staggered
+amplitude. In this way, the critical states both, at (0, π)
+and (−π/2, π/2) get effectively gapped out for at least a
+fraction of the runs, resulting in a suppression of peaks
+in the power spectrum at ω = 0 and ω = π, respectively
+(cf. Fig. 3). To further illustrate this point, we repeated
+numerical simulations of ensembles of 500 random bi-
+nary angle configurations a large number of times. In the
+right-lower panel of Fig. 3 we show the distributions of
+peak values S(ω = 0, π) in the power spectrum, resulting
+from 1000 repetitions of the previously described proce-
+dure. That is, each point here presents the average over
+an ensemble of 500 angle realizations (the ensembles used
+in experiment are indicated by the squares). As antici-
+pated above, the distribution for static disordered is cen-
+tered around large peak values, either at ω = 0, at ω = π,
+or at both. The distribution for dynamic disorder, on the
+other hand, is always dominated by small peak values at
+both frequencies. The average power spectrum for the
+entire ensemble of 1000 × 500 angle configurations (cf.
+Supplemental Material) then shows the expected two-
+peak structure, with peaks at ω = 0, π dominating over
+an otherwise approximately flat background, confirming
+thus that upon simulating larger runs, fluctuation effects
+diminish.
+Conclusion:—We have realized an optical linear net-
+work simulator of one-dimensional topological quantum
+criticality. The simulator implements the photonic quan-
+tum walk of a spin-1/2 particle with chiral symmetry.
+Its optical network architecture allows us to fully ac-
+cess and monitor the state’s internal degree of freedom.
+Upon tuning to the critical point separating two topo-
+logical Anderson insulating phases, we observe a time-
+staggered spin-polarization recently suggested as a smok-
+ing gun signature of quantum critical dynamics. Exter-
+nally imposed time dependent noise, or ‘dephasing disor-
+der’, destroys the signal, revealing its origin in quantum
+coherence. A similar destruction takes place upon break-
+ing the chiral symmetry of the walk, and along with it
+
+5
+the transition between two symmetry protected topolog-
+ical phases. Ideally, one would like to monitor scaling
+phenomena induced by such type of symmetry breaking,
+however, the currently realizable signal times are still too
+short for such type of statistics. Irrespective of such lim-
+itations, we believe that fully programmable quantum
+networks promise interesting perspectives for the simu-
+lation of topological quantum matter in the presence of
+engineered randomness.
+The Integrated Quantum Optics group acknowledges
+support by the ERC project QuPoPCoRN (Grant no.
+725366).
+T. M. acknowledges financial support by
+Brazilian agencies CNPq and FAPERJ. A. A. and
+D. B. acknowledge partial support from the Deutsche
+Forschungsgemeinschaft (DFG) within the CRC network
+TR 183 (project grant 277101999) as part of projects A01
+and A03. K.W.K. acknowledges financial support by Ba-
+sic Science Research Program through the National Re-
+search Foundation of Korea (NRF) funded by the Min-
+istry of Education (No.2021R1F1A1055797) and Korea
+government(MSIT) (No.2020R1A5A1016518).
+[1] P. W. Anderson, Absence of diffusion in certain random
+lattices, Phys. Rev. 109, 1492 (1958).
+[2] E. Abrahams, P. W. Anderson, D. C. Licciardello, and
+T. V. Ramakrishnan, Scaling theory of localization: Ab-
+sence of quantum diffusion in two dimensions, Phys. Rev.
+Lett. 42, 673 (1979).
+[3] F. Evers and A. D. Mirlin, Anderson transitions, Rev.
+Mod. Phys. 80, 1355 (2008).
+[4] A. Altland, D. Bagrets, and A. Kamenev, Topology ver-
+sus anderson localization: Nonperturbative solutions in
+one dimension, Phys. Rev. B 91, 085429 (2015).
+[5] L. Balents and M. P. A. Fisher, Delocalization transition
+via supersymmetry in one dimension, Phys. Rev. B 56,
+12970 (1997).
+[6] J. Chab´e, G. Lemari´e, B. Gr´emaud, D. Delande, P. Szrift-
+giser, and J. C. Garreau, Experimental Observation of
+the Anderson Metal-Insulator Transition with Atomic
+Matter Waves, Phys. Rev. Lett. 101, 255702 (2008).
+[7] C. Hainaut, I. Manai, J.-F. Cl´ement, J. C. Garreau,
+P. Szriftgiser, G. Lemari´e, N. Cherroret, D. Delande,
+and R. Chicireanu, Controlling symmetry and localiza-
+tion with an artificial gauge field in a disordered quantum
+system, Nature Communications 9, 1382 (2018).
+[8] J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht,
+P. Lugan, D. Cl´ement, L. Sanchez-Palencia, P. Bouyer,
+and A. Aspect, Direct observation of Anderson localiza-
+tion of matter waves in a controlled disorder, Nature 453,
+891 (2008).
+[9] D. Bagrets, K. W. Kim, S. Barkhofen, S. De, J. Sper-
+ling, C. Silberhorn, A. Altland, and T. Micklitz, Probing
+the topological anderson transition with quantum walks,
+Phys. Rev. Research 3, 023183 (2021).
+[10] We have U(ˆσ2 |ψ⟩)
+=
+ˆσ2U † |ψ⟩
+=
+e−iϵˆσ2 |ψ⟩, and
+U ˆS |ψ⟩ = − ˆSU |ψ⟩ = eiπ+iϵ |ψ⟩.
+[11] The associated topological invariants may be identified
+by analysis of the auxiliary ‘Hamiltonian’ ˆH = −i ln ˆU.
+However, we will not need this underlying structure [19,
+20] throughout.
+[12] To probe only critical states, one may prepare initial
+states extended over several sites with fixed phase re-
+lations, as discussed in Ref. [9]. Their experimental re-
+alization is, however, challenging and we thus focus on
+initial states localized on single sites, probing the entire
+quasi-energy spectrum.
+[13] A. Schreiber, K. N. Cassemiro, V. Potoˇcek, A. G´abris,
+P. J. Mosley, E. Andersson, I. Jex, and C. Silberhorn,
+Photons Walking the Line: A Quantum Walk with Ad-
+justable Coin Operations, Phys. Rev. Lett. 104, 050502
+(2010).
+[14] A. Schreiber, K. N. Cassemiro, V. Potoˇcek, A. G´abris,
+I. Jex, and C. Silberhorn, Decoherence and Disorder in
+Quantum Walks: From Ballistic Spread to Localization,
+Physical Review Letters 106, 180403 (2011).
+[15] T.
+Nitsche,
+S.
+Barkhofen,
+R.
+Kruse,
+L.
+Sansoni,
+M. ˇStefaˇn´ak, A. G´abris, V. Potoˇcek, T. Kiss, I. Jex,
+and C. Silberhorn, Probing measurement-induced effects
+in quantum walks via recurrence, Science Advances 4,
+eaar6444 (2018).
+[16] S. Barkhofen, T. Nitsche, F. Elster, L. Lorz, A. G´abris,
+I. Jex, and C. Silberhorn, Measuring topological invari-
+ants in disordered discrete-time quantum walks, Phys.
+Rev. A 96, 033846 (2017).
+[17] S. Barkhofen, L. Lorz, T. Nitsche, C. Silberhorn, and
+H. Schomerus, Supersymmetric polarization anomaly in
+photonic discrete-time quantum walks, Phys. Rev. Lett.
+121, 260501 (2018).
+[18] T. Nitsche,
+S. De,
+S. Barkhofen,
+E. Meyer-Scott,
+J. Tiedau, J. Sperling, A. G´abris, I. Jex, and C. Sil-
+berhorn, Local versus global two-photon interference in
+quantum networks, Phys. Rev. Lett. 125, 213604 (2020).
+[19] J. K. Asb´oth, Symmetries, topological phases, and bound
+states in the one-dimensional quantum walk, Phys. Rev.
+B 86, 195414 (2012).
+[20] J. K. Asb´oth and H. Obuse, Bulk-boundary correspon-
+dence for chiral symmetric quantum walks, Phys. Rev. B
+88, 121406 (2013).
+Probability distributions at longer times
+As discussed in the main text, all states in the
+quasienergy spectrum contribute to the probability dis-
+tribution of the walker. Since most of the states are non-
+critical, the probability distribution for static binary dis-
+order thus follows |ψ(q)|2 ∼ e−|q|/λ, as shown in Fig. 4
+(left panel) for numerical simulations of different time
+steps t = 15, 23, 31, 39. The q-dependence of − ln |ψ(q)|2
+is plotted in Fig. 4 (right-top panel), clearly showing a
+linear dependence on position. When the binary disor-
+der also fluctuates in time, quantum phase coherence
+is destroyed turning dynamics thus into classical diffu-
+sion, |ψ(q)|2 ∼ e−q2/σ2
+t , with σ2
+t ∼ t.
+The numerical
+simulation, Fig. 4 (center and right-bottom panel), con-
+firms the diffusive behavior and corresponding scaling
+of − ln |ψ(q)|2. Notice that the boundary peaks in the
+probability distributions are due to a small fraction of
+
+6
+quasi-ballistically propagating states. As shown in Fig. 4,
+their contribution disappears as longer times t ≳ 20 are
+probed.
+Statistics of the power spectrum S(ω)
+The relatively small number of time steps measured
+in the experiment implies that the signature of critical
+states, viz. the time-staggered spin polarization, is sub-
+ject to large statistical fluctuations.
+From our numer-
+ical simulations we find that it is necessary to average
+over many more than 500 disorder realizations to clearly
+observe the two-peak structure in the power spectrum.
+Figure 3 (bottom-right panel) in the main text shows the
+distribution of 1000 peak heights at frequencies ω = 0, π
+in the power spectrum for static (purple) and dephas-
+ing disorder (blue).
+Each of the 1000 shown points is
+obtained from averaging the power spectrum over 500
+random disorder realizations. The average and standard
+deviation of the entire power spectrum S(ω) after aver-
+aging over the total ensemble of 1000 × 500 realizations
+is plotted in Fig. 5 below.
+
+7
+FIG. 4.
+Probability distributions after t = 15, 23, 31, 39 time steps for static (left panel) and dephasing (center panel) disorder.
+The boundary peak visible at t = 15 disappears for larger time steps. Right panels show − ln(|ψ(q)|2) for static disorder (top)
+and dephasing disorder (bottom).
+FIG. 5.
+Average and standard deviation of the power spec-
+trum of the spin polarization for an ensemble of 1000 × 500
+disorder realizations.
+
+static
+dephasing
+static
+0.12
+0.1 
+6
+- t=15
+t=15
+135(q)/2
+ t=23
+t=23
+0.1
+t=31
+0.08
+t=31
+4
+- t=39
+ t=39
+0.08
+2
+0.06
+2
+2
+0
+20
+40
+I(b) /
+q
+dephasing
+0.04
+0.04
+6
+125(q)/2
+0.02
+0.02
+4
+0
+0
+2
+4
+8
+12
+20
+24
+28
+32 
+36
+0
+16
+0
+4
+8
+12
+20
+24
+28
+32
+36
+0
+20
+40
+q
+q
+q0.6
+dephasing
+0.5
+static
+0.4
+0.3
+3
+S
+0.2
+0.1
+0
+-0.1
+0.2
+0.4
+0.6
+0.8
+1
+w (2元)
\ No newline at end of file
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new file mode 100644
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+page_content='Experimental Observation of Topological Quantum Criticality  Sonja Barkhofen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
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+page_content='1 Christine Silberhorn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
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+page_content='3 Kun Woo Kim,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
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+page_content=' Kharagpur 721302,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' India  3Institut f¨ur Theoretische Physik,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Universit¨at zu K¨oln,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Z¨ulpicher Straße 77,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' 50937 K¨oln,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Germany  4Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Chung-Ang University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' 06974 Seoul,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Republic of Korea  5Centro Brasileiro de Pesquisas F´ısicas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Rua Xavier Sigaud 150,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' 22290-180,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Rio de Janeiro,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Brazil  (Dated: January 16,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' 2023)  We report on the observation of quantum criticality forming at the transition point between  topological Anderson insulator phases in a one-dimensional photonic quantum walk with spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The  walker’s probability distribution reveals a time-staggered profile of the dynamical spin-susceptibility,  recently suggested as a smoking gun signature for topological Anderson criticality in the chiral  symmetry class AIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Controlled breaking of phase coherence removes the signal, revealing its origin  in quantum coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  Introduction:—The  presence  of  disorder  in  one-  dimensional (1d) systems generically causes Anderson lo-  calization of single particle states at microscopically short  length scales [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The single known exception to this  rule is quantum criticality between different symmetry  protected topological phases [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  At criticality, the  number of topological boundary states changes, and the  only way to do so is by hybridization through the bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  This topologically enforced delocalization trumps Ander-  son localization and leads to the transient formation of an  extended quantum critical state whose exotic properties  include the extremely (logarithmically) slow spreading of  wave packages, or vanishing typical (but finite ensemble  averaged) conductance [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' In this paper, we report on  the experimental observation of such criticality between  topological phases in symmetry class AIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  The experimental realization of this setting is challeng-  ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' It requires precision control over an internal degree  of freedom, or ‘spin’, difficult to achieve in ultracold atom  setups, otherwise tailored to the observation of Anderson  localization [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Second, the identification of reluctant  (logarithmically slow) delocalization appears to require  signal observation over exponentially long time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  However, the toolbox of quantum optics experimen-  tation turns out sufficiently versatile to overcome these  challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' In a recent work, we proposed the blueprint of  a photonic quantum simulator of the extended state, and  a delicate time-staggered signature in the spin suscepti-  bility as smoking gun evidence for criticality already on  short time scales [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' We here report on the experimen-  tal realization of this proposal within a tunable optical  linear network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  Quantum walk protocol:—A schematic of the photonic  quantum simulator is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The optical linear  network simulates a one-dimensional quantum walk of a  spin-1/2 particle with single time-step Floquet evolution,  ˆU( ˆϕ, ˆθ) = R3( ˆϕ  2 )R1(  ˆθ  2) ˆT R1(  ˆθ  2)R3( ˆϕ  2 ),  (1)  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Experimental setup based on an unbalanced Mach-  Zehnder interferometer with dynamic coin and deterministic  in/out coupling (see text for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  and step- and coin-operators,  ˆT =  �  q  (|q + 1, ↑⟩⟨↑, q| + |q − 1, ↓⟩⟨↓, q|) ,  (2)  Rj(ˆα) =  �  q,σσ′  |q, σ⟩  �  e−iαq ˆσi�  σσ′ ⟨q, σ′|,  j = 1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  (3)  Here the sums are over lattice sites (with unit spacing)  q ∈ Z, and spin-orientations σ, σ′ ∈ {↑, ↓} parametriz-  ing the walker’s internal degrees of freedom, with Pauli  matrices ˆσi, i = 1, 2, 3 operating on the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Through-  out the work we denote eigenstates of ˆσ3 and ˆσ2 by ↑, ↓  and +, −, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The latter will play a major role,  and are also referred to as circular right (R)/left (L) po-  larized states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Static disorder is introduced by drawing  site-dependent angles ϕq, θq from some distribution (see  also below), which leaves the average values ¯ϕ, ¯θ as tun-  ing parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  The operator  ˆU  possesses the (chiral) symmetry  ˆσ2 ˆU ˆσ2 = ˆU †, putting it into class AIII of the classi-  fication scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  Eigenstates, |ψ⟩, of Floquet systems  with chiral symmetries come in pairs with quasi-energies  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='05428v1  [quant-ph]  13 Jan 2023  EOM2  laser  H  EOM3  H  EOM12  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Experimental (left panel) and numerical (center panel) probability distributions for static (purple/red) and dephasing  (orange/brown) disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The distributions are averaged over 500 disorder realizations detected in the circular R/L basis with  circular R input state, γ = θEOM = π/8 for step 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Right panel: Both, in experiment and numerical simulation, distributions  are well described by exponential profiles for static, respectively, Gaussian profiles for dephasing disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Dotted lines with  slope 1 (red) and 2 (blue) here are introduced for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  ϵ0 ± ω, mirror symmetric around center energies, ϵ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  Presently, we have four of those, ϵ0 = 0, π, ±π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The  first emerges as a direct consequence of chirality: if |ψ⟩  is a state with quasi-energy ϵ, then ˆσ2|ψ⟩ is the one  with energy −ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  The remaining three originate in a  second symmetry, our walk operator anticommutes with  ˆS ≡ �  q |q⟩(−1)q⟨q|, which is to say that it hops between  neighboring sites, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' q ←→ (q + 1) ←→ (q + 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='. It is  straightforward to check that ˆS|ψ⟩ is an eigenstate with  energy ϵ+π[10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Hence, the entire spectrum is π-shift in-  variant, explaining the second mirror symmetry 0 → π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  The remaining pair is best understood by defining an  auxiliary operator V = iU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' With (ˆσ2 ˆS)V (ˆσ2 ˆS) = V †, it  follows that 0 and π are mirror energies of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' However,  V and U have the same eigenstates, with quasi-energies  shifted by a factor i = eiπ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' This explains why the (0, π)  pair of V becomes the (−π/2, π/2) pair of our operator U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  At critical points separating topologically distinct An-  derson insulating phases, gaps in the quasienergy spec-  trum of the clean system close and delocalized states at  the above quasienergies emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  Considering the walk  (1), we find that it contains a critical point for the av-  erage disorder value ¯θ = 0 [9] at which all four energies  simultaneously are critical [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Finite constant values  ¯θ gap out the pair (0, π) while a staggered configura-  tion θs = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' × (−)q gaps out (−π/2, π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' However,  throughout we will keep the system at ¯θ = 0 and identify  signatures of the esnuing delocalized states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  Time-staggered spin-polarization:—Anderson critical  states retain ‘memory’ of the anti-unitary symmetries  defining them [5]: consider the average probability for  a walker initially prepared on site q = 0 in spin-state σ  to be found after t time-steps at a distance q in spin-state  σ′,  Pσ′σ(t, q) = ⟨|⟨q, σ′| ˆU t|0, σ⟩|2⟩θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  (4)  Our linear optical network gives direct experimental ac-  cess to this observable, where the average ⟨.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' ⟩θ is over  multiple runs, each for a randomly drawn binary config-  uration θq ∈ {±θ} at constant ϕq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Preparing the  walker in a ˆσ2-eigenstate σ ∈ {+, −}, the probabilities  P+,σ and P−,σ differ [9], reflecting the origin of critical-  ity in a symmetry involving ˆσ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' This asymmetry, absent  in Anderson localized phases, motivates the introduction  of the spin-polarization  ∆P(t) ≡  �  q  (P−−(t, q) − P+−(t, q)) ,  (5)  sampled over all sites as a unique diagnostic of topological  quantum criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  What makes the measurement of ∆P challenging is  that all states in the quasienergy spectrum contribute to  P, while only the states of distinguished quasi-energies  (0, π) and (−π/2, π/2) contribute to the asymmetry [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  More precisely, the states defined by ˆσ2 yield a signal with  smooth time dependence, while those with associated to  ˆσ2 ˆS yield a staggered signal ∆P(t) = (−1)t|∆P(t)|, cor-  responding to the sign alternation of the ˆS-operator as  introduce above [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The added contribution of all states  is thus expected to show spectral peaks at ω = 0, π in the  Fourier transform of ∆P(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' This is our principal exper-  imental signature of AIII topological quantum criticality  in the walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  Simulator and experimental realization:—Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' 1 shows  the experimental setup for our quantum simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' At its  core is a Mach-Zehnder interferometer with a feedback  loop realising the chiral translation ˆT, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' (2), via time-  multiplexing: A laser pulse is split into horizontal and  experiment  numerics  experiment  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='12  In(-In (q)/ 5(0)2)  0  dephasing  I dephasing  static  I static  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='1  2  dephasing  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='08  static  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='4  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='5  1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='5  In q  numerics  0  1(o)/(b)u-)ul  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='04  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='02  dephasing  static  0  0  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='4  12  8  4  0  4  8  12  12  8  4  0  4  8  12  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='5  2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='5  1  q  q  In g3  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  Comparison of experimental results and numerical simulations for critical quantum walks with static (upper panels)  and dephasing (left-bottom panel) disorder and up to t = 15 time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Results are averaged over 500 different realisations of  the quantum walk and error bars indicate the statistical error of the 500 measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Top-left: Average spin-polarization  ⟨∆P(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Top-right: Corresponding power spectra, S(ω), of the static binary disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Bottom-left: The power spectra of  quantum walk with dephasing disorder are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Bottom-right: The distribution of power spectrum S(ω = 0) and S(ω = π)  is plotted by sampling 1000 points where each data point is obtained from the spin polarization averaged over 500 random  disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  vertical polarization orientations (the ‘spin’-components  ↑, ↓) and send through fiber lines of different lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' In  this way positions on the lattice are mapped onto time  domain, with time delay between vertically/horizontally  polarized pulses defining the lattice constant [13–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  The network architecture offers full control over the dy-  namic coin operations ˆR, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' (3) via polarization rota-  tions [14, 16], and allows to measure the spin resolved  probability distributions Pσ′σ defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' (4) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' It  thus provides the key ingredients of our proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' In the  following, we discuss the concrete realization of this pro-  tocol in the quantum optical simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Readers primar-  ily interested in results are invited to continue reading in  the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  We realize the quantum walker by a weak coherent  laser pulse at telecom wavelength and its polarisation  acts as the internal degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The initial polar-  isation is set at quarter wave plate QWP1 at -45◦, such  that left circular light L enters the setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The dynamic  coin operation is accomplished by an EOM, implement-  ing the voltage dependent polarisation rotation  R1(θU) =  � cos(θU)  −i sin(θU)  −i sin(θU)  cos(θU)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  (6)  As the EOM switches fast enough to address each pulse  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' each quantum walk position) individually it is ca-  pable of realising both static and dynamic disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Its  programmability makes the recording of hundreds of pat-  terns in short time without manual setup of parameters  possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' After the coin operation, the light pulses are  split according to their polarisation at a polarising beam  splitter (PBS1) and are routed into the two arms of the  interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The pulses propagating in the upper arm  are retarded by a time delay of ∆τ ≈ 105 ns relative to  those in the lower arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The conditional routing of the  photons through the long or short fibre realises the chi-  ral translation operator, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The time delay ∆τ and  roundtrip time τRT here define the lattice spacing and  single time step duration, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' After PBS2 the  static disorder  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='02  <()  P  0  V  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='04  numeric  experiment  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='06  5  6  7  8  9  10  11  12  13  14  qstatic disorder  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='7  numeric  experiment  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='5  (3)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='4  S  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='8  1  w (2元)dephasing disorder  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='7  numerics  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='6  experiment  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='4  (3)  S  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='8  1  w (2元)Distribution  static(num)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='8  dephasing(num)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='7  static(exp)  dephasing(exp)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='5  3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='4  S  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='1  0  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='8  S(w =0)4  pulses are feedbacked in the loop and the dynamics con-  tinues until the desired final step, in which the train of  pulses is deterministically coupled out by the EOMs 2  and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Given that these EOMs flip the polarization for  the in and outcoupling the full dynamics of an N step  evolution is described by  ˆUfull = ˆσ1(T −1R1(θU))N ˆσ1T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  The application of the operator T describes the first tran-  sition through the fibre arms in which the role of the  horizontally and the vertically polarized light is swapped  with respect to the following roundtrips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The consecutive  ˆσ1 matrices signify the polarization flip by the EOMs 2  and 3 for in and outcoupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' As ˆσ1 commutes with R1  and using T = ˆσ1T −1ˆσ1 we obtain for the full evolution  ˆUfull = (TR1(θU))NT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  which satisfies the chiral symmetries introduced above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  The deterministic in- and outcoupling of the light, as  already introduced in [15, 18], minimizes the roundtrip  losses with respect to the probabilistic version used ear-  lier [13] and thus enables the recording of longer dynam-  ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The detection units consists of two superconducting  nanowire single photon detectors (SNSPDs) with dead  times well below the pulse separation, which in combina-  tion with PBS3 and a QWP at 45◦ enable the measure-  ment of each pulse intensity in the circular R/L basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  Results:—We obtain the spin-resolved probability dis-  tribution by recording measurements for 500 different re-  alisations of polarization rotations simulating the static  binary disorder discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' To define a reference  frame without Anderson localization to compare to, we  additionally simulate dynamic disorder for which local-  ization is absent due to the destruction of phase coher-  ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The latter is realized via application of the polar-  ization rotations randomly distributed in time, at other-  wise identical system parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  The comparison between the two recordings is pre-  sented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' 2, at time step t = 14, traced over polari-  sation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' At these values, the Anderson localized and the  diffusively spreading wave package do not yet differ by  much in absolute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Instead, the difference shows in  the shape of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' That is, the convex profile  of an exponential distribution in the Anderson localized  case, compared to the concave form of a Gaussian diffu-  sive profile, as verified in the right panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' (The boundary  peaks in the diffusive case represent a small fraction of  quasi-ballistically propagating contributions which dis-  appear in the numerical simulation of longer runs, see  Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=')  For each random realization of binary angles we extract  the spatially integrated spin-polarization, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' (5), which  we then average over different angle realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  The  upper left panel in Figure 3 shows experimental data,  and corresponding numerical simulations, for ⟨∆P(t)⟩  between t = 5 and 14 time steps, averaged over 500  realizations of static binary disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  The upper-right  panel exhibits the resulting power spectrum, S(ω) =  | ¯P(ω)|2/ �  ω | ¯P(ω)|2, where ¯P(ω) = �14  t=5 eiωt⟨∆P(t)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  For comparison, we show in the lower left panel the  corresponding power spectrum for dynamic binary dis-  order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  The latter has a structureless random pattern  with contributions of the same order from all frequen-  cies, as expected for a noisy spin polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' In con-  trast, the spin polarization for static disorder shows the  time-staggering predicted for the topological quantum  critical state, which is also witnessed by the pronounced  peak at ω = π in the power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The relatively  small number of time-steps probed in the experiment im-  plies sizeable fluctuations of random parameters, which  need to be taken into account in the interpretation of  data: for a given run, the average value of tmax ran-  domly drawn angular parameters scales as ∼ O(1/  √  t),  and a value of comparable magnitude for the staggered  amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' In this way, the critical states both, at (0, π)  and (−π/2, π/2) get effectively gapped out for at least a  fraction of the runs, resulting in a suppression of peaks  in the power spectrum at ω = 0 and ω = π, respectively  (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' To further illustrate this point, we repeated  numerical simulations of ensembles of 500 random bi-  nary angle configurations a large number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' In the  right-lower panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' 3 we show the distributions of  peak values S(ω = 0, π) in the power spectrum, resulting  from 1000 repetitions of the previously described proce-  dure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' That is, each point here presents the average over  an ensemble of 500 angle realizations (the ensembles used  in experiment are indicated by the squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' As antici-  pated above, the distribution for static disordered is cen-  tered around large peak values, either at ω = 0, at ω = π,  or at both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The distribution for dynamic disorder, on the  other hand, is always dominated by small peak values at  both frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The average power spectrum for the  entire ensemble of 1000 × 500 angle configurations (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  Supplemental Material) then shows the expected two-  peak structure, with peaks at ω = 0, π dominating over  an otherwise approximately flat background, confirming  thus that upon simulating larger runs, fluctuation effects  diminish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  Conclusion:—We have realized an optical linear net-  work simulator of one-dimensional topological quantum  criticality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The simulator implements the photonic quan-  tum walk of a spin-1/2 particle with chiral symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  Its optical network architecture allows us to fully ac-  cess and monitor the state’s internal degree of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  Upon tuning to the critical point separating two topo-  logical Anderson insulating phases, we observe a time-  staggered spin-polarization recently suggested as a smok-  ing gun signature of quantum critical dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Exter-  nally imposed time dependent noise, or ‘dephasing disor-  der’, destroys the signal, revealing its origin in quantum  coherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' A similar destruction takes place upon break-  ing the chiral symmetry of the walk, and along with it  5  the transition between two symmetry protected topolog-  ical phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Ideally, one would like to monitor scaling  phenomena induced by such type of symmetry breaking,  however, the currently realizable signal times are still too  short for such type of statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Irrespective of such lim-  itations, we believe that fully programmable quantum  networks promise interesting perspectives for the simu-  lation of topological quantum matter in the presence of  engineered randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  The Integrated Quantum Optics group acknowledges  support by the ERC project QuPoPCoRN (Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  725366).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' acknowledges financial support by  Brazilian agencies CNPq and FAPERJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
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+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' and  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
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+page_content=' acknowledge partial support from the Deutsche  Forschungsgemeinschaft (DFG) within the CRC network  TR 183 (project grant 277101999) as part of projects A01  and A03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
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+page_content=' Altland, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Micklitz, Probing  the topological anderson transition with quantum walks,  Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Research 3, 023183 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  [10] We have U(ˆσ2 |ψ⟩)  =  ˆσ2U † |ψ⟩  =  e−iϵˆσ2 |ψ⟩, and  U ˆS |ψ⟩ = − ˆSU |ψ⟩ = eiπ+iϵ |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  [11] The associated topological invariants may be identified  by analysis of the auxiliary ‘Hamiltonian’ ˆH = −i ln ˆU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  However, we will not need this underlying structure [19,  20] throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  [12] To probe only critical states, one may prepare initial  states extended over several sites with fixed phase re-  lations, as discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Their experimental re-  alization is, however, challenging and we thus focus on  initial states localized on single sites, probing the entire  quasi-energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Schreiber, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Cassemiro, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Potoˇcek, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' G´abris,  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Mosley, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Andersson, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Jex, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Silberhorn,  Photons Walking the Line: A Quantum Walk with Ad-  justable Coin Operations, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
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+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Cassemiro, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Potoˇcek, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' G´abris,  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
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+page_content=' G´abris,  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Jex, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
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+page_content=' Asb´oth, Symmetries, topological phases, and bound  states in the one-dimensional quantum walk, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  B 86, 195414 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
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+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Asb´oth and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Obuse, Bulk-boundary correspon-  dence for chiral symmetric quantum walks, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' B  88, 121406 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  Probability distributions at longer times  As discussed in the main text, all states in the  quasienergy spectrum contribute to the probability dis-  tribution of the walker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Since most of the states are non-  critical, the probability distribution for static binary dis-  order thus follows |ψ(q)|2 ∼ e−|q|/λ, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' 4  (left panel) for numerical simulations of different time  steps t = 15, 23, 31, 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The q-dependence of − ln |ψ(q)|2  is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' 4 (right-top panel), clearly showing a  linear dependence on position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' When the binary disor-  der also fluctuates in time, quantum phase coherence  is destroyed turning dynamics thus into classical diffu-  sion, |ψ(q)|2 ∼ e−q2/σ2  t , with σ2  t ∼ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  The numerical  simulation, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' 4 (center and right-bottom panel), con-  firms the diffusive behavior and corresponding scaling  of − ln |ψ(q)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Notice that the boundary peaks in the  probability distributions are due to a small fraction of  6  quasi-ballistically propagating states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' 4,  their contribution disappears as longer times t ≳ 20 are  probed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  Statistics of the power spectrum S(ω)  The relatively small number of time steps measured  in the experiment implies that the signature of critical  states, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' the time-staggered spin polarization, is sub-  ject to large statistical fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  From our numer-  ical simulations we find that it is necessary to average  over many more than 500 disorder realizations to clearly  observe the two-peak structure in the power spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  Figure 3 (bottom-right panel) in the main text shows the  distribution of 1000 peak heights at frequencies ω = 0, π  in the power spectrum for static (purple) and dephas-  ing disorder (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  Each of the 1000 shown points is  obtained from averaging the power spectrum over 500  random disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' The average and standard  deviation of the entire power spectrum S(ω) after aver-  aging over the total ensemble of 1000 × 500 realizations  is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' 5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  7  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  Probability distributions after t = 15, 23, 31, 39 time steps for static (left panel) and dephasing (center panel) disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  The boundary peak visible at t = 15 disappears for larger time steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' Right panels show − ln(|ψ(q)|2) for static disorder (top)  and dephasing disorder (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/jNE5T4oBgHgl3EQfGA7Z/content/2301.05428v1.pdf'}
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+arXiv:2301.00464v1  [math.DS]  1 Jan 2023
+On rationally integrable planar dual multibilliards
+and piecewise smooth projective billiards
+Alexey Glutsyuk∗†‡§
+January 3, 2023
+Abstract
+A planar projective billiard is a planar curve C equipped with a
+transversal line field. It defines reflection of lines from C. Its projec-
+tive dual is a dual billiard: a curve γ ⊂ RP2 equipped with a family
+of non-trivial projective involutions acting on its projective tangent
+lines and fixing the tangency points.
+Projective and dual billiards
+were introduced by S.Tabachnikov. He stated the following conjecture
+generalizing the famous Birkhoff Conjecture on integrable billiards to
+dual and projective billiards. Let a dual billiard γ be strictly convex
+and closed, and let its outer neighborhood admit a foliation by closed
+curves (including γ) such that the involution of each tangent line to γ
+permutes its intersection points with every leaf. Then γ and the leaves
+are conics forming a pencil. In a recent paper the author proved this
+conjecture under the rational integrability assumption: existence of a
+non-constant rational function (integral) whose restriction to tangent
+lines is invariant under their involutions. He has also shown that if γ is
+not closed, then it is still a conic, but the dual billiard structure needs
+not be defined by a pencil. He classified all the rationally integrable
+dual billiard structures (with singularities) on conic. In the present
+paper we give classification of rationally integrable dual multibilliards:
+collections of dual billiards and points Qj (called vertices) equipped
+with a family of projective involutions acting on lines through Qj from
+an open subset in RP1. As an application, we get classification of piece-
+wise smooth projective billiards whose billiard flow has a non-constant
+first integral that is a rational 0-homogeneous function of the velocity.
+∗CNRS, UMR 5669 (UMPA, ENS de Lyon), France. E-mail: aglutsyu@ens-lyon.fr
+†HSE University, Moscow, Russia
+‡Kharkevich Institute for Information Transmission Problems (IITP, RAS), Moscow
+§Supported by part by RFBR grant 20-01-00420
+1
+
+Contents
+1
+Introduction
+3
+1.1
+Introduction, brief description of main results and plan of the
+paper
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+3
+1.2
+Previous results 1: classification of real and complex ratio-
+nally integral dual billiards on one curve . . . . . . . . . . . .
+9
+1.3
+Previous results 2: classification of rationally 0-homogeneously
+integrable projective billiards on one curve . . . . . . . . . . .
+11
+1.4
+Main results:
+classification of rationally integrable planar
+dual multibilliards with C4-smooth curves . . . . . . . . . . .
+14
+1.5
+Application: classification of rationally 0-homogeneously in-
+tegrable piecewise C4-smooth projective billiards . . . . . . .
+22
+1.6
+Plan of proofs of main results . . . . . . . . . . . . . . . . . .
+29
+1.7
+Historical remarks
+. . . . . . . . . . . . . . . . . . . . . . . .
+32
+2
+Rationally integrable dual multibilliards.
+Proofs of Theo-
+rems 1.25, 1.26, 1.31, 1.27
+34
+2.1
+Rational integrability of pencil type multibilliards . . . . . . .
+34
+2.2
+Foliation by level curves of rational integral. Proof of Theo-
+rems 1.25 and 1.26 . . . . . . . . . . . . . . . . . . . . . . . .
+38
+2.3
+Dual billiard structures at vertices. Birationality and types
+of involutions . . . . . . . . . . . . . . . . . . . . . . . . . . .
+44
+2.4
+Pencil case. Proof of Theorem 1.26 . . . . . . . . . . . . . . .
+51
+2.5
+Exotic multibilliards. Proof of Theorem 1.31
+. . . . . . . . .
+54
+2.6
+Admissible vertices of real pencils of conics. Proof of Propo-
+sition 1.24 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+59
+3
+Rationally 0-homogeneously integrable piecewise smooth pro-
+jective billiards. Proof of Theorems 1.38, 1.39, 1.45
+61
+3.1
+Duality between projective billiards and dual multibilliards.
+Correspondence between integrals . . . . . . . . . . . . . . . .
+61
+3.2
+Case of dual pencil. Proof of Theorems 1.38, 1.39, 1.40 . . . .
+63
+3.3
+Exotic projective billiards. Proof of Theorem 1.45
+. . . . . .
+64
+4
+Integrals of dual pencil type billiards: examples of degrees
+4 and 12. Proof of Theorems 1.28, 1.41 and Lemma 1.42
+67
+4.1
+Multibilliards of pencil type. Proof of Theorem 1.28
+. . . . .
+67
+4.2
+Dual pencil type projective billiards. Proof of Theorem 1.41
+and Lemma 1.42
+. . . . . . . . . . . . . . . . . . . . . . . . .
+70
+2
+
+4.3
+Generic dual pencil type projective billiards with integrals of
+degrees 4 and 12
+. . . . . . . . . . . . . . . . . . . . . . . . .
+72
+4.4
+Semi-(pseudo-) Euclidean billiards with integrals of different
+degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+74
+1
+Introduction
+1.1
+Introduction, brief description of main results and plan
+of the paper
+Consider a planar billiard Ω ⊂ R2 bounded by a C2-smooth strictly convex
+closed curve.
+Recall that its caustic is a curve S ⊂ R2 such that each
+tangent line to S is reflected from the boundary ∂Ω to a line tangent to
+S. A billiard is Birkhoff caustic-integrable, if some inner neighborhood of
+its boundary is foliated by closed caustics, with boundary being a leaf of
+the foliation. This is the case in an elliptic billiard, where confocal ellipses
+form a foliation by closed caustics of a domain adjacent to the boundary
+ellipse. The famous open Birkhoff Conjecture states that the only integrable
+billiards are ellipses. See its brief survey in Subsection 1.7. S.Tabachnikov
+suggested its generalization to projective billiards introduced by himself in
+1997 in [35]. See the following definition and conjecture.
+Definition 1.1 [35] A projective billiard is a smooth planar curve C ⊂ R2
+equipped with a transversal line field N. For every Q ∈ C the projective
+billiard reflection involution at Q acts on the space of lines through Q as
+the affine involution R2 → R2 that fixes the points of the tangent line to C
+at Q, preserves the line N(Q) and acts on N(Q) as the central symmetry
+with respect to the point1 Q. In the case, when C is a strictly convex closed
+curve, the projective billiard map acts on the phase cylinder: the space of
+oriented lines intersecting C. It sends an oriented line to its image under
+the above reflection involution at its last point of intersection with C in the
+sense of orientation. See Fig. 1.
+Example 1.2 A usual Euclidean planar billiard is a projective billiard with
+transversal line field being normal line field. Each billiard in a complete
+Riemannian surface Σ of non-zero constant curvature (i.e., in sphere S2 and
+1In other words, two lines a, b through Q are permuted by reflection at Q, if and only
+if the quadruple of lines TQC, N(Q), a, b is harmonic: there exists a projective involution
+of the space RP1 of lines through Q that fixes TQC, N(Q) and permutes a, b.
+3
+
+Figure 1: The projective billiard reflection.
+in hyperbolic plane H2) also can be seen as a projective billiard, see [35].
+Namely, consider Σ = S2 as the unit sphere in the Euclidean space R3, and
+Σ = H2 as the semi-pseudo-sphere {x2
+1 + x2
+2 − x2
+3 = −1, x3 > 0} in the
+Minkovski space R3 equipped with the form dx2
+1 + dx2
+2 − dx2
+3. The billiard
+in a domain Ω ⊂ Σ+ := Σ ∩ {x3 > 0} is defined by reflection of geodesics
+from its boundary. The tautological projection π : R3 \ {0} → RP2 sends
+Ω diffeomorphically to a domain in the affine chart {x3 = 1}.
+It sends
+billiard orbits in Ω to orbits of the projective billiard on C = π(∂Ω) with
+the transversal line field N on C being the image of the normal line field to
+∂Ω under the differential dπ.
+The notion of caustic (integrability) of a projective billiard repeats the
+above notions for the usual billiards.
+Tabachnikov Conjecture states
+that if a projective billiard is integrable, then the billiard boundary and the
+caustics are conics, whose dual conics form a pencil. To study it, Tabach-
+nikov introduced the dual objects to projective billiards, the so-called dual
+billiards, and stated the dual version of his conjecture for them, see the next
+definition and conjecture.
+Definition 1.3 [27, definitions 1.6, 1.17] A real (complex) dual billiard is
+a smooth (holomorphic) curve γ ⊂ RP2(CP2) where for each point P ∈ γ
+the real (complex) projective line LP tangent to γ at P is equipped with
+a projective involution σP : LP → LP fixing P; the family of involutions
+(called the dual billiard structure) is parametrized by tangency points P.
+The dual Tabachnikov Conjecture deals with a strictly convex closed
+curve γ ⊂ RP2 equipped with a dual billiard structure such that an outer
+neighborhood of the curve γ admits a foliation by strictly convex closed
+4
+
+curves, including γ, such that each involution σP , P ∈ γ, permutes the
+intersection points of the line LP with each individual leaf. It states that
+under this assumption (called integrability condition) the curve γ and the
+leaves of the foliation are conics forming a pencil. It implies that the dual
+billiard structure on γ is of pencil type, see the next definition.
+Definition 1.4 [27, example 1.14] A dual billiard is of pencil type, if the
+underlying curve γ is a (punctured) conic and there exists a pencil of conics
+containing γ such that for every P ∈ γ the involution σP permutes the
+intersection points of the line LP with each conic of the pencil (or fixes
+the intersection point, if it is unique). As was observed by S.Tabachnikov,
+conversely, for every conic γ and every pencil containing γ, for every P in
+the conic γ punctured in at most 4 complex base points of the pencil there
+exists a projective involution σP : LP → LP satisfying the above condition,
+and thus, a well-defined pencil type dual billiard on the punctured conic γ.
+The dual Tabachnikov Conjecture is open. It would imply his above con-
+jecture on integrable projective billiards, and hence, the Birkhoff Conjecture
+and its versions for billiards on surfaces of constant curvature and for outer
+planar billiards.
+In the previous paper by the author [27] the dual Tabachnikov Conjecture
+was proved under the additional assumption that the foliation in question
+admits a non-constant rational first integral. This assumption is equivalent
+to the existence of a non-constant rational function R whose restriction to
+each tangent line LP to γ is invariant under the corresponding involution:
+(R ◦ σP)|LP = R|LP .
+(1.1)
+Definition 1.5 [27, definition 1.12] A dual billiard for which there exists
+a non-constant rational function (called integral) satisfying (1.1) is called
+rationally integrable.
+Example 1.6 If a dual billiard on a nonlinear curve has a polynomial in-
+tegral, then it is an outer billiard, that is, the corresponding projective
+involution of each tangent line is its central symmetry with respect to the
+tangency point (see [27, example 1.13]). Each pencil type dual billiard is
+rationally integrable with a quadratic integral (Tabachnikov’s observation,
+see [27, example 1.14]).
+5
+
+It was shown in [27, theorem 1.16] that if a dual billiard is rationally
+integrable, but the underlying curve γ is not necessarily closed, then
+- the curve γ is still a conic;
+- the dual billiard structure extends to a global analytic dual billiard
+structure on the whole conic with at most four points deleted;
+- but the dual billiard is not necessary of pencil type.
+(Singular) rationally integrable dual billiard structures on conic were
+classified by [27, theorem 1.16 and its addendum], with explicit formulas for
+rational integrals. These results are recalled in Subsection 1.2 as Theorem
+1.11 and its addendum. The dual version of Theorem 1.11 yields classifi-
+cation of those projective billiards with underlying curve being C4-smooth
+and connected that are rationally 0-homogeneously integrable, i.e., whose
+flow admits a nontrivial first integral that is a rational 0-homogeneous func-
+tion of the velocity [27, theorem 1.26 and its addendum]. These results are
+recalled in Subsection 1.3 as Theorem 1.16 and its addendum.
+The main result of the present paper is the classification of rationally
+0-homogeneously integrable projective billiards with piecewise C4-smooth
+boundary that contains a nonlinear arc and maybe also straightline segments
+(results stated in Subsection 1.5). To classify them, we introduce a gener-
+alized dual object to projective billiards with piecewise smooth boundary:
+the so-called dual multibilliards, which are collections of curves and points
+equipped with dual billiard structures. We obtain classification of rationally
+integrable dual multibilliards containing a nonlinear arc (results stated in
+Subsection 1.4) and then deduce classification of rationally 0-homogeneously
+integrable projective billiards by using projective duality.
+The projective duals to the curvilinear pieces of the boundary of a pro-
+jective billiard are planar curves equipped with dual billiard structure. The
+projective dual to each straightline piece of its boundary is a point. The
+projective billiard structure on the straightline piece (which is an open sub-
+set of a projective line) is transformed by duality to dual billiard structure
+at a point, see the next definition.
+Definition 1.7 A dual billiard structure at a point Q ∈ RP2(CP2) is a fam-
+ily of projective involutions σQ,ℓ : ℓ → ℓ acting on real (complex) projective
+lines ℓ through Q. It is assumed that σQ,ℓ are defined on an open subset
+U ⊂ RP1(CP1) of the space of lines through Q. No regularity of the family
+σQ,ℓ is assumed.
+Definition 1.8 A real (complex) dual multibilliard is a (may be infinite)
+collection of smooth (holomorphic) nonlinear connected curves γj and points
+6
+
+Qs in RP2(CP2) (called vertices), where each curve γj and each point Qs are
+equipped with a dual billiard structure.
+Definition 1.9 A dual multibilliard is rationally integrable, if there exists
+a non-constant rational function on RP2(CP2) whose restriction to each tan-
+gent line to every curve γj is invariant under the corresponding involution,
+and the same statement holds for its restriction to each line ℓ through any
+vertex Q, where the corresponding involution σQ,ℓ is defined.
+The main results on classification of rationally integrable real and com-
+plex dual multibilliards are given by Theorems 1.25, 1.26 and 1.31 in Sub-
+section 1.4. They deal with the case, when the multibilliard is not reduced
+to one curve (without vertices). In the case, when it contains at least two
+curves, Theorems 1.25, 1.26 together state that it is rationally integrable,
+if and only if it is of so-called pencil type. This means that all its curves
+are conics lying in one pencil and equipped with the dual billiard struc-
+ture defined by this pencil; its vertices belong to an explicit list of so-called
+admissible vertices for the given pencil; the collection of vertices of the multi-
+billiard satisfies additional conditions given in Definition 1.23. Theorem 1.26
+also yields analogous result in the case, when the multibilliard consists of
+a single curve equipped with a dual billiard structure of pencil type and
+maybe some vertices. Theorems 1.25 and 1.11 together imply that every
+other a priori possible rationally integrable dual multibilliard, not covered
+by Theorems 1.25, 1.26 is a so-called exotic multibilliard: it is formed by
+conic equipped with an exotic (i.e., non-pencil) dual billiard structure from
+Theorem 1.11 and maybe some vertices. Theorem 1.31 yields classification
+of rationally integrable exotic multibilliards. It implies that such a multibil-
+liard may contain at most three vertices, and the dual billiard structure at
+each vertex is given by a global projective involution fixing the conic.
+Recall that a dual multibilliard formed by just a finite collection of conics
+from the same pencil P, with dual billiard structures defined by P, always
+has a quadratic rational integral, see Example 1.6. If one adds to it appro-
+priate vertex collection (from the finite list of so-called admissible vertices
+for the pencil P) so that the dual multibilliard thus obtained be of pencil
+type, then it will be still rationally integrable, by Theorem 1.26. However,
+Theorem 1.27 shows that the minimal degree of its rational integral may be
+bigger: it may be equal to 2, 4 or 12.
+The dual to a pencil type multibilliard defined by a pencil P is a projec-
+tive billiard of the so-called dual pencil type. This means that its boundary
+is piecewise-smooth and consists of arcs of conics from the dual pencil P∗
+7
+
+equipped with projective billiard structures having conical caustics from
+the same pencil P∗, and maybe segments of so-called admissible lines for P∗
+equipped with appropriate projective billiard structures defined by P∗. The-
+orems 1.38, 1.39 (dual to Theorems 1.25, 1.26) together imply that the dual
+pencil type projective billiards are rationally 0-homogeneously integrable
+and the only integrable billiards that are not of pencil type are the so-called
+exotic ones, with nonlinear part of boundary lying in one conic equipped
+with an exotic projective billiard structure from Theorem 1.16. The inte-
+grable exotic billiards are classified by Theorem 1.45 (dual to Theorem 1.31).
+Theorem 1.40 (the dual to Theorem 1.27) implies that the minimal degree of
+rational 0-homogeneous integral of a dual pencil type projective billiard with
+piecewise C4-smooth boundary may be equal to 2, 4 or 12. See formulas for
+integrals of degree 12 in Theorems 1.28, 1.41 and Lemma 1.42.
+Remark 1.10 The flow of a Euclidean planar billiard with boundary con-
+taining a curvilinear arc admits the trivial first integral: the squared module
+of the velocity ||v||2 = v2
+1 + v2
+2. It is known that it admits a non-trivial inte-
+gral polynomial in the velocity (that is, nonconstant along the unit velocity
+hypersurface {||v||2 = 1}, if and only if it is of confocal dual pencil type.
+This was proved in particular case in [14]; this statement in full generality
+is a joint result of M.Bialy, A.E.Mironov and the author [8, 25]. Together
+with results of [14], it implies that the minimal degree of non-trivial poly-
+nomial integral (if it exists) of an Euclidean billiard is equal to either 2, or
+4. A nontrivial polynomial integral Ix(v), which can be chosen of even de-
+gree 2n, generates a non-trivial rational 0-homogeneous integral Ix(v)
+||v||n . This
+also implies that the minimal degree of non-trivial rational 0-homogeneous
+integral of an Euclidean billiard is also equal to either 2, or 4. Similar state-
+ments hold for billiards on the other surfaces of constant curvature, that is,
+the round sphere and the hyperbolic plane, and for the projective billiards
+equivalent to them from Example 1.2. See [14, 8, 9, 25].
+Thus, rationally 0-homogeneously integrable projective billiards of dual
+pencil type with integrals of degree 12 presented and classified in the present
+paper form an essentially new class of rationally integrable projective bil-
+liards of dual pencil type, not covered by the known list of polynomially
+integrable billiards on surfaces of constant curvature.
+Plan of proof of main results is presented in Subsection 1.6. A historical
+survey is given in Subsection 1.7. The main results are proved in Sections
+2 (for multibilliards) and 3 (for projective billiards). In Section 4 we prove
+formulas for degree 12 integrals (Theorems 1.28, 1.41 and Lemma 1.42) and
+8
+
+present examples of projective billiards with integrals of degree 4 and 12.
+1.2
+Previous results 1: classification of real and complex ra-
+tionally integral dual billiards on one curve
+Theorem 1.11 [27, theorem 1.16] Let γ ⊂ R2 ⊂ RP2 be a C4-smooth
+connected non-linear (germ of) curve equipped with a rationally integrable
+dual billiard structure. Then γ is a conic, and the dual billiard structure has
+one of the three following types (up to real-projective equivalence):
+1) The dual billiard is of conical pencil type and has a quadratic integral.
+2) There exists an affine chart R2
+z,w ⊂ RP2 in which γ = {w = z2} and
+such that for every P = (z0, w0) ∈ γ the involution σP : LP → LP is given
+by one of the following formulas:
+a) In the coordinate
+ζ := z
+z0
+σP : ζ �→ ηρ(ζ) := (ρ − 1)ζ − (ρ − 2)
+ρζ − (ρ − 1)
+,
+ρ = 2 −
+2
+2N + 1,
+or ρ = 2 −
+1
+N + 1
+for some N ∈ N.
+(1.2)
+b) In the coordinate
+u := z − z0
+σP : u �→ −
+u
+1 + f(z0)u,
+(1.3)
+f = fb1(z) :=
+5z − 3
+2z(z − 1) (type 2b1)),
+or f = fb2(z) :=
+3z
+z2 + 1 (type 2b2)).
+(1.4)
+c) In the above coordinate u the involution σP takes the form (1.3) with
+f = fc1(z) :=
+4z2
+z3 − 1 (type 2c1)),
+or f = fc2(z) :=
+8z − 4
+3z(z − 1) (type 2c2)).
+(1.5)
+d) In the above coordinate u the involution σP takes the form (1.3) with
+f = fd(z) = 4
+3z +
+1
+z − 1 =
+7z − 4
+3z(z − 1)
+(type 2d).
+(1.6)
+9
+
+Addendum to Theorem 1.11. Every dual billiard structure on γ of
+type 2a) has a rational first integral R(z, w) of the form
+R(z, w) =
+(w − z2)2N+1
+�N
+j=1(w − cjz2)2 ,
+cj = −4j(2N + 1 − j)
+(2N + 1 − 2j)2 ,
+for ρ = 2 −
+2
+2N + 1;
+(1.7)
+R(z, w) =
+(w − z2)N+1
+z �N
+j=1(w − cjz2)
+,
+cj = −j(2N + 2 − j)
+(N + 1 − j)2 ,
+for ρ = 2 −
+1
+N + 1.
+(1.8)
+The dual billiards of types 2b1) and 2b2) have respectively the integrals
+Rb1(z, w) =
+(w − z2)2
+(w + 3z2)(z − 1)(z − w),
+(1.9)
+Rb2(z, w) =
+(w − z2)2
+(z2 + w2 + w + 1)(z2 + 1).
+(1.10)
+The dual billiards of types 2c1), 2c2) have respectively the integrals
+Rc1(z, w) =
+(w − z2)3
+(1 + w3 − 2zw)2 ,
+(1.11)
+Rc2(z, w) =
+(w − z2)3
+(8z3 − 8z2w − 8z2 − w2 − w + 10zw)2 .
+(1.12)
+The dual billiard of type 2d) has the integral
+Rd(z, w) =
+(w − z2)3
+(w + 8z2)(z − 1)(w + 8z2 + 4w2 + 5wz2 − 14zw − 4z3).
+(1.13)
+Theorem 1.12 [27, theorem 1.18 and its addendum]. Every regular (germ
+of) connected holomorphic curve in CP2 (different from a straight line)
+equipped with a rationally integrable complex dual billiard structure is a
+conic. Up to complex-projective equivalence, the corresponding billiard struc-
+ture has one of the types 1), 2a), 2b1), 2c1), 2d) listed in Theorem 1.11, with
+a rational integral as in its addendum. The billiards of types 2b1), 2b2), see
+(1.4), are complex-projectively equivalent, and so are billiards 2c1), 2c2).
+10
+
+1.3
+Previous results 2: classification of rationally 0-homogeneously
+integrable projective billiards on one curve
+Consider a domain Ω ⊂ R2
+x1,x2 with smooth boundary ∂Ω equipped with
+a projective billiard structure (transverse line field). The projective billiard
+flow, see [35], acts on TR2|Ω. It moves a point (Q, v) ∈ TR2, Q = (x1, x2) ∈
+Ω, v = (v1, v2) ∈ TQR2 so that v remains constant and Q moves along the
+straight line directed by v with uniform velocity v, until it hits the boundary
+∂Ω at some point H. Let v∗ ∈ THR2 denote the image of the velocity vector
+v (translated to H) under the projective billiard reflection from the tangent
+line TH∂Ω.
+Afterwards it moves (H, v∗) so that H moves with the new
+uniform velocity v∗ until it hits the boundary again etc. See Fig. 2.
+Figure 2: Projective billiard flow
+Each Euclidean planar billiard flow always has the trivial first integral
+||v||2. But this is not true for a generic projective billiard. It is a well-known
+folklore fact that Birkhoff integrability of a Euclidean planar billiard with
+strictly convex closed boundary is equivalent to the existence of a non-trivial
+first integral of the billiard flow independent with ||v||2 on a neighborhood
+of the unit tangent bundle to ∂Ω in TR2|Ω.
+Definition 1.13 A projective billiard is rationally 0-homogeneously inte-
+grable, if its flow admits a first integral that depends on the velocity as a
+non-constant rational 0-homogeneous function of degree uniformly bounded
+by some number n: a function Ψ(Q, v) =
+P (v)
+T(v), where P and T are ho-
+mogeneous polynomials in v of degree no greater than n with coefficients
+depending on the position of the point Q. The maximal degree of the latter
+rational function through all Q is called the degree of the rational integral.
+Example 1.14 The projective billiard structure on (an arc of) a regular
+conic C is of dual pencil type, if it has a regular conical caustic.
+More
+11
+
+H
+Q1precisely, if there exists a conic Γ such that for every point Q ∈ C the
+complex tangent lines through Q to the complexified conic Γ are permuted by
+the projective billiard reflection at Q. Consider now the pencil P containing
+the dual conics C∗ and Γ∗. Let P∗ denote its dual, consisting of conics dual
+to the conics from the pencil P: it contains the conics C and Γ. Then the
+latter, caustic statement automatically holds for Γ being replaced by any
+other conic from the dual pencil P∗. See [27, proposition 1.27, remark 1.28].
+A dual pencil type projective billiard on a conic is known to be rationally 0-
+homogeneously integrable with a quadratic 0-homogeneous rational integral.
+This is the statement dual to the similar statement for pencil type dual
+billiards, see Example 1.6.
+Remark 1.15 The notion of rationally 0-homogeneously integrable projec-
+tive billiard also makes sense for a projective billiard structure on an arc of
+planar curve C (or a germ of curve), with projective billiard flow defined in
+a (germ of) domain adjacent to C. A rational 0-homogeneous integral of
+degree n is always a rational 0-homogeneous function of degree n in three
+variables: v1, v2 and the moment ∆ := x1v2 − x2v1. See analogous state-
+ment for polynomial integrals of the usual planar billiards in [14] and the
+statement for projective billiards in full generality in [27, proposition 1.23,
+statement 1)]. The property of rational 0-homogeneous integrability of a
+projective billiard on a curve C is independent on the side from C on which
+the billiard domain is chosen: an integral for one side is automatically an
+integral for the other side. See [27, proposition 1.23, statement 2)].
+Theorem 1.16 Let C ⊂ R2
+x1,x2 be a non-linear C4-smooth germ of curve
+equipped with a transversal line field N.
+Let the corresponding germ of
+projective billiard be 0-homogeneously rationally integrable.
+Then C is a
+conic; the line field N extends to a global analytic transversal line field on the
+whole conic C punctured in at most four points; the corresponding projective
+billiard has one of the following types up to projective equivalence.
+1) A dual pencil type projective billiard.
+2) C = {x2 = x2
+1} ⊂ R2
+x1,x2 ⊂ RP2, and the line field N is directed by
+one of the following vector fields at points of the conic C:
+2a)
+( ˙x1, ˙x2) = (ρ, 2(ρ − 2)x1),
+ρ = 2 −
+2
+2N + 1 (case 2a1), or
+ρ = 2 −
+1
+N + 1 (case 2a2),
+N ∈ N,
+the vector field 2a) has quadratic first integral Qρ(x1, x2) := ρx2 −(ρ−2)x2
+1.
+2b1) ( ˙x1, ˙x2) = (5x1 + 3, 2(x2 − x1)),
+2b2) ( ˙x1, ˙x2) = (3x1, 2x2 − 4),
+12
+
+2c1) ( ˙x1, ˙x2) = (x2, x1x2 − 1),
+2c2) ( ˙x1, ˙x2) = (2x1 + 1, x2 − x1).
+2d) ( ˙x1, ˙x2) = (7x1 + 4, 2x2 − 4x1).
+Addendum to Theorem 1.16.
+The projective billiards from Theorem
+1.16 have the following 0-homogeneous rational integrals:
+Case 1): A ratio of two homogeneous quadratic polynomials in (v1, v2, ∆),
+∆ := x1v2 − x2v1.
+Case 2a1), ρ = 2 −
+2
+2N+1:
+Ψ2a1(x1, x2, v1, v2) :=
+(4v1∆ − v2
+2)2N+1
+v2
+1
+�N
+j=1(4v1∆ − cjv2
+2)2 .
+(1.14)
+Case 2a2), ρ = 2 −
+1
+N+1:
+Ψ2a2(x1, x2, v1, v2) =
+(4v1∆ − v2
+2)N+1
+v1v2
+�N
+j=1(4v1∆ − cjv2
+2)
+.
+(1.15)
+The cj in (1.14), (1.15) are the same, as in (1.7) and (1.8) respectively.
+Case 2b1):
+Ψ2b1(x1, x2, v1, v2) =
+(4v1∆ − v2
+2)2
+(4v1∆ + 3v2
+2)(2v1 + v2)(2∆ + v2).
+(1.16)
+Case 2b2):
+Ψ2b2(x1, x2, v1, v2) =
+(4v1∆ − v2
+2)2
+(v2
+2 + 4∆2 + 4v1∆ + 4v2
+1)(v2
+2 + 4v2
+1).
+(1.17)
+Case 2c1):
+Ψ2c1(x1, x2, v1, v2) =
+(4v1∆ − v2
+2)3
+(v3
+1 + ∆3 + v1v2∆)2 .
+(1.18)
+Case 2c2):
+Ψ2c2(x1, x2, v1, v2) =
+(4v1∆ − v2
+2)3
+(v3
+2 + 2v2
+2v1 + (v2
+1 + 2v2
+2 + 5v1v2)∆ + v1∆2)2 . (1.19)
+Case 2d): Ψ2d(x1, x2, v1, v2)
+=
+(4v1∆ − v2
+2)3
+(v1∆ + 2v2
+2)(2v1 + v2)(8v1v2
+2 + 2v3
+2 + (4v2
+1 + 5v2
+2 + 28v1v2)∆ + 16v1∆2).
+(1.20)
+13
+
+1.4
+Main results: classification of rationally integrable pla-
+nar dual multibilliards with C4-smooth curves
+Each curve of a rationally integrable dual multibilliard is a conic, being itself
+an integrable dual billiard, see Theorem 1.11. The first results on classifi-
+cation of rationally integrable dual multibilliards presented below deal with
+those multibilliards whose curves are conics lying in one pencil, equipped
+with dual billiard structure defined by the same pencil. They state that its
+vertices should be admissible for the pencil. To define admissible vertices,
+let us first introduce the following definition.
+Definition 1.17 A projective angular symmetry centered at a point A ∈
+CP2 is a non-trivial projective involution σA : CP2 → CP2 fixing A and each
+line through A. It is known to have a fixed point line Λ ⊂ CP2 disjoint from
+A. Its restrictions to lines throughs A define a dual billiard structure at A.
+Example 1.18 Let now A ∈ CP2 and let S ⊂ CP2 be a (may be singular)
+conic disjoint from A. There exists a projective angular symmetry centered
+at A and permuting the intersection points with S of each line through A,
+called S-angular symmetry, see [25, definition 2.4].
+Definition 1.19 Let now A ∈ CP2, S ⊂ CP2 be a regular conic through
+A, and LA the projective tangent line to S at A. The degenerate S-angular
+symmetry centered at A is the involution σA = σS
+A acting on the complement
+CP2 \ (LA \ {A}) that fixes A, fixes each line ℓ ̸= LA through A and whose
+restriction to ℓ is the projective involution fixing A and the other point of
+the intersection ℓ ∩ S. It is known to be a birational map CP2 → CP2.
+Definition 1.20 A dual billiard structure at a point A ∈ CP2 is called global
+(quasi-global) if it is given by a projective angular symmetry (respectively,
+degenerate S-angular symmetry) centered at A.
+Definition 1.21 Consider a complex pencil of conics in CP2.
+A vertex,
+i.e., a point of the ambient plane equipped with a complex dual billiard
+structure, is called admissible for the pencil, if it belongs to the following
+list of vertices split in two types: standard, or skew.
+Case a): a pencil of conics through 4 distinct points A, B, C, D; Fig. 3.
+a1) The standand vertices: M1 = AB ∩ CD, M2 = AD ∩ BC, M3 =
+AC ∩ BD equipped with the global dual billiard structure given by the
+projective angular symmetry σMj = σMiMk
+Mj
+, i, k ̸= j, i ̸= k, centered at Mj
+with fixed point line MiMk.
+14
+
+Figure 3:
+a2) The skew vertices KEL, E, L ∈ {A, B, C, D}, E ̸= L: KEL is the
+intersection point of the line EL with the line MiMj such that Mi, Mj /∈ EL.
+The involution σKEL is the projective angular symmetry P2 → P2 centered
+at KEL with fixed point line ST, {S, T} = {A, B, C, D} \ {E, L}.
+Case b): a pencil of conics through 3 points A, B, C tangent at the point
+C to the same line L. See Fig. 4.
+b1) One standard vertex M = AB ∩ L, equipped with the projective
+angular symmetry σM : P2 → P2 centered at M with fixed point line CKAB.
+The point KAB is defined as follows.
+b2) The skew vertex KAB ∈ AB such that the projective involution
+AB → AB fixing M and KAB permutes A and B. That is, the points M,
+KAB, A, B form a harmonic quadruple. The dual billiard structure at KAB
+is given by the projective angular symmetry σKAB : P2 → P2 centered at
+KAB with fixed point line L.
+b3) The skew vertex C equipped with the projective angular symmetry
+σC = σAB
+C
+: P2 → P2 centered at C with fixed point line AB.
+b4) The skew vertex C equipped with a degenerate S-angular symmetry
+σC = σS
+C centered at C, defined by arbitrary given regular conic S of the
+pencil; see Definition 1.19. This yields a one-parametric family of quasi-
+global dual billiard structures at C.
+Case c): a pencil of conics through two given points A and C that are
+tangent at them to two given lines LA and LC respectively. See Fig. 20.
+c1) Standard vertices: M = LA ∩ LC and any point M′ ∈ AC, M′ ̸=
+A, C.
+The vertex M is equipped with the projective angular symmetry
+σM centered at M with fixed point line AC. The vertex M′ is equipped
+15
+
+Figure 4:
+Figure 5:
+with the (LA ∪ LC)-angular symmetry centered at M′, which permutes the
+intersection points of each line through M′ with the lines LA and LC.
+c2) Skew vertices equipped with global dual billiard structures: the points
+A and C. The dual billiard structure at A (C) is the projective angular
+symmetry centered at A (C) with fixed point line LC (respectively, LA).
+c3) Skew vertices A and C; A (C) being equipped with a degenerate SA
+(SC)-angular symmetry centered at A (C), defined by any regular conic SA
+(SC) of the pencil. This yields a one-parametric family of quasi-global dual
+billiard structures at each one of the vertices A, C, as in b4).
+Case d): pencil of conics through two distinct points A and B, tangent
+to each other at A with contact of order three; let L denote their common
+tangent line at A. See Fig. 6.
+d1) The skew vertex A, equipped with a quasi-global dual billiard struc-
+ture: a degenerate S-angular symmetry σS
+A centered at A defined by any
+regular conic S from the pencil.
+d2) Any point C ∈ L \ {A}, called a skew vertex, equipped with a
+projective angular symmetry σC centered at C with fixed point line AB.
+Case e): pencil of conics through one given point A, tangent to each
+other with contact of order four. See Fig. 7. Let L denote their common
+tangent line at A.
+e1) The skew vertex A equipped with a degenerate S-angular symmetry
+σS
+A centered at A defined by any given regular conic S of the pencil.
+e2) Any point C ∈ L \ {A}, called a standard vertex, equipped with a
+projective angular symmetry σC centered at C. Its fixed point line is the
+set of those points D ∈ P2 for which the line CD is tangent to the conic of
+16
+
+Figure 6:
+Figure 7:
+the pencil through D at D (including D = A).
+The definition of real (standard or skew) admissible vertex for a real pencil
+of conics in RP2 is analogous.
+Definition 1.22 Consider a dual multibilliard formed by some conics and
+maybe by some vertices. Let the dual billiard structure at each vertex (if
+any) be either global, or quasi-global, and that on each conic be either of
+pencil type, or as in Theorem 1.11, Case 2). We say that its two conics
+(vertices) are distinct, if they either are geometrically distinct, or coincide
+as conics (vertices) but have different dual billiard structures.
+Definition 1.23 A (real or complex) dual multibilliard is said to be of
+pencil type, if the following conditions hold.
+1) All its curves are conics lying in one pencil, and their dual billiard
+structures are defined by the same pencil. (Case of one conic equipped with
+a dual billiard structure of pencil type is possible.)
+2) All its vertices are admissible for the pencil.
+3) If the multibilliard contains a skew vertex equipped with a quasi-
+global dual billiard structure, then it contains no other skew vertex, with
+the following exceptions:
+- in Case c) the skew vertex collection is allowed to be the pair of vertices
+A and C equipped with quasi-global structures defined by one and the same
+(but arbitrary) regular conic S = SA = SC of the pencil.
+17
+
+s
+AT
+cB
+s
+c.
+L
+A- in Case d) the skew vertex collection is allowed to be a pair of vertices
+A and C defined by any given regular conic S of the pencil: the vertex A is
+equipped with the quasi-global S-dual billiard structure; the vertex C is the
+intersection point of the line L with the line tangent to S at B, equipped
+with the projective angular symmetry with fixed point line AB (it coincides
+with the S-angular symmetry centered at C).
+4) In Case d) the multibilliard may contain at most one vertex C ∈
+L \ {A}.
+5) Each skew admissible vertex that a priori admits several possible dual
+billiard structures listed above is allowed to be included in the multibilliard
+with no more than one dual billiard structure.
+Well-definedness of the above notion of admissible vertex and pencil type
+dual multibilliard in the real case is implied by the following proposition.
+Proposition 1.24 1) Consider a real pencil of conics in RP2 whose com-
+plexifications pass through four distinct but maybe complex points in CP2:
+pencil of type a). At least one vertex Mj from Definition 1.21 is real, and
+in this case the involution σMj is also real. Hovewer in general Mj (KEL)
+are not necessarily all real.
+2) Consider a real pencil of conics whose complexifications form a pencil
+of type b). All its admissible vertices C, M, KAB are always real, and so
+are the corresponding global projective involutions. In the case, when C is
+equipped with a quasi-global structure defined by a real conic, the correspond-
+ing involution σC is real.
+3) For a real pencil with complexification of type c) the admissible vertex
+M and the corresponding projective involution σM are both real. But the
+vertices A, C, M′ are not necessarily real. If M′ is real, then so is σM′.
+4) For a real pencil with complexification of type d) or e) the admissible
+vertex A is always real. The corresponding involution σA = σA,S is real, if
+and only if so is the conic S defining it.
+Theorem 1.25 Let a (real or complex) dual multibilliard on a collection of
+real C4-smooth (or holomorphic) nonlinear connected curves γj and some
+vertices be rationally integrable. Then the following statements hold.
+1) Each curve γj is a conic equipped with a dual billiard structure either
+of pencil type, or as in Theorem 1.11, Case 2).
+2) If the multibilliard contains at least two distinct conics (in the sense
+of Definition 1.22), then all the conics γj lie in the same pencil, and the
+dual billiard structures on them are defined by the same pencil.
+18
+
+Theorem 1.26 Let in a dual multibilliard all the curves be conics lying in
+the same pencil. Let they be equipped with the dual billiard structure defined
+by the same pencil.
+(Case of one conic equipped with a pencil type dual
+billiard structure is possible.) Then the multibilliard is rationally integrable,
+if and only if it is of pencil type, see Definition 1.23.
+Theorem 1.27 The minimal degree of a rational integral of a pencil type
+multibilliard is
+(i) degree two, if it contains no skew vertices;
+(ii) degree 12, if the pencil has type a) and the multibilliard contains
+some two non-opposite skew vertices, i.e., a pair of vertices of type KEL and
+KES for some distinct E, L, S ∈ {A, B, C, D}.
+(iii) degree four in any other case.
+The next theorem yields a formula for integral of degree 12 of pencil type
+multibilliards for pencils of type a). To state it, let us introduce the fol-
+lowing notations.
+Let RP2
+[y1:y2:y3] denote the ambient projective plane of
+the multibilliard, considered as the projectivization of the space R3
+y1,y2,y3.
+For every projective line X let π−1(X) ⊂ R3 denote the corresponding
+two-dimensional subspace. Let ξX(Y ) denote a non-zero linear functional
+vanishing on π−1(X). It is well-defined up to constant factor.
+Theorem 1.28 Consider a pencil of conics through four distinct base points
+A, B, C, D. Set M1 = AB ∩ CD, M2 = BC ∩ AD, M3 = AC ∩ BD.
+1) The functionals ξEL corresponding to the lines EL through distinct
+points E, L ∈ {A, B, C, D} can be normalized by constant factors so that
+ξABξCD + ξBCξAD + ξADξBC = 0,
+(1.21)
+2) If (1.21) holds, then for every µ ∈ R\0 the degree 12 rational function
+�
+{EL;F N}̸={E′L′;F ′N′}
+� ξELξF N
+ξE′L′ξF ′N′ (Y ) + µ)
+�
+(1.22)
+is a first integral of every pencil type multibilliard defined by the given pen-
+cil. Here the product is taken over ordered pairs of two-line sets {EL; FN},
+{E′L′; F ′N ′} with {E, L, F, N} = {E′, L′, F ′, N ′} = {A, B, C, D}. In Theo-
+rem 1.27, Case (ii) this is a minimal degree integral.
+Definition 1.29 A rationally integrable real (complex) dual billiard struc-
+ture on conic that is not of pencil type, see Theorem 1.11, Case 2), will be
+19
+
+called exotic. The singular points of the dual billiard structure (which are
+exactly the indeterminacy points of the corresponding integral R from the
+Addendum to Theorem 1.11) will be called the base points.
+Corollary 1.30 Let a rationally integrable real (complex) multibilliard be
+not of pencil type. Then it contains only one curve, namely, a conic equipped
+with an exotic rationally integrable dual billiard structure from Theorem
+1.11, Case 2), and maybe some vertices.
+Theorem 1.31 A (real or complex) multibilliard consisting of one conic γ
+equipped with an exotic dual billiard structure from Theorem 1.11, Case 2),
+and maybe some vertices is rationally integrable, if and only if the collection
+of vertices either is empty, or consists of the so-called admissible vertices
+Q defined below, being equipped with the γ-angular symmetry σQ:
+(i) Case of type 2a) dual billiard on γ. The unique admissible vertex
+is the intersection point Q = [1 : 0 : 0] of the z-axis and the infinity line; one
+has σQ(z, w) = (−z, w) in the chart (z, w). See Fig. 8. In Subcase 2a1),
+when ρ = 2 −
+2
+2N+1, the function R(z, w) from (1.7) is a rational integral of
+the multibilliard (γ, (Q, σQ)) of minimal degree: deg R = 4N +2. In Subcase
+2a2), when ρ = 2−
+1
+N+1, the function R2(z, w) with R the same, as in (1.8),
+is a rational integral of (γ, (Q, σQ)) of minimal degree: deg R2 = 4N + 4.
+(ii) Case of type 2b1) or 2b2). There are three base points. One of them,
+denoted X, is the intersection point of two lines contained in the polar locus
+R = ∞. The unique admissible vertex Q is the intersection point of
+two lines: the tangent line to γ at X and the line through the two other
+base points.
+In Case 2b1) one has Q = (0, −1).
+In Case 2b2) one has
+Q = [1 : 0 : 0], σQ(z, w) = (−z, w). The corresponding rational function
+R, see (1.9), (1.10) is a rational integral of the multibilliard (γ, (Q, σQ)) of
+minimal degree: deg R = 4. See Fig. 9.
+(iii) Case of type 2c1) or 2c2). There are three complex base points.
+There are three admissible vertices. Each of them is the intersection
+point of a line through two base points and the tangent line to γ at the other
+one. In Case 2c1) the point (0, −1) is the unique real admissible vertex.
+In Case 2c2) all the admissible vertices are real: they are (0, −1), (1, 0),
+[1 : 1 : 0], see Fig. 9. The function R is a degree 6 rational integral of the
+multibilliard formed by the conic γ and arbitrary admissible vertex collection.
+(iv) Case of type 2d). No admissible vertices.
+20
+
+Figure 8: The only admissible vertex in Case 2a) is the infinite point Q =
+[1 : 0 : 0].
+Figure 9: The admissible vertices in Cases 2b1) and 2c2) are marked in bold.
+21
+
+Case 2c2)
+W
+[1:1:0]
+R=80
+(1,1)
+0
+7
+(1,0)
+*(0,-1)Case 2b1)
+8
+W
+(1,1)
+0
+*Q=(0,-1)Case 2a)
+Q=[1:0:0]
+W
+0
+zProposition 1.32 1) The two multibilliards of type (ii) (Cases 2b1), 2b2))
+with one admissible vertex are complex-projectively equivalent. 2) Two multi-
+billiards of type (iii) (either of different subtypes 2c1), 2c2), or of the same
+subtype) are complex-projectively equivalent, if and only if they have the
+same number of vertices. 3) Two real multibilliards of type (iii) are real-
+projectively equivalent, if and only if they have either both subtype 2c1) and
+one real admissible vertex, or both subtype 2c2) and the same number (ar-
+bitrary, from 1 to 3) of real admissible vertices.
+Proposition 1.32 follows from the last statement of Theorem 1.12 and
+the fact that the dual billiard of type 2c2) has real order three projective
+symmetry cyclically permuting admissible points.
+1.5
+Application: classification of rationally 0-homogeneously
+integrable piecewise C4-smooth projective billiards
+Let us recall the following definition.
+Definition 1.33 The central projective billiard structure on a planar curve
+C with center O ∈ R2 is the field of lines on C passing through O.
+Everywhere below for two projective lines e and f by ef we will denote
+their intersection point.
+Definition 1.34 Consider a complex dual pencil of conics: a family of con-
+ics whose dual form a pencil. Let ℓ be a line equipped with a projective
+billiard structure; the corresponding line field is well-defined either on all
+of ℓ, or on ℓ punctured at one point called singular. The line ℓ is said to
+be admissible for the dual pencil, if it belongs to the following list of lines
+equipped with projective billiard structures, called either standard, or skew.
+Case a): dual pencil of conics tangent to four distinct lines a, b, c, d.
+a1) The standard admissible lines are the lines m1, m2, m3 through the
+points ab and cd, the points bc and ad, the points ac and bd respectively.
+The line m1 is equipped with projective billiard structure centered at m2m3,
+and the projective billiard structures on m2, m3 are defined analogously.
+a2) Let kbc denote the line through the points m1m3 and bc, equipped
+with the projective billiard structure centered at ad: the field of lines through
+ad. Let kad be the line through m1m3 and ad, equipped with the projective
+billiard structure centered at bc. The other lines kef, e, f ∈ {a, b, c, d}, e ̸= f,
+equipped with central projective billiard structures are defined analogously.
+22
+
+We identify kef with kfe. All the six lines kef thus constructed are called
+skew admissible lines. See Fig. 10.
+Case b): dual pencil of conics tangent to three distinct lines a, b, c and
+having common tangency point C with c. See Fig. 11.
+b1) The skew line c equipped with the field of lines through the point
+ab.
+b2) The skew line k such that the quadruple of lines a, b, m, k through
+the point ab is harmonic. It is equipped with the field of lines through C.
+Here m is the line through C and ab.
+b3) The standard line m with the field of lines through the point ck.
+b4) For arbitrary given conic S of the dual pencil the line c equipped
+with the field of lines tangent to S is a skew line.
+Case c): dual pencil of conics tangent to each other at two points A and
+B. Led a and b denote the corresponding tangent lines. See Fig. 12.
+c1) The standard line m = AB with the field of lines through ab.
+c2) The skew lines a and b equipped with the fields of lines through B
+and A respectively.
+c3) Fix arbitrary line c ̸= a, b through ab. Let Z ∈ m denote the point
+such that the quadruple of points cm, Z, B, A ∈ m is harmonic. The line c
+equipped with the field of lines through Z is called a skew line.
+c4) Fix a regular conic S from the dual pencil. The lines a and b, each
+being equipped with the field of lines tangent to S at points distinct from
+A and B respectively are called skew lines.
+Case d): dual pencil of conics tangent to a given line a at a given point
+A, having triple contact between each other at A, and tangent to another
+given line b ̸= a. See Fig. 13.
+d1) The skew line a equipped with the line field tangent to a given
+(arbitrary) regular conic S from the pencil.
+d2) Any line c ̸= a through A called skew, equipped with the field of
+lines through the point ab.
+Case e): dual pencil of conics tangent to each other at a point A with
+order 4 contact. Let a denote their common tangent line at A. See Fig. 14.
+e1) The skew line a equipped with the line field tangent to a given (ar-
+bitrary) regular conic S from the pencil.
+e2) Any line b through A called skew, equipped with the field of lines
+tangent to the conics of the pencil at points of the line b: these tangent lines
+pass through the same point C = C(b) ∈ a.
+Proposition 1.35 In Case a) for every distinct e, f, g ∈ {a, b, c, d} the lines
+23
+
+Figure 10: Dual pencil of type a). The standard admissible lines are m1,
+m2, m3. The skew admissible lines kef are marked in bold.
+Figure 11: Dual pencil of type b): i) one standard line m and two skew lines
+c, k; ii) skew line c with another line field, tangent to a given conic S.
+24
+
+i)
+ii)
+s
+ck
+Ic
+c
+Cbd
+d
+b
+m
+d
+ad
+a
+ac
+m1
+m2
+Kab
+Ibc
+cd
+m2m3Figure 12: Dual pencil of type c): i) one standard line m and two skew lines
+a, b equipped with central projective billiard structures; ii) arbitrary line
+c ̸= a, b through ab (called skew) with field of lines through Z, and the skew
+lines a, b with fields of lines tangent to a given conic S from the pencil.
+Figure 13: Dual pencil of type d): the skew line a and an arbitrary skew
+line c ̸= a.
+Figure 14: Dual pencil of type e): the skew line a and a standard line b ̸= a.
+25
+
+s
+b
+A
+c
+ab
+S
+ab
+a
+ATi)
+cml
+i)
+b
+m
+B
+b
+S
+z
+ab
+ab
+m
+a
+a
+AT
+IAkef, kfg, kge pass through the same point. In particular, the line kab passes
+through the intersection points kbd ∩ kad and kac ∩ kbc, see Fig. 10.
+Proof The latter intersection points and the point m2m3 lie on one line, by
+the dual Desargues Theorem applied to the triangles (bd, ad, kbd ∩ kad) and
+(ac, bc, kac ∩ kbc). The point ab lies on the same line, by Pappus Theorem
+applied to the triples of points bd, m1m3, ac and ad, m1m2, bc.
+✷
+Definition 1.36 A projective billiard with piecewise-C4-smooth boundary
+having at least one nonlinear smooth arc is said to be of dual pencil type, if
+it satisfies the following conditions:
+1) Each C4-smooth arc of the boundary is either a conical arc, or a
+segment. All the conical arcs lie in the same dual pencil and are equipped
+with the projective billiard structure defined by the same pencil.
+2) The segments in the boundary are contained in lines admissible for
+the pencil and are equipped with the projective billiard structures of the
+ambient admissible lines.
+3) If the boundary contains a skew line segment whose projective billiard
+structure is not a central one (i.e., given by a field of lines tangent to a conic
+of the pencil), then the boundary contains no segments of other skew lines
+with the following exceptions of possible of ambient skew line collections:
+- in Case c) the skew line collection is allowed to be the pair of lines a
+and b equipped with the fields of lines tangent to one and the same (but
+arbitrary) conic of the pencil;
+- in Case d) the skew line collection is allowed to be a pair of lines a and
+c: a being equipped with the field of lines tangent to a given conic S from
+the pencil; c is the line through the point A and the tangency point of the
+conic S with the line b, equipped with the field of lines throgh the point ab.
+4) In Case d) the boundary may contain a segment of at most one line
+c ̸= a.
+5) Each ambient skew line of a boundary segment that a priori admits
+several possible projective structures listed above is allowed to be included
+in the boundary with no more than one projective billiard structure.
+Remark 1.37 The notion of admissible line for dual pencil is dual to that
+of admissible vertex of a pencil.
+Theorem 1.38 Let a planar projective billiard with piecewise C4-smooth
+boundary containing a nonlinear arc be rationally integrable. Then the fol-
+lowing statements hold.
+26
+
+1) All the nonlinear arcs of the boundary are conical. Different arcs of
+the same conic are equipped with the restriction to them of one and the same
+projective billiard structure on the ambient conic: either of dual pencil type,
+or a one from Theorem 1.16, Case 2).
+2) If the boundary contains at least two arcs of two distinct regular conics,
+then all the ambient conics of nonlinear arcs lie in the same dual pencil and
+their projective billiard structures are defined by the same dual pencil.
+Theorem 1.39 Let a planar projective billiard have piecewise C4-smooth
+boundary whose all nonlinear C4-smooth pieces are conical arcs lying in the
+same dual pencil and equipped with projective billiard structures defined by
+the same pencil. Then the billiard is 0-homogeneously rationally integrable,
+if and only if it is of dual pencil type.
+Theorem 1.40 The minimal degree of 0-homogeneous rational integral of
+a dual pencil type projective billiard is
+(i) degree two, if its boundary contains no skew line segment;
+(ii)) degree 12, if the dual pencil has type a) and the billiard bound-
+ary contains segments of some two skew admissible lines kef, kfs for some
+distinct e, f, s ∈ {a, b, s, d};
+(iii) degree four in any other case.
+Theorem 1.41 Consider a type a) dual pencil of conics tangent to given
+four distinct lines a, b, c, d. Let us consider the ambient plane R2
+x1,x2 as the
+horizontal plane {x3 = 1} ⊂ R3
+x1,x2,x3. Set
+r := (x1, x2, 1) ∈ R3, v = (v1, v2, 0) for every (v1, v2) ∈ T(x1,x2)R2,
+M = M(r, v) := [r, v] = (−v2, v1, ∆), ∆ := x1v2 − x2v1.
+(1.23)
+In the above notations for intersection points em of lines e and m set
+r(em) = (x1(em), x2(em), 1).
+There exists a collection of three numbers χem;fn ∈ R, {e, m, f, n} = {a, b, c, d},
+indexed by unordered pairs of intersection points em = e ∩ m, fn = f ∩ n
+((em; fn) = (fn; em), by definition) such that
+�
+(em;fn)
+χem;fn < r(em), M >< r(fn), M >= 0
+(1.24)
+27
+
+Figure 15: Pencil of type a). Distances between intersection points.
+as a quadratic form in M. This collection is unique up to common constant
+factor. For every µ ∈ R \ {0} the corresponding expression
+�
+(em;fn)̸=(e′m′;f′n′)
+�
+χem;fn < r(em), M >< r(fn), M >
+χe′m′;f′n′ < r(e′m′), M >< r(f ′n′), M > + µ
+�
+(1.25)
+is a degree 12 first integral of every projective billiard of dual pencil type
+defined by the pencil in question. Here the product is taken over ordered ”big”
+pairs: any two indices ((em; fn); (e′m′; f ′n′)) and ((e′m′; f ′n′), (em; fn))
+that differ by permutation correspond to two distinct factors in the product.
+Lemma 1.42 Consider the following segment lengths, see Fig. 15:
+ρ := |bc − ab|, t := |ab − bd|, τ := |ad − ab|, s := |ab − ac|.
+(1.26)
+Here the lengths are oriented: lengths of two adjacent aligned segments (say,
+s and τ), are taken with the same sign, if their common end separates them
+(ab lies between the points ac and ad), and with opposite signs otherwise.
+Relation (1.24) (and hence, the statements of Theorem 1.41) hold for
+(χab;cd, χbc;ad, χac;bd) =
+� st
+ρτ − 1, − st
+ρτ , 1
+�
+.
+(1.27)
+Definition 1.43 A rationally 0-homogeneously integrable projective bil-
+liard structure on a conic γ that is not of dual pencil type (i.e., any projec-
+tive billiard structure from Theorem 1.16, Case 2)) will be called exotic. Its
+singular points (which are the points, where the corresponding line field is
+either undefined, or tangent to γ) will be called the base points.
+28
+
+cd
+b
+bc
+ad
+0
+ab
+h
+s
+a
+ac
+t
+bd
+IdCorollary 1.44 Let a rationally integrable projective billiard with piecewise
+C4-smooth boundary be not of dual pencil type. Let its boundary contain at
+least one nonlinear arc. Then all its nonlinear arcs lie in one conic, equipped
+with an exotic projective billiard structure from Theorem 1.16, Case 2).
+Theorem 1.45 Let a projective bulliard has piecewise smooth boundary
+consisting of arcs of one and the same conic γ equipped with an exotic
+projective billiard structure from Theorem 1.16, Case 2), and maybe some
+straightline segments.
+The billiard is rationally integrable, if and only if
+the collection of ambient lines of the boundary segments either is empty, or
+consists of the following admissible lines equipped with central-projective
+billiard structures:
+(i) Case of type 2a) projective billiard structure on γ; ρ = 2− 2
+m, m ∈ N,
+m ≥ 3. The unique admissible line is the vertical x2-axis, equipped with
+the normal (i.e., horizontal) line field. See Fig. 16. The projective billiard
+bounded by a half of γ and the x2-axis has a rational 0-homogeneous integral
+of minimal degree 2m: the function Ψ2a1 from (1.14) for odd m; the function
+Ψ2
+2a2 with Ψ2a2 the same, as in (1.15), for even m. See Fig. 16.
+(ii) Case of type 2b1). The unique admissible line is the line {x2 = 1}
+equipped with the field of lines through the point (0, −1).
+(iii) Case of type 2b2).
+The unique admissible line is the Ox2-axis
+equipped with the normal (horizontal) line field. See Fig. 17. In both cases
+2b1), 2b2) the functions Ψ2b1, Ψ2b2 from (1.16) and (1.17) are integrals of
+minimal degree 4 for each billiard bounded by γ and the admissible line.
+(iv) Case of type 2c1). The unique admissible line is the line {x2 = 1}
+equipped with the field of lines through the point (0, −1).
+(v) Case of type 2c2). There are three admissible lines:
+- the line {x2 = 1}, with the field of lines through the point (0, −1);
+- the line {x1 = − 1
+2}, with the line field parallel to the vector (−1, 1);
+- the line {x2 = −2x1}, with the field of lines through the point (−1, 0).
+See Fig. 18. In both cases 2c1), 2c2) the corresponding functions Ψ2c1,
+Ψ2c2 from (1.18) and (1.19) are integrals of minimal degree 6 for each billiard
+bounded by γ and segments of admissible lines.
+(vi) Case of type 2d). No admissible lines.
+1.6
+Plan of proofs of main results
+Step 1.
+In Subsection 2.1 we prove rational integrability of pencil type
+complex multibilliard.
+(This implies analogous result in the real case.)
+To do this, we show that for every pencil all the involutions associated
+29
+
+Figure 16: The only admissible line in Case 2a) is the Ox2-axis.
+Figure 17: The only admissible line in Case 2b1) is the line {x2 = 1}. The
+only admissible line in Case 2b2) is the x2-axis.
+Figure 18: The only admissible line in Case 2c1) is the line {x2 = 1}. In Case
+2c2) there are three admissible lines: {x2 = 1}; {x1 = − 1
+2}, {x2 = −2x1}.
+30
+
+2c2)
+X2 =-2x,
+Y
+0
+1
+-1/2
+-12c1)
+Y
+0
+-1
+12b2)
+Y
+02b1)
+Y
+0
+-1
+12a)
+Y
+(2-p)X1
+￥1
+0to all the corresponding admissible vertices preserve the pencil and act
+on its parameter space C by conformal involutions (Proposition 2.1). We
+fix an arbitrary collection of admissible vertices and consider the subgroup
+G ⊂ Aut(C) = PSL2(C) generated by the corresponding conformal involu-
+tions. We show that finiteness of the group G is equivalent to the system of
+Conditions 3)–5) of Definition 1.23 of pencil type multibilliard (Proposition
+2.4), and in this case G is either trivial, or isomorphic to either Z2, or S3.
+We then deduce rational integrability of every pencil type multibilliard with
+integral of degree 2|G| ∈ {2, 4, 12}.
+To classify rationally integrable dual multibilliards, in what follows we
+consider an arbitrary dual multibilliard with a rational integral Ψ. Each its
+curve is already known to be a conic equipped with either a pencil type dual
+billiard structure, or an exotic billiard structure from Theorem 1.11, Case
+2). We fix some its conic S and consider the canonical integral R of its dual
+billiard structure: either a quadratic integral in the case of pencil; of the
+corresponding integral from the Addendum to Theorem 1.11.
+Step 2. In Subsection 2.2 we show that the singular foliations Ψ = const
+and R = const on CP2 coincide.
+We show that a generic level curve of
+the integral R is irreducible, of the same degree d = deg R, and thus, R is a
+rational first integral of minimal degree for the above foliation. In the exotic
+case we also show that the conic S is its unique level curve of multiplicity d,
+which means that the irreducible level curves of the function R accumulating
+to S converge to d
+2[S] as divisors: the intersection of a small cross-section to
+S with a level curve close to S consists of d
+2 points.
+Step 3. We then deduce (in Subsection 2.2) that if on some conic of
+the multibilliard the dual billiard structure is defined by a pencil (or if the
+multibilliard contains at least two distinct conics), then the above foliation
+coincides with the pencil and the dual billiard structures on all the other
+conics are defined by the same pencil. This will prove Theorem 1.25. Results
+of Step 1 together with constance of integral on the conics of the pencil (given
+by Step 2) imply Theorem 1.27.
+Step 4. In Subsection 2.3 we study vertices of the multibilliard. First we
+show that the family of involutions σA,ℓ : ℓ → ℓ associated to each vertex A
+is given by the restrictions to the lines ℓ through A of a birational involution
+σA : CP2 → CP2 preserving the foliation Ψ = const. Then we deduce that
+each σA is either a projective angular symmetry, or a degenerate angular
+symmetry defined by a regular conic S through A. We show that in the
+latter case the foliation Ψ = const is a pencil of conics containing S.
+Step 5. In Subsection 2.4 we prove Theorem 1.26. It deals with the case,
+31
+
+when the foliation Ψ = const is a pencil of conics. We show that each vertex
+of the multibilliard is admissible. Each level curve of the function Ψ is a
+collection of at most deg Ψ
+2
+conics of the pencil, and it is invariant under the
+involutions defining the dual billiard structures at the vertices. This implies
+finiteness of the group G generated by the conformal involutions correspond-
+ing to the vertices. Together with the results of Step 1 (Subsection 2.1), this
+implies that the multibilliard is of pencil type.
+In Subsection 2.5 we prove Theorem 1.31 on classification of rationally
+integrable multibilliards consisting of a conic S with an exotic dual billiard
+structure and (may be) some vertices. We describe the corresponding ad-
+missible vertices using the result of Step 4 stating that the corresponding
+involutions are projective angular symmetries.
+In Subsection 2.6 we prove Proposition 1.24.
+In Section 3 we prove the main results on classification of rationally 0-
+homogeneously integrable projective billiards with C4-smooth boundaries
+(Theorems 1.38, 1.39, 1.40, 1.45). We reduce them to the main results on
+dual multibilliards.
+Namely, we consider the projective duality given by
+orthogonal polarity, which transforms a projective billiard to a dual multi-
+billiard. We consider the ambient plane R2
+x1,x2 of the projective billiard as
+the horizontal plane {x3 = 1} ⊂ R3
+x1,x2,x3, set r = (x1, x2, x3). We use the
+fact that a rational 0-homogeneous integral of a projective billiard can be
+written as a rational 0-homogeneous function R(M) of the moment vector
+M = [r, v], where v is the velocity, and R(M) is a rational integral of the
+corresponding dual multibilliard in RP2
+[M1:M2:M3]. We show that this yields
+a bijective correspondence between rational 0-homogeneous integrals of the
+projective billiard and rational integrals of the corresponding dual multibil-
+liard. This together with the results from [27] on duality between exotic dual
+billiards from Theorem 1.11 and exotic projective billiards from Theorem
+1.16 and the results of the present paper on dual multibilliards will imply
+the main results on projective billiards.
+1.7
+Historical remarks
+Existence of a continuum of closed caustics in every strictly convex bounded
+planar billiard with sufficiently smooth boundary was proved by V.F.Lazutkin
+[31]. Existence of continuum of foliations by (non-closed) caustics in open
+billiards was proved by the author [26]. H.Poritsky [33] (and later E.Amiran
+[2]) proved the Birkhoff Conjecture under the additional assumption that for
+every two caustics the smaller one is a caustic for the bigger one. M.Bialy
+[3] proved that if the phase cylinder is foliated by non-contractible invariant
+32
+
+curves for the billiard map, then the billiard table is a disk. See also [42],
+where another proof of Bialy’s result was given, and Bialy’s papers [4, 5] for
+similar results on billiards on constant curvature surfaces and on magnetic
+billiards on these surfaces. D.V.Treschev conjectured existence of billiards
+where the squared billiard map has fixed point where its germ is analyti-
+cally conjugated to rotation and confirmed this by numerical experiments:
+in two dimensions [37, 38] and in higher dimensions [39]. V.Kaloshin and
+A.Sorrentino [28] proved that any integrable deformation of an ellipse is an
+ellipse. For ellipses with small excentricities this result was earlier proved by
+A.Avila, V.Kaloshin and J. De Simoi [1]. Recently M.Bialy and A.B.Mironov
+proved the Birkhoff Conjecture for centrally-symmetric billiards admitting
+a continuous family of caustics extending up to a caustic of 4-periodic or-
+bits [12]. For a dynamical entropic version of the Birkhoff Conjecture and
+related results see [32]. For a survey on the Birkhoff Conjecture and results
+see [28, 29, 12] and references therein.
+A.P.Veselov proved a series of complete integrability results for billiards
+bounded by confocal quadrics in space forms of any dimension and described
+billiard orbits there in terms of a shift of the Jacobi variety corresponding
+to an appropriate hyperelliptic curve [40, 41]. Dynamics in (not necessarily
+convex) billiards of this type was also studied in [15, 16, 17, 18, 19].
+The Polynomial Birkhoff Conjecture together with its generalization to
+piecewise smooth billiards on surfaces of constant curvature was stated by
+S.V.Bolotin and partially studied by himself, see [13], [14, section 4], and
+by M.Bialy and A.E.Mironov [6]. Its complete solution is a joint result of
+M.Bialy, A.E.Mironov and the author given in the series of papers [8, 9, 24,
+25]. It implies that if a polynomial integral of a piecewise smooth billiard
+exists, then its minimal degree is equal to either two, or four. For a survey
+of Bolotin’s Polynomial Birkhoff Conjecture and of its version for magnetic
+billiards (an open conjecture, with a substantial progress made in [7, 10])
+and related results see [30, 29, 8, 9, 7, 10, 11] and references therein.
+The generalization of the Birkhoff Conjecture to dual billiards was stated
+by S.Tabachnikov in [36]. Its rationally integrable version was solved by the
+author of the present paper in [27]. Its polynomially integrable version for
+outer billiards was stated and partially studied in [36] and solved completely
+in [23].
+Projective billiards were introduced by S.Tabachnikov [35].
+He
+had shown in the same paper that if a projective billiard on circle has an
+invariant area form smooth up to the boundary of the phase cylinder, then
+it is integrable.
+A series of results on the analogue of Ivrii Conjecture on periodic orbits in
+billiard (stating that their Lebesgue measure is zero) for projective billiards
+33
+
+was obtained by C.Fierobe [20, 21, ?].
+2
+Rationally integrable dual multibilliards. Proofs
+of Theorems 1.25, 1.26, 1.31, 1.27
+2.1
+Rational integrability of pencil type multibilliards
+Proposition 2.1 Consider a complex pencil of conics and the correspond-
+ing admissible vertices. For every standard vertex the corresponding invo-
+lution leaves invariant each conic of the pencil. For every skew vertex the
+corresponding involution permutes conics of the pencil non-trivially: it acts
+as a conformal involution of the parameter space C of the complex pencil.
+Proof Case a): pencil of conics through four distinct basic points A, B, C,
+D, see Fig. 3. It is well-known that in this case no three of them lie on the
+same line. This implies that the three vertices Mj are well-defined, distinct,
+do not lie on the same line and different from the basic points, and so are
+the vertices KEL, and the latter are distinct from the vertices Mj. Set
+Γ1 := AB ∪ CD, Γ2 := BC ∪ AD, Γ3 = AC ∪ BD.
+(2.1)
+Let σM1 : CP2 → CP2 be the Γ2-angular symmetry centered at M1: the
+projective involution fixing each line through ℓ and permuting its intersection
+points with the lines AD and BC. It permutes the points A and B, C and
+D. Hence, it preserves the pencil. It fixes the line M2M3, which passes
+through the points AD ∩ BC and AC ∩ BD. Hence, it fixes each its point
+X, since it fixes the line M1X. The pencil is parametrized by a parameter
+λ ∈ C, and σM1 acts on Cλ by conformal automorphism. Let λ1, λ2, λ3 ∈ C
+denote the parameter values corresponding to the singular conics Γ1, Γ2,
+Γ3 respectively. Each Γj is σM1-invariant, by construction. Therefore, the
+conformal automorphism C → C induced by σM1 fixes three distinct points
+λ1, λ2, λ3. Hence, it is identity, and σM1 preserves each conic of the pencil.
+This proof is valid for the other vertices Mj.
+The involution σKBC is the projective angular symmetry centered at
+KBC with fixed point line AD. Hence, it fixes M2.
+Claim 1. The involution σKBC permutes B and C. Or equivalently, the
+quadruple of points KBC, M2, B, C on the line BC is harmonic.
+Proof The restriction of the involution σKBC to the line BC coincides with
+the involution σM2, since both of them are non-trivial projective involutions
+of the line BC fixing KBC and M2. The involution σM2 permutes B and C,
+as in the above discussion on σM1. Hence, so does σKBC.
+✷
+34
+
+Corollary 2.2 Each one of the involutions σKBC, σAD fixes Γ2 and per-
+mutes Γ1, Γ3. Hence, it yields a non-trivial conformal involution Cλ → Cλ
+of the parameter space of the pencil, fixing λ2 and permuting λ1, λ3.
+Proof The involution σKBC fixes A, D and permutes B, C. Similarly, the
+involution σKAD fixes B, C and permutes A, D.
+✷
+Case b): pencil of conics through three distinct points A, B, C tangent
+at the point C to the same line L, see Fig. 4. The involution σM fixes
+the points C, KAB, the line L and permutes A and B, by definition and
+harmonicity of the quadruple M, KAB, A, B. Therefore, it preserves the
+pencil. Similarly, the involution σKAB preserves the pencil. And so does
+the involution σC : CP2 → CP2 defined to fix C and each point of the line
+AB. Now the pencil, parametrized by a parameter λ ∈ C, contains just two
+singular conics:
+Γ1 := AB ∪ L, Γ2 := AC ∪ BC,
+(2.2)
+corresponding to some parameter values λ1 and λ2.
+Claim 2. The involution σM and the composition σC ◦ σKAB preserve
+each conic of the pencil. Each one of the involutions σC, σKAB fixes only
+the conics Γ1, Γ2 of the pencil.
+Proof
+The pencil in question is the limit of a family of pencils of conics
+through A, B, Cµ, Dµ with basic points Cµ, Dµ depending on small pa-
+rameter µ, confluenting to C, as µ → 0, so that the line CµDµ pass through
+M = M1 and tends to the tangent line L, as µ → 0. Then M2 = M2(µ) → C,
+M3 = M3(µ) → C, and the involutions σM1 = σM1(µ) corresponding to the
+perturbed pencil, with µ ̸= 0, converge to σM, as µ → 0. The involution
+σM1(µ) preserves each conic of the pencil for µ ̸= 0. Hence, so does its limit
+σM. The involutions at the vertices KCµDµ, KAB converge to σC and σKAB,
+by construction. They act on the perturbed pencil as non-trivial involutions,
+permuting conics in the same way (Corollary 2.2). Hence, this statement
+remains valid for their limits σC and σKAB. The claim is proved.
+✷
+Proposition 2.3 Consider a pencil of complex conics that are tangent to
+each other at a point C. Let S be its regular conic, and let C be equipped with
+the quasi-global dual billiard structure defined by S. Then the correspond-
+ing involution σC preserves the pencil and induces a nontrivial conformal
+involution C → C of its parameter space.
+Proof Let L denote the common projective tangent line at C to the regular
+conics of the pencil. Let us take an affine chart Cz,w = CP2 \ L so that C is
+35
+
+the intersection point of the w-axis with the infinity line L. Then the conics
+of the pencil are parabolas Sλ := {w = (a1z2+b1z+c1)+λ(a2z2+b2z+c2)}.
+Let us normalize the parameter λ so that S0 = S. Then in the affine chart
+(z, w) one has
+σC(z, w) = (z, 2(a1z2 + b1z + c1) − y).
+Hence, σC(Sλ) = S−λ. The proposition is proved.
+✷
+Case c): pencil of conics through two distinct points A and C tangent
+to two given lines LA and LC through them; LA, LC ̸= AC. See Fig. 5. Fix
+an arbitrary point M′ ∈ AC \ {A, C}.
+Claim 3. The projective angular symmetries σA, σC with fixed point
+lines LC and LA respectively preserve the pencil. The involutions σM, σM′
+and the composition σA ◦ σC preserve each conic of the pencil.
+The claim is proved analogously to the above discussion, by considering
+the pencil in question as the limit of the family of pencils through points
+Aµ, Bµ, Cµ, Dµ, Aµ, Bµ → A, Cµ, Dµ → C, as µ → 0 so that AµBµ = LA,
+CµDµ = LC, and the lines AµCµ, BµDµ are intersected at M′. Similarly to
+the above discussion, the involutions corresponding to KAµBµ and KCµDµ
+converge to σA and σC respectively. This implies the statement of the claim
+on the involutions σM, σA, σC. It remains to prove its statement on the
+vertex M′. The intersection point M2(µ) of the lines BµCµ and AµDµ, the
+point M′, and the intersection points of the line M2(µ)M′ with lines LA,
+LC form a harmonic tuple of points on the line M2(µ)M′, as in Claim 1.
+Hence, the involution σM′,µ corresponding to the vertex M′ and the per-
+turbed pencil, with µ ̸= 0, fixes M′ and each line through M′ and permutes
+its intersection points with the lines LA and LC. Thus, it coincides with
+the involution σM′ corresponding to the nonperturbed pencil. Hence, σM′
+preserves each conic of the nonperturbed pencil, as of the perturbed one.
+Consider the skew vertices in Cases c), d), e) equipped with quasi-global
+dual billiard structures. The corresponding involutions preserve the pencil
+and induce nontrivial conformal involutions of Cλ, by Proposition 2.3.
+Consider now the vertices C in Cases d) and e). In Case d) the involution
+σC preserves the singular conic L ∪ AB of the pencil and the conic tangent
+to BC at B, as σM in Claim 3. Hence, it preserves the pencil. It does not
+preserve other conics, since their tangent lines at B are not σC-invariant.
+In Case e) σC preserves each conic of the pencil. This can be seen in the
+affine chart (z, w) for which C, A are the intersection point of the infinity
+line with the z- and w-axes respectively, and the conics are the parabolas
+w = z2 + λ: σC(z, w) = σC(−z, w). Proposition 2.1 is proved.
+✷
+36
+
+Proposition 2.4 Let in a complex dual multibilliard all the curves be con-
+ics lying in a pencil, and their dual billiard structures be defined by the same
+pencil. Let all its vertices be admissible for the pencil. Let G ⊂ PSL2(C) =
+Aut(C) denote the group generated by conformal transformations of the pa-
+rameter space C of the pencil induced by the involutions assotiated to the
+vertices, see Proposition 2.1. Then the following statements are equivalent:
+(i) The group G is finite.
+(ii) The vertex collection satisfies Conditions 3)–5) of Definition 1.23.
+If the group G is finite, then it is either trivial (if and only if the multibilliard
+contains no skew vertex), or isomorphic to Z2 or S3. One has G = S3, if
+and only if the pencil has type a) and the multibilliard contains some two
+skew vertices KEX, KEY with base point pairs having one common point.
+Proof Let us first prove equivalence of statements (i) and (ii).
+Case of pencil of type a). Then Conditions 3)–5) of Definition 1.21 im-
+pose no restriction on admissible vertex collection. The involution defining
+the dual billiard structure at each admissible vertex preserves the triple of
+the singular conics Γ1, Γ2, Γ3 of the pencil, see (2.1). Therefore, the confor-
+mal involutions C → C of the parameter space defined by the skew vertices
+permute the corresponding parameter values λ1, λ2, λ3, and hence, gen-
+erate a finite group G ⊂ PSL2(C) isomorphic to a subgroup of S3. The
+conformal involution corresponding to a standard vertex is trivial.
+Each
+one of the involutions σKBC, σAD fixes the singular conic Γ2 and permutes
+Γ1, Γ3 (Corollary 2.2). See the above proof of Proposition 2.1. These two
+statements and the versions of the latter statement for the other base points
+together imply the statements of Proposition 2.4.
+Case of pencil of type b). For every skew vertex equipped with a projec-
+tive angular symmetry the latter symmetry fixes only the parameter values
+λ1, λ2 corresponding to the singular conics Γ1, Γ2 from (2.2), see Claim 2
+in the proof of Proposition 2.1.
+Suppose the multibilliard contains only vertices of the above type. Then
+the group G is either trivial (if the skew vertex subset is empty), or isomor-
+phic to Z2 (if it is non-empty), by the above statement.
+Let now the multibilliard contain the skew vertex C equipped with a
+degenerate S-angular symmetry defined by a regular conic S of the pencil.
+Let λS denote the parameter value corresponding to S.
+The conformal
+involution corresponding to the vertex C fixes only λ2 and λS. Therefore, if
+the multibilliard contains no other skew vertices, then G ≃ Z2. If it contains
+another skew vertex, then G is generated by two involutions having only one
+common fixed point λ2. Their composition is a parabolic transformation
+37
+
+with the unique fixed point λ2. It has infinite order. Hence, G is infinite.
+Case of pencil of type c) is treated analogously.
+Case of pencil of type d). The involution σC corresponding to a skew
+vertex C ∈ L \ {A} is a projective angular symmetry fixing two conics: the
+singular conic L ∪ AB and the regular conic S of the pencil that is tangent
+to the line CB at B. The correspondence S �→ C is bijective. This implies
+that G is finite, if and only if the involution corresponding to any other skew
+vertex of the multibilliard fixes the same conic S, as in the above discussion.
+This holds, if and only if Conditions 3)–5) of Definition 1.21 hold.
+Case of pencil of type e) is treated analogously, with the singular conic
+now being the double line L. Proposition 2.4 is proved.
+✷
+Proposition 2.5 Every multibilliard of pencil type is rationally integrable,
+with integral of minimal degree 2|G| ∈ {2, 4, 12}, where |G| is the cardinality
+of the group G.
+Proof
+The group G is finite, by Proposition 2.4 and since the multibil-
+liard is of pencil type (hence, satisfying Conditions 3)–5) of Definition 1.21).
+Let F be a quadratic first integral of the pencil: the ratio of two quadratic
+polynomials defining two its conics. Its constant value on each conic co-
+incides with the corresponding parameter λ (after replacing F by its post-
+composition with conformal automorphism C → C). The product �
+g∈G g◦F
+is a rational first integral of the multibilliard, since it is invariant under the
+involutions corresponding to the vertices (by definition) and the dual bil-
+liard involution of each tangent line to a multibilliard conic permutes its
+intersection points with each conic of the pencil. Proposition 2.5 is proved.
+✷
+2.2
+Foliation by level curves of rational integral.
+Proof of
+Theorems 1.25 and 1.26
+Definition 2.6 Consider a rationally integrable dual billiard structure on
+a complex conic γ (which belongs to the list given by Theorem 1.11). In
+the case, when it is defined by a pencil of conics, its canonical integral is a
+quadratic rational function constant on each conic of the pencil that vanishes
+on γ. In the case, when it is exotic, its canonical integral is the one given
+by the Addendum to Theorem 1.11 (whose zero locus is γ).
+38
+
+Proposition 2.7 Every rational integral of a rationally integrable dual bil-
+liard on a conic is constant on each irreducible component of each level curve
+of its canonical integral.
+Proof
+Let γ be the conic in question, Ψ be a rational integral of the dual
+billiard, and let R be its canonical integral. We have to show that Ψ ≡ const
+along the leaves of the foliation R = const (which are, by definition, the
+irreducible components of level curves of the function R with its critical and
+indeterminacy points deleted). It suffices to prove the above statement in a
+small neighborhood of the conic γ. Fix a point P ∈ γ such that it is a regular
+point for the foliation and the dual billiard involution σP is defined there.
+(In fact, σP is well-defined whenever P is regular for the foliation. But we
+will not use this.) Let U ⊂ CP2 be a small neighborhood of the point P that
+is a flowbox for the foliation R = const and whose closure is disjoint from
+singular points of the foliation and indeterminacy points for the involution
+family σt, t ∈ γ. We equip it with biholomorphic coordinates (x, y), where
+the local leaves of the flowbox are the horizontal fibers y = const. Fix a
+point P0 /∈ γ close to P. Take a tangent line ℓ0 to γ through P0; let Q0
+denote the tangency point. Set P1 = σQ0(P0). Let ℓ1 be the tangent line to
+γ through P1 distinct from ℓ0, and let Q1 be their tangency point. Set
+P2 = σQ1(P1),
+etc. PN = σQN−1(PN−1);
+xj = x(Pj).
+Here N is the biggest number such that P1, . . . , PN, Q0, . . . , QN−1 ∈ U. We
+claim that as P0 → P, the cardinality N = N(P0) of the above sequence
+tends to infinity. This follows from the fact the involutions σQ|LQ, Q ∈ γ∩U
+are uniformly asymptotic to the central symmetries x �→ 2x(Q) − x with
+respect to the points x(Q), as x−x(Q) → 0 and Q ∈ U: they are non-trivial
+conformal involutions of the lines LQ with fixed points Q. Therefore, N is
+bigger than the product of the degrees deg Ψ deg R, whenever P0 is close
+enough to P. One has
+Ψ(P0) = · · · = Ψ(PN),
+R(P0) = · · · = R(PN),
+since both Ψ and R are integrals. This together with Bezout Theorem and
+the above inequality implies that Ψ ≡ const along each leaf of the foliation
+R = const. Proposition 2.7 is proved.
+✷
+Lemma 2.8 Let R be a rational first integral of an exotic dual billiard struc-
+ture from Theorem 1.11 given by the corresponding formula in its addendum.
+39
+
+1) For all but a finite number of values of λ ∈ C the complex level curve
+Γλ := {R = λ}
+is irreducible of degree d = deg R. In the case, when R is given by (1.7) or
+(1.8), the curve Γλ is irreducible for every λ ̸= 0, ∞.
+2) The (punctured) curve γ = {w = z2} is a multiplicity d
+2 leaf of the
+foliation R = const, which means that each small transversal cross-section
+to γ intersects each leaf close enough to γ (dependently on cross-section)
+transversely at d
+2 distinct points; or equivalently, [Γλ] → d
+2[γ] as divisors, as
+λ → 0.
+3) The curve γ is the unique nonlinear multiplicity d
+2 leaf.
+Proof
+Let us prove Statement 1), on irreducibility. Let us first consider
+Case 2a1), when R(z, w) =
+(w−z2)2N+1
+�N
+j=1(w−cjz2)2 , see (1.7).
+Claim 4.
+The germ of the curve Γλ at the point Q = [0 : 1 : 0] ∈
+CP2 (i.e., at the intersection point of the infinity line with the w-axis) is
+irreducible, whenever λ ̸= 0, ∞.
+Proof Let λ ̸= 0, ∞. In the affine chart (�z, �w) = ( z
+w, 1
+w) centered at Q one
+has
+Γλ = {( �w − �z2)2N+1 − λ �w2
+N
+�
+j=1
+( �w − cj�z2)2}.
+In the new coordinates (�z, u), u := �w − �z2, Γλ is the zero locus of the
+polynomial
+Pλ(�z, u) := u2N+1 − λ(u + �z2)2
+N
+�
+j=1
+(u + (1 − cj)�z2)2.
+(2.3)
+It suffices to show that the germ of the polynomial Pλ at the origin is
+irreducible.
+To do this, we will deal with its Newton diagram.
+Namely,
+consider the bidegrees (m, n) ∈ (R2
+≥0)x,y of all the monomials �zmun entering
+Pλ. Consider the convex hull of the union of the corresponding quadrants
+(m, n)+R2
+≥0. The union ND of its boundary edges except for the coordinate
+axes is called the Newton diagram. We claim that the Newton diagram of
+the polynomial Pλ is one edge E = [(4N + 4, 0), (0, 2N + 1)]. Indeed, the
+bidegrees of the monomials entering Pλ are (0, 2N + 1) and a collection of
+bidegrees lying in the line {2y + x = 4N + 4}, since the multiplier at λ in
+(2.3) is a (2, 1)-quasihomogeneous polynomial. But the bidegrees lying in
+the latter line lie above the edge E, except for its vertex (4N + 4, 0). This
+proves that ND = E.
+40
+
+Suppose the contrary: the germ of the polynomial Pλ is not irreducible.
+Then it is the product of two germs of analytic functions with Newton
+diagrams being edges parallel to E whose endpoints lie in the lattice Z2.
+The latter edges should be closer to the origin than E and have smaller
+lengths. But E is the edge of smallest length among all the above edges,
+since E contains no integer points in its interior: the numbers 4N + 4 and
+2N + 1 are coprime. The contradiction thus obtained proves irreducibility
+of the germ of the polynomial Pλ and hence, Claim 4.
+✷
+Recall that a germ of analytic curve is irreducible, if and only if it is a
+parametrized curve. This together with Claim 4 implies that for every fixed
+λ ̸= 0, ∞ there exists a neighborhood U = U(Q) ⊂ CP2 (depending on λ)
+such that the intersection Γλ,U := Γλ ∩(U \{Q}) is a connected submanifold
+in U \ {Q} and every line L close enough to the w-axis intersects Γλ,U
+at 2N + 1 distinct points. Therefore, all the latter points lie in the same
+irreducible component of the curve Γλ, as Γλ,U. Hence, the latter component
+has degree at least 2N + 1. But the ambient curve Γλ has degree at most
+2N + 1 = deg R. Therefore, Γλ coincides with its irreducible component in
+question, and hence, is irreducible.
+Case of integral R given by (1.8) is treated analogously with the following
+modification: in the above coordinates (�z, u) the Newton diagram of the new
+polynomial Pλ is [(2N + 3, 0), (0, N + 1)]; 2N + 3, N + 1 are again coprime.
+For the proof of Statement 1) of the lemma for the other integrals from
+the Addendum to Theorem 1.11 it suffices to prove irreducibility of level
+curve {R = λ} for an open subset of values λ. We will prove this for generic
+small λ: for an open set of values λ accumulating to zero. Indeed, it is well-
+known that if the level curve {R = λ} of a rational function is irreducible for
+an open subset of values λ, then it is irreducible for all but a finite number
+of λ. This is implied by the two following statements:
+- each indeterminacy point can be resolved by a sequence of blow-ups, so
+that the function in question becomes a well-defined C-valued holomorphic
+funciton on a new connected compact manifold, a blown-up CP2;
+- every non-constant holomorphic C-valued function on a connected com-
+pact complex manifold has finite number of critical values.
+The other canonical rational integrals have degrees 4 or 6 and the type
+R(z, w) = (w − z2)m
+Φ(z, w) , Φ is a polynomial, deg Φ = 2m, m ∈ {2, 3}.
+(2.4)
+41
+
+Proposition 2.9 Let R be as in (2.4). Let there exist a sequence of values
+λ converging to zero for which the curve Γλ := {R = λ} is not irreducible.
+Then the foliation R = const is a pencil of conics.
+Proof
+Passing to a subsequence we can and will consider that one of the
+following statements holds for all above λ:
+(i) m = 2 and Γλ is a union of two regular conics C1,λ, C2,λ;
+(ii) Γλ contains a line;
+(iii) m = 3 and Γλ is a union of two regular cubics C1,λ, C2,λ;
+(iv) m = 3 and Γλ is a union of three regular conics.
+Statement (ii) cannot hold: the contrary would imply that the limit conic
+Γ0 = γ = {w = z2} = limλ→0 Γλ contains a line, which is not true. Suppose
+(iii) holds. Then each cubic considered as a divisor of degree three converge
+to an integer multiple of the divisor [γ] of degree two: thus, to a divisor
+of even degree. This is obviously impossible. Therefore, the only possible
+cases are (i) and (iv). The a priori possible intersection points of the conics
+from (i), (iv) lie in the finite set of indeterminacy and critical points of the
+rational function R. Therefore, passing to a subsequence one can and will
+achieve that a family of conics Cλ ⊂ Γλ lies in a pencil. The function R is
+constant on them for infinite number of values of λ. Therefore, it is constant
+on each conic of the pencil, since the set of those parameters of the pencil
+for which R = const on the corresponding conics is finite (being algebraic).
+Finally, the foliation R = const is a pencil of conics.
+✷
+Let R be a degree four integral given by (1.9) or (1.10). We treat only
+case (1.9), since the integrals (1.9) and (1.9) are obtained one from the other
+(up to constant factor) by complex projective transformation fixing the conic
+γ = {w = z2}. Thus,
+R = Rb1(z, w) =
+(w − z2)2
+(w + 3z2)(z − 1)(z − w).
+Suppose the contrary: the curve Γλ := {R = λ} is not irreducible for a
+sequence of numbers λ converging to zero. Then the foliation R = const is a
+pencil of conics, by Proposition 2.9. It contains the conics γ and {w +3z2 =
+0}, which are tangent to each other at the origin and at infinity. Therefore,
+the pencil consists of conics tangent to them at these points. On the other
+hand, the line {z = 1} lies in the polar locus {R = ∞}. Hence, it should lie in
+a conic from the pencil. But this is obviously impossible, – a contradiction.
+Let now R be a degree 6 integral from the Addendum to Theorem 1.11,
+Cases 2c) or 2d). Supposing the contrary to irreducibility, we similarly get
+42
+
+that the foliation R = const is a pencil of conics. But in both Cases 2c)
+and 2d) the polar locus {R = ∞} contains an irreducible cubic, see [27,
+subsections 7.5, 7.6]. This contradiction proves Statement 1) of Lemma 2.8.
+Statement 2) of Lemma 2.8 follows from Statement 1) and the fact that
+γ is a multiplicity d
+2 zero curve of the integral R.
+Let us prove Statement 3). Suppose the contrary: there exists another
+leaf α of multiplicity d
+2 and degree µ ≥ 2. Then for every given line L that is
+transversal to α and does not pass through singularities of the foliation each
+leaf close enough to α intersects L in at least µ d
+2 ≥ d points. The number
+of intersection points cannot be greater than d. Hence, µ = 2 and α is a
+conic. Let us renormalize the integral R by postcomposition with M¨obius
+transformation ν to an integral �R = ν ◦ R so that �R|γ = 0, �R|α = ∞. Let
+Y (z, w) be a quadratic polynomial vanishing on α. Then
+�R =
+� z − w2
+Y (z, w)
+� d
+2
+,
+up to constant factor, by construction and multiplicity assumption. There-
+fore, the foliation �R = const is a pencil of conics containing γ and α, and so
+is R = const. But this is not the case, since its generic leaves are punctured
+irreducible algebraic curves Γλ of degree d ≥ 4.
+The contradiction thus
+obtained proves Lemma 2.8.
+✷
+Proof
+of Theorem 1.25. Consider a rationally integrable dual multibil-
+liard with integral Ψ ̸≡ const. Then the dual billiard on each its curve γj
+is rationally integrable with integral Ψ. Hence, each γj is a conic equipped
+with either pencil type, or exotic dual billiard structure, by Theorem 1.11,
+and Ψ|γj ≡ const, by [27, proposition 1.35] (or by Proposition 2.7).
+Case 1). Let some two conics γ1, γ2 be the same conic γ equipped with
+two distinct dual billiard structures, given by projective involution families
+σP,j : LP → LP, j = 1, 2. Here P lies outside a finite set: the union of the
+indeterminacy loci of families σP,j, which are finite by Theorem 1.11. The
+product g := σP,1 ◦ σP,2 is a parabolic transformation LP → LP , having
+unique fixed point P. The integral Ψ is g-invariant: Ψ ◦ g = Ψ along each
+line LP. But each non-fixed point of a parabolic transformation has infinite
+orbit. Therefore, Ψ ≡ const along each line tangent to γ. But we know
+that Ψ is constant along the curve γ, as noted above. Therefore, Ψ ≡ const,
+by the two latter statements and since the union of lines tangent to γ at
+points lying in an open subset in γ contains an open subset in CP2. The
+contradiction thus obtained proves that Case 1) is impossible.
+43
+
+Case 2): there are at least two geometrically distinct conics, say, γ1, γ2.
+For every j = 1, 2 let Rj denote the canonical integral of the corresponding
+dual billiard structure. We have to prove the two following statements:
+1) the dual billiard structure on each γj is defined by a pencil of conics,
+that is, the degree dj := deg Rj is equal to 2;
+2) the latter pencil is the same for j = 1, 2, and it contains both γj.
+Let F denote the foliation Ψ = const. For every j for all but a finite
+number of values λ ∈ C the complex level curve {Rj = λ} is irreducible
+of degree dj, by Lemma 2.8, and Ψ ≡ const along it (Proposition 2.7).
+Hence, each foliation Rj = const coincides with F. This together with the
+previous statement implies that all the degrees dj are equal, set d = dj, and
+both (punctured) conics γ1, γ2 are leaves of the same multiplicity d
+2 for the
+foliation F. Therefore, the foliation F is a pencil of conics containing γ1 and
+γ2, by Statement 3) of Lemma 2.8. Hence, all the conics of the multibilliard
+lie in this pencil, and d = 2 (since d is the degree of irreducible level curve of
+the function R1). Thus, each Rj is a ratio of two quadratic polynomials, and
+its level curves are conics from the pencil. Hence, the dual billiard structure
+on γj is given by the same pencil. Theorem 1.25 is proved.
+✷
+Proof of Theorem 1.27. A rational first integral of a pencil type multi-
+billiard is constant on each conic of the pencil (Proposition 2.7). Moreover,
+it is constant on every union of those conics whose parameter values λ lie in
+the same G-orbit. Here G is the group from Proposition 2.4. The cardinality
+of a generic G-orbit is equal to the cardinality |G| of the group G, since a
+generic point in C has trivial stabilizer in G. Thus, the minimal degree of
+the integral (which is achieved, by Proposition 2.5) is 2|G| ∈ {2, 4, 12}. This
+together with Proposition 2.4 implies the statement of Theorem 1.27.
+✷
+2.3
+Dual billiard structures at vertices.
+Birationality and
+types of involutions
+Proposition 2.10 Let A be a point in RP2 (CP2) equipped with real (com-
+plex) dual billiard structure given by involution family σA,ℓ that has a real
+(complex) rational first integral Ψ ̸≡ const: Ψ ◦ σA,ℓ = Ψ on each line ℓ
+through A on which the involution is defined. Let the foliation Ψ = const
+be not the family of lines through A.
+Then σA,ℓ coincide (up to correc-
+tion at a finite number of lines ℓ through A) with a birational involution
+σA : CP2 → CP2 fixing each line through A and holomorphic and bijec-
+tive on the complement to a finite number of lines through A. The rational
+integral Ψ and the corresponding foliation Ψ = const are σA-invariant.
+44
+
+Proof
+Let σA,ℓ : ℓ → ℓ be the corresponding projective involution family
+acting on lines ℓ through A. They are defined on lines ℓ through A from an
+open subset U ∈ CP1 in complex case (U ⊂ RP1 in real case).
+Fix a non-linear complex level curve X := {Ψ = λ}. Fix an ℓ0 ∈ U
+(consider it as a complex line) such that the points of intersection X ∩ ℓ0
+distinct from A are regular points of the curve X, the intersections are
+transversal, and the multiplicity of the intersection X ∩ ℓ0 at A is minimal.
+There exists a simply connected neighborhood V = V (ℓ0) ⊂ CP1 such that
+for every ℓ ∈ V the number of geometrically distinct points of the set Xℓ :=
+(X ∩ℓ)\{A} ⊂ CP2 is the same (let us denote their number by d), and they
+depend holomorphically on ℓ (Implicit Function Theorem). We numerate
+these holomorphic intersection point families by indices 1, . . . , d. For every
+ℓ ∈ V the involution σA,ℓ makes a (ℓ-dependent) permutation of the latter
+intersection points, which is identified with a permutation of indices 1, . . . , d:
+an element in Sd. There exists a permutation α ∈ Sd realized by σA,ℓ for a
+continuum cardinality subset Y ⊂ V of lines. Let us fix it.
+Claim 6. There exists a projective involution family �σA,ℓ : ℓ → ℓ de-
+pending holomorphically on the parameter ℓ ∈ V that makes the permuta-
+tion α on Xℓ for every ℓ ∈ V . The rational function Ψ|ℓ is �σA,ℓ-invariant:
+Ψ ◦ �σA,ℓ = Ψ on every ℓ ∈ V .
+Proof Consider first the case, when Xℓ is just one point. For every ℓ ∈ V
+set �σA,ℓ : ℓ → ℓ to be the nontrivial conformal involution fixing the points Xℓ
+and A. It depends holomorphically on ℓ ∈ V . It preserves Ψ|ℓ: Ψ ◦ �σA,ℓ = Ψ
+on every ℓ ∈ Y , and the latter relation holds for every ℓ ∈ V , since Y is of
+cardinality continuum and by uniqueness of analytic extension.
+Let now Xℓ consists of at least two points. Let us define �σA,ℓ to be the
+unique projective transformation ℓ → ℓ fixing A and sending the points in
+Xℓ with indices 1, 2 to the points with indices α(1), α(2) respectively. For
+every ℓ ∈ V this is an involution preserving Ψ|ℓ, since this is true for every
+ℓ ∈ Y and by uniqueness of analytic extension. The claim is proved.
+✷
+Claim 7. The involution family �σA,ℓ extends holomorphically to a finitely
+punctured space CP1 of lines through A. It coincides with σA,ℓ on all the
+lines ℓ ∈ U except maybe for a finite number of them, on which Ψ = const.
+Proof We can extend the involution family �σA,ℓ analytically in the param-
+eter ℓ along each path avoiding a finite number of lines ℓ for which either
+some of the points in Xℓ are not transversal intersections, or the index of
+intersection ℓ ∩ X at A is not the minimal possible. This follows from the
+previous claim and its proof. Extension along a closed path does not change
+holomorphic branch. Indeed, otherwise there would exist its another holo-
+45
+
+morphic branch over a domain W ⊂ V : an involution family HA,ℓ : ℓ → ℓ
+depending holomorphically on ℓ ∈ W, HA,ℓ ̸= �σA,ℓ, which preserves the in-
+tegral Ψ. The product Fℓ := �σA,ℓ ◦ HA,ℓ : ℓ → ℓ is a parabolic projective
+transformation, with A being its unique fixed point, for every ℓ ∈ W. Its or-
+bits are infinite, and Ψ should be constant along each of them. This implies
+that Ψ = const along each line ℓ ∈ W. Hence, the foliation Ψ = const is the
+family of lines though A, which is forbidden by our assumption. The con-
+tradiction thus obtained proves unique definedness of analytic extensions of
+the involution family �σA,ℓ along paths and the first statement of the claim.
+Its second statement follows from the fact that for those ℓ ∈ U for which
+�σA,ℓ ̸= σA,ℓ, one has Ψ ≡ const along ℓ: see the above argument, now with
+the parabolic transformation �σA,ℓ ◦ σA,ℓ. The claim is proved.
+✷
+Without loss of generality we consider that σA,ℓ = �σA,ℓ, correcting σA,ℓ
+at a finite number of lines. The latter equality defines analytic extension of
+the involution family σA,ℓ to all but a finite number of lines ℓ through A.
+The invariance condition Ψ ◦ σ|ℓ = Ψ|ℓ is a system of algebraic equations on
+the pairs (ℓ, σ), where ℓ is a projective line through A and σ : ℓ → ℓ is a
+nontrivial projective involution fixing A. For every line ℓ through A (except
+for a finite set of lines, including those along which Ψ ≡ const) its solu-
+tion space is finite, and its cardinality is bounded from above. This implies
+that the family σA,ℓ is a connected open subset in an algebraic subset of a
+smooth algebraic manifold and all σA,ℓ paste together to a global birational
+automorphism CP2 → CP2 acting as a holomorphic involution on the com-
+plement to a finite number of lines through A. It preserves Ψ, and hence,
+the foliation Ψ = const, by construction. Proposition 2.10 is proved.
+✷
+Lemma 2.11 The dual billiard structure at each vertex A of any rationally
+integrable dual multibilliard is either global, or quasi-global. In the case of
+quasi-global structure the foliation by level curves of rational integral is a
+pencil of conics, and the conic of fixed points of the corresponding involution
+σA lies in the same pencil.
+Proof Consider first the case, when the foliation by level curves of integral
+is a pencil of conics. It is invariant under the birational involution σA from
+Proposition 2.10. Therefore, σA acts on its parameter space C as a conformal
+involution with at least two fixed points. Thus, σA fixes at least two distinct
+conics of the pencil. Fix one of them that is not a pair of lines intersecting
+at A, let us denote it by Γ. It is possible, since a pencil cannot contain
+two singular conics, each of them being a pair of lines, so that all the four
+lines forming them pass through the same point A. Indeed, otherwise A
+46
+
+would be the unique base point of the pencil, and the pencil would have
+type e). Thus, its only singular conic would be the double line tangent to
+all its regular conics at A, – a contradiction.
+Subcase 1.1): Γ is disjoint from A. Then σA is a projective involution,
+by Example 1.17, part 2).
+Subcase 1.2): Γ passes through A. If Γ is a union of lines through A,
+some of its lines, let us denote it by L, does not pass through A. Then σA
+is the projective involution that fixes each point of the line L, by definition.
+Similarly, if Γ is a regular conic, then σA fixes each its point. Hence, it
+defines a quasi-global dual billiard structure.
+Consider now the case, when the foliation by level curves of the integral is
+not a pencil. Then the multibilliard contains just one conic, let us denote it
+by γ, equipped with an exotic dual billiard structure. Let R be its canonical
+integral, d = deg R. The foliation Ψ = const coincides with R = const,
+by Proposition 2.7. The (punctured) curve γ being a leaf of multiplicity
+d
+2, its (punctured) image γ′ := σA(γ) is also a multiplicity
+d
+2 leaf, since
+Ψ ◦ σA = Ψ and multiplicity is invariant under birational automorphism of
+foliation. Hence, γ′ coincides with γ, if it is not a line.
+Subcase 2.1): A /∈ γ. Then a generic line through A intersects γ at two
+points distinct from A. Hence, the same holds for the image γ′. Thus, it is
+not a line, and σA(γ) = γ. Therefore, σA is a global projective transforma-
+tion, the γ-angular symmetry.
+Subcase 2.2): A ∈ γ and γ′ ̸= γ. Let us show that this case is impossible.
+Indeed, then γ′ is a line, see the above discussion, and R|γ′ ̸≡ 0. Therefore,
+the points of intersection γ′ ∩ γ are indeterminacy points for the function
+R.
+In Case 2a) the only indeterminacy points are the origin O and the
+infinity. Therefore, γ′ is some of the following lines: the Ow-axis (which
+passes through both latter points), the Oz-axis or the infinity line (which
+are tangent to γ at O and at infinity respectively). But each of the latter
+lines satisfies at least one of the following statements:
+- either R is non-constant there;
+- or R has a pole of multiplicity less than d there.
+See the two first formulas for the integrals in the addendum to Theorem
+1.11. Therefore, the (punctured) line γ′ cannot be a multiplicity d leaf of
+the foliation R = const. This contradiction proves that the case under con-
+sideration is impossible. The other Cases 2c), 2d) are treated analogously.
+Subcase 2.3): A ∈ γ = σA(γ). Therefore, for every line ℓ through A
+distinct from the line L tangent to γ at A the involution σA fixes the point
+of intersection ℓ ∩ γ distinct from A. Thus, it is the degenerate γ-angular
+symmetry. In the chart (x, y) where γ = {y = x2}, A is the point of the
+47
+
+parabola γ at infinity and the line L tangent to γ at A is the infinity line,
+σA acts as
+σA : (x, y) �→ (x, 2x2 − y).
+In the coordinates (x, y) one has
+R = R(x, y) = (y − x2)m
+F(x, y) , F(x, y) is a polynomial, deg F ≤ 2m.
+(2.5)
+To treate the case in question we use the following proposition.
+Claim 8. The point A is an indeterminacy point of the function R.
+Proof
+Let us first consider the case, when L lies in a level curve Sλ :=
+{R = λ}. Then λ ̸= 0, since the zero locus Sλ coincides with γ. Thus, A lies
+in two distinct level curves, and hence, is an indeterminacy point. Let us
+now suppose that R|L ̸≡ const. As a line ℓ through A tends to the tangent
+line L to γ, its only intersection point B(ℓ) with γ distinct from A tends to
+A. Therefore, the involution (σA)|ℓ, which fixes the confluenting points A
+and B(ℓ), tends to the constant map L → A uniformly on compact subsets
+in L \ {A}. Suppose the contrary: A is not an indeterminacy point. Fix a
+λ ̸= 0. The image Sλ′ := σA(Sλ) is a level curve of the function R, λ′ ̸= 0,
+hence A /∈ Sλ, Sλ′. Therefore, the points of intersections ℓ∩Sλ, ℓ∩Sλ′ do not
+accumulate to A, as ℓ → L. But the points of the subset σA(ℓ ∩ Sλ) ⊂ Sλ′
+converge to A, and hence, A ∈ Sλ′. This contradiction proves the claim. ✷
+Claim 9. Let F be the same, as in (2.5). Then
+F ◦ σA(x, y) = (−1)m+1F(x, y) + a(x − y2)m,
+a = const ∈ C.
+(2.6)
+Proof The involution σA is birational, and it permutes leaves of the folia-
+tion R = const. All but a finite number of leaves are punctured level curves
+of the function R, since all but a finite number of level curves are irreducible
+(Lemma 2.8). Therefore σA permutes level curves of the function R and
+acts on its values by conformal involution C → C. The latter involution
+preserves zero, which corresponds to the σA-invariant curve γ. Hence, its
+action on values of the function 1
+R is either identity, or an affine involution
+µ �→ −µ + b, b = const. This together with the fact that σA changes sign of
+the polynomial y − x2 implies the statement of the claim.
+✷
+We consider the rational integrals R from the addendum to Theorem
+1.11, all their indeterminacy points A and the corresponding involutions σA
+fixing points of γ.
+For every pair (R, A), assuming σA-invariance of the
+foliation R = const, we will arrive to contradiction.
+48
+
+Everywhere below by F we denote the denominator of the rational func-
+tion R (written in the coordinates (z, w) or (x, y) under consideration).
+2a1) The integral R given by (1.7). Let A be the infinity point of the
+parabola γ = {w = z2}. Then σA(z, w) = (z, 2z2 − w). The functions R
+and R ◦ σA have the same foliation by level curves. Therefore, their ratio,
+which is equal to ± F ◦σA
+F
+, is constant along each leaf. But the latter ratio
+is constant on the w-axis, since σA preserves the degree of higher purely
+w-term wk, and z is σA-invariant. On the other hand, the w-axis is not a
+leaf, since R(0, w) = w2. Thus, the above ratio is globally constant, and
+F ◦ σA = F
+up to constant factor.
+(2.7)
+But F(z, w) = �N
+j=1(w −cjz2)2, F ◦σA(z, w) = �(w −(2−cj)z2), the coef-
+ficients cj in F are negative, while the latter coefficients 2 − cj are positive.
+Therefore, equality (2.7) cannot hold, – a contradiction.
+Let A = O = (0, 0). Consider the chart (x, y) = ( z
+w, 1
+w), in which A = ∞,
+R(x, y) = (y − x2)2N+1
+F(x, y)
+, F(x, y) = y2
+N
+�
+j=1
+(y − cjx2)2, cj < 0.
+Equality (2.7) is proved analogously.
+The coefficients cj at x2 in F are
+negative. But F ◦σA(x, y) = (y −2z2) �(y −(2−cj)x2), and the coefficients
+at x2 are positive there. Hence, equality (2.7) cannot hold, – a contradiction.
+2a2) Case of integral (1.8). Treated analogously to the above case.
+Cases 2b1) and 2b2) have the same complexification; thus we treat only
+Case 2b2), when the integral R is given by (1.10). There are three indeter-
+minacy points: the infinity and (±i, −1). Let A ∈ γ be the infinite point,
+hence σA(z, w) = (z, 2z2 − w). The denominator F is the product of the
+σA-invariant quadratic polynomial z2 +1 and another quadratic polynomial
+Φ(z, w) = z2 + w2 + w + 1. Therefore, the polynomial F ◦ σA is divisible by
+z2+1, and hence is equal to ±F, by (2.6). This implies that Φ◦σA = ±Φ is a
+quadratic polynomial, while it has obviously degree four, – a contradiction.
+Let now A = (±i, −1). Let B denote the infinite point of the parabola γ.
+Let us choose an affine chart (x, y) centered at B so that the line tangent
+to γ at A is the infinity line and the line {z = z(A)} is the Oy-axis. In the
+new coordinates one has R(x, y) = (y−x2)2
+xΦ(x,y) , where Φ is a cubic polynomial
+coprime with y − x2. Analogously we get that Φ ◦ σA = ±Φ. If Φ contains a
+monomial divisible by y2, then deg(Φ ◦σA) ≥ 4, and we get a contradiction.
+Otherwise Φ(x, y) = c(y + Ψ(x)), where Ψ is a polynomial; Φ is coprime
+with y − x2, hence, Ψ(x) ̸= −x2. But then Φ ◦ σA ̸= ±Φ, – a contradiction.
+49
+
+Cases 2c1) and 2c2). Note that the integral R from 2c1) is invariant
+under the order 3 group generated by the symmetry (z, w) �→ (εz, ε2w),
+where ε is a cubic root of unity. This group acts transitively on the set of
+three indeterminacy points. Thus, it suffices to treat the case of just one
+indeterminacy point A.
+Again it suffices to treat only Case 2c2), which
+has the same complexification, as Case 2c1), with A being the infinite
+point of the parabola γ.
+To do this, let us first recall that the (2, 1)-
+quasihomogeneous degree of a monomial zmwn is the number m+2n. A poly-
+nomial is (2, 1)-quasihomogeneous, if all its monomials have the same (2, 1)-
+quasihomogeneous degree.
+Each polynomial in two variables is uniquely
+presented as a finite sum of (2, 1)-quasihomogeneous polynomials of distinct
+quasihomogeneous degrees. The indeterminacy points of the integral R given
+by (1.12) are O = (0, 0), (1, 1) and the infinity point of the parabola γ.
+Taking composition with σA(z, w) = (z, 2z2 − w) preserves the quasiho-
+mogeneous degrees. The lower quasihomogeneous part of the denominator
+F in (1.12) is the polynomial (w + 8z2)2 of quasihomogeneous degree 4.
+The numerator (w − z2)3 is quasihomogeneous of degree 6. Therefore, the
+lower quasihomogeneous part of the polynomial F ◦ σA is the polynomial
+(w − 10z2)2 ̸= ±(w + 8z2)2. The two latter statement together imply that
+formula (2.6) cannot hold, – a contradiction.
+Case 2d).
+Then we have three indeterminacy points: the origin, the
+infinity point and the point (1, 1). The case, when A is the infinity point
+of the parabola γ, is treated analogously to Case 2b2). Let us consider the
+case, when A is the origin. In the coordinates (x, y) = ( z
+w, 1
+w) the function
+R takes the form R(x, y) = (y−x2)3
+F (x,y) ,
+F(x, y) = (y + 8x2)(x − y)(y2 + 8x2y + 4y + 5x2 − 14xy − 4x3).
+The lower (2, 1)-quasihomogeneous part of the polynomial F is V (x, y) :=
+x(y+8x2)(4y+5x2). It has quasihomogeneous degree 5, while the numerator
+in R has quasihomogeneous degree 6. This together with (2.6) implies that
+σA multiplies the lower quasihomogeneous part by ±1. But V ◦ σA(x, y) =
+x(y − 10x2)(4y − 13x2) ̸= ±V (x, y), – a contradiction.
+Let us now consider the case, when A = (1, 1). Take the affine coordi-
+nates (x, y) centered at the infinite point of the parabola γ such that the
+complement to the affine chart (x, y) is the tangent line L to γ at A and the
+y-axis is the zero line {z = 1} of the denominator (which passes through A).
+The rational function R takes the form
+R(x, y) = (y − x2)3
+F(x, y) , F(x, y) = xG2(x, y)G3(x, y), Gj(0, y) ̸≡ 0, deg Gj = j,
+50
+
+G2(x, y) = w + z2,
+G3(x, y) = w + 8z2 + 4w2 + 5wz2 − 14zw − 4z3.
+In the chart (x, y) one has σA(x, y) = (x, 2x2 − y). The factor x in F is
+σA-invariant. Therefore, F ◦σA = ±F, by (2.6), and σA leaves invariant the
+zero locus Z = {G2G3 = 0}. The latter zero locus is a union of the conic
+{G2 = 0} disjoint from A and a cubic {G3 = 0}. The latter intersects a
+generic line ℓ through A at a unique point distinct from A, since it has a cusp
+at A, see [27, subsection 7.6, claim 9]. Thus, for a generic line ℓ through
+A, the intersection Z ∩ (ℓ \ {A}) consists of three distinct points disjoint
+from γ and it is invariant under the involution σA. This implies that one of
+them is fixed; let us denote it by B. Hence, for a generic ℓ the projective
+involution (σA)|ℓ has three distinct fixed points: A, B and the unique point
+of the intersection γ ∩ (ℓ \ {A}). Therefore, it is identity, – a contradiction.
+Thus, we have checked that a rationally integrable dual multibilliard with
+exotic foliation R = const cannot have a vertex A whose involution σA is
+not a global projective transformation. This proves Lemma 2.11.
+✷
+2.4
+Pencil case. Proof of Theorem 1.26
+We already know that if a dual multibilliard is of pencil type, then it is
+rationally integrable (Proposition 2.5). Consider now an arbitrary rationally
+integrable dual multibilliard where all the curves are conics equipped with a
+dual billiard structure defined by the same pencil (containing each conic of
+the multibilliard). Let us show that the multibilliard is of pencil type, that
+is, its vertices (if any) are from the list given by Definition 1.21 and their
+collection satisfies the conditions of Definition 1.23.
+Proposition 2.12 Let a rationally integrable multibilliard consist of conics
+lying in a pencil, equipped with dual billiard structures defined by the same
+pencil, and some vertices. Then each its vertex is admissible for the pencil.
+Proof
+Let Ψ be a rational integral of the multibilliard, Ψ ̸≡ const. The
+foliation Ψ = const coincides with the pencil under consideration, by Propo-
+sition 2.7. Let K be a vertex of the multibilliard. Then its dual billiard
+structure is given by an involution σK : CP2 → CP2 preserving the pencil
+that is either a global projective involution, or an involution fixing points of
+a regular conic α passing through K and lying in the pencil (Lemma 2.11).
+Let us first treat the second case, when σK fixes points of a regular conic
+α from the pencil. Let L denote the tangent line to α at K. Consider the
+affine chart (z, w) on the complement CP2 \ L in which α = {w = z2}; the
+point K is the intersection of the infinity line with the w-axis. One has
+51
+
+σK(z, w) = (z, 2z2 − w). Each regular conic β of the pencil is given by a
+quadratic equation {w + Φ2(z) = 0}, where Φ2 is a quadratic polynomial.
+Indeed, the quadratic equation on β contain neither w2, nor wz terms, since
+σK transforms them to polynomials of degrees four and three respectively,
+while it should send β to a conic of the pencil. This implies that all the
+regular conics of the pencil are tangent to each other at K, and we get that
+K is an admissible vertex from Definition 1.21.
+Let us now treat the first case, when σK is a global projective involution.
+Claim 10. Let K be not a base point of the pencil. Then the conic S
+through K is a pair of lines.
+Proof The conic S is fixed by σK, as is K. If it were regular, then it would
+intersect a generic line ℓ through K at a point distinct from K, and the
+involution σK would fix each point in S. Thus, it would not be a projective
+transformation, – a contradiction.
+✷
+Subcase 1): K is not a base point of the pencil, and the conic S through
+A is a pair of distinct lines L1, L2, both passing through K. In the case,
+when the pencil consists of conics passing through four distinct base points,
+there are two base points Vj1, Vj2 in each line Lj. Therefore, K is a standard
+vertex Ms from Definition 1.21, and the projective involution σK permutes
+Vj1, Vj2 for every j = 1, 2. Hence, σK coincides with the involution σMs from
+Definition 1.21, Case a). In the case, when the pencil has three distinct base
+points, two of them lie in one of the lines, say L1, the third one (denoted
+by C) lies in L2, and L2 is tangent at C to all the regular conics of the
+pencil. We get analogously that K = M and σK = σM, see Definition 1.21,
+Case b). Case c), when the conics of the pencil are tangent to each other
+at two base points, is treated analogously. Case e) is impossible, since then
+the pencil contains no distinct line pair. In Case d) it contains the unique
+pair of distinct lines. They intersect at a base point: the common tangenty
+point of conics. Hence, this case is impossible.
+Subcase 2): K is not a base point, the conic S consists of a line L1
+through K and a line L2 that does not pass through K. Then σK fixes K
+and each point of the line L2, and hence, the intersection point M = L1∩L2.
+In Case a) there are two base points in each line Lj. Those lying in L1 should
+be permuted by σK. Therefore, the points K, M and the two base points
+in L1 form a harmonic quadruple. Hence K is one of the skew vertices from
+Definition 1.21, Case a). In Case b) there are three distinct base points.
+The case, when two of them lie in L1, is treated as above. In the case, when
+L1 contains only one base point C, it should be tangent there to the regular
+conics of the pencil, and C should be fixed by σK, as is M.
+Therefore,
+52
+
+C = M, since the projective involution (σK)|L1 cannot have three distinct
+fixed points C, M, K. Finally, all the three base points are contained in the
+line L2, which is obviously impossible. Similarly in Case c) we get that K
+is a point of the line through the two base points A, C. Indeed, the other
+a priori possible subcase is when K lies in a line tangent to the conics at a
+base point (say, A). Then σK would fix three points of the latter line: K,
+A and the intersection point of the tangent lines to conics at A and C. The
+contradiction thus obtained shows that this subcase is impossible.
+In Case d) (conics tangent at a point A with triple contact and passing
+through another point B) the point K should lie in the line L tangent to
+the conics at A. (This realizes the vertex C from Subcase d2) in Definition
+1.21.) Indeed, otherwise K would lie in AB \{A, B}, and the involution σK
+would fix three distinct points A, B, K ∈ AB, – a contradiction. Case e) is
+impossible, since then the pencil contains no distinct line pair.
+Subcase 3): K is not a base point, and the conic S is a double line L
+through K. Then
+- either the pencil is of type c) and L passes through the two base points;
+the involution σK should permute them and the tangent lines at them to
+the conics of the pencil;
+- or the pencil is of type e) and L is tangent to its regular conics at the
+unique base point; the involution should fix K and those points where the
+tangent lines to conics pass through K; the latter points form a line through
+the base point.
+Both cases are realized by vertices from Definition 1.21.
+Subcase 4): K is a base point. Then each line L passing through K and
+another base point A contains no more base points. Thus, the involution
+(σK)|L should fix both K and A. Finally, σK fixes each base point. There-
+fore, the base points different from K lie on the fixed point line Λ of the
+projective involution σK. Let the pencil be of type a). Then Λ contains
+three base points, – a contradiction. Let now the pencil have type b). Then
+Λ contains two base points. If the conics of the pencil are tangent to each
+other at K, the vertex K has type b3) from Definition 1.21. Consider the
+opposite case, when the common tangency point C of the conics lies in Λ.
+Let H denote their tangent line at C. Then H contains no other base point,
+thus, H ̸= KC, Λ. Hence, the restriction of the involution σK to each line
+ℓ ̸= KC through K fixes three distinct points: K, ℓ ∩ H, ℓ ∩ Λ. The contra-
+diction thus obtained proves that the case under consideration is impossible.
+Case of pencil of type c) is treated analogously: K is a vertex of type c2).
+Consider the case of pencil of type d).
+Let K be the base point of
+transversal intersection of conics. Then the involution σK, which preserves
+53
+
+the pencil, should fix K and each point of the common tangent line L to the
+conics at the other base point A. In an affine chart (z, w) centered at A, in
+which K is the intersection point of the infinity line with the Ow-axis and L
+is the z-axis, the involution σK is the symmetry with respect to the z-axis.
+The latter symmetry changes the 2-jet of a regular conic of the pencil at
+A. But its conics should have the same 2nd jet at A. Hence, σK cannot
+preserve the pencil, – a contradiction. Therefore, K is the base point where
+the conics have a common tangent line L, and the other base point B (of
+transversal intersection) lies in Λ. Let us choose an affine chart (z, w) so
+that L is the infinity line, K is its intersection with the Ow-axis, and Λ is
+the z-axis. Then σK is again the symmetry as above, and it changes 2nd jets
+of conics (which are parabolas) at their infinity point K. Therefore, Case
+d) is impossible. Case e) is treated analogously to the latter discussion.
+Finally, all the possible vertices listed above belong to the list of admis-
+sible vertices from Definition 1.21. Proposition 2.12 is proved.
+✷
+The rational integral of the multibilliard is constant on each union of
+conics of the pencil whose parameter values form a G-orbit, and the double
+cardinality 2|G| is no greater than the degree of the integral: see the proof of
+Theorem 1.27 at the end of Subsection 2.2. Therefore, G is finite. Hence, the
+multibilliard is of pencil type, by Proposition 2.4. Theorem 1.26 is proved.
+2.5
+Exotic multibilliards. Proof of Theorem 1.31
+Proposition 2.13 Each multibilliard from Theorem 1.31 is rationally inte-
+grable. The corresponding rational function R from the addendum to Theo-
+rem 1.11 is its integral of minimal degree, except for the subcase in Case (i),
+when ρ = 2 −
+1
+N+1; in this subcase R2 is a first integral of minimal degree.
+Proof In the case, when there are no vertices, rational integrability follows
+by Theorem 1.11. Let us consider that the multibilliard contains at least
+one admissible vertex.
+Case (i).
+The function R (respectively, R2) is a rational integral of
+minimal degree, since R is even (odd) in z.
+Case (ii) is treated analogously to Case (i). It suffices to treat the sub-
+case 2b2), since the multibilliards of types 2b1), 2b2) containing the unique
+admissible vertex are projectively isomorphic (Proposition 1.32). In subcase
+2b2) the function R is even in z, and hence, invariant under the correspond-
+ing admissible vertex involution (z, w) �→ (−z, w).
+Case (iii). Let us show that the γ-angular symmetry σQ centered at each
+admissible vertex Q preserves the integral R. Cases 2c1) and 2c2) being
+54
+
+complex-projectively isomorphic and invariant under order three symmetry
+cyclically permuting the three singular points, we treat Case 2c2), with
+Q = (0, −1) being the intersection point of the w-axis (i.e., the line through
+two indeterminacy points: O and ∞) and the line tangent to γ at the
+indeterminacy point (1, 1). We use the following two claims and proposition.
+Claim 11. The polar locus
+S := {P(z, w) = 8z3 − 8z2w − 8z2 − w2 − w + 10zw = 0}
+passes through Q = (0, −1) and has an inflection point there.
+Proof
+One has Q = (0, −1) ∈ S (straighforward calculation). To show
+that Q is an inflection point, it suffices to show that ∇P(Q) ̸= 0 and the
+Hessian form of the function P evaluated on the skew gradient (∂P
+∂w, − ∂P
+∂z )
+(which is a function of Q denoted by H(P)(Q)) vanishes at Q. Indeed,
+∂P
+∂z (Q) = −10, ∂P
+∂w (Q) = 2 − 1 = 1,
+∂2P
+∂z2 (Q) = 0, ∂2P
+∂w2 (Q) = −2,
+∂2P
+∂z∂w (Q) = 10,
+H(P) = ∂2P
+∂z2
+�∂P
+∂w
+�2
++ ∂2P
+∂w2
+�∂P
+∂z
+�2
+− 2 ∂2P
+∂z∂w
+∂P
+∂w
+∂P
+∂z = 0 − 200 + 200 = 0
+at the point Q. The claim is proved.
+✷
+Proposition 2.14 Let a cubic S ⊂ CP2 have an inflection point Q. Then
+there exists a projective infolution σQ,S : CP2 → CP2 that fixes each line
+through Q and permutes its intersection points with S distinct from Q (for
+ℓ being not the tangent line to S at Q).
+Proof The above involution is well-defined on each line ℓ through Q distinct
+from the tangent line Λ to S at Q as a projective involution σQ,S,ℓ : ℓ → ℓ
+depending holomorphically on ℓ ̸= Λ. It suffices to show that the involution
+family thus obtained extends holomorphically to ℓ = Λ. This will imply
+that σQ,S is a well-defined global holomorphic involution CP2 → CP2, and
+hence, a projective transformation. Indeed, let us take an affine chart (x, y)
+centered at Q and adapted to S, so that Λ is the Ox-axis and the germ of
+the cubic S is the graph of a germ of holomorphic function:
+y = f(x), f(x) = ax3 + (b + o(1))x4;
+S = {y = f(x)}.
+55
+
+A line ℓδ := {y = δx} with small slope δ intersects S at two points distinct
+from Q = (0, 0) with x-coordinates x0, x1 satisfying the equation
+ax2
+0 + (b + o(1))x3
+0 = ax2
+1 + (b + o(1))x3
+1 = δ.
+(2.8)
+Taking square root and expressing x1 as an implicit function −x0(1 + o(1))
+of x0 yields
+x1 = −x0 + (c + o(1))x2
+0,
+c = − b
+a.
+Writing the projective involution σQ,S : ℓδ → ℓδ fixing the origin and per-
+muting the above intersection points as a fractional-linear transformation in
+the coordinate x, we get a transformation
+x �→ −
+x
+1 + ν(δ)x.
+(2.9)
+Substituting x0 to (2.9) yields
+−
+x0
+1 + ν(δ)x0
+= −x0 + (c + o(1))x2
+0;
+(1 + ν(δ)x0)(1 − (c + o(1))x0) = 1, x0 = x0(δ) → 0,
+as δ → 0. Therefore, ν(δ) = c + o(1) → c, and the one-parametric holomor-
+phic family of transformation (2.9) extends holomorphically to δ = 0 as the
+projective transfomation x �→ −
+x
+1+cx (continuity and Erasing Singularity
+Theorem). The proposition is proved.
+✷
+Claim 12. The polar cubic S is σQ-invariant.
+Proof
+The projective involution σQ,S from Proposition 2.14 fixes S and
+each line through Q. It preserves the conic γ, which is the unique regular
+conic tangent to S at the three indeterminacy points of the integral R.
+Indeed, if there were two such distinct conics, then their total intersection
+index at the three latter points would be no less than 6, – a contradiction
+to B´ezout Theorem. Therefore, σQ,S is the γ-angular symmetry, and hence,
+it coincides with σQ. This implies that σQ(S) = S. The claim is proved. ✷
+Claim 12 together with σQ-invariance of the zero locus γ of the rational
+function R implies that R ◦ σQ = ±R. The restriction of the integral R to
+the line ℓ through Q tangent to the conic γ at the point (1, 1) is holomorphic
+and nonconstant at (1, 1), since its numerator being restricted to ℓ has order
+6 zero at (1, 1), and so does its denominator. The point (1, 1) is fixed by
+σQ. Therefore, the above equality holds with sign ”+” near the point (1, 1),
+and hence, everywhere. Case iv) is treated. The proposition is proved.
+✷
+56
+
+Recall that in our case of a multibilliard containing a conic with an
+exotic dual billiard structure, the involution associated to each vertex is
+a projective angular symmetry (Lemma 2.11). As it is shown below, this
+together with the next proposition implies Theorem 1.31.
+Proposition 2.15 Consider an exotic rationally integrable dual billiard on
+a conic γ.
+Let R be its canonical integral.
+Let A ∈ CP2, and let σA :
+CP2 → CP2 be a projective angular symmetry centered at A that preserves
+the foliation R = const. Then σA is the γ-angular symmetry centered at A,
+and A belongs to the list from Theorem 1.31.
+Proof Set d = deg R ∈ 2N. The (punctured) conic γ is the only conical leaf
+of multiplicity d
+2 of the foliation R = const, see the proof of Lemma 2.11.
+Therefore, σA(γ) = γ. Hence, A /∈ γ, since σA is a projective involution.
+Thus, its restriction to each line ℓ through A permutes its intersection points
+with γ. Hence, σA is the γ-angular symmetry centered at A. It preserves
+the set of indeterminacy points of the integral R, whose number is either
+two, or three, since it preserves the foliation R = const. Hence, it preserves
+the union of lines tangent to γ at the indeterminacy points.
+Consider first Case 2a), where there are two intederminacy points: in
+the affine chart (z, w) these are the origin O and the infinity point B of
+the parabola γ. Let LO, LB denote the lines tangent to γ at O and at B
+respectively, and let Q be their intersection point. Let us show that A = Q:
+this is Case (i) from Theorem 1.31. The infinity line LB is a leaf of the
+foliation R = const, while LO isn’t. Therefore, the lines LB and LO are
+σA-invariant. Suppose the contrary: A ̸= Q. Say, A /∈ LO; the case A /∈ LB
+is treated analogously. Then the restriction of the involution σA to each line
+ℓ ̸= LB, OB through A fixes A and its intersection points distinct from A
+with the lines LO, LB and OB. The number of the latter points is at least
+two, unless A is one of their pairwise intersection points Q, O, B ∈ γ. But
+A ̸= O, B, since A /∈ γ, and A ̸= Q by assumption, – a contradiction. Thus,
+A = Q.
+Case 2b). Then there are three indeterminacy points, and we can name
+them by X, Y , Z so that R|Y Z ≡ const ̸= ∞, while R|XY , R|XZ ≡ ∞. In
+Subcase 2b1) X = (1, 1), and Y , Z are the origin and the infinity point
+of the parabola γ. The involution σA should fix one of the indeterminacy
+points and permute two other ones. We claim that it fixes X and permutes
+Y and Z. Indeed, suppose the contrary: σA fixes, say, Y and permutes X
+and Z. Then σA(XY ) = Y Z and σA(XZ) = XZ. On the other hand,
+σA should send level sets of the integral R to its level sets, since this is
+57
+
+true for generic, irreducible level sets of degree deg R, and remains valid
+after passing to limit. Thus, σA should preserve the infinity level set, since
+σA(XZ) = XZ. On the other hand, it should permute it with a finite level
+set containing the line Y Z, since σA(XY ) = Y Z. The contradiction thus
+obtained proves that σA(X) = X and σA(Y ) = Z. This implies that A is
+the intersection point of the line Y Z with the tangent line to γ at X, and
+the corresponding involution σA permutes the intersection point of each line
+through A with the lines XY and XZ. Thus, the pair (A, σA) is the same,
+as in the Cases (ii) of Theorem 1.31. These cases are obtained one from the
+other by complex projective transformation, as in Theorem 1.12.
+Case 2c1). (Case 2c2) is obtained from it by projective transformation.)
+The integral R has three indeterminacy points lying on the conic γ. Let
+us denote them by X, Y , Z. Then one of them, say, X, is fixed by the
+γ-angular symmetry σA, and the two other ones are permuted. Indeed, if
+all of them were fixed, then the three distinct tangent lines to γ at them
+would intersect at the point Q, which is impossible: through every point
+lying outside the conic γ there are only two tangent lines to γ. This implies
+that A is the intersection point of the line tangent to γ at X and the line
+Y Z. Thus, it is admissible in the sense of Theorem 1.31, case (iii).
+In Case 2c2) all the admissible vertices are real, since so are the inde-
+terminacy points. In Case 2c1) X = (1, 1) is the unique real indeterminacy
+point; the other ones are Y = (ε, ¯ε) and Z = (¯ε, ε), where ε = e
+2πi
+3 . The
+intersection of the line {w = 2z − 1} tangent to γ at (1, 1) and the line
+Y Z is the admissible vertex (0, −1). Each one of the complex lines XZ,
+XY has non-real slope, and hence, X is its unique real point. Therefore,
+the admissible vertex lying there is not real. Thus, in Case 2c1) the point
+(0, −1) is the unique real admissible vertex.
+Case 2d). The corresponding integral R = Rd has three indeterminacy
+points: the origin, the point (1, 1) and the infinity point of the conic γ.
+The line {z = 1} through the two latter indeterminacy points lies in a level
+curve of the integral R: namely, in its polar locus.
+On the other hand,
+R is non-constant on the lines {z = 0} and {z = w} passing through the
+origin and the other indeterminacy points. This implies that every projective
+transformation preserving the foliation R = const should fix the origin, and
+hence, the Oz-axis: the corresponding tangent line to γ. Let us show that it
+cannot be a γ-angular symmetry. Suppose the contrary: it is the γ-angular
+symmetry σA centered at a point A. Then σA has to fix the origin and to
+permute the two other indeterminacy points, as in the case discussed above.
+Therefore, A is the intersection point of the line {z = 1} through them
+58
+
+and the Oz-axis: thus, A = (1, 0). Thus, the involution σA fixes the line
+{z = 1}, which lies in the polar locus of the integral R. Hence, it preserves
+the whole polar locus, as in the above Case 2b). The polar locus consists
+of the above line, the regular conic α := {w = −8z2} and an irreducible
+rational cubic, see [27, proposition 7.15]. Hence, σA fixes the conic α. Hence,
+it permutes its infinite point (coinciding with that of γ) and its other, finite
+intersection point (1, −8) with the line {z = 1}.
+On the other hand, it
+should send the infinite point to the other point (1, 1) ∈ γ ∩ {z = 1}, since
+σA is the γ-angular symmetry. The contradiction thus obtained proves that
+if a multibilliard contains a conic with exotic dual billiard structure of type
+2d), then it contains no vertices. The proof of Theorem 1.31 is complete. ✷
+Proof
+of Proposition 1.32. The complex projective equivalence of bil-
+liards of type (ii) is obvious. Let us prove the analogous second statement
+of Proposition 1.32 on billiards of type (iii). The dual billiard structure on
+γ of type 2c1) admits the order 3 symmetry (z, w) �→ (εz, ¯εw) cyclically per-
+muting the indeterminacy points of the integral. Therefore, it also permutes
+cyclically admissible vertices and hence, acts transitively on them and on
+their unordered pairs. The same statement holds for type 2c2), since the
+dual billiards 2c1) and 2c2) are complex-projectively isomorphic. This im-
+plies the second statement of Proposition 1.32. In the case of type 2c2) the
+indeterminacy points are real, and hence, so are the admissible vertices, and
+the order 3 symmetry is a real projective transformation. This together with
+the above discussion implies the third statement of Proposition 1.32.
+✷
+2.6
+Admissible vertices of real pencils of conics.
+Proof of
+Proposition 1.24
+Proof
+of Proposition 1.24. The ambient projective plane CP2 is the
+projectivization of a three-dimensional complex space C3.
+The complex
+conjugation involution acting on C3 induces its action on CP2, which will
+be also called conjugation. It sends projective lines to projective lines and
+preserves the complexification of every real pencil of conics.
+Case a): pencil of real conics through four distinct (may be complex)
+points A, B, C, D.
+The conjugation permutes the points A, B, C, D.
+Therefore, it permutes vertices M1, M2 and M3.
+Hence, at least one of
+them is fixed (say, M1), or equivalently, real. The line through the other
+points M2, M3 should be fixed, and thus, real, since the union {M2, M3} is
+invariant. Finally, the involution σM1 is real.
+Let now the point KAB be real. Then the ambient line AB is invariant
+59
+
+under the conjugation, since the collection of complex lines through pairs of
+permuted basic points of the pencil is invariant. Therefore, it is a real line,
+the union {A, B} is conjugation invariant. Hence, so is {C, D}, and the line
+CD is real. Thus, the involution σKAB is also real.
+Note that the case, when C, D are real and A, B aren’t is possible. In
+this case A and B are permuted by the conjugation. Hence, the points M2
+and M3 are also permuted, and thus, they are not real. Similarly, , KBC
+and KAC are permuted, KBD and KAD are permuted, and hence, they are
+not real.
+Case b): real pencil of conics through 3 points A, B, C tangent at the
+point C to the same line L.
+See Fig.
+4.
+The point C and the line L
+should be obviously fixed by conjugation, and hence, real. Therefore, the
+points A, B are either both fixed, or permuted. Hence, the line AB is real,
+and so is the intersection point M = AB ∩ L. The point KAB ∈ AB is
+also real, since complex conjugation acting on complex projective line sends
+harmonic quadruples to harmonic quadruples and the harmonicity property
+of a quadruple of points is invariant under two transpositions: one permuting
+its two first points; the other one permuting its two last points. Therefore,
+the line CKAB is real, and so is σM. The global projective involutions σC
+and σKAB are both real, since so are the lines AB and L. In the case, when
+the dual billiard structure at C is quasi-global and is defined by a real conic,
+the corresponding involution σC is real.
+Case c): real pencil of conics through two distinct points A, C tangent at
+them to two given lines LA and LC. See Fig. 5. A priori, the points A and
+C need not be real. For example, a pencil of concentric circles satisfies the
+above statements with A = [1 : i : 0], C = [1 : −i : 0]: the so-called isotropic
+points at infinity. The line AC is real, and so is M = LA ∩ LC, since the
+complex conjugation either permutes A and C (and hence, the lines LA and
+LC), or fixes them. Therefore, the involution σM is real. The point M′,
+which is an arbitrary point of the complex line AC needs not be real. But
+if it is real, then so is the involution σM′. Indeed, σM′ can be equivalently
+defined to fix M′ and the line through M and the point K ∈ AC for which
+the quadruple M′, K, A, C is harmonic. If M′ is real, then so is K, as in the
+above case. Hence, the line MK is real and so is σM′.
+Cases d) and e). Reality of the vertex A, is obvious. Equivalence of
+reality of the involution σA,S and reality of the conic S, follows from reality
+of the vertex A and from the fact that a projective involition of a complex
+line having at least one real fixed point is real, if and only if its other fixed
+point is also real. Proposition 1.24 is proved.
+✷
+60
+
+3
+Rationally 0-homogeneously integrable piecewise
+smooth projective billiards. Proof of Theorems
+1.38, 1.39, 1.45
+The first step of the above-mentioned classification is the following lemma.
+Lemma 3.1 Let a planar projective billiard with piecewise C4-smooth bound-
+ary containing a nonlinear arc be rationally 0-homogeneously integrable.
+Then the C4-smooth pieces of its boundary are conical arcs and straight-
+line segments. The projective billiard structure of each conical arc is either
+of dual pencil type, or an exotic one from Statement 2) of Theorem 1.16.
+Proof
+A rational 0-homogeneous integral of the billiard is automatically
+such an integral for the projective billiard on each nonlinear arc. Hence, by
+Theorem 1.16, the nonlinear arcs are conics, and each of them is equipped
+with a projective billiard structure either of dual pencil type, or exotic.
+✷
+Below we describe the possible combinations of conical arcs and straight-
+line segments equipped with projective billiard structures that yield alto-
+gether a rationally integrable projective billiard. We reduce this description
+to the classification of rationally integrable dual multibilliards. To do this,
+we use the projective duality given by orthogonal polarity, see Subsection
+3.1, which transforms a projective billiard to a dual multibilliard. We show
+that the former is rationally 0-homogeneously integrable, if and only if the
+latter is rationally integrable. We present a one-to-one correspondence be-
+tween rational 0-homogeneous integrals of the former and rational integrals
+of the latter. Afterwards the main results on classification of rationally 0-
+homogeneously integrable projective billiards follow immediately by duality
+from those on dual multibilliards.
+3.1
+Duality between projective billiards and dual multibil-
+liards. Correspondence between integrals
+Consider the orthogonal polarity, which sends each two-dimensional sub-
+space in the Euclidean space R3
+x1,x2,x3 to its orthogonal one-dimensional
+subspace. The projectivization of this correspondence is a projective dual-
+ity that sends each projective line in RP2
+[x1:x2:x3] to a point in RP2, called
+its dual point. It sends an immersed smooth2 strictly convex curve α to its
+2Note that in general, the straightforward parametrization of the dual curve α∗ induced
+by a Cm-smooth parametrization of the curve α is not Cm-smooth. But one can choose
+the parameter of the dual curve to make it Cm-smooth.
+61
+
+dual: the immersed smooth strictly convex curve α∗, whose points are dual
+to the projective lines tangent to α.
+Consider a projective billiard in R2
+x1,x2 with piecewise C4-smooth bound-
+ary such that each its C4-smooth arc is either strictly convex, or a straight-
+line segment. This holds automatically, if all the nonlinear boundary arcs
+are conical, as in Lemma 3.1. Consider the ambient plane R2
+x1,x2 as the
+horizontal plane {x3 = 1} ⊂ R3
+x1,x2,x3. We identify it with the standard
+affine chart {x3 = 1} ∈ RP2
+[x1:x2:x3] by tautological projection. The above
+duality sends each nonlinear C4-smooth boundary arc α to the dual curve
+α∗ ⊂ RP2. For every point Q ∈ α let LQ denote the projective line tangent
+to α at Q. Let Q∗ denote the line dual to Q. It is tangent to the dual curve
+α∗ at the point P = L∗
+Q dual to LQ. The projective billiard reflection at Q is
+the nontrivial affine involution acting on TQR2, which fixes the points of the
+line LQ and fixes the line N(Q) of the transversal line field. Its projectiviza-
+tion acts as an involution RP1 → RP1 on the space RP1 of lines through Q.
+The duality conjugates the latter involution acting on lines to a projective
+involution σP acting on the projective line Q∗ = LP tangent to α∗ at P and
+fixing the points P and N ∗(Q). The projective involution family (σP )P ∈α∗
+thus obtained is a dual billiard structure on the curve α∗. It will be called
+the dual billiard structure dual to the projective billiard on α.
+Definition 3.2 Consider a projective billiard as above. Its dual multibil-
+liard is the collection of curves α∗ in RP2 dual to its C4-smooth nonlinear
+boundary arcs α, equipped with the dual billiard structure defined above,
+and the points A (called vertices) dual to the ambient lines a of the straight-
+line billiard boundary segments. Each vertex A is equipped with the fol-
+lowing dual billiard structure. Let U denote the union of all the billiard
+boundary intervals lying in the line a. Each point Q ∈ U is dual to a line q
+through A. The projective billiard reflection involution acting on the space
+RP1 of lines through Q is conjugated by duality to a projective involution
+σA,q : q → q fixing A. The family of involutions σA,q, Q ∈ U, yields the
+prescribed dual billiard structure at the point A.
+For every (x1, x2) ∈ R2 and (v1, v2) ∈ T(x1,x2)R2 set
+r := (x1, x2, 1),
+v = (v1, v2, 0) ∈ R3,
+M = M(r, v) := [r, v] = (−v2, v1, ∆),
+∆ = ∆(x1, x2, v) = x1v2 − x2v1.
+(3.1)
+For every fixed r the map M is a linear operator sending the space T(x1,x2)R2
+isomorphically onto the orthogonal complement r⊥.
+Its projectivization
+62
+
+sends the space of lines in R2 through r onto the projective line dual to
+the point (x1, x2) = [x1 : x2 : 1] ∈ R2 ⊂ RP2.
+Each line through r is
+sent onto its dual point. Therefore, the projectivized map M yields a well-
+defined map from the space of projective lines to RP2 that coincides with the
+above projective duality. In particular, it conjugates the projective billiard
+reflections at billiard boundary points to the corresponding dual multibil-
+liard involutions. Therefore, we can consider that the dual multibilliard lies
+in the projective plane RP2 with homogeneous coordinates [M1 : M2 : M3].
+Proposition 3.3 1) A projective billiard is rationally 0-homogeneously in-
+tegrable, if and only if its dual multibilliard is rationally integrable.
+2) Each rational 0-homogeneous integral of degree n (if any) of the pro-
+jective billiard is a rational 0-homogeneous function of the moment vector
+M = (M1, M2, M3), see (3.1), of the same degree n.
+3) Let R be a rational integral of the dual multibilliard written in ho-
+mogeneous coordinates [M1 : M2 : M3] as a ratio of two homogeneous
+polynomials of degree n. Then R[r, v] is a rational 0-homogeneous integral
+of the projective billiard of the same degree n.
+Proof
+The statements of the proposition extend [27, propositions 1.23,
+1.24] (formulated for a projective billiard on a connected curve) to projec-
+tive billiards with piecewise C4-smooth boundary. The proofs given in [27,
+subsections 9.1, 9.2] remain valid in this more general case.
+✷
+Corollary 3.4 The minimal degree of rational 0-homogeneous integral of a
+projective billiard is equal to the minimal degree of rational integral of its
+dual multibilliard.
+3.2
+Case of dual pencil. Proof of Theorems 1.38, 1.39, 1.40
+We use the following proposition.
+Proposition 3.5 Let P be a pencil of conics, P∗ be its dual pencil.
+1) Let α ⊂ R2 ⊂ RP2 be a conical arc whose ambient conic lies in P∗,
+equipped with the projective billiard structure defined by P∗. Then its dual is
+the dual conical arc α∗ equipped by the dual billiard structure of pencil type,
+defined by the pencil P. The converse statement also holds.
+2) A planar projective billiard is of dual pencil type, defined by the dual
+pencil P∗, if and only if its dual multibilliard is of pencil type, defined by P.
+63
+
+Proof
+Statement 1) of the proposition follows from definition. The def-
+initions of dual multibilliard of pencil type and projective billiard of dual
+pencil type are dual to each other: the standard (skew) admissible lines for
+the dual pencil P∗ are dual to the standard (skew) admissible vertices for
+the pencil P. This implies Statement 2).
+✷
+Proof
+of Theorem 1.38. Let a projective billiard with piecewise C4-
+smooth boundary containing a nonlinear arc be rationally integrable. Then
+its dual multibilliard is rationally integrable (Proposition 3.3). Hence, all
+its curves are conics, and the nonlinear arcs of projective billiard boundary
+are conical.
+Let there be at least two arcs of two distinct conics.
+Then
+the multibilliard contains their dual conics, which are also distinct. Hence,
+they are equipped with the dual billiard structure defined by the pencil P
+containing them, each conic of the multibilliard lies in the same pencil P and
+is equipped with the dual billiard structure defined by P (Theorem 1.25).
+Thus, all the conical arcs of the projective billiard boundary lie in the dual
+pencil P∗ and are equipped with the projective billiard structures defined
+by P∗, by Proposition 3.5, Statement 1). Theorem 1.38 is proved.
+✷
+Proof
+of Theorem 1.39. Let in a projective billiard all the nonlinear
+boundary arcs be conics lying in the same dual pencil P∗, equipped with
+the projective billiard structure defined by P∗. Then the curves of the dual
+multibilliard are conics lying in the pencil P, equipped with the dual billiard
+structure defined by P. The projective billiard is rationally 0-homogeneously
+integrable, if and only if the dual multibilliard is rationally integrable, by
+Proposition 3.3, Statement 1). The latter holds, if and only if the multibil-
+liard is of pencil type (Theorem 1.26). Or equivalently, if and only if the
+projective billiard is of dual pencil type, see Proposition 3.5, Statement 2).
+Theorem 1.39 is proved.
+✷
+Theorem 1.40 follows immediately from Theorem 1.27 and Corollary 3.4
+by duality, since (neighbor) skew admissible lines for a dual pencil P∗ are
+dual to (neighbor) skew admissible vertices for the pencil P and vice versa.
+3.3
+Exotic projective billiards. Proof of Theorem 1.45
+Let a projective billiard has boundary that consists of conical arcs of one
+and the same conic equipped with an exotic dual billiard structure from
+Theorem 1.16, Case 2), and maybe some straightline segments. It is ra-
+tionally 0-homogeneously integrable, if and only if the corresponding dual
+multibilliard is rationally integrable, by Proposition 3.3, Statement 1). In
+appropriate coordinates the dual multibilliard consists of one conic γ =
+64
+
+{w = z2} ⊂ R2
+z,w = {t = 1} ⊂ RP2
+[z:w:t] equipped with the corresponding
+exotic dual billiard structure from Theorem 1.11 and maybe some vertices.
+It is rationally integrable, if and only if either it has no vertices, or each its
+vertex is admissible in the sense of Theorem 1.31. This holds if and only
+if the ambient lines of the projective billiard boundary segments are dual
+to the admissible vertices, and their corresponding projective billiard struc-
+tures are dual to the dual billiard structures at the vertices. The lines dual
+to the admissible vertices, equipped with the corresponding dual projective
+billiard structures, will be called admissible. Let us find the admissible lines
+case by case. To do this, we use the following proposition.
+Proposition 3.6 Consider the above parabola γ equipped with an exotic
+dual billiard structure from Theorem 1.11, Case 2). Let C ⊂ RP2
+[z:w:t] denote
+the conic orthogonal-polar-dual to γ.
+1) The projectivization [F] : RP2
+[z:w:t] → RP2
+[x1:x2:x3] of the linear map
+F : (z, w, t) �→ (x1, x2, x3) := (z
+2, t, w)
+(3.2)
+sends C to the parabola {x2x3 = x2
+1}, which will be now referred to, as C,
+C ∩ {x3 = 1} = {x2 = x2
+1},
+equipped with the corresponding projective billiard structure from Theorem
+1.16, Case 2).
+2) For every point (z0, z2
+0) ∈ γ the corresponding point of the dual curve
+C has [x1 : x2 : x3]-coordinates [−z0 : z2
+0 : 1].
+3) The points in C corresponding to O = (0, 0), (1, 1), ∞ ∈ γ are respec-
+tively [0 : 0 : 1] = (0, 0), [−1 : 1 : 1] = (−1, 1), [0 : 1 : 0] = ∞ in the
+coordinates [x1 : x2 : x3] and in the coordinates (x1, x2) in the affine chart
+R2
+x1,x2 = {x3 = 1}.
+Proof Statements 1) and 2) follow from [27, claim 14, subsection 9.4] and
+the discussion after it. Statement 3) follows from Statement 2).
+✷
+Case 2a). The only admissible vertex of the dual billiard on γ is the
+intersection point Q = [1 : 0 : 0] of the tangent lines to γ at the origin and
+the infinity. It is equipped with the projective involution [z : w : t] �→ [−z :
+w : t] fixing the points of the line Ow through the origin and the infinity.
+The duality sends the above tangent lines to the origin and to the infinity
+respectively in the coordinates (x1, x2). Thus, the dual line Q∗ is the line
+through the origin and the infinity, equipped with the field of lines through
+the point [1 : 0 : 0]: the horizontal line field orthogonal to it.
+65
+
+Case 2b1) (Case 2b2) is treated analogously). The only admissible vertex
+Q = (0, −1) is the intersection point of the tangent line to γ at the point
+(1, 1) and the line Ow through the origin and the infinity. The dual point
+to the above tangent line is (−1, 1), by Proposition 3.6, Statement 3). The
+dual to the Ow-axis is the infinity point [1 : 0 : 0]: the intersection point
+of the tangent lines at the origin and at the infinity. Therefore, the line Q∗
+dual to Q is the line {x2 = 1} through the points (−1, 1) and [1 : 0 : 0]. Let
+us find the corresponding dual projective billiard structure on it. The fixed
+point line of the involution σQ is the line L = {w = 1}. Indeed, σQ fixes
+γ, and hence, the tangency points of the lines through Q tangent to γ. The
+latter tangency points are (±1, 1). Hence, the fixed point line is the line L
+through them. The line L intersects γ at the points (±1, 1). Their dual lines
+are tangent to C at the points (±1, 1), by Proposition 3.6, Statement 3).
+Therefore, the dual point L∗ is the intersection point (0, −1) of the latter
+tangent lines. Finally, the admissible line Q∗ = {x2 = 1} is equipped with
+the field of lines through the point (0, −1).
+Case 2c2).
+The dual billiard structure on the conic γ has three base
+(indeterminacy) points: (0, 0), (1, 1), ∞.
+Each admissible vertex is the
+intersection point of a line tangent to γ at one of them and the line through
+two other ones. The admissible vertices are (1, 0), (0, −1) and [1 : 1 : 0]. Let
+us find their dual lines and the projective billiard structures on them. The
+point Q = (1, 0) is the intersection point of the Oz-axis (which is tangent
+to γ at (0, 0)) and the line {z = 1} (which is the line through the points
+(1, 1) and ∞). The dual point to the Oz-axis is the origin (0, 0). The dual
+point to the line {z = 1} is the point [−1 : 2 : 0]. Indeed, it is the point of
+intersection of the lines tangent to C at the points (−1, 1) and infinity, by
+Proposition 3.6, Statement 3). The latter intersection point is [−1 : 2 : 0],
+since the line tangent to C at (−1, 1) has slope −2. Finally, the admissible
+line Q∗ dual to Q is the line {x2 = −2x1} through the origin and the point
+[−1 : 2 : 0]. Let us find its projective billiard structure. The lines through Q
+tangent to γ are the Oz-axis and the line {x2 = 2(x1 − 1)}, with tangency
+points (0, 0) and (2, 4) respectively. Therefore, the fixed point line of the
+involution σQ is the line L = {w = 2z} through them. Its dual point L∗
+is the intersection of the lines dual to (0, 0), (2, 4) ∈ γ. The latter lines are
+tangent to C at the points (0, 0) and (−2, 4), by Proposition 3.6, Statement
+3), and they intersect at (−1, 0). Thus, L∗ = (−1, 0), and the admissible line
+Q∗ = {x2 = −2x1} is equipped with the field of lines through L∗ = (−1, 0).
+The line dual to the admissible vertex (0, −1) is the line {x2 = 1}
+equipped with the field of lines through the point (0, −1), as in Case 2b1).
+The admissible vertex [1 : 1 : 0] is the intersection point of the line
+66
+
+tangent to γ at infinity and the line through the points (0, 0) and (1, 1).
+Therefore, its dual line passes through infinity (i.e., is parallel to the Ox2-
+axis) and the intersection point of the tangent lines to C at the corresponding
+points (0, 0) and (−1, 1). The latter intersection point is (− 1
+2, 0). Hence,
+the line dual to [1 : 1 : 0] is {x1 = − 1
+2}. It intersects the conic C at two
+points: the infinity and the point (− 1
+2, 1
+4). Its projective billiard structure
+is the field of lines through the intersection point of the tangent lines to C
+at the two latter points, as in the above cases. Their intersection point is
+[−1 : 1 : 0], since the slope of the tangent line to C at (− 1
+2, 1
+4) is equal to
+−1. Finally, the admissible line [1 : 1 : 0]∗ = {x1 = − 1
+2} is equipped with
+the line field parallel to the vector (−1, 1).
+Case 2c1).
+Then (0, −1) is the unique admissible vertex for the dual
+billiard. The only admissible line is its dual line {x2 = 1} equipped with
+the field of lines through the point (0, −1), as in Cases 2c2) and 2b1).
+Case 2d). There are no admissible lines, since the dual billiard has no
+admissible vertices (Theorem 1.31). Theorem 1.45 is proved.
+4
+Integrals of dual pencil type billiards: examples
+of degrees 4 and 12.
+Proof of Theorems 1.28,
+1.41 and Lemma 1.42
+First we prove Theorems 1.28, 1.41 and Lemma 1.42. Then we provide ex-
+amples of dual pencil type projective billiards with integrals of degrees 4 and
+12. Afterwards we discuss their realization by the so-called semi-(pseudo-)
+Euclidean billiards, with nonlinear part of boundary being equipped with
+normal line field for the standard (pseudo-) Euclidean form.
+4.1
+Multibilliards of pencil type. Proof of Theorem 1.28
+For every admissible vertex V from Definition 1.21, Case a) (Mj or KEL)
+equipped with the corresponding projective involution σV , let �V : R3 → R3
+denote the linear involution whose projectivization is σV .
+We normalize
+it to fix the points of the two-dimensional subspace projected to the fixed
+point line of σV and to act as the central symmetry α �→ −α on the one-
+dimensional subspace projected to V . Let �V ∗ denote its conjugate, acting on
+the space R3∗ of linear functionals on R∗. The symmetric square Sym2(R3∗)
+is identified with the space of homogeneous quadratic polynomials on R3.
+The operators �V ∗ lifted to Sym2(R3∗) will be also denoted by �V ∗. In the
+proof of Theorem 1.28 we use the two following propositions.
+67
+
+Proposition 4.1 Let a pencil have type 2a): conics through four different
+points A, B, C, D. One has
+( �KEL �KLF )3 = Id
+for every three distinct E, L, F ∈ {A, B, C, D}.
+(4.1)
+Proof
+Let N ∈ {A, B, C, D} be the point distinct from E, L, F.
+The
+involutions σKEL, σKLF fix N; σKEL fixes F and permutes E, L; σKLF fixes
+E and permutes L, F. Hence, their product fixes N and makes an order three
+cyclic permutation of the points E, L, F. Thus, Π := (σKEL ◦ σKLF )3 fixes
+all the four points A, B, C, D ∈ RP2, hence Π = Id. Thus, ( �KEL �KLF )3 = Id
+up to constant factor. The latter constant factor should be equal to one,
+since the operator in question has unit determinant, being a product of six
+involutions �KST , each with determinant −1. This proves (4.1).
+✷
+Recall that for every line X ⊂ RP2 by ξX ∈ R3∗ we denote a linear
+functional vanishing on the two-dimensional subspace in R3 projected to X.
+Proposition 4.2 1) The subspace W ⊂ Sym2(R3∗) generated by the prod-
+ucts ξEL(Y )ξ(EL)′(Y ) with (EL)′ being the line through the pair of points
+{E′, L′} := {A, B, C, D} \ {E, L}, is two-dimensional and �V ∗-invariant for
+every admissible vertex V . Each operator �V ∗ corresponding to a standard
+admissible vertex acts on W as the identity up to constant factor.
+2) The above functionals ξEL can be normalized so that
+�K∗
+AB(ξABξCD) = −ξABξCD, �K∗
+AB(ξBCξAD) = −ξACξBD,
+(4.2)
+and so that analogous formulas hold for the other operators �K∗
+EL.
+Proof The zero conics of the polynomials ξEL(Y )ξ(EL)′(Y ) are the singular
+conics AB ∪ CD, BC ∪ AD, AC ∪ BD. They lie in the pencil of conics
+through A, B, C, D. Hence the space W spanned by these polynomials is
+two-dimensional. Its �V ∗-invariance follows from σV -invariance of the pencil.
+For every V ∈ {M1, M2, M3} the involution σV fixes the three above conics,
+and hence, each conic of the pencil. Thus �V ∗|W = Id up to constant factor.
+Let us prove the first formula in (4.2) for arbitrary normalization of the
+functionals ξAB and ξCD. Every vector v ∈ R3 \{0} with π(v) /∈ AB ∪CD is
+sent by �KAB to the opposite side from the hyperplane π−1(CD) and to the
+same side from the hyperplane π−1(AB), by definition: �KAB fixes the points
+of the former hyperplane and acts as central symmetry on its complementary
+invariant subspace π−1(KAB), which lies in the latter hyperplane. Therefore,
+it keeps the sign of the functional ξAB and changes the sign of ξCD. Hence,
+it multiplies their product by −1, being an involution.
+68
+
+The operator �KAB permutes the conics BC ∪AD and AC ∪BD. There-
+fore, the functionals ξBC, ξAD, ξAC, ξBD can be normalized so that the
+corresponding products ξBCξAD and ξACξBD be permuted by �K∗
+AB with
+change of sign. Formula (4.2) is proved. Let us normalize ξAB and ξCD by
+constant factors (this does not change formula (4.2)) so that the analogue
+of the second formula in (4.2) holds for �K∗
+BC:
+�K∗
+BC(ξACξBD) = −ξABξCD.
+(4.3)
+This together with the second formula in (4.2) and involutivity of the oper-
+ators �V ∗ imply that
+�K∗
+BC �K∗
+AB(ξACξBD) = − �K∗
+BC(ξBCξAD) = ξBCξAD.
+(4.4)
+Replacing the right-hand side in (4.3) by �K∗
+AB(ξABξCD), applying �K∗
+BC to
+both sides, denoting H := �K∗
+BC �K∗
+AB, together with (4.4) yield
+H(ξABξCD) = ξACξBD, H(ξACξBD) = ξBCξAD.
+(4.5)
+One also has
+H(ξBCξAD) = ξABξCD,
+(4.6)
+since H3 = Id, by (4.1). Therefore,
+ξACξBD + ξBCξAD + ξABξCD = 0
+(4.7)
+since the terms in the latter sum form an orbit of order three two-dimensional
+operator acting on W. Let us now prove the analogues of formula (4.2) for
+the other KEL. To this end, let us show that
+�K∗
+CD = �K∗
+AB on the space W.
+(4.8)
+Indeed, the composition σKCD ◦ σKAB fixes the three singular conics (and
+hence, each conic of the pencil), by definition. Therefore, �K∗
+CD = �K∗
+AB on
+W, up to constant factor. The latter constant factor is equal to one, since
+the operators in question take equal value at ξABξCD, by the first formula in
+(4.2) (which holds for KAB replaced by KCD). This proves (4.8). Formula
+(4.8) together with the other similar formulas, and already proved formula
+(4.2) for the operators �K∗
+AB, �K∗
+BC imply the analogues of (4.2) for �K∗
+CD,
+�K∗
+DA. Let us prove its analogue for �K∗
+AC. One has
+�K∗
+BC �K∗
+AC(ξACξBD) = − �K∗
+BC(ξACξBD) = ξABξCD,
+(4.9)
+69
+
+by formula (4.2) for KBC.
+Therefore, ξACξBD, ξABξCD together with a
+third vector �K∗
+BC �K∗
+AC(ξABξCD) form the orbit of order three linear operator
+�K∗
+BC �K∗
+AC, see (4.1). The sum of the vectors in the orbit should be equal to
+zero. This together with (4.7) implies that
+�K∗
+BC �K∗
+AC(ξABξCD) = ξBCξAD.
+Applying �K∗
+BC to this equality yields �K∗
+AC(ξABξCD) = −ξBCξAD. Formula
+(4.2) for KAC is proved. For KBD if follows from its version for KAC as
+above. Formula (4.2) is proved for all KEL. Proposition 4.2 is proved.
+✷
+Claim.
+For the normalization chosen as in Proposition 4.2 formula
+(1.21), i.e., (4.7) holds.
+Conversely, if (4.7) holds, then the statements
+of Proposition 4.2 also hold.
+Relation (4.7) determines the collection of
+products ξELξF N uniquely up to common constant factor.
+Proof The first statement of the claim is already proved above. Its third,
+uniqueness statement follows from two-dimensionality of the space W. These
+two statements together imply the second statement of the claim.
+✷
+Proof
+of Theorem 1.28. Let the linear functionals ξEL be normalized
+to satisfy (1.21), which is possible by Proposition 4.2 and the above claim.
+Then they satisfy (4.2), by the claim. Projective transformations of RP2
+act on rational functions on RP2 (which can be represented as rational 0-
+homogeneous functions of Y = (y1, y2, y3)). The ratio ξABξCD
+ξBCξAD is sent by
+σKAB to ξABξCD
+ξACξBD etc., by (4.2). This implies invariance of the degree 12 ra-
+tional function (1.22) under all the involutions σKEL. Its invariance under
+the involutions corresponding to the standard vertices follows from Proposi-
+tion 4.2, Statement 1). Thus, the integral in question is invariant under the
+involutions of all the admissible vertices. It is invariant under the involution
+of tangent line to a conic of the multibilliard, since so are its factors, which
+are constant on the conics of the pencil. Theorem 1.28 is proved.
+✷
+4.2
+Dual pencil type projective billiards. Proof of Theorem
+1.41 and Lemma 1.42
+Proof
+of Theorem 1.41. Consider a dual pencil type projective billiard
+given by a dual pencil of conics tangent to four distinct lines a, b, c, d.
+Consider the corresponding dual multibilliard in RP2
+[M1:M2:M3] obtained by
+orthogonal polarity duality. It is of pencil type, defined by the pencil of
+conics through the points A, B, C, D dual to the latter lines. Expression
+(1.25) considered as a rational 0-homogeneous function of M is a well-defined
+70
+
+function on RP2. It is a rational integral of the multibilliard, provided that
+relation (1.24) holds. There indeed exist χem;fn satisfying (1.24), by Theo-
+rem 1.28 and since each scalar product < r(en), M > is a linear functional
+vanishing on the two-dimensional subspace π−1(EN) ⊂ R3 corresponding to
+the line EN: r(en) ⊥ π−1(EN), by orthogonal polarity duality. Therefore,
+substituting M = [r, v] to (1.25) yields an integral of the projective billiard,
+by Proposition 3.3, Statement 3). Theorem 1.41 is proved.
+✷
+Proof of Lemma 1.42. Let us find χem;fn from linear equations implied
+by (1.24). Substituting M = (0, 0, 1) to (1.24) yields
+�
+χem;fn = 0.
+(4.10)
+For every intersection point em set
+x(em) := (x1(em), x2(em)) ∈ R2.
+Let now M ⊥ r(ab). Then there exists a v = (v1, v2) = (v1, v2, 0) such that
+M = [r(ab), v] = (−v2, v1, ∆ab), ∆ab := [x(ab), v] = x1(ab)v2 − x2(ab)v1,
+(4.11)
+since v �→ [r, v] is a linear isomorphism R2 → r⊥. Substituting (4.11) to
+(1.24) yields
+χbc;ad[x(bc) − x(ab), v][x(ad) − x(ab), v]
++ χac;bd[x(ac) − x(ab), v][x(bd) − x(ab), v] = 0.
+(4.12)
+The vector differences (x(bc)−x(ab), x(bd)−x(ab) in (4.12) are proportional
+(being parallel to the line b), and so are the other vector differences (parallel
+to a). Therefore, equality (4.12) is equivalent to the relation
+χbc;adρτ + χac;bdst = 0.
+(4.13)
+Combining (4.13) with (4.10) and normalizing the χem;fn so that χac;bd = 1
+yields (1.27). Lemma 1.42 is proved.
+✷
+Remark 4.3 The χem;fn satisfying (1.24) can be also found by all the pos-
+sible substitutions M ⊥ r(em) for em = ab, bc, cd, ac, bd, ad.
+This yields
+a system of linear equations. It appears that their matrix has rank two,
+so that there exists a unique common non-zero solution. Namely, all the
+3x3-minors vanish. This follows from two-dimensionality of the subspace W
+generated by the three quadratic forms < r(em), M >< r(fn), M >, which
+in its turn follows from the fact that the three singular conics AB ∪ CD,
+71
+
+Figure 19:
+AC ∪ BD, AD ∪ BC formed by the lines EM dual to the points em lie in
+the same pencil of conics through the points A, B, C, D. On the other
+hand, a direct calculation of 3x3-minors of the matrix of linear equations
+and equating these minors to zero yields relations on the (oriented) lengths
+s, τ, ρ, t, p = |cd − bc|, u = |bc − ac|, q = |cd − ad|, h = |ad − bd|, see Fig.
+15. These relations are given by the following geometric theorem, which can
+also be deduced from Sine Theorem. The author believes that this theorem
+is well-known, but he did not found a reference to it.
+Theorem 4.4 In a triangle XZT, see Fig. 19, let us take arbitrary points
+Y , I on its sides ZT and XZ respectively. Let V denote the intersection
+point of the lines XY and TI. Set
+ρ := |Y V |, t := |V X|, s := |TV |, τ := |V I|,
+u := |TY |, p := |Y Z|, q := |IZ|, h := |XI|.
+Then
+pq
+(p + u)(q + h) = ρτ
+st ,
+tp
+(ρ + t)(p + u) =
+τ
+s + τ ,
+sq
+(s + τ)(q + h) =
+ρ
+ρ + t.
+4.3
+Generic dual pencil type projective billiards with inte-
+grals of degrees 4 and 12
+Let us construct explicit examples of dual pencil type projective billiards
+with minimal degree of integral being equal to 4 and 12, with non-degenerate
+dual pencil. Consider a dual pencil of conics tangent to four given distinct
+lines: a, b, c, d.
+Fix some its conic γ.
+We consider that it is a closed
+curve in R2. Let us equip it with the projective billiard structure defined
+by the pencil: the conics of the pencil are its complex caustics.
+Let us
+construct the corresponding admissible lines m1, m2, m3 and kef, e, f ∈
+{a, b, c, d}, e ̸= f, equipped with their central projective billiard structures.
+We consider that the intersection points ab, bc, cd, da of the tangent lines
+72
+
+Sform a convex quadrilateral in which γ is inscribed. Then the lines kac and
+kbd both intersect the convex domain bounded by γ, see Figures 20 and 21.
+Example 4.5 of projective billiards with integral of degree 4. The
+line kac cuts the domain bounded by γ into two pieces. Each of them is a
+projective billiard bounded by an arc of the curve γ and a segment of the
+line kac, both equipped with the corresponding projective billiard structures.
+Both these projective billiards are rationally integrable with minimal degree
+of integral equal to four (Theorems 1.39 and 1.40). See Fig. 20. However
+the tangency points (marked in bold) of the curve γ with the lines a, b, c,
+d are indeterminacy points of the projective billiard structure on γ. But
+in each piece we can split its boundary arc lying in γ into open subarcs
+separated by the tangency points. The projective billiard structure is well-
+defined on the latter subarcs, and we can consider them as smaller smooth
+pieces of the boundary. A way to exclude the indeterminacy points from the
+boundary is to cut by the line m2 and to consider a smaller domain, bounded
+by segments of the lines kac, m2 and an arc of the curve γ, now without
+indeterminacies on smooth boundary arcs (except for the ”corners”). This
+yields a curvilinear triangle equipped with a projective billiard structure,
+also admitting a rational integral of minimal degree 4.
+Figure 20: Three projective billiards (with boundaries marked in bold) with
+rational integral of minimal degree 4. The indeterminacies of the projective
+billiard structure on γ are marked in bold.
+Example 4.6 with integral of degree 12. Consider the two curvilinear
+quadrilaterals with boundaries marked in bold at Fig.
+21 as projective
+73
+
+m2
+d
+m1
+Y
+m3
+a
+Cbilliards. The first one is bounded by segments of the lines kad, kbd, m2 and
+an arc of the conic γ. The second one is bounded by segments of the lines
+kbd, kab, kac and an arc of the conic γ. (We need to note that the boundary
+arc in γ in at least some of them contains (at least one) tangency point,
+which is an indeterminacy point of the projective billiard structure.) Both
+quadrilaterals considered as projective billiards are rationally integrable with
+minimal degree of integral being equal to 12, by Theorems 1.39 and 1.40.
+Figure 21: Two projective billiards (with boundaries marked in bold) with
+integral of minimal degree 12.
+4.4
+Semi-(pseudo-) Euclidean billiards with integrals of dif-
+ferent degrees
+Definition 4.7 A projective billiard in R2
+x1,x2 with piecewise smooth bound-
+ary is called semi-Euclidean (semi-pseudo-Euclidean), if the nonlinear part
+of the boundary, i.e., its complement to the union of straightline intervals
+contained there, is equipped with normal line field for the standard Eu-
+clidean metric dx2
+1 + dx2
+2 (respectively, for the standard pseudo-Euclidean
+metric dx2
+1 − dx3
+2).
+Theorem 4.8 A semi-Euclidean billiard is rationally 0-homogeneously in-
+tegrable, if and only if the nonlinear part of its boundary is a finite union
+of confocal conical arcs and segments of some of the admissible real lines
+(listed below) for the corresponding confocal pencil of conics:
+Case 1), pencil of confocal ellipses and hyperbolas:
+- the two symmetry axes of the ellipses, equipped with normal line field;
+74
+
+Kab
+Kbd
+Kad
+d
+m
+Y
+m3
+Kac
+c- the lines L1, L2 through the foci F1, F2, orthogonal to the line F1F2,
+each Lj is equipped with the field of lines through the other focus F2−j.
+The billiard has quadratic integral, if and only if its boundary contains no
+segments of lines L1,2; otherwise the minimal degree of integral is four.
+Case 2), pencil of confocal parabolas:
+- the common axis of the parabolas;
+- the line L through the focus that is orthogonal to the axis.
+Both lines are equipped with the normal line field. The billiard has quadratic
+integral, if and only if its boundary contains no segments of the line L;
+otherwise the minimal degree of integral is four.
+Proof
+The other admissible lines from Definition 1.34 are not finite real
+lines. For example, in Case 1) the dual pencil of confocal conics consists of
+conics tangent to two given pairs of lines through the two isotropic points
+[1 : ±i : 0] at infinity. In this case the only real skew admissible lines are
+the lines L1 and L2, and they are opposite as skew admissible lines: they
+correspond to two opposite intersection points of the above tangent lines,
+namely, the foci F1 and F2. Similarly in Case 2) the only real skew admissible
+line is L. This together with Theorem 1.40 proves Theorem 4.8.
+✷
+Example 4.9 Consider an ellipse and a line L1 through its left focus F1
+that is orthogonal to the foci line. See Fig.
+22, the left part. Consider
+the dashed domain bounded by the intersection segment of the line L1 with
+the ellipse interior and the left elliptic arc; the latter arc is equipped with
+normal line field, and the segment with the field of lines through the other
+focus F2. This projective billiard admits a rational 0-homogeneous integral
+of minimal degree four. As the second focus F2 tends to infinity so that
+the ellipse tends to a parabola with the focus F = F1, the above billiard
+converges to a usual billiard (with normal line field) bounded by a segment
+of the line L through F orthogonal to the parabola axis and by a parabola
+arc. See the right part of Fig. 22. The latter parabolic billiard is known
+to have a polynomial integral of minimal degree four (and hence, a rational
+0-homogeneous integral of the same degree). It was first discovered in [34].
+Proposition 4.10 The projective billiards with rational 0-homogeneous in-
+tegral of minimal degrees 4 and 12 presented at Fig. 20 and 21 respectively
+are realized by semi-pseudo-Euclidean billiards in (R2, dx2
+1 − dx2
+2).
+Proof Take the points bd = b ∩ d and ac = a ∩ c to be the isotropic points
+at infinity [1 : ±1 : 0].
+✷
+75
+
+Figure 22: Billiards (dashed) with degree 4 integrals. On the left: the semi-
+Euclidean billiard bounded by a segment of the line L1 and an elliptic arc.
+As the ellipse degenerates to a horizontal parabola, it tends to the Euclidean
+billiard on the right discovered in [34], with degree 4 polynomial integral.
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+
diff --git a/qdAyT4oBgHgl3EQfl_jn/content/tmp_files/load_file.txt b/qdAyT4oBgHgl3EQfl_jn/content/tmp_files/load_file.txt
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf,len=2890
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='00464v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='DS]  1 Jan 2023  On rationally integrable planar dual multibilliards  and piecewise smooth projective billiards  Alexey Glutsyuk∗†‡§  January 3, 2023  Abstract  A planar projective billiard is a planar curve C equipped with a  transversal line field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It defines reflection of lines from C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Its projec-  tive dual is a dual billiard: a curve γ ⊂ RP2 equipped with a family  of non-trivial projective involutions acting on its projective tangent  lines and fixing the tangency points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Projective and dual billiards  were introduced by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Tabachnikov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' He stated the following conjecture  generalizing the famous Birkhoff Conjecture on integrable billiards to  dual and projective billiards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let a dual billiard γ be strictly convex  and closed, and let its outer neighborhood admit a foliation by closed  curves (including γ) such that the involution of each tangent line to γ  permutes its intersection points with every leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then γ and the leaves  are conics forming a pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In a recent paper the author proved this  conjecture under the rational integrability assumption: existence of a  non-constant rational function (integral) whose restriction to tangent  lines is invariant under their involutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' He has also shown that if γ is  not closed, then it is still a conic, but the dual billiard structure needs  not be defined by a pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' He classified all the rationally integrable  dual billiard structures (with singularities) on conic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In the present  paper we give classification of rationally integrable dual multibilliards:  collections of dual billiards and points Qj (called vertices) equipped  with a family of projective involutions acting on lines through Qj from  an open subset in RP1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' As an application, we get classification of piece-  wise smooth projective billiards whose billiard flow has a non-constant  first integral that is a rational 0-homogeneous function of the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ∗CNRS, UMR 5669 (UMPA, ENS de Lyon), France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' E-mail: aglutsyu@ens-lyon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='fr  †HSE University, Moscow, Russia  ‡Kharkevich Institute for Information Transmission Problems (IITP, RAS), Moscow  §Supported by part by RFBR grant 20-01-00420  1  Contents  1  Introduction  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1  Introduction, brief description of main results and plan of the  paper  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content='  3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2  Previous results 1: classification of real and complex ratio-  nally integral dual billiards on one curve .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content='  9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3  Previous results 2: classification of rationally 0-homogeneously  integrable projective billiards on one curve .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  11  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4  Main results:  classification of rationally integrable planar  dual multibilliards with C4-smooth curves .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content='  14  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='5  Application: classification of rationally 0-homogeneously in-  tegrable piecewise C4-smooth projective billiards .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  22  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6  Plan of proofs of main results .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content='  29  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7  Historical remarks  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  32  2  Rationally integrable dual multibilliards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proofs of Theo-  rems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='26, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='31, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='27  34  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1  Rational integrability of pencil type multibilliards .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content='  34  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2  Foliation by level curves of rational integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proof of Theo-  rems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='25 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='26 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content='  38  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3  Dual billiard structures at vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Birationality and types  of involutions .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content='  44  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4  Pencil case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='26 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  51  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='5  Exotic multibilliards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='31  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content='  54  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6  Admissible vertices of real pencils of conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proof of Propo-  sition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='24 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content='  59  3  Rationally 0-homogeneously integrable piecewise smooth pro-  jective billiards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proof of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='38, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='39, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='45  61  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1  Duality between projective billiards and dual multibilliards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Correspondence between integrals .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content='  61  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2  Case of dual pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proof of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='38, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='39, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='40 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content='  63  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3  Exotic projective billiards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='45  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content='  64  4  Integrals of dual pencil type billiards: examples of degrees  4 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proof of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='28, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='41 and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='42  67  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1  Multibilliards of pencil type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='28  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content='  67  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2  Dual pencil type projective billiards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='41  and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  74  1  Introduction  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1  Introduction, brief description of main results and plan  of the paper  Consider a planar billiard Ω ⊂ R2 bounded by a C2-smooth strictly convex  closed curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Recall that its caustic is a curve S ⊂ R2 such that each  tangent line to S is reflected from the boundary ∂Ω to a line tangent to  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' A billiard is Birkhoff caustic-integrable, if some inner neighborhood of  its boundary is foliated by closed caustics, with boundary being a leaf of  the foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This is the case in an elliptic billiard, where confocal ellipses  form a foliation by closed caustics of a domain adjacent to the boundary  ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The famous open Birkhoff Conjecture states that the only integrable  billiards are ellipses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See its brief survey in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Tabachnikov  suggested its generalization to projective billiards introduced by himself in  1997 in [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See the following definition and conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1 [35] A projective billiard is a smooth planar curve C ⊂ R2  equipped with a transversal line field N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' For every Q ∈ C the projective  billiard reflection involution at Q acts on the space of lines through Q as  the affine involution R2 → R2 that fixes the points of the tangent line to C  at Q, preserves the line N(Q) and acts on N(Q) as the central symmetry  with respect to the point1 Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In the case, when C is a strictly convex closed  curve, the projective billiard map acts on the phase cylinder: the space of  oriented lines intersecting C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It sends an oriented line to its image under  the above reflection involution at its last point of intersection with C in the  sense of orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2 A usual Euclidean planar billiard is a projective billiard with  transversal line field being normal line field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Each billiard in a complete  Riemannian surface Σ of non-zero constant curvature (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=', in sphere S2 and  1In other words, two lines a, b through Q are permuted by reflection at Q, if and only  if the quadruple of lines TQC, N(Q), a, b is harmonic: there exists a projective involution  of the space RP1 of lines through Q that fixes TQC, N(Q) and permutes a, b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  3  Figure 1: The projective billiard reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  in hyperbolic plane H2) also can be seen as a projective billiard, see [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Namely, consider Σ = S2 as the unit sphere in the Euclidean space R3, and  Σ = H2 as the semi-pseudo-sphere {x2  1 + x2  2 − x2  3 = −1, x3 > 0} in the  Minkovski space R3 equipped with the form dx2  1 + dx2  2 − dx2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The billiard  in a domain Ω ⊂ Σ+ := Σ ∩ {x3 > 0} is defined by reflection of geodesics  from its boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The tautological projection π : R3 \\ {0} → RP2 sends  Ω diffeomorphically to a domain in the affine chart {x3 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  It sends  billiard orbits in Ω to orbits of the projective billiard on C = π(∂Ω) with  the transversal line field N on C being the image of the normal line field to  ∂Ω under the differential dπ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The notion of caustic (integrability) of a projective billiard repeats the  above notions for the usual billiards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Tabachnikov Conjecture states  that if a projective billiard is integrable, then the billiard boundary and the  caustics are conics, whose dual conics form a pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' To study it, Tabach-  nikov introduced the dual objects to projective billiards, the so-called dual  billiards, and stated the dual version of his conjecture for them, see the next  definition and conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3 [27, definitions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='17] A real (complex) dual billiard is  a smooth (holomorphic) curve γ ⊂ RP2(CP2) where for each point P ∈ γ  the real (complex) projective line LP tangent to γ at P is equipped with  a projective involution σP : LP → LP fixing P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' the family of involutions  (called the dual billiard structure) is parametrized by tangency points P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The dual Tabachnikov Conjecture deals with a strictly convex closed  curve γ ⊂ RP2 equipped with a dual billiard structure such that an outer  neighborhood of the curve γ admits a foliation by strictly convex closed  4  curves, including γ, such that each involution σP , P ∈ γ, permutes the  intersection points of the line LP with each individual leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It states that  under this assumption (called integrability condition) the curve γ and the  leaves of the foliation are conics forming a pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It implies that the dual  billiard structure on γ is of pencil type, see the next definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4 [27, example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='14] A dual billiard is of pencil type, if the  underlying curve γ is a (punctured) conic and there exists a pencil of conics  containing γ such that for every P ∈ γ the involution σP permutes the  intersection points of the line LP with each conic of the pencil (or fixes  the intersection point, if it is unique).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' As was observed by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Tabachnikov,  conversely, for every conic γ and every pencil containing γ, for every P in  the conic γ punctured in at most 4 complex base points of the pencil there  exists a projective involution σP : LP → LP satisfying the above condition,  and thus, a well-defined pencil type dual billiard on the punctured conic γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The dual Tabachnikov Conjecture is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It would imply his above con-  jecture on integrable projective billiards, and hence, the Birkhoff Conjecture  and its versions for billiards on surfaces of constant curvature and for outer  planar billiards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  In the previous paper by the author [27] the dual Tabachnikov Conjecture  was proved under the additional assumption that the foliation in question  admits a non-constant rational first integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This assumption is equivalent  to the existence of a non-constant rational function R whose restriction to  each tangent line LP to γ is invariant under the corresponding involution:  (R ◦ σP)|LP = R|LP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1)  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='5 [27, definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='12] A dual billiard for which there exists  a non-constant rational function (called integral) satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1) is called  rationally integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6 If a dual billiard on a nonlinear curve has a polynomial in-  tegral, then it is an outer billiard, that is, the corresponding projective  involution of each tangent line is its central symmetry with respect to the  tangency point (see [27, example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Each pencil type dual billiard is  rationally integrable with a quadratic integral (Tabachnikov’s observation,  see [27, example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  5  It was shown in [27, theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='16] that if a dual billiard is rationally  integrable, but the underlying curve γ is not necessarily closed, then  the curve γ is still a conic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  the dual billiard structure extends to a global analytic dual billiard  structure on the whole conic with at most four points deleted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  but the dual billiard is not necessary of pencil type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (Singular) rationally integrable dual billiard structures on conic were  classified by [27, theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='16 and its addendum], with explicit formulas for  rational integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' These results are recalled in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2 as Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11 and its addendum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The dual version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11 yields classifi-  cation of those projective billiards with underlying curve being C4-smooth  and connected that are rationally 0-homogeneously integrable, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=', whose  flow admits a nontrivial first integral that is a rational 0-homogeneous func-  tion of the velocity [27, theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='26 and its addendum].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' These results are  recalled in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3 as Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='16 and its addendum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The main result of the present paper is the classification of rationally  0-homogeneously integrable projective billiards with piecewise C4-smooth  boundary that contains a nonlinear arc and maybe also straightline segments  (results stated in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' To classify them, we introduce a gener-  alized dual object to projective billiards with piecewise smooth boundary:  the so-called dual multibilliards, which are collections of curves and points  equipped with dual billiard structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We obtain classification of rationally  integrable dual multibilliards containing a nonlinear arc (results stated in  Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4) and then deduce classification of rationally 0-homogeneously  integrable projective billiards by using projective duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The projective duals to the curvilinear pieces of the boundary of a pro-  jective billiard are planar curves equipped with dual billiard structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The  projective dual to each straightline piece of its boundary is a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The  projective billiard structure on the straightline piece (which is an open sub-  set of a projective line) is transformed by duality to dual billiard structure  at a point, see the next definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7 A dual billiard structure at a point Q ∈ RP2(CP2) is a fam-  ily of projective involutions σQ,ℓ : ℓ → ℓ acting on real (complex) projective  lines ℓ through Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It is assumed that σQ,ℓ are defined on an open subset  U ⊂ RP1(CP1) of the space of lines through Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' No regularity of the family  σQ,ℓ is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='8 A real (complex) dual multibilliard is a (may be infinite)  collection of smooth (holomorphic) nonlinear connected curves γj and points  6  Qs in RP2(CP2) (called vertices), where each curve γj and each point Qs are  equipped with a dual billiard structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='9 A dual multibilliard is rationally integrable, if there exists  a non-constant rational function on RP2(CP2) whose restriction to each tan-  gent line to every curve γj is invariant under the corresponding involution,  and the same statement holds for its restriction to each line ℓ through any  vertex Q, where the corresponding involution σQ,ℓ is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The main results on classification of rationally integrable real and com-  plex dual multibilliards are given by Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='26 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='31 in Sub-  section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' They deal with the case, when the multibilliard is not reduced  to one curve (without vertices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In the case, when it contains at least two  curves, Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='26 together state that it is rationally integrable,  if and only if it is of so-called pencil type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This means that all its curves  are conics lying in one pencil and equipped with the dual billiard struc-  ture defined by this pencil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' its vertices belong to an explicit list of so-called  admissible vertices for the given pencil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' the collection of vertices of the multi-  billiard satisfies additional conditions given in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='26  also yields analogous result in the case, when the multibilliard consists of  a single curve equipped with a dual billiard structure of pencil type and  maybe some vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='25 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11 together imply that every  other a priori possible rationally integrable dual multibilliard, not covered  by Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='26 is a so-called exotic multibilliard: it is formed by  conic equipped with an exotic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=', non-pencil) dual billiard structure from  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11 and maybe some vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='31 yields classification  of rationally integrable exotic multibilliards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It implies that such a multibil-  liard may contain at most three vertices, and the dual billiard structure at  each vertex is given by a global projective involution fixing the conic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Recall that a dual multibilliard formed by just a finite collection of conics  from the same pencil P, with dual billiard structures defined by P, always  has a quadratic rational integral, see Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' If one adds to it appro-  priate vertex collection (from the finite list of so-called admissible vertices  for the pencil P) so that the dual multibilliard thus obtained be of pencil  type, then it will be still rationally integrable, by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' However,  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='27 shows that the minimal degree of its rational integral may be  bigger: it may be equal to 2, 4 or 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The dual to a pencil type multibilliard defined by a pencil P is a projec-  tive billiard of the so-called dual pencil type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This means that its boundary  is piecewise-smooth and consists of arcs of conics from the dual pencil P∗  7  equipped with projective billiard structures having conical caustics from  the same pencil P∗, and maybe segments of so-called admissible lines for P∗  equipped with appropriate projective billiard structures defined by P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The-  orems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='38, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='39 (dual to Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='26) together imply that the dual  pencil type projective billiards are rationally 0-homogeneously integrable  and the only integrable billiards that are not of pencil type are the so-called  exotic ones, with nonlinear part of boundary lying in one conic equipped  with an exotic projective billiard structure from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The inte-  grable exotic billiards are classified by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='45 (dual to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='40 (the dual to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='27) implies that the minimal degree of  rational 0-homogeneous integral of a dual pencil type projective billiard with  piecewise C4-smooth boundary may be equal to 2, 4 or 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See formulas for  integrals of degree 12 in Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='28, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='41 and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='10 The flow of a Euclidean planar billiard with boundary con-  taining a curvilinear arc admits the trivial first integral: the squared module  of the velocity ||v||2 = v2  1 + v2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It is known that it admits a non-trivial inte-  gral polynomial in the velocity (that is, nonconstant along the unit velocity  hypersurface {||v||2 = 1}, if and only if it is of confocal dual pencil type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  This was proved in particular case in [14];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' this statement in full generality  is a joint result of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Bialy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Mironov and the author [8, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Together  with results of [14], it implies that the minimal degree of non-trivial poly-  nomial integral (if it exists) of an Euclidean billiard is equal to either 2, or  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' A nontrivial polynomial integral Ix(v), which can be chosen of even de-  gree 2n, generates a non-trivial rational 0-homogeneous integral Ix(v)  ||v||n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This  also implies that the minimal degree of non-trivial rational 0-homogeneous  integral of an Euclidean billiard is also equal to either 2, or 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Similar state-  ments hold for billiards on the other surfaces of constant curvature, that is,  the round sphere and the hyperbolic plane, and for the projective billiards  equivalent to them from Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See [14, 8, 9, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Thus, rationally 0-homogeneously integrable projective billiards of dual  pencil type with integrals of degree 12 presented and classified in the present  paper form an essentially new class of rationally integrable projective bil-  liards of dual pencil type, not covered by the known list of polynomially  integrable billiards on surfaces of constant curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Plan of proof of main results is presented in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' A historical  survey is given in Subsection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The main results are proved in Sections  2 (for multibilliards) and 3 (for projective billiards).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In Section 4 we prove  formulas for degree 12 integrals (Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='28, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='41 and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='42) and  8  present examples of projective billiards with integrals of degree 4 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2  Previous results 1: classification of real and complex ra-  tionally integral dual billiards on one curve  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11 [27, theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='16] Let γ ⊂ R2 ⊂ RP2 be a C4-smooth  connected non-linear (germ of) curve equipped with a rationally integrable  dual billiard structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then γ is a conic, and the dual billiard structure has  one of the three following types (up to real-projective equivalence):  1) The dual billiard is of conical pencil type and has a quadratic integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  2) There exists an affine chart R2  z,w ⊂ RP2 in which γ = {w = z2} and  such that for every P = (z0, w0) ∈ γ the involution σP : LP → LP is given  by one of the following formulas:  a) In the coordinate  ζ := z  z0  σP : ζ �→ ηρ(ζ) := (ρ − 1)ζ − (ρ − 2)  ρζ − (ρ − 1)  ,  ρ = 2 −  2  2N + 1,  or ρ = 2 −  1  N + 1  for some N ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2)  b) In the coordinate  u := z − z0  σP : u �→ −  u  1 + f(z0)u,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3)  f = fb1(z) :=  5z − 3  2z(z − 1) (type 2b1)),  or f = fb2(z) :=  3z  z2 + 1 (type 2b2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4)  c) In the above coordinate u the involution σP takes the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3) with  f = fc1(z) :=  4z2  z3 − 1 (type 2c1)),  or f = fc2(z) :=  8z − 4  3z(z − 1) (type 2c2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='5)  d) In the above coordinate u the involution σP takes the form (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3) with  f = fd(z) = 4  3z +  1  z − 1 =  7z − 4  3z(z − 1)  (type 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6)  9  Addendum to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Every dual billiard structure on γ of  type 2a) has a rational first integral R(z, w) of the form  R(z, w) =  (w − z2)2N+1  �N  j=1(w − cjz2)2 ,  cj = −4j(2N + 1 − j)  (2N + 1 − 2j)2 ,  for ρ = 2 −  2  2N + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7)  R(z, w) =  (w − z2)N+1  z �N  j=1(w − cjz2)  ,  cj = −j(2N + 2 − j)  (N + 1 − j)2 ,  for ρ = 2 −  1  N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='8)  The dual billiards of types 2b1) and 2b2) have respectively the integrals  Rb1(z, w) =  (w − z2)2  (w + 3z2)(z − 1)(z − w),  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='9)  Rb2(z, w) =  (w − z2)2  (z2 + w2 + w + 1)(z2 + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='10)  The dual billiards of types 2c1), 2c2) have respectively the integrals  Rc1(z, w) =  (w − z2)3  (1 + w3 − 2zw)2 ,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11)  Rc2(z, w) =  (w − z2)3  (8z3 − 8z2w − 8z2 − w2 − w + 10zw)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='12)  The dual billiard of type 2d) has the integral  Rd(z, w) =  (w − z2)3  (w + 8z2)(z − 1)(w + 8z2 + 4w2 + 5wz2 − 14zw − 4z3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='13)  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='12 [27, theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='18 and its addendum].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Every regular (germ  of) connected holomorphic curve in CP2 (different from a straight line)  equipped with a rationally integrable complex dual billiard structure is a  conic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Up to complex-projective equivalence, the corresponding billiard struc-  ture has one of the types 1), 2a), 2b1), 2c1), 2d) listed in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11, with  a rational integral as in its addendum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The billiards of types 2b1), 2b2), see  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4), are complex-projectively equivalent, and so are billiards 2c1), 2c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  10  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3  Previous results 2: classification of rationally 0-homogeneously  integrable projective billiards on one curve  Consider a domain Ω ⊂ R2  x1,x2 with smooth boundary ∂Ω equipped with  a projective billiard structure (transverse line field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The projective billiard  flow, see [35], acts on TR2|Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It moves a point (Q, v) ∈ TR2, Q = (x1, x2) ∈  Ω, v = (v1, v2) ∈ TQR2 so that v remains constant and Q moves along the  straight line directed by v with uniform velocity v, until it hits the boundary  ∂Ω at some point H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let v∗ ∈ THR2 denote the image of the velocity vector  v (translated to H) under the projective billiard reflection from the tangent  line TH∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Afterwards it moves (H, v∗) so that H moves with the new  uniform velocity v∗ until it hits the boundary again etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Figure 2: Projective billiard flow  Each Euclidean planar billiard flow always has the trivial first integral  ||v||2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But this is not true for a generic projective billiard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It is a well-known  folklore fact that Birkhoff integrability of a Euclidean planar billiard with  strictly convex closed boundary is equivalent to the existence of a non-trivial  first integral of the billiard flow independent with ||v||2 on a neighborhood  of the unit tangent bundle to ∂Ω in TR2|Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='13 A projective billiard is rationally 0-homogeneously inte-  grable, if its flow admits a first integral that depends on the velocity as a  non-constant rational 0-homogeneous function of degree uniformly bounded  by some number n: a function Ψ(Q, v) =  P (v)  T(v), where P and T are ho-  mogeneous polynomials in v of degree no greater than n with coefficients  depending on the position of the point Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The maximal degree of the latter  rational function through all Q is called the degree of the rational integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='14 The projective billiard structure on (an arc of) a regular  conic C is of dual pencil type, if it has a regular conical caustic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  More  11  H  Q1precisely, if there exists a conic Γ such that for every point Q ∈ C the  complex tangent lines through Q to the complexified conic Γ are permuted by  the projective billiard reflection at Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Consider now the pencil P containing  the dual conics C∗ and Γ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let P∗ denote its dual, consisting of conics dual  to the conics from the pencil P: it contains the conics C and Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then the  latter, caustic statement automatically holds for Γ being replaced by any  other conic from the dual pencil P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See [27, proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='27, remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  A dual pencil type projective billiard on a conic is known to be rationally 0-  homogeneously integrable with a quadratic 0-homogeneous rational integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  This is the statement dual to the similar statement for pencil type dual  billiards, see Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='15 The notion of rationally 0-homogeneously integrable projec-  tive billiard also makes sense for a projective billiard structure on an arc of  planar curve C (or a germ of curve), with projective billiard flow defined in  a (germ of) domain adjacent to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' A rational 0-homogeneous integral of  degree n is always a rational 0-homogeneous function of degree n in three  variables: v1, v2 and the moment ∆ := x1v2 − x2v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See analogous state-  ment for polynomial integrals of the usual planar billiards in [14] and the  statement for projective billiards in full generality in [27, proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='23,  statement 1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The property of rational 0-homogeneous integrability of a  projective billiard on a curve C is independent on the side from C on which  the billiard domain is chosen: an integral for one side is automatically an  integral for the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See [27, proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='23, statement 2)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='16 Let C ⊂ R2  x1,x2 be a non-linear C4-smooth germ of curve  equipped with a transversal line field N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let the corresponding germ of  projective billiard be 0-homogeneously rationally integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Then C is a  conic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' the line field N extends to a global analytic transversal line field on the  whole conic C punctured in at most four points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' the corresponding projective  billiard has one of the following types up to projective equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  1) A dual pencil type projective billiard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  2) C = {x2 = x2  1} ⊂ R2  x1,x2 ⊂ RP2, and the line field N is directed by  one of the following vector fields at points of the conic C:  2a)  ( ˙x1, ˙x2) = (ρ, 2(ρ − 2)x1),  ρ = 2 −  2  2N + 1 (case 2a1), or  ρ = 2 −  1  N + 1 (case 2a2),  N ∈ N,  the vector field 2a) has quadratic first integral Qρ(x1, x2) := ρx2 −(ρ−2)x2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  2b1) ( ˙x1, ˙x2) = (5x1 + 3, 2(x2 − x1)),  2b2) ( ˙x1, ˙x2) = (3x1, 2x2 − 4),  12  2c1) ( ˙x1, ˙x2) = (x2, x1x2 − 1),  2c2) ( ˙x1, ˙x2) = (2x1 + 1, x2 − x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  2d) ( ˙x1, ˙x2) = (7x1 + 4, 2x2 − 4x1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Addendum to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The projective billiards from Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='16 have the following 0-homogeneous rational integrals:  Case 1): A ratio of two homogeneous quadratic polynomials in (v1, v2, ∆),  ∆ := x1v2 − x2v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case 2a1), ρ = 2 −  2  2N+1:  Ψ2a1(x1, x2, v1, v2) :=  (4v1∆ − v2  2)2N+1  v2  1  �N  j=1(4v1∆ − cjv2  2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='14)  Case 2a2), ρ = 2 −  1  N+1:  Ψ2a2(x1, x2, v1, v2) =  (4v1∆ − v2  2)N+1  v1v2  �N  j=1(4v1∆ − cjv2  2)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='15)  The cj in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='14), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='15) are the same, as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='8) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case 2b1):  Ψ2b1(x1, x2, v1, v2) =  (4v1∆ − v2  2)2  (4v1∆ + 3v2  2)(2v1 + v2)(2∆ + v2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='16)  Case 2b2):  Ψ2b2(x1, x2, v1, v2) =  (4v1∆ − v2  2)2  (v2  2 + 4∆2 + 4v1∆ + 4v2  1)(v2  2 + 4v2  1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='17)  Case 2c1):  Ψ2c1(x1, x2, v1, v2) =  (4v1∆ − v2  2)3  (v3  1 + ∆3 + v1v2∆)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='18)  Case 2c2):  Ψ2c2(x1, x2, v1, v2) =  (4v1∆ − v2  2)3  (v3  2 + 2v2  2v1 + (v2  1 + 2v2  2 + 5v1v2)∆ + v1∆2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='19)  Case 2d): Ψ2d(x1, x2, v1, v2)  =  (4v1∆ − v2  2)3  (v1∆ + 2v2  2)(2v1 + v2)(8v1v2  2 + 2v3  2 + (4v2  1 + 5v2  2 + 28v1v2)∆ + 16v1∆2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='20)  13  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4  Main results: classification of rationally integrable pla-  nar dual multibilliards with C4-smooth curves  Each curve of a rationally integrable dual multibilliard is a conic, being itself  an integrable dual billiard, see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The first results on classifi-  cation of rationally integrable dual multibilliards presented below deal with  those multibilliards whose curves are conics lying in one pencil, equipped  with dual billiard structure defined by the same pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' They state that its  vertices should be admissible for the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' To define admissible vertices,  let us first introduce the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='17 A projective angular symmetry centered at a point A ∈  CP2 is a non-trivial projective involution σA : CP2 → CP2 fixing A and each  line through A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It is known to have a fixed point line Λ ⊂ CP2 disjoint from  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Its restrictions to lines throughs A define a dual billiard structure at A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='18 Let now A ∈ CP2 and let S ⊂ CP2 be a (may be singular)  conic disjoint from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' There exists a projective angular symmetry centered  at A and permuting the intersection points with S of each line through A,  called S-angular symmetry, see [25, definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='19 Let now A ∈ CP2, S ⊂ CP2 be a regular conic through  A, and LA the projective tangent line to S at A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The degenerate S-angular  symmetry centered at A is the involution σA = σS  A acting on the complement  CP2 \\ (LA \\ {A}) that fixes A, fixes each line ℓ ̸= LA through A and whose  restriction to ℓ is the projective involution fixing A and the other point of  the intersection ℓ ∩ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It is known to be a birational map CP2 → CP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='20 A dual billiard structure at a point A ∈ CP2 is called global  (quasi-global) if it is given by a projective angular symmetry (respectively,  degenerate S-angular symmetry) centered at A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='21 Consider a complex pencil of conics in CP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  A vertex,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=', a point of the ambient plane equipped with a complex dual billiard  structure, is called admissible for the pencil, if it belongs to the following  list of vertices split in two types: standard, or skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case a): a pencil of conics through 4 distinct points A, B, C, D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  a1) The standand vertices: M1 = AB ∩ CD, M2 = AD ∩ BC, M3 =  AC ∩ BD equipped with the global dual billiard structure given by the  projective angular symmetry σMj = σMiMk  Mj  , i, k ̸= j, i ̸= k, centered at Mj  with fixed point line MiMk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  14  Figure 3:  a2) The skew vertices KEL, E, L ∈ {A, B, C, D}, E ̸= L: KEL is the  intersection point of the line EL with the line MiMj such that Mi, Mj /∈ EL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The involution σKEL is the projective angular symmetry P2 → P2 centered  at KEL with fixed point line ST, {S, T} = {A, B, C, D} \\ {E, L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case b): a pencil of conics through 3 points A, B, C tangent at the point  C to the same line L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  b1) One standard vertex M = AB ∩ L, equipped with the projective  angular symmetry σM : P2 → P2 centered at M with fixed point line CKAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The point KAB is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  b2) The skew vertex KAB ∈ AB such that the projective involution  AB → AB fixing M and KAB permutes A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' That is, the points M,  KAB, A, B form a harmonic quadruple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The dual billiard structure at KAB  is given by the projective angular symmetry σKAB : P2 → P2 centered at  KAB with fixed point line L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  b3) The skew vertex C equipped with the projective angular symmetry  σC = σAB  C  : P2 → P2 centered at C with fixed point line AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  b4) The skew vertex C equipped with a degenerate S-angular symmetry  σC = σS  C centered at C, defined by arbitrary given regular conic S of the  pencil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This yields a one-parametric family of quasi-  global dual billiard structures at C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case c): a pencil of conics through two given points A and C that are  tangent at them to two given lines LA and LC respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  c1) Standard vertices: M = LA ∩ LC and any point M′ ∈ AC, M′ ̸=  A, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The vertex M is equipped with the projective angular symmetry  σM centered at M with fixed point line AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The vertex M′ is equipped  15  Figure 4:  Figure 5:  with the (LA ∪ LC)-angular symmetry centered at M′, which permutes the  intersection points of each line through M′ with the lines LA and LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  c2) Skew vertices equipped with global dual billiard structures: the points  A and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The dual billiard structure at A (C) is the projective angular  symmetry centered at A (C) with fixed point line LC (respectively, LA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  c3) Skew vertices A and C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' A (C) being equipped with a degenerate SA  (SC)-angular symmetry centered at A (C), defined by any regular conic SA  (SC) of the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This yields a one-parametric family of quasi-global dual  billiard structures at each one of the vertices A, C, as in b4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case d): pencil of conics through two distinct points A and B, tangent  to each other at A with contact of order three;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' let L denote their common  tangent line at A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  d1) The skew vertex A, equipped with a quasi-global dual billiard struc-  ture: a degenerate S-angular symmetry σS  A centered at A defined by any  regular conic S from the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  d2) Any point C ∈ L \\ {A}, called a skew vertex, equipped with a  projective angular symmetry σC centered at C with fixed point line AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case e): pencil of conics through one given point A, tangent to each  other with contact of order four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let L denote their common  tangent line at A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  e1) The skew vertex A equipped with a degenerate S-angular symmetry  σS  A centered at A defined by any given regular conic S of the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  e2) Any point C ∈ L \\ {A}, called a standard vertex, equipped with a  projective angular symmetry σC centered at C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Its fixed point line is the  set of those points D ∈ P2 for which the line CD is tangent to the conic of  16  Figure 6:  Figure 7:  the pencil through D at D (including D = A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The definition of real (standard or skew) admissible vertex for a real pencil  of conics in RP2 is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='22 Consider a dual multibilliard formed by some conics and  maybe by some vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let the dual billiard structure at each vertex (if  any) be either global, or quasi-global, and that on each conic be either of  pencil type, or as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11, Case 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We say that its two conics  (vertices) are distinct, if they either are geometrically distinct, or coincide  as conics (vertices) but have different dual billiard structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='23 A (real or complex) dual multibilliard is said to be of  pencil type, if the following conditions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  1) All its curves are conics lying in one pencil, and their dual billiard  structures are defined by the same pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' (Case of one conic equipped with  a dual billiard structure of pencil type is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=')  2) All its vertices are admissible for the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  3) If the multibilliard contains a skew vertex equipped with a quasi-  global dual billiard structure, then it contains no other skew vertex, with  the following exceptions:  in Case c) the skew vertex collection is allowed to be the pair of vertices  A and C equipped with quasi-global structures defined by one and the same  (but arbitrary) regular conic S = SA = SC of the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  17  s  AT  cB  s  c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  L  A- in Case d) the skew vertex collection is allowed to be a pair of vertices  A and C defined by any given regular conic S of the pencil: the vertex A is  equipped with the quasi-global S-dual billiard structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' the vertex C is the  intersection point of the line L with the line tangent to S at B, equipped  with the projective angular symmetry with fixed point line AB (it coincides  with the S-angular symmetry centered at C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  4) In Case d) the multibilliard may contain at most one vertex C ∈  L \\ {A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  5) Each skew admissible vertex that a priori admits several possible dual  billiard structures listed above is allowed to be included in the multibilliard  with no more than one dual billiard structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Well-definedness of the above notion of admissible vertex and pencil type  dual multibilliard in the real case is implied by the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='24 1) Consider a real pencil of conics in RP2 whose com-  plexifications pass through four distinct but maybe complex points in CP2:  pencil of type a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' At least one vertex Mj from Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='21 is real, and  in this case the involution σMj is also real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hovewer in general Mj (KEL)  are not necessarily all real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  2) Consider a real pencil of conics whose complexifications form a pencil  of type b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' All its admissible vertices C, M, KAB are always real, and so  are the corresponding global projective involutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In the case, when C is  equipped with a quasi-global structure defined by a real conic, the correspond-  ing involution σC is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  3) For a real pencil with complexification of type c) the admissible vertex  M and the corresponding projective involution σM are both real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But the  vertices A, C, M′ are not necessarily real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' If M′ is real, then so is σM′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  4) For a real pencil with complexification of type d) or e) the admissible  vertex A is always real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The corresponding involution σA = σA,S is real, if  and only if so is the conic S defining it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='25 Let a (real or complex) dual multibilliard on a collection of  real C4-smooth (or holomorphic) nonlinear connected curves γj and some  vertices be rationally integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  1) Each curve γj is a conic equipped with a dual billiard structure either  of pencil type, or as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11, Case 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  2) If the multibilliard contains at least two distinct conics (in the sense  of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='22), then all the conics γj lie in the same pencil, and the  dual billiard structures on them are defined by the same pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  18  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='26 Let in a dual multibilliard all the curves be conics lying in  the same pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let they be equipped with the dual billiard structure defined  by the same pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (Case of one conic equipped with a pencil type dual  billiard structure is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=') Then the multibilliard is rationally integrable,  if and only if it is of pencil type, see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='27 The minimal degree of a rational integral of a pencil type  multibilliard is  (i) degree two, if it contains no skew vertices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (ii) degree 12, if the pencil has type a) and the multibilliard contains  some two non-opposite skew vertices, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=', a pair of vertices of type KEL and  KES for some distinct E, L, S ∈ {A, B, C, D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (iii) degree four in any other case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The next theorem yields a formula for integral of degree 12 of pencil type  multibilliards for pencils of type a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' To state it, let us introduce the fol-  lowing notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let RP2  [y1:y2:y3] denote the ambient projective plane of  the multibilliard, considered as the projectivization of the space R3  y1,y2,y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  For every projective line X let π−1(X) ⊂ R3 denote the corresponding  two-dimensional subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let ξX(Y ) denote a non-zero linear functional  vanishing on π−1(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It is well-defined up to constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='28 Consider a pencil of conics through four distinct base points  A, B, C, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Set M1 = AB ∩ CD, M2 = BC ∩ AD, M3 = AC ∩ BD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  1) The functionals ξEL corresponding to the lines EL through distinct  points E, L ∈ {A, B, C, D} can be normalized by constant factors so that  ξABξCD + ξBCξAD + ξADξBC = 0,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='21)  2) If (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='21) holds, then for every µ ∈ R\\0 the degree 12 rational function  �  {EL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='F N}̸={E′L′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='F ′N′}  � ξELξF N  ξE′L′ξF ′N′ (Y ) + µ)  �  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='22)  is a first integral of every pencil type multibilliard defined by the given pen-  cil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Here the product is taken over ordered pairs of two-line sets {EL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' FN},  {E′L′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' F ′N ′} with {E, L, F, N} = {E′, L′, F ′, N ′} = {A, B, C, D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In Theo-  rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='27, Case (ii) this is a minimal degree integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='29 A rationally integrable real (complex) dual billiard struc-  ture on conic that is not of pencil type, see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11, Case 2), will be  19  called exotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The singular points of the dual billiard structure (which are  exactly the indeterminacy points of the corresponding integral R from the  Addendum to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11) will be called the base points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='30 Let a rationally integrable real (complex) multibilliard be  not of pencil type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then it contains only one curve, namely, a conic equipped  with an exotic rationally integrable dual billiard structure from Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11, Case 2), and maybe some vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='31 A (real or complex) multibilliard consisting of one conic γ  equipped with an exotic dual billiard structure from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11, Case 2),  and maybe some vertices is rationally integrable, if and only if the collection  of vertices either is empty, or consists of the so-called admissible vertices  Q defined below, being equipped with the γ-angular symmetry σQ:  (i) Case of type 2a) dual billiard on γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The unique admissible vertex  is the intersection point Q = [1 : 0 : 0] of the z-axis and the infinity line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' one  has σQ(z, w) = (−z, w) in the chart (z, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In Subcase 2a1),  when ρ = 2 −  2  2N+1, the function R(z, w) from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7) is a rational integral of  the multibilliard (γ, (Q, σQ)) of minimal degree: deg R = 4N +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In Subcase  2a2), when ρ = 2−  1  N+1, the function R2(z, w) with R the same, as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='8),  is a rational integral of (γ, (Q, σQ)) of minimal degree: deg R2 = 4N + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (ii) Case of type 2b1) or 2b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' There are three base points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' One of them,  denoted X, is the intersection point of two lines contained in the polar locus  R = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The unique admissible vertex Q is the intersection point of  two lines: the tangent line to γ at X and the line through the two other  base points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  In Case 2b1) one has Q = (0, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  In Case 2b2) one has  Q = [1 : 0 : 0], σQ(z, w) = (−z, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The corresponding rational function  R, see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='9), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='10) is a rational integral of the multibilliard (γ, (Q, σQ)) of  minimal degree: deg R = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (iii) Case of type 2c1) or 2c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' There are three complex base points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  There are three admissible vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Each of them is the intersection  point of a line through two base points and the tangent line to γ at the other  one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In Case 2c1) the point (0, −1) is the unique real admissible vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  In Case 2c2) all the admissible vertices are real: they are (0, −1), (1, 0),  [1 : 1 : 0], see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The function R is a degree 6 rational integral of the  multibilliard formed by the conic γ and arbitrary admissible vertex collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (iv) Case of type 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' No admissible vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  20  Figure 8: The only admissible vertex in Case 2a) is the infinite point Q =  [1 : 0 : 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Figure 9: The admissible vertices in Cases 2b1) and 2c2) are marked in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  21  Case 2c2)  W  [1:1:0]  R=80  (1,1)  0  7  (1,0)  (0,-1)Case 2b1)  8  W  (1,1)  0  Q=(0,-1)Case 2a)  Q=[1:0:0]  W  0  zProposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='32 1) The two multibilliards of type (ii) (Cases 2b1), 2b2))  with one admissible vertex are complex-projectively equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 2) Two multi-  billiards of type (iii) (either of different subtypes 2c1), 2c2), or of the same  subtype) are complex-projectively equivalent, if and only if they have the  same number of vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 3) Two real multibilliards of type (iii) are real-  projectively equivalent, if and only if they have either both subtype 2c1) and  one real admissible vertex, or both subtype 2c2) and the same number (ar-  bitrary, from 1 to 3) of real admissible vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='32 follows from the last statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='12 and  the fact that the dual billiard of type 2c2) has real order three projective  symmetry cyclically permuting admissible points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='5  Application: classification of rationally 0-homogeneously  integrable piecewise C4-smooth projective billiards  Let us recall the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='33 The central projective billiard structure on a planar curve  C with center O ∈ R2 is the field of lines on C passing through O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Everywhere below for two projective lines e and f by ef we will denote  their intersection point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='34 Consider a complex dual pencil of conics: a family of con-  ics whose dual form a pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let ℓ be a line equipped with a projective  billiard structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' the corresponding line field is well-defined either on all  of ℓ, or on ℓ punctured at one point called singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The line ℓ is said to  be admissible for the dual pencil, if it belongs to the following list of lines  equipped with projective billiard structures, called either standard, or skew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case a): dual pencil of conics tangent to four distinct lines a, b, c, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  a1) The standard admissible lines are the lines m1, m2, m3 through the  points ab and cd, the points bc and ad, the points ac and bd respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The line m1 is equipped with projective billiard structure centered at m2m3,  and the projective billiard structures on m2, m3 are defined analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  a2) Let kbc denote the line through the points m1m3 and bc, equipped  with the projective billiard structure centered at ad: the field of lines through  ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let kad be the line through m1m3 and ad, equipped with the projective  billiard structure centered at bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The other lines kef, e, f ∈ {a, b, c, d}, e ̸= f,  equipped with central projective billiard structures are defined analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  22  We identify kef with kfe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' All the six lines kef thus constructed are called  skew admissible lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case b): dual pencil of conics tangent to three distinct lines a, b, c and  having common tangency point C with c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  b1) The skew line c equipped with the field of lines through the point  ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  b2) The skew line k such that the quadruple of lines a, b, m, k through  the point ab is harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It is equipped with the field of lines through C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Here m is the line through C and ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  b3) The standard line m with the field of lines through the point ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  b4) For arbitrary given conic S of the dual pencil the line c equipped  with the field of lines tangent to S is a skew line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case c): dual pencil of conics tangent to each other at two points A and  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Led a and b denote the corresponding tangent lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  c1) The standard line m = AB with the field of lines through ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  c2) The skew lines a and b equipped with the fields of lines through B  and A respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  c3) Fix arbitrary line c ̸= a, b through ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let Z ∈ m denote the point  such that the quadruple of points cm, Z, B, A ∈ m is harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The line c  equipped with the field of lines through Z is called a skew line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  c4) Fix a regular conic S from the dual pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The lines a and b, each  being equipped with the field of lines tangent to S at points distinct from  A and B respectively are called skew lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case d): dual pencil of conics tangent to a given line a at a given point  A, having triple contact between each other at A, and tangent to another  given line b ̸= a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  d1) The skew line a equipped with the line field tangent to a given  (arbitrary) regular conic S from the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  d2) Any line c ̸= a through A called skew, equipped with the field of  lines through the point ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case e): dual pencil of conics tangent to each other at a point A with  order 4 contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let a denote their common tangent line at A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  e1) The skew line a equipped with the line field tangent to a given (ar-  bitrary) regular conic S from the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  e2) Any line b through A called skew, equipped with the field of lines  tangent to the conics of the pencil at points of the line b: these tangent lines  pass through the same point C = C(b) ∈ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='35 In Case a) for every distinct e, f, g ∈ {a, b, c, d} the lines  23  Figure 10: Dual pencil of type a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The standard admissible lines are m1,  m2, m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The skew admissible lines kef are marked in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Figure 11: Dual pencil of type b): i) one standard line m and two skew lines  c, k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' ii) skew line c with another line field, tangent to a given conic S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  24  i)  ii)  s  ck  Ic  c  Cbd  d  b  m  d  ad  a  ac  m1  m2  Kab  Ibc  cd  m2m3Figure 12: Dual pencil of type c): i) one standard line m and two skew lines  a, b equipped with central projective billiard structures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' ii) arbitrary line  c ̸= a, b through ab (called skew) with field of lines through Z, and the skew  lines a, b with fields of lines tangent to a given conic S from the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Figure 13: Dual pencil of type d): the skew line a and an arbitrary skew  line c ̸= a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Figure 14: Dual pencil of type e): the skew line a and a standard line b ̸= a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  25  s  b  A  c  ab  S  ab  a  ATi)  cml  i)  b  m  B  b  S  z  ab  ab  m  a  a  AT  IAkef, kfg, kge pass through the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In particular, the line kab passes  through the intersection points kbd ∩ kad and kac ∩ kbc, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof The latter intersection points and the point m2m3 lie on one line, by  the dual Desargues Theorem applied to the triangles (bd, ad, kbd ∩ kad) and  (ac, bc, kac ∩ kbc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The point ab lies on the same line, by Pappus Theorem  applied to the triples of points bd, m1m3, ac and ad, m1m2, bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='36 A projective billiard with piecewise-C4-smooth boundary  having at least one nonlinear smooth arc is said to be of dual pencil type, if  it satisfies the following conditions:  1) Each C4-smooth arc of the boundary is either a conical arc, or a  segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' All the conical arcs lie in the same dual pencil and are equipped  with the projective billiard structure defined by the same pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  2) The segments in the boundary are contained in lines admissible for  the pencil and are equipped with the projective billiard structures of the  ambient admissible lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  3) If the boundary contains a skew line segment whose projective billiard  structure is not a central one (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=', given by a field of lines tangent to a conic  of the pencil), then the boundary contains no segments of other skew lines  with the following exceptions of possible of ambient skew line collections:  in Case c) the skew line collection is allowed to be the pair of lines a  and b equipped with the fields of lines tangent to one and the same (but  arbitrary) conic of the pencil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  in Case d) the skew line collection is allowed to be a pair of lines a and  c: a being equipped with the field of lines tangent to a given conic S from  the pencil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' c is the line through the point A and the tangency point of the  conic S with the line b, equipped with the field of lines throgh the point ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  4) In Case d) the boundary may contain a segment of at most one line  c ̸= a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  5) Each ambient skew line of a boundary segment that a priori admits  several possible projective structures listed above is allowed to be included  in the boundary with no more than one projective billiard structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='37 The notion of admissible line for dual pencil is dual to that  of admissible vertex of a pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='38 Let a planar projective billiard with piecewise C4-smooth  boundary containing a nonlinear arc be rationally integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then the fol-  lowing statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  26  1) All the nonlinear arcs of the boundary are conical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Different arcs of  the same conic are equipped with the restriction to them of one and the same  projective billiard structure on the ambient conic: either of dual pencil type,  or a one from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='16, Case 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  2) If the boundary contains at least two arcs of two distinct regular conics,  then all the ambient conics of nonlinear arcs lie in the same dual pencil and  their projective billiard structures are defined by the same dual pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='39 Let a planar projective billiard have piecewise C4-smooth  boundary whose all nonlinear C4-smooth pieces are conical arcs lying in the  same dual pencil and equipped with projective billiard structures defined by  the same pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then the billiard is 0-homogeneously rationally integrable,  if and only if it is of dual pencil type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='40 The minimal degree of 0-homogeneous rational integral of  a dual pencil type projective billiard is  (i) degree two, if its boundary contains no skew line segment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (ii)) degree 12, if the dual pencil has type a) and the billiard bound-  ary contains segments of some two skew admissible lines kef, kfs for some  distinct e, f, s ∈ {a, b, s, d};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (iii) degree four in any other case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='41 Consider a type a) dual pencil of conics tangent to given  four distinct lines a, b, c, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us consider the ambient plane R2  x1,x2 as the  horizontal plane {x3 = 1} ⊂ R3  x1,x2,x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Set  r := (x1, x2, 1) ∈ R3, v = (v1, v2, 0) for every (v1, v2) ∈ T(x1,x2)R2,  M = M(r, v) := [r, v] = (−v2, v1, ∆), ∆ := x1v2 − x2v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='23)  In the above notations for intersection points em of lines e and m set  r(em) = (x1(em), x2(em), 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  There exists a collection of three numbers χem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='fn ∈ R, {e, m, f, n} = {a, b, c, d},  indexed by unordered pairs of intersection points em = e ∩ m, fn = f ∩ n  ((em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' fn) = (fn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' em), by definition) such that  �  (em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='fn)  χem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='fn < r(em), M >< r(fn), M >= 0  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='24)  27  Figure 15: Pencil of type a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Distances between intersection points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  as a quadratic form in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This collection is unique up to common constant  factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' For every µ ∈ R \\ {0} the corresponding expression  �  (em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='fn)̸=(e′m′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='f′n′)  �  χem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='fn < r(em), M >< r(fn), M >  χe′m′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='f′n′ < r(e′m′), M >< r(f ′n′), M > + µ  �  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='25)  is a degree 12 first integral of every projective billiard of dual pencil type  defined by the pencil in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Here the product is taken over ordered ”big”  pairs: any two indices ((em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' fn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' (e′m′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' f ′n′)) and ((e′m′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' f ′n′), (em;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' fn))  that differ by permutation correspond to two distinct factors in the product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='42 Consider the following segment lengths, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 15:  ρ := |bc − ab|, t := |ab − bd|, τ := |ad − ab|, s := |ab − ac|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='26)  Here the lengths are oriented: lengths of two adjacent aligned segments (say,  s and τ), are taken with the same sign, if their common end separates them  (ab lies between the points ac and ad), and with opposite signs otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Relation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='24) (and hence, the statements of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='41) hold for  (χab;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='cd, χbc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='ad, χac;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='bd) =  � st  ρτ − 1, − st  ρτ , 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='27)  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='43 A rationally 0-homogeneously integrable projective bil-  liard structure on a conic γ that is not of dual pencil type (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=', any projec-  tive billiard structure from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='16, Case 2)) will be called exotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Its  singular points (which are the points, where the corresponding line field is  either undefined, or tangent to γ) will be called the base points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  28  cd  b  bc  ad  0  ab  h  s  a  ac  t  bd  IdCorollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='44 Let a rationally integrable projective billiard with piecewise  C4-smooth boundary be not of dual pencil type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let its boundary contain at  least one nonlinear arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then all its nonlinear arcs lie in one conic, equipped  with an exotic projective billiard structure from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='16, Case 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='45 Let a projective bulliard has piecewise smooth boundary  consisting of arcs of one and the same conic γ equipped with an exotic  projective billiard structure from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='16, Case 2), and maybe some  straightline segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The billiard is rationally integrable, if and only if  the collection of ambient lines of the boundary segments either is empty, or  consists of the following admissible lines equipped with central-projective  billiard structures:  (i) Case of type 2a) projective billiard structure on γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' ρ = 2− 2  m, m ∈ N,  m ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The unique admissible line is the vertical x2-axis, equipped with  the normal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=', horizontal) line field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The projective billiard  bounded by a half of γ and the x2-axis has a rational 0-homogeneous integral  of minimal degree 2m: the function Ψ2a1 from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='14) for odd m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' the function  Ψ2  2a2 with Ψ2a2 the same, as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='15), for even m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (ii) Case of type 2b1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The unique admissible line is the line {x2 = 1}  equipped with the field of lines through the point (0, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (iii) Case of type 2b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The unique admissible line is the Ox2-axis  equipped with the normal (horizontal) line field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In both cases  2b1), 2b2) the functions Ψ2b1, Ψ2b2 from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='16) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='17) are integrals of  minimal degree 4 for each billiard bounded by γ and the admissible line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (iv) Case of type 2c1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The unique admissible line is the line {x2 = 1}  equipped with the field of lines through the point (0, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (v) Case of type 2c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' There are three admissible lines:  the line {x2 = 1}, with the field of lines through the point (0, −1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  the line {x1 = − 1  2}, with the line field parallel to the vector (−1, 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  the line {x2 = −2x1}, with the field of lines through the point (−1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In both cases 2c1), 2c2) the corresponding functions Ψ2c1,  Ψ2c2 from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='18) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='19) are integrals of minimal degree 6 for each billiard  bounded by γ and segments of admissible lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (vi) Case of type 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' No admissible lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6  Plan of proofs of main results  Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  In Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1 we prove rational integrability of pencil type  complex multibilliard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (This implies analogous result in the real case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=')  To do this, we show that for every pencil all the involutions associated  29  Figure 16: The only admissible line in Case 2a) is the Ox2-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Figure 17: The only admissible line in Case 2b1) is the line {x2 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The  only admissible line in Case 2b2) is the x2-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Figure 18: The only admissible line in Case 2c1) is the line {x2 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In Case  2c2) there are three admissible lines: {x2 = 1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' {x1 = − 1  2}, {x2 = −2x1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  30  2c2)  X2 =-2x,  Y  0  1  1/2  12c1)  Y  0  1  12b2)  Y  02b1)  Y  0  1  12a)  Y  (2-p)X1  ￥1  0to all the corresponding admissible vertices preserve the pencil and act  on its parameter space C by conformal involutions (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We  fix an arbitrary collection of admissible vertices and consider the subgroup  G ⊂ Aut(C) = PSL2(C) generated by the corresponding conformal involu-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We show that finiteness of the group G is equivalent to the system of  Conditions 3)–5) of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='23 of pencil type multibilliard (Proposition  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4), and in this case G is either trivial, or isomorphic to either Z2, or S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  We then deduce rational integrability of every pencil type multibilliard with  integral of degree 2|G| ∈ {2, 4, 12}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  To classify rationally integrable dual multibilliards, in what follows we  consider an arbitrary dual multibilliard with a rational integral Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Each its  curve is already known to be a conic equipped with either a pencil type dual  billiard structure, or an exotic billiard structure from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11, Case  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We fix some its conic S and consider the canonical integral R of its dual  billiard structure: either a quadratic integral in the case of pencil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' of the  corresponding integral from the Addendum to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2 we show that the singular foliations Ψ = const  and R = const on CP2 coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  We show that a generic level curve of  the integral R is irreducible, of the same degree d = deg R, and thus, R is a  rational first integral of minimal degree for the above foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In the exotic  case we also show that the conic S is its unique level curve of multiplicity d,  which means that the irreducible level curves of the function R accumulating  to S converge to d  2[S] as divisors: the intersection of a small cross-section to  S with a level curve close to S consists of d  2 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We then deduce (in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2) that if on some conic of  the multibilliard the dual billiard structure is defined by a pencil (or if the  multibilliard contains at least two distinct conics), then the above foliation  coincides with the pencil and the dual billiard structures on all the other  conics are defined by the same pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This will prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Results  of Step 1 together with constance of integral on the conics of the pencil (given  by Step 2) imply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3 we study vertices of the multibilliard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' First we  show that the family of involutions σA,ℓ : ℓ → ℓ associated to each vertex A  is given by the restrictions to the lines ℓ through A of a birational involution  σA : CP2 → CP2 preserving the foliation Ψ = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then we deduce that  each σA is either a projective angular symmetry, or a degenerate angular  symmetry defined by a regular conic S through A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We show that in the  latter case the foliation Ψ = const is a pencil of conics containing S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4 we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It deals with the case,  31  when the foliation Ψ = const is a pencil of conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We show that each vertex  of the multibilliard is admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Each level curve of the function Ψ is a  collection of at most deg Ψ  2  conics of the pencil, and it is invariant under the  involutions defining the dual billiard structures at the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This implies  finiteness of the group G generated by the conformal involutions correspond-  ing to the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Together with the results of Step 1 (Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1), this  implies that the multibilliard is of pencil type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  In Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='5 we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='31 on classification of rationally  integrable multibilliards consisting of a conic S with an exotic dual billiard  structure and (may be) some vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We describe the corresponding ad-  missible vertices using the result of Step 4 stating that the corresponding  involutions are projective angular symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  In Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6 we prove Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  In Section 3 we prove the main results on classification of rationally 0-  homogeneously integrable projective billiards with C4-smooth boundaries  (Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='38, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='39, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='40, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We reduce them to the main results on  dual multibilliards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Namely, we consider the projective duality given by  orthogonal polarity, which transforms a projective billiard to a dual multi-  billiard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We consider the ambient plane R2  x1,x2 of the projective billiard as  the horizontal plane {x3 = 1} ⊂ R3  x1,x2,x3, set r = (x1, x2, x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We use the  fact that a rational 0-homogeneous integral of a projective billiard can be  written as a rational 0-homogeneous function R(M) of the moment vector  M = [r, v], where v is the velocity, and R(M) is a rational integral of the  corresponding dual multibilliard in RP2  [M1:M2:M3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We show that this yields  a bijective correspondence between rational 0-homogeneous integrals of the  projective billiard and rational integrals of the corresponding dual multibil-  liard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This together with the results from [27] on duality between exotic dual  billiards from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11 and exotic projective billiards from Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='16 and the results of the present paper on dual multibilliards will imply  the main results on projective billiards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7  Historical remarks  Existence of a continuum of closed caustics in every strictly convex bounded  planar billiard with sufficiently smooth boundary was proved by V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Lazutkin  [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Existence of continuum of foliations by (non-closed) caustics in open  billiards was proved by the author [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Poritsky [33] (and later E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Amiran  [2]) proved the Birkhoff Conjecture under the additional assumption that for  every two caustics the smaller one is a caustic for the bigger one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Bialy  [3] proved that if the phase cylinder is foliated by non-contractible invariant  32  curves for the billiard map, then the billiard table is a disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See also [42],  where another proof of Bialy’s result was given, and Bialy’s papers [4, 5] for  similar results on billiards on constant curvature surfaces and on magnetic  billiards on these surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Treschev conjectured existence of billiards  where the squared billiard map has fixed point where its germ is analyti-  cally conjugated to rotation and confirmed this by numerical experiments:  in two dimensions [37, 38] and in higher dimensions [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Kaloshin and  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Sorrentino [28] proved that any integrable deformation of an ellipse is an  ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' For ellipses with small excentricities this result was earlier proved by  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Avila, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Kaloshin and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' De Simoi [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Recently M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Bialy and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Mironov  proved the Birkhoff Conjecture for centrally-symmetric billiards admitting  a continuous family of caustics extending up to a caustic of 4-periodic or-  bits [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' For a dynamical entropic version of the Birkhoff Conjecture and  related results see [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' For a survey on the Birkhoff Conjecture and results  see [28, 29, 12] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Veselov proved a series of complete integrability results for billiards  bounded by confocal quadrics in space forms of any dimension and described  billiard orbits there in terms of a shift of the Jacobi variety corresponding  to an appropriate hyperelliptic curve [40, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Dynamics in (not necessarily  convex) billiards of this type was also studied in [15, 16, 17, 18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The Polynomial Birkhoff Conjecture together with its generalization to  piecewise smooth billiards on surfaces of constant curvature was stated by  S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Bolotin and partially studied by himself, see [13], [14, section 4], and  by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Bialy and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Mironov [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Its complete solution is a joint result of  M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Bialy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Mironov and the author given in the series of papers [8, 9, 24,  25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It implies that if a polynomial integral of a piecewise smooth billiard  exists, then its minimal degree is equal to either two, or four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' For a survey  of Bolotin’s Polynomial Birkhoff Conjecture and of its version for magnetic  billiards (an open conjecture, with a substantial progress made in [7, 10])  and related results see [30, 29, 8, 9, 7, 10, 11] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The generalization of the Birkhoff Conjecture to dual billiards was stated  by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Tabachnikov in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Its rationally integrable version was solved by the  author of the present paper in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Its polynomially integrable version for  outer billiards was stated and partially studied in [36] and solved completely  in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Projective billiards were introduced by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Tabachnikov [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  He  had shown in the same paper that if a projective billiard on circle has an  invariant area form smooth up to the boundary of the phase cylinder, then  it is integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  A series of results on the analogue of Ivrii Conjecture on periodic orbits in  billiard (stating that their Lebesgue measure is zero) for projective billiards  33  was obtained by C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='Fierobe [20, 21, ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  2  Rationally integrable dual multibilliards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proofs  of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='26, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='31, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='27  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1  Rational integrability of pencil type multibilliards  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1 Consider a complex pencil of conics and the correspond-  ing admissible vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' For every standard vertex the corresponding invo-  lution leaves invariant each conic of the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' For every skew vertex the  corresponding involution permutes conics of the pencil non-trivially: it acts  as a conformal involution of the parameter space C of the complex pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof Case a): pencil of conics through four distinct basic points A, B, C,  D, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It is well-known that in this case no three of them lie on the  same line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This implies that the three vertices Mj are well-defined, distinct,  do not lie on the same line and different from the basic points, and so are  the vertices KEL, and the latter are distinct from the vertices Mj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Set  Γ1 := AB ∪ CD, Γ2 := BC ∪ AD, Γ3 = AC ∪ BD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1)  Let σM1 : CP2 → CP2 be the Γ2-angular symmetry centered at M1: the  projective involution fixing each line through ℓ and permuting its intersection  points with the lines AD and BC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It permutes the points A and B, C and  D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, it preserves the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It fixes the line M2M3, which passes  through the points AD ∩ BC and AC ∩ BD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, it fixes each its point  X, since it fixes the line M1X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The pencil is parametrized by a parameter  λ ∈ C, and σM1 acts on Cλ by conformal automorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let λ1, λ2, λ3 ∈ C  denote the parameter values corresponding to the singular conics Γ1, Γ2,  Γ3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Each Γj is σM1-invariant, by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, the  conformal automorphism C → C induced by σM1 fixes three distinct points  λ1, λ2, λ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, it is identity, and σM1 preserves each conic of the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  This proof is valid for the other vertices Mj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The involution σKBC is the projective angular symmetry centered at  KBC with fixed point line AD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, it fixes M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The involution σKBC permutes B and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Or equivalently, the  quadruple of points KBC, M2, B, C on the line BC is harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof The restriction of the involution σKBC to the line BC coincides with  the involution σM2, since both of them are non-trivial projective involutions  of the line BC fixing KBC and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The involution σM2 permutes B and C,  as in the above discussion on σM1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, so does σKBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  34  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2 Each one of the involutions σKBC, σAD fixes Γ2 and per-  mutes Γ1, Γ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, it yields a non-trivial conformal involution Cλ → Cλ  of the parameter space of the pencil, fixing λ2 and permuting λ1, λ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof The involution σKBC fixes A, D and permutes B, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Similarly, the  involution σKAD fixes B, C and permutes A, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Case b): pencil of conics through three distinct points A, B, C tangent  at the point C to the same line L, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The involution σM fixes  the points C, KAB, the line L and permutes A and B, by definition and  harmonicity of the quadruple M, KAB, A, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, it preserves the  pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Similarly, the involution σKAB preserves the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' And so does  the involution σC : CP2 → CP2 defined to fix C and each point of the line  AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Now the pencil, parametrized by a parameter λ ∈ C, contains just two  singular conics:  Γ1 := AB ∪ L, Γ2 := AC ∪ BC,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2)  corresponding to some parameter values λ1 and λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The involution σM and the composition σC ◦ σKAB preserve  each conic of the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Each one of the involutions σC, σKAB fixes only  the conics Γ1, Γ2 of the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof  The pencil in question is the limit of a family of pencils of conics  through A, B, Cµ, Dµ with basic points Cµ, Dµ depending on small pa-  rameter µ, confluenting to C, as µ → 0, so that the line CµDµ pass through  M = M1 and tends to the tangent line L, as µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then M2 = M2(µ) → C,  M3 = M3(µ) → C, and the involutions σM1 = σM1(µ) corresponding to the  perturbed pencil, with µ ̸= 0, converge to σM, as µ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The involution  σM1(µ) preserves each conic of the pencil for µ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, so does its limit  σM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The involutions at the vertices KCµDµ, KAB converge to σC and σKAB,  by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' They act on the perturbed pencil as non-trivial involutions,  permuting conics in the same way (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, this statement  remains valid for their limits σC and σKAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3 Consider a pencil of complex conics that are tangent to  each other at a point C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let S be its regular conic, and let C be equipped with  the quasi-global dual billiard structure defined by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then the correspond-  ing involution σC preserves the pencil and induces a nontrivial conformal  involution C → C of its parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof Let L denote the common projective tangent line at C to the regular  conics of the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us take an affine chart Cz,w = CP2 \\ L so that C is  35  the intersection point of the w-axis with the infinity line L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then the conics  of the pencil are parabolas Sλ := {w = (a1z2+b1z+c1)+λ(a2z2+b2z+c2)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let us normalize the parameter λ so that S0 = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then in the affine chart  (z, w) one has  σC(z, w) = (z, 2(a1z2 + b1z + c1) − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Hence, σC(Sλ) = S−λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The proposition is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Case c): pencil of conics through two distinct points A and C tangent  to two given lines LA and LC through them;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' LA, LC ̸= AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Fix  an arbitrary point M′ ∈ AC \\ {A, C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The projective angular symmetries σA, σC with fixed point  lines LC and LA respectively preserve the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The involutions σM, σM′  and the composition σA ◦ σC preserve each conic of the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The claim is proved analogously to the above discussion, by considering  the pencil in question as the limit of the family of pencils through points  Aµ, Bµ, Cµ, Dµ, Aµ, Bµ → A, Cµ, Dµ → C, as µ → 0 so that AµBµ = LA,  CµDµ = LC, and the lines AµCµ, BµDµ are intersected at M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Similarly to  the above discussion, the involutions corresponding to KAµBµ and KCµDµ  converge to σA and σC respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This implies the statement of the claim  on the involutions σM, σA, σC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It remains to prove its statement on the  vertex M′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The intersection point M2(µ) of the lines BµCµ and AµDµ, the  point M′, and the intersection points of the line M2(µ)M′ with lines LA,  LC form a harmonic tuple of points on the line M2(µ)M′, as in Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Hence, the involution σM′,µ corresponding to the vertex M′ and the per-  turbed pencil, with µ ̸= 0, fixes M′ and each line through M′ and permutes  its intersection points with the lines LA and LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, it coincides with  the involution σM′ corresponding to the nonperturbed pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, σM′  preserves each conic of the nonperturbed pencil, as of the perturbed one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Consider the skew vertices in Cases c), d), e) equipped with quasi-global  dual billiard structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The corresponding involutions preserve the pencil  and induce nontrivial conformal involutions of Cλ, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Consider now the vertices C in Cases d) and e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In Case d) the involution  σC preserves the singular conic L ∪ AB of the pencil and the conic tangent  to BC at B, as σM in Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, it preserves the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It does not  preserve other conics, since their tangent lines at B are not σC-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  In Case e) σC preserves each conic of the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This can be seen in the  affine chart (z, w) for which C, A are the intersection point of the infinity  line with the z- and w-axes respectively, and the conics are the parabolas  w = z2 + λ: σC(z, w) = σC(−z, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  36  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4 Let in a complex dual multibilliard all the curves be con-  ics lying in a pencil, and their dual billiard structures be defined by the same  pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let all its vertices be admissible for the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let G ⊂ PSL2(C) =  Aut(C) denote the group generated by conformal transformations of the pa-  rameter space C of the pencil induced by the involutions assotiated to the  vertices, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then the following statements are equivalent:  (i) The group G is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (ii) The vertex collection satisfies Conditions 3)–5) of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  If the group G is finite, then it is either trivial (if and only if the multibilliard  contains no skew vertex), or isomorphic to Z2 or S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' One has G = S3, if  and only if the pencil has type a) and the multibilliard contains some two  skew vertices KEX, KEY with base point pairs having one common point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof Let us first prove equivalence of statements (i) and (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case of pencil of type a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then Conditions 3)–5) of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='21 im-  pose no restriction on admissible vertex collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The involution defining  the dual billiard structure at each admissible vertex preserves the triple of  the singular conics Γ1, Γ2, Γ3 of the pencil, see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, the confor-  mal involutions C → C of the parameter space defined by the skew vertices  permute the corresponding parameter values λ1, λ2, λ3, and hence, gen-  erate a finite group G ⊂ PSL2(C) isomorphic to a subgroup of S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The  conformal involution corresponding to a standard vertex is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Each  one of the involutions σKBC, σAD fixes the singular conic Γ2 and permutes  Γ1, Γ3 (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See the above proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' These two  statements and the versions of the latter statement for the other base points  together imply the statements of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case of pencil of type b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' For every skew vertex equipped with a projec-  tive angular symmetry the latter symmetry fixes only the parameter values  λ1, λ2 corresponding to the singular conics Γ1, Γ2 from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2), see Claim 2  in the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Suppose the multibilliard contains only vertices of the above type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then  the group G is either trivial (if the skew vertex subset is empty), or isomor-  phic to Z2 (if it is non-empty), by the above statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let now the multibilliard contain the skew vertex C equipped with a  degenerate S-angular symmetry defined by a regular conic S of the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let λS denote the parameter value corresponding to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The conformal  involution corresponding to the vertex C fixes only λ2 and λS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, if  the multibilliard contains no other skew vertices, then G ≃ Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' If it contains  another skew vertex, then G is generated by two involutions having only one  common fixed point λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Their composition is a parabolic transformation  37  with the unique fixed point λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It has infinite order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, G is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case of pencil of type c) is treated analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case of pencil of type d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The involution σC corresponding to a skew  vertex C ∈ L \\ {A} is a projective angular symmetry fixing two conics: the  singular conic L ∪ AB and the regular conic S of the pencil that is tangent  to the line CB at B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The correspondence S �→ C is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This implies  that G is finite, if and only if the involution corresponding to any other skew  vertex of the multibilliard fixes the same conic S, as in the above discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  This holds, if and only if Conditions 3)–5) of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='21 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case of pencil of type e) is treated analogously, with the singular conic  now being the double line L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='5 Every multibilliard of pencil type is rationally integrable,  with integral of minimal degree 2|G| ∈ {2, 4, 12}, where |G| is the cardinality  of the group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof  The group G is finite, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4 and since the multibil-  liard is of pencil type (hence, satisfying Conditions 3)–5) of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let F be a quadratic first integral of the pencil: the ratio of two quadratic  polynomials defining two its conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Its constant value on each conic co-  incides with the corresponding parameter λ (after replacing F by its post-  composition with conformal automorphism C → C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The product �  g∈G g◦F  is a rational first integral of the multibilliard, since it is invariant under the  involutions corresponding to the vertices (by definition) and the dual bil-  liard involution of each tangent line to a multibilliard conic permutes its  intersection points with each conic of the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='5 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2  Foliation by level curves of rational integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof of  Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='25 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='26  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6 Consider a rationally integrable dual billiard structure on  a complex conic γ (which belongs to the list given by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In  the case, when it is defined by a pencil of conics, its canonical integral is a  quadratic rational function constant on each conic of the pencil that vanishes  on γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In the case, when it is exotic, its canonical integral is the one given  by the Addendum to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11 (whose zero locus is γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  38  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7 Every rational integral of a rationally integrable dual bil-  liard on a conic is constant on each irreducible component of each level curve  of its canonical integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof  Let γ be the conic in question, Ψ be a rational integral of the dual  billiard, and let R be its canonical integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We have to show that Ψ ≡ const  along the leaves of the foliation R = const (which are, by definition, the  irreducible components of level curves of the function R with its critical and  indeterminacy points deleted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It suffices to prove the above statement in a  small neighborhood of the conic γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Fix a point P ∈ γ such that it is a regular  point for the foliation and the dual billiard involution σP is defined there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (In fact, σP is well-defined whenever P is regular for the foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But we  will not use this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=') Let U ⊂ CP2 be a small neighborhood of the point P that  is a flowbox for the foliation R = const and whose closure is disjoint from  singular points of the foliation and indeterminacy points for the involution  family σt, t ∈ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We equip it with biholomorphic coordinates (x, y), where  the local leaves of the flowbox are the horizontal fibers y = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Fix a  point P0 /∈ γ close to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Take a tangent line ℓ0 to γ through P0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' let Q0  denote the tangency point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Set P1 = σQ0(P0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let ℓ1 be the tangent line to  γ through P1 distinct from ℓ0, and let Q1 be their tangency point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Set  P2 = σQ1(P1),  etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' PN = σQN−1(PN−1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  xj = x(Pj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Here N is the biggest number such that P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' , PN, Q0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' , QN−1 ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We  claim that as P0 → P, the cardinality N = N(P0) of the above sequence  tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This follows from the fact the involutions σQ|LQ, Q ∈ γ∩U  are uniformly asymptotic to the central symmetries x �→ 2x(Q) − x with  respect to the points x(Q), as x−x(Q) → 0 and Q ∈ U: they are non-trivial  conformal involutions of the lines LQ with fixed points Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, N is  bigger than the product of the degrees deg Ψ deg R, whenever P0 is close  enough to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' One has  Ψ(P0) = · · · = Ψ(PN),  R(P0) = · · · = R(PN),  since both Ψ and R are integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This together with Bezout Theorem and  the above inequality implies that Ψ ≡ const along each leaf of the foliation  R = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='8 Let R be a rational first integral of an exotic dual billiard struc-  ture from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11 given by the corresponding formula in its addendum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  39  1) For all but a finite number of values of λ ∈ C the complex level curve  Γλ := {R = λ}  is irreducible of degree d = deg R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In the case, when R is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7) or  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='8), the curve Γλ is irreducible for every λ ̸= 0, ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  2) The (punctured) curve γ = {w = z2} is a multiplicity d  2 leaf of the  foliation R = const, which means that each small transversal cross-section  to γ intersects each leaf close enough to γ (dependently on cross-section)  transversely at d  2 distinct points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' or equivalently, [Γλ] → d  2[γ] as divisors, as  λ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  3) The curve γ is the unique nonlinear multiplicity d  2 leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof  Let us prove Statement 1), on irreducibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us first consider  Case 2a1), when R(z, w) =  (w−z2)2N+1  �N  j=1(w−cjz2)2 , see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The germ of the curve Γλ at the point Q = [0 : 1 : 0] ∈  CP2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=', at the intersection point of the infinity line with the w-axis) is  irreducible, whenever λ ̸= 0, ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof Let λ ̸= 0, ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In the affine chart (�z, �w) = ( z  w, 1  w) centered at Q one  has  Γλ = {( �w − �z2)2N+1 − λ �w2  N  �  j=1  ( �w − cj�z2)2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  In the new coordinates (�z, u), u := �w − �z2, Γλ is the zero locus of the  polynomial  Pλ(�z, u) := u2N+1 − λ(u + �z2)2  N  �  j=1  (u + (1 − cj)�z2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3)  It suffices to show that the germ of the polynomial Pλ at the origin is  irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  To do this, we will deal with its Newton diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Namely,  consider the bidegrees (m, n) ∈ (R2  ≥0)x,y of all the monomials �zmun entering  Pλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Consider the convex hull of the union of the corresponding quadrants  (m, n)+R2  ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The union ND of its boundary edges except for the coordinate  axes is called the Newton diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We claim that the Newton diagram of  the polynomial Pλ is one edge E = [(4N + 4, 0), (0, 2N + 1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Indeed, the  bidegrees of the monomials entering Pλ are (0, 2N + 1) and a collection of  bidegrees lying in the line {2y + x = 4N + 4}, since the multiplier at λ in  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3) is a (2, 1)-quasihomogeneous polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But the bidegrees lying in  the latter line lie above the edge E, except for its vertex (4N + 4, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This  proves that ND = E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  40  Suppose the contrary: the germ of the polynomial Pλ is not irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Then it is the product of two germs of analytic functions with Newton  diagrams being edges parallel to E whose endpoints lie in the lattice Z2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The latter edges should be closer to the origin than E and have smaller  lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But E is the edge of smallest length among all the above edges,  since E contains no integer points in its interior: the numbers 4N + 4 and  2N + 1 are coprime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The contradiction thus obtained proves irreducibility  of the germ of the polynomial Pλ and hence, Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Recall that a germ of analytic curve is irreducible, if and only if it is a  parametrized curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This together with Claim 4 implies that for every fixed  λ ̸= 0, ∞ there exists a neighborhood U = U(Q) ⊂ CP2 (depending on λ)  such that the intersection Γλ,U := Γλ ∩(U \\{Q}) is a connected submanifold  in U \\ {Q} and every line L close enough to the w-axis intersects Γλ,U  at 2N + 1 distinct points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, all the latter points lie in the same  irreducible component of the curve Γλ, as Γλ,U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, the latter component  has degree at least 2N + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But the ambient curve Γλ has degree at most  2N + 1 = deg R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, Γλ coincides with its irreducible component in  question, and hence, is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case of integral R given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='8) is treated analogously with the following  modification: in the above coordinates (�z, u) the Newton diagram of the new  polynomial Pλ is [(2N + 3, 0), (0, N + 1)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 2N + 3, N + 1 are again coprime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  For the proof of Statement 1) of the lemma for the other integrals from  the Addendum to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11 it suffices to prove irreducibility of level  curve {R = λ} for an open subset of values λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We will prove this for generic  small λ: for an open set of values λ accumulating to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Indeed, it is well-  known that if the level curve {R = λ} of a rational function is irreducible for  an open subset of values λ, then it is irreducible for all but a finite number  of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This is implied by the two following statements:  each indeterminacy point can be resolved by a sequence of blow-ups, so  that the function in question becomes a well-defined C-valued holomorphic  funciton on a new connected compact manifold, a blown-up CP2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  every non-constant holomorphic C-valued function on a connected com-  pact complex manifold has finite number of critical values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The other canonical rational integrals have degrees 4 or 6 and the type  R(z, w) = (w − z2)m  Φ(z, w) , Φ is a polynomial, deg Φ = 2m, m ∈ {2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4)  41  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='9 Let R be as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let there exist a sequence of values  λ converging to zero for which the curve Γλ := {R = λ} is not irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Then the foliation R = const is a pencil of conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof  Passing to a subsequence we can and will consider that one of the  following statements holds for all above λ:  (i) m = 2 and Γλ is a union of two regular conics C1,λ, C2,λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (ii) Γλ contains a line;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (iii) m = 3 and Γλ is a union of two regular cubics C1,λ, C2,λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (iv) m = 3 and Γλ is a union of three regular conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Statement (ii) cannot hold: the contrary would imply that the limit conic  Γ0 = γ = {w = z2} = limλ→0 Γλ contains a line, which is not true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Suppose  (iii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then each cubic considered as a divisor of degree three converge  to an integer multiple of the divisor [γ] of degree two: thus, to a divisor  of even degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This is obviously impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, the only possible  cases are (i) and (iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The a priori possible intersection points of the conics  from (i), (iv) lie in the finite set of indeterminacy and critical points of the  rational function R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, passing to a subsequence one can and will  achieve that a family of conics Cλ ⊂ Γλ lies in a pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The function R is  constant on them for infinite number of values of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, it is constant  on each conic of the pencil, since the set of those parameters of the pencil  for which R = const on the corresponding conics is finite (being algebraic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Finally, the foliation R = const is a pencil of conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Let R be a degree four integral given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='9) or (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We treat only  case (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='9), since the integrals (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='9) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='9) are obtained one from the other  (up to constant factor) by complex projective transformation fixing the conic  γ = {w = z2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus,  R = Rb1(z, w) =  (w − z2)2  (w + 3z2)(z − 1)(z − w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Suppose the contrary: the curve Γλ := {R = λ} is not irreducible for a  sequence of numbers λ converging to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then the foliation R = const is a  pencil of conics, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It contains the conics γ and {w +3z2 =  0}, which are tangent to each other at the origin and at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore,  the pencil consists of conics tangent to them at these points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' On the other  hand, the line {z = 1} lies in the polar locus {R = ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, it should lie in  a conic from the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But this is obviously impossible, – a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let now R be a degree 6 integral from the Addendum to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11,  Cases 2c) or 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Supposing the contrary to irreducibility, we similarly get  42  that the foliation R = const is a pencil of conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But in both Cases 2c)  and 2d) the polar locus {R = ∞} contains an irreducible cubic, see [27,  subsections 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='5, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This contradiction proves Statement 1) of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Statement 2) of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='8 follows from Statement 1) and the fact that  γ is a multiplicity d  2 zero curve of the integral R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let us prove Statement 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Suppose the contrary: there exists another  leaf α of multiplicity d  2 and degree µ ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then for every given line L that is  transversal to α and does not pass through singularities of the foliation each  leaf close enough to α intersects L in at least µ d  2 ≥ d points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The number  of intersection points cannot be greater than d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, µ = 2 and α is a  conic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us renormalize the integral R by postcomposition with M¨obius  transformation ν to an integral �R = ν ◦ R so that �R|γ = 0, �R|α = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let  Y (z, w) be a quadratic polynomial vanishing on α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then  �R =  � z − w2  Y (z, w)  � d  2  ,  up to constant factor, by construction and multiplicity assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' There-  fore, the foliation �R = const is a pencil of conics containing γ and α, and so  is R = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But this is not the case, since its generic leaves are punctured  irreducible algebraic curves Γλ of degree d ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The contradiction thus  obtained proves Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Proof  of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Consider a rationally integrable dual multibil-  liard with integral Ψ ̸≡ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then the dual billiard on each its curve γj  is rationally integrable with integral Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, each γj is a conic equipped  with either pencil type, or exotic dual billiard structure, by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11,  and Ψ|γj ≡ const, by [27, proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='35] (or by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let some two conics γ1, γ2 be the same conic γ equipped with  two distinct dual billiard structures, given by projective involution families  σP,j : LP → LP, j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Here P lies outside a finite set: the union of the  indeterminacy loci of families σP,j, which are finite by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The  product g := σP,1 ◦ σP,2 is a parabolic transformation LP → LP , having  unique fixed point P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The integral Ψ is g-invariant: Ψ ◦ g = Ψ along each  line LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But each non-fixed point of a parabolic transformation has infinite  orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, Ψ ≡ const along each line tangent to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But we know  that Ψ is constant along the curve γ, as noted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, Ψ ≡ const,  by the two latter statements and since the union of lines tangent to γ at  points lying in an open subset in γ contains an open subset in CP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The  contradiction thus obtained proves that Case 1) is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  43  Case 2): there are at least two geometrically distinct conics, say, γ1, γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  For every j = 1, 2 let Rj denote the canonical integral of the corresponding  dual billiard structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We have to prove the two following statements:  1) the dual billiard structure on each γj is defined by a pencil of conics,  that is, the degree dj := deg Rj is equal to 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  2) the latter pencil is the same for j = 1, 2, and it contains both γj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let F denote the foliation Ψ = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' For every j for all but a finite  number of values λ ∈ C the complex level curve {Rj = λ} is irreducible  of degree dj, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='8, and Ψ ≡ const along it (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Hence, each foliation Rj = const coincides with F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This together with the  previous statement implies that all the degrees dj are equal, set d = dj, and  both (punctured) conics γ1, γ2 are leaves of the same multiplicity d  2 for the  foliation F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, the foliation F is a pencil of conics containing γ1 and  γ2, by Statement 3) of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, all the conics of the multibilliard  lie in this pencil, and d = 2 (since d is the degree of irreducible level curve of  the function R1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, each Rj is a ratio of two quadratic polynomials, and  its level curves are conics from the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, the dual billiard structure  on γj is given by the same pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='25 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' A rational first integral of a pencil type multi-  billiard is constant on each conic of the pencil (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Moreover,  it is constant on every union of those conics whose parameter values λ lie in  the same G-orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Here G is the group from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The cardinality  of a generic G-orbit is equal to the cardinality |G| of the group G, since a  generic point in C has trivial stabilizer in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, the minimal degree of  the integral (which is achieved, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='5) is 2|G| ∈ {2, 4, 12}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This  together with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4 implies the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3  Dual billiard structures at vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Birationality and  types of involutions  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='10 Let A be a point in RP2 (CP2) equipped with real (com-  plex) dual billiard structure given by involution family σA,ℓ that has a real  (complex) rational first integral Ψ ̸≡ const: Ψ ◦ σA,ℓ = Ψ on each line ℓ  through A on which the involution is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let the foliation Ψ = const  be not the family of lines through A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Then σA,ℓ coincide (up to correc-  tion at a finite number of lines ℓ through A) with a birational involution  σA : CP2 → CP2 fixing each line through A and holomorphic and bijec-  tive on the complement to a finite number of lines through A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The rational  integral Ψ and the corresponding foliation Ψ = const are σA-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  44  Proof  Let σA,ℓ : ℓ → ℓ be the corresponding projective involution family  acting on lines ℓ through A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' They are defined on lines ℓ through A from an  open subset U ∈ CP1 in complex case (U ⊂ RP1 in real case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Fix a non-linear complex level curve X := {Ψ = λ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Fix an ℓ0 ∈ U  (consider it as a complex line) such that the points of intersection X ∩ ℓ0  distinct from A are regular points of the curve X, the intersections are  transversal, and the multiplicity of the intersection X ∩ ℓ0 at A is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  There exists a simply connected neighborhood V = V (ℓ0) ⊂ CP1 such that  for every ℓ ∈ V the number of geometrically distinct points of the set Xℓ :=  (X ∩ℓ)\\{A} ⊂ CP2 is the same (let us denote their number by d), and they  depend holomorphically on ℓ (Implicit Function Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We numerate  these holomorphic intersection point families by indices 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' For every  ℓ ∈ V the involution σA,ℓ makes a (ℓ-dependent) permutation of the latter  intersection points, which is identified with a permutation of indices 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' , d:  an element in Sd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' There exists a permutation α ∈ Sd realized by σA,ℓ for a  continuum cardinality subset Y ⊂ V of lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us fix it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' There exists a projective involution family �σA,ℓ : ℓ → ℓ de-  pending holomorphically on the parameter ℓ ∈ V that makes the permuta-  tion α on Xℓ for every ℓ ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The rational function Ψ|ℓ is �σA,ℓ-invariant:  Ψ ◦ �σA,ℓ = Ψ on every ℓ ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof Consider first the case, when Xℓ is just one point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' For every ℓ ∈ V  set �σA,ℓ : ℓ → ℓ to be the nontrivial conformal involution fixing the points Xℓ  and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It depends holomorphically on ℓ ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It preserves Ψ|ℓ: Ψ ◦ �σA,ℓ = Ψ  on every ℓ ∈ Y , and the latter relation holds for every ℓ ∈ V , since Y is of  cardinality continuum and by uniqueness of analytic extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let now Xℓ consists of at least two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us define �σA,ℓ to be the  unique projective transformation ℓ → ℓ fixing A and sending the points in  Xℓ with indices 1, 2 to the points with indices α(1), α(2) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' For  every ℓ ∈ V this is an involution preserving Ψ|ℓ, since this is true for every  ℓ ∈ Y and by uniqueness of analytic extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Claim 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The involution family �σA,ℓ extends holomorphically to a finitely  punctured space CP1 of lines through A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It coincides with σA,ℓ on all the  lines ℓ ∈ U except maybe for a finite number of them, on which Ψ = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof We can extend the involution family �σA,ℓ analytically in the param-  eter ℓ along each path avoiding a finite number of lines ℓ for which either  some of the points in Xℓ are not transversal intersections, or the index of  intersection ℓ ∩ X at A is not the minimal possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This follows from the  previous claim and its proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Extension along a closed path does not change  holomorphic branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Indeed, otherwise there would exist its another holo-  45  morphic branch over a domain W ⊂ V : an involution family HA,ℓ : ℓ → ℓ  depending holomorphically on ℓ ∈ W, HA,ℓ ̸= �σA,ℓ, which preserves the in-  tegral Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The product Fℓ := �σA,ℓ ◦ HA,ℓ : ℓ → ℓ is a parabolic projective  transformation, with A being its unique fixed point, for every ℓ ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Its or-  bits are infinite, and Ψ should be constant along each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This implies  that Ψ = const along each line ℓ ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, the foliation Ψ = const is the  family of lines though A, which is forbidden by our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The con-  tradiction thus obtained proves unique definedness of analytic extensions of  the involution family �σA,ℓ along paths and the first statement of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Its second statement follows from the fact that for those ℓ ∈ U for which  �σA,ℓ ̸= σA,ℓ, one has Ψ ≡ const along ℓ: see the above argument, now with  the parabolic transformation �σA,ℓ ◦ σA,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Without loss of generality we consider that σA,ℓ = �σA,ℓ, correcting σA,ℓ  at a finite number of lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The latter equality defines analytic extension of  the involution family σA,ℓ to all but a finite number of lines ℓ through A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The invariance condition Ψ ◦ σ|ℓ = Ψ|ℓ is a system of algebraic equations on  the pairs (ℓ, σ), where ℓ is a projective line through A and σ : ℓ → ℓ is a  nontrivial projective involution fixing A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' For every line ℓ through A (except  for a finite set of lines, including those along which Ψ ≡ const) its solu-  tion space is finite, and its cardinality is bounded from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This implies  that the family σA,ℓ is a connected open subset in an algebraic subset of a  smooth algebraic manifold and all σA,ℓ paste together to a global birational  automorphism CP2 → CP2 acting as a holomorphic involution on the com-  plement to a finite number of lines through A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It preserves Ψ, and hence,  the foliation Ψ = const, by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='10 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11 The dual billiard structure at each vertex A of any rationally  integrable dual multibilliard is either global, or quasi-global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In the case of  quasi-global structure the foliation by level curves of rational integral is a  pencil of conics, and the conic of fixed points of the corresponding involution  σA lies in the same pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof Consider first the case, when the foliation by level curves of integral  is a pencil of conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It is invariant under the birational involution σA from  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, σA acts on its parameter space C as a conformal  involution with at least two fixed points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, σA fixes at least two distinct  conics of the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Fix one of them that is not a pair of lines intersecting  at A, let us denote it by Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It is possible, since a pencil cannot contain  two singular conics, each of them being a pair of lines, so that all the four  lines forming them pass through the same point A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Indeed, otherwise A  46  would be the unique base point of the pencil, and the pencil would have  type e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, its only singular conic would be the double line tangent to  all its regular conics at A, – a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Subcase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1): Γ is disjoint from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then σA is a projective involution,  by Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='17, part 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Subcase 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2): Γ passes through A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' If Γ is a union of lines through A,  some of its lines, let us denote it by L, does not pass through A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then σA  is the projective involution that fixes each point of the line L, by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Similarly, if Γ is a regular conic, then σA fixes each its point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, it  defines a quasi-global dual billiard structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Consider now the case, when the foliation by level curves of the integral is  not a pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then the multibilliard contains just one conic, let us denote it  by γ, equipped with an exotic dual billiard structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let R be its canonical  integral, d = deg R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The foliation Ψ = const coincides with R = const,  by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The (punctured) curve γ being a leaf of multiplicity  d  2, its (punctured) image γ′ := σA(γ) is also a multiplicity  d  2 leaf, since  Ψ ◦ σA = Ψ and multiplicity is invariant under birational automorphism of  foliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, γ′ coincides with γ, if it is not a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1): A /∈ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then a generic line through A intersects γ at two  points distinct from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, the same holds for the image γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, it is  not a line, and σA(γ) = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, σA is a global projective transforma-  tion, the γ-angular symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2): A ∈ γ and γ′ ̸= γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us show that this case is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Indeed, then γ′ is a line, see the above discussion, and R|γ′ ̸≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore,  the points of intersection γ′ ∩ γ are indeterminacy points for the function  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  In Case 2a) the only indeterminacy points are the origin O and the  infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, γ′ is some of the following lines: the Ow-axis (which  passes through both latter points), the Oz-axis or the infinity line (which  are tangent to γ at O and at infinity respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But each of the latter  lines satisfies at least one of the following statements:  either R is non-constant there;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  or R has a pole of multiplicity less than d there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  See the two first formulas for the integrals in the addendum to Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, the (punctured) line γ′ cannot be a multiplicity d leaf of  the foliation R = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This contradiction proves that the case under con-  sideration is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The other Cases 2c), 2d) are treated analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Subcase 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3): A ∈ γ = σA(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, for every line ℓ through A  distinct from the line L tangent to γ at A the involution σA fixes the point  of intersection ℓ ∩ γ distinct from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, it is the degenerate γ-angular  symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In the chart (x, y) where γ = {y = x2}, A is the point of the  47  parabola γ at infinity and the line L tangent to γ at A is the infinity line,  σA acts as  σA : (x, y) �→ (x, 2x2 − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  In the coordinates (x, y) one has  R = R(x, y) = (y − x2)m  F(x, y) , F(x, y) is a polynomial, deg F ≤ 2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='5)  To treate the case in question we use the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Claim 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The point A is an indeterminacy point of the function R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof  Let us first consider the case, when L lies in a level curve Sλ :=  {R = λ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then λ ̸= 0, since the zero locus Sλ coincides with γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, A lies  in two distinct level curves, and hence, is an indeterminacy point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us  now suppose that R|L ̸≡ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' As a line ℓ through A tends to the tangent  line L to γ, its only intersection point B(ℓ) with γ distinct from A tends to  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, the involution (σA)|ℓ, which fixes the confluenting points A  and B(ℓ), tends to the constant map L → A uniformly on compact subsets  in L \\ {A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Suppose the contrary: A is not an indeterminacy point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Fix a  λ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The image Sλ′ := σA(Sλ) is a level curve of the function R, λ′ ̸= 0,  hence A /∈ Sλ, Sλ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, the points of intersections ℓ∩Sλ, ℓ∩Sλ′ do not  accumulate to A, as ℓ → L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But the points of the subset σA(ℓ ∩ Sλ) ⊂ Sλ′  converge to A, and hence, A ∈ Sλ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This contradiction proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' ✷  Claim 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let F be the same, as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then  F ◦ σA(x, y) = (−1)m+1F(x, y) + a(x − y2)m,  a = const ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6)  Proof The involution σA is birational, and it permutes leaves of the folia-  tion R = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' All but a finite number of leaves are punctured level curves  of the function R, since all but a finite number of level curves are irreducible  (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore σA permutes level curves of the function R and  acts on its values by conformal involution C → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The latter involution  preserves zero, which corresponds to the σA-invariant curve γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, its  action on values of the function 1  R is either identity, or an affine involution  µ �→ −µ + b, b = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This together with the fact that σA changes sign of  the polynomial y − x2 implies the statement of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  We consider the rational integrals R from the addendum to Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11, all their indeterminacy points A and the corresponding involutions σA  fixing points of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  For every pair (R, A), assuming σA-invariance of the  foliation R = const, we will arrive to contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  48  Everywhere below by F we denote the denominator of the rational func-  tion R (written in the coordinates (z, w) or (x, y) under consideration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  2a1) The integral R given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let A be the infinity point of the  parabola γ = {w = z2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then σA(z, w) = (z, 2z2 − w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The functions R  and R ◦ σA have the same foliation by level curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, their ratio,  which is equal to ± F ◦σA  F  , is constant along each leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But the latter ratio  is constant on the w-axis, since σA preserves the degree of higher purely  w-term wk, and z is σA-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' On the other hand, the w-axis is not a  leaf, since R(0, w) = w2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, the above ratio is globally constant, and  F ◦ σA = F  up to constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7)  But F(z, w) = �N  j=1(w −cjz2)2, F ◦σA(z, w) = �(w −(2−cj)z2), the coef-  ficients cj in F are negative, while the latter coefficients 2 − cj are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Therefore, equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7) cannot hold, – a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let A = O = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Consider the chart (x, y) = ( z  w, 1  w), in which A = ∞,  R(x, y) = (y − x2)2N+1  F(x, y)  , F(x, y) = y2  N  �  j=1  (y − cjx2)2, cj < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7) is proved analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The coefficients cj at x2 in F are  negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But F ◦σA(x, y) = (y −2z2) �(y −(2−cj)x2), and the coefficients  at x2 are positive there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, equality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7) cannot hold, – a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  2a2) Case of integral (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Treated analogously to the above case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Cases 2b1) and 2b2) have the same complexification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' thus we treat only  Case 2b2), when the integral R is given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' There are three indeter-  minacy points: the infinity and (±i, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let A ∈ γ be the infinite point,  hence σA(z, w) = (z, 2z2 − w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The denominator F is the product of the  σA-invariant quadratic polynomial z2 +1 and another quadratic polynomial  Φ(z, w) = z2 + w2 + w + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, the polynomial F ◦ σA is divisible by  z2+1, and hence is equal to ±F, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This implies that Φ◦σA = ±Φ is a  quadratic polynomial, while it has obviously degree four, – a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let now A = (±i, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let B denote the infinite point of the parabola γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let us choose an affine chart (x, y) centered at B so that the line tangent  to γ at A is the infinity line and the line {z = z(A)} is the Oy-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In the  new coordinates one has R(x, y) = (y−x2)2  xΦ(x,y) , where Φ is a cubic polynomial  coprime with y − x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Analogously we get that Φ ◦ σA = ±Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' If Φ contains a  monomial divisible by y2, then deg(Φ ◦σA) ≥ 4, and we get a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Otherwise Φ(x, y) = c(y + Ψ(x)), where Ψ is a polynomial;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Φ is coprime  with y − x2, hence, Ψ(x) ̸= −x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But then Φ ◦ σA ̸= ±Φ, – a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  49  Cases 2c1) and 2c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Note that the integral R from 2c1) is invariant  under the order 3 group generated by the symmetry (z, w) �→ (εz, ε2w),  where ε is a cubic root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This group acts transitively on the set of  three indeterminacy points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, it suffices to treat the case of just one  indeterminacy point A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Again it suffices to treat only Case 2c2), which  has the same complexification, as Case 2c1), with A being the infinite  point of the parabola γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  To do this, let us first recall that the (2, 1)-  quasihomogeneous degree of a monomial zmwn is the number m+2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' A poly-  nomial is (2, 1)-quasihomogeneous, if all its monomials have the same (2, 1)-  quasihomogeneous degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Each polynomial in two variables is uniquely  presented as a finite sum of (2, 1)-quasihomogeneous polynomials of distinct  quasihomogeneous degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The indeterminacy points of the integral R given  by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='12) are O = (0, 0), (1, 1) and the infinity point of the parabola γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Taking composition with σA(z, w) = (z, 2z2 − w) preserves the quasiho-  mogeneous degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The lower quasihomogeneous part of the denominator  F in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='12) is the polynomial (w + 8z2)2 of quasihomogeneous degree 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The numerator (w − z2)3 is quasihomogeneous of degree 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, the  lower quasihomogeneous part of the polynomial F ◦ σA is the polynomial  (w − 10z2)2 ̸= ±(w + 8z2)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The two latter statement together imply that  formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6) cannot hold, – a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Then we have three indeterminacy points: the origin, the  infinity point and the point (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The case, when A is the infinity point  of the parabola γ, is treated analogously to Case 2b2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us consider the  case, when A is the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In the coordinates (x, y) = ( z  w, 1  w) the function  R takes the form R(x, y) = (y−x2)3  F (x,y) ,  F(x, y) = (y + 8x2)(x − y)(y2 + 8x2y + 4y + 5x2 − 14xy − 4x3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The lower (2, 1)-quasihomogeneous part of the polynomial F is V (x, y) :=  x(y+8x2)(4y+5x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It has quasihomogeneous degree 5, while the numerator  in R has quasihomogeneous degree 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6) implies that  σA multiplies the lower quasihomogeneous part by ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But V ◦ σA(x, y) =  x(y − 10x2)(4y − 13x2) ̸= ±V (x, y), – a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let us now consider the case, when A = (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Take the affine coordi-  nates (x, y) centered at the infinite point of the parabola γ such that the  complement to the affine chart (x, y) is the tangent line L to γ at A and the  y-axis is the zero line {z = 1} of the denominator (which passes through A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The rational function R takes the form  R(x, y) = (y − x2)3  F(x, y) , F(x, y) = xG2(x, y)G3(x, y), Gj(0, y) ̸≡ 0, deg Gj = j,  50  G2(x, y) = w + z2,  G3(x, y) = w + 8z2 + 4w2 + 5wz2 − 14zw − 4z3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  In the chart (x, y) one has σA(x, y) = (x, 2x2 − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The factor x in F is  σA-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, F ◦σA = ±F, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6), and σA leaves invariant the  zero locus Z = {G2G3 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The latter zero locus is a union of the conic  {G2 = 0} disjoint from A and a cubic {G3 = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The latter intersects a  generic line ℓ through A at a unique point distinct from A, since it has a cusp  at A, see [27, subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6, claim 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, for a generic line ℓ through  A, the intersection Z ∩ (ℓ \\ {A}) consists of three distinct points disjoint  from γ and it is invariant under the involution σA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This implies that one of  them is fixed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' let us denote it by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, for a generic ℓ the projective  involution (σA)|ℓ has three distinct fixed points: A, B and the unique point  of the intersection γ ∩ (ℓ \\ {A}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, it is identity, – a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Thus, we have checked that a rationally integrable dual multibilliard with  exotic foliation R = const cannot have a vertex A whose involution σA is  not a global projective transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This proves Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4  Pencil case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='26  We already know that if a dual multibilliard is of pencil type, then it is  rationally integrable (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Consider now an arbitrary rationally  integrable dual multibilliard where all the curves are conics equipped with a  dual billiard structure defined by the same pencil (containing each conic of  the multibilliard).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us show that the multibilliard is of pencil type, that  is, its vertices (if any) are from the list given by Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='21 and their  collection satisfies the conditions of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='12 Let a rationally integrable multibilliard consist of conics  lying in a pencil, equipped with dual billiard structures defined by the same  pencil, and some vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then each its vertex is admissible for the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof  Let Ψ be a rational integral of the multibilliard, Ψ ̸≡ const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The  foliation Ψ = const coincides with the pencil under consideration, by Propo-  sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let K be a vertex of the multibilliard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then its dual billiard  structure is given by an involution σK : CP2 → CP2 preserving the pencil  that is either a global projective involution, or an involution fixing points of  a regular conic α passing through K and lying in the pencil (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let us first treat the second case, when σK fixes points of a regular conic  α from the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let L denote the tangent line to α at K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Consider the  affine chart (z, w) on the complement CP2 \\ L in which α = {w = z2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' the  point K is the intersection of the infinity line with the w-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' One has  51  σK(z, w) = (z, 2z2 − w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Each regular conic β of the pencil is given by a  quadratic equation {w + Φ2(z) = 0}, where Φ2 is a quadratic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Indeed, the quadratic equation on β contain neither w2, nor wz terms, since  σK transforms them to polynomials of degrees four and three respectively,  while it should send β to a conic of the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This implies that all the  regular conics of the pencil are tangent to each other at K, and we get that  K is an admissible vertex from Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let us now treat the first case, when σK is a global projective involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Claim 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let K be not a base point of the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then the conic S  through K is a pair of lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof The conic S is fixed by σK, as is K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' If it were regular, then it would  intersect a generic line ℓ through K at a point distinct from K, and the  involution σK would fix each point in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, it would not be a projective  transformation, – a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Subcase 1): K is not a base point of the pencil, and the conic S through  A is a pair of distinct lines L1, L2, both passing through K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In the case,  when the pencil consists of conics passing through four distinct base points,  there are two base points Vj1, Vj2 in each line Lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, K is a standard  vertex Ms from Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='21, and the projective involution σK permutes  Vj1, Vj2 for every j = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, σK coincides with the involution σMs from  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='21, Case a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In the case, when the pencil has three distinct base  points, two of them lie in one of the lines, say L1, the third one (denoted  by C) lies in L2, and L2 is tangent at C to all the regular conics of the  pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We get analogously that K = M and σK = σM, see Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='21,  Case b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Case c), when the conics of the pencil are tangent to each other  at two base points, is treated analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Case e) is impossible, since then  the pencil contains no distinct line pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In Case d) it contains the unique  pair of distinct lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' They intersect at a base point: the common tangenty  point of conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, this case is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Subcase 2): K is not a base point, the conic S consists of a line L1  through K and a line L2 that does not pass through K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then σK fixes K  and each point of the line L2, and hence, the intersection point M = L1∩L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  In Case a) there are two base points in each line Lj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Those lying in L1 should  be permuted by σK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, the points K, M and the two base points  in L1 form a harmonic quadruple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence K is one of the skew vertices from  Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='21, Case a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In Case b) there are three distinct base points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The case, when two of them lie in L1, is treated as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In the case, when  L1 contains only one base point C, it should be tangent there to the regular  conics of the pencil, and C should be fixed by σK, as is M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Therefore,  52  C = M, since the projective involution (σK)|L1 cannot have three distinct  fixed points C, M, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Finally, all the three base points are contained in the  line L2, which is obviously impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Similarly in Case c) we get that K  is a point of the line through the two base points A, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Indeed, the other  a priori possible subcase is when K lies in a line tangent to the conics at a  base point (say, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then σK would fix three points of the latter line: K,  A and the intersection point of the tangent lines to conics at A and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The  contradiction thus obtained shows that this subcase is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  In Case d) (conics tangent at a point A with triple contact and passing  through another point B) the point K should lie in the line L tangent to  the conics at A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' (This realizes the vertex C from Subcase d2) in Definition  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=') Indeed, otherwise K would lie in AB \\{A, B}, and the involution σK  would fix three distinct points A, B, K ∈ AB, – a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Case e) is  impossible, since then the pencil contains no distinct line pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Subcase 3): K is not a base point, and the conic S is a double line L  through K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then  either the pencil is of type c) and L passes through the two base points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  the involution σK should permute them and the tangent lines at them to  the conics of the pencil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  or the pencil is of type e) and L is tangent to its regular conics at the  unique base point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' the involution should fix K and those points where the  tangent lines to conics pass through K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' the latter points form a line through  the base point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Both cases are realized by vertices from Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Subcase 4): K is a base point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then each line L passing through K and  another base point A contains no more base points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, the involution  (σK)|L should fix both K and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Finally, σK fixes each base point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' There-  fore, the base points different from K lie on the fixed point line Λ of the  projective involution σK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let the pencil be of type a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then Λ contains  three base points, – a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let now the pencil have type b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then  Λ contains two base points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' If the conics of the pencil are tangent to each  other at K, the vertex K has type b3) from Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Consider the  opposite case, when the common tangency point C of the conics lies in Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let H denote their tangent line at C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then H contains no other base point,  thus, H ̸= KC, Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, the restriction of the involution σK to each line  ℓ ̸= KC through K fixes three distinct points: K, ℓ ∩ H, ℓ ∩ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The contra-  diction thus obtained proves that the case under consideration is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case of pencil of type c) is treated analogously: K is a vertex of type c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Consider the case of pencil of type d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let K be the base point of  transversal intersection of conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then the involution σK, which preserves  53  the pencil, should fix K and each point of the common tangent line L to the  conics at the other base point A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In an affine chart (z, w) centered at A, in  which K is the intersection point of the infinity line with the Ow-axis and L  is the z-axis, the involution σK is the symmetry with respect to the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The latter symmetry changes the 2-jet of a regular conic of the pencil at  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But its conics should have the same 2nd jet at A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, σK cannot  preserve the pencil, – a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, K is the base point where  the conics have a common tangent line L, and the other base point B (of  transversal intersection) lies in Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us choose an affine chart (z, w) so  that L is the infinity line, K is its intersection with the Ow-axis, and Λ is  the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then σK is again the symmetry as above, and it changes 2nd jets  of conics (which are parabolas) at their infinity point K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, Case  d) is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Case e) is treated analogously to the latter discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Finally, all the possible vertices listed above belong to the list of admis-  sible vertices from Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='12 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  The rational integral of the multibilliard is constant on each union of  conics of the pencil whose parameter values form a G-orbit, and the double  cardinality 2|G| is no greater than the degree of the integral: see the proof of  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='27 at the end of Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, G is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, the  multibilliard is of pencil type, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='26 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='5  Exotic multibilliards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='31  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='13 Each multibilliard from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='31 is rationally inte-  grable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The corresponding rational function R from the addendum to Theo-  rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11 is its integral of minimal degree, except for the subcase in Case (i),  when ρ = 2 −  1  N+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' in this subcase R2 is a first integral of minimal degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof In the case, when there are no vertices, rational integrability follows  by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us consider that the multibilliard contains at least  one admissible vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The function R (respectively, R2) is a rational integral of  minimal degree, since R is even (odd) in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case (ii) is treated analogously to Case (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It suffices to treat the sub-  case 2b2), since the multibilliards of types 2b1), 2b2) containing the unique  admissible vertex are projectively isomorphic (Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In subcase  2b2) the function R is even in z, and hence, invariant under the correspond-  ing admissible vertex involution (z, w) �→ (−z, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us show that the γ-angular symmetry σQ centered at each  admissible vertex Q preserves the integral R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Cases 2c1) and 2c2) being  54  complex-projectively isomorphic and invariant under order three symmetry  cyclically permuting the three singular points, we treat Case 2c2), with  Q = (0, −1) being the intersection point of the w-axis (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=', the line through  two indeterminacy points: O and ∞) and the line tangent to γ at the  indeterminacy point (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We use the following two claims and proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Claim 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The polar locus  S := {P(z, w) = 8z3 − 8z2w − 8z2 − w2 − w + 10zw = 0}  passes through Q = (0, −1) and has an inflection point there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof  One has Q = (0, −1) ∈ S (straighforward calculation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' To show  that Q is an inflection point, it suffices to show that ∇P(Q) ̸= 0 and the  Hessian form of the function P evaluated on the skew gradient (∂P  ∂w, − ∂P  ∂z )  (which is a function of Q denoted by H(P)(Q)) vanishes at Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Indeed,  ∂P  ∂z (Q) = −10, ∂P  ∂w (Q) = 2 − 1 = 1,  ∂2P  ∂z2 (Q) = 0, ∂2P  ∂w2 (Q) = −2,  ∂2P  ∂z∂w (Q) = 10,  H(P) = ∂2P  ∂z2  �∂P  ∂w  �2  + ∂2P  ∂w2  �∂P  ∂z  �2  − 2 ∂2P  ∂z∂w  ∂P  ∂w  ∂P  ∂z = 0 − 200 + 200 = 0  at the point Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='14 Let a cubic S ⊂ CP2 have an inflection point Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then  there exists a projective infolution σQ,S : CP2 → CP2 that fixes each line  through Q and permutes its intersection points with S distinct from Q (for  ℓ being not the tangent line to S at Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof The above involution is well-defined on each line ℓ through Q distinct  from the tangent line Λ to S at Q as a projective involution σQ,S,ℓ : ℓ → ℓ  depending holomorphically on ℓ ̸= Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It suffices to show that the involution  family thus obtained extends holomorphically to ℓ = Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This will imply  that σQ,S is a well-defined global holomorphic involution CP2 → CP2, and  hence, a projective transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Indeed, let us take an affine chart (x, y)  centered at Q and adapted to S, so that Λ is the Ox-axis and the germ of  the cubic S is the graph of a germ of holomorphic function:  y = f(x), f(x) = ax3 + (b + o(1))x4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  S = {y = f(x)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  55  A line ℓδ := {y = δx} with small slope δ intersects S at two points distinct  from Q = (0, 0) with x-coordinates x0, x1 satisfying the equation  ax2  0 + (b + o(1))x3  0 = ax2  1 + (b + o(1))x3  1 = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='8)  Taking square root and expressing x1 as an implicit function −x0(1 + o(1))  of x0 yields  x1 = −x0 + (c + o(1))x2  0,  c = − b  a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Writing the projective involution σQ,S : ℓδ → ℓδ fixing the origin and per-  muting the above intersection points as a fractional-linear transformation in  the coordinate x, we get a transformation  x �→ −  x  1 + ν(δ)x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='9)  Substituting x0 to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='9) yields  −  x0  1 + ν(δ)x0  = −x0 + (c + o(1))x2  0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (1 + ν(δ)x0)(1 − (c + o(1))x0) = 1, x0 = x0(δ) → 0,  as δ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, ν(δ) = c + o(1) → c, and the one-parametric holomor-  phic family of transformation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='9) extends holomorphically to δ = 0 as the  projective transfomation x �→ −  x  1+cx (continuity and Erasing Singularity  Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The proposition is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Claim 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The polar cubic S is σQ-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof  The projective involution σQ,S from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='14 fixes S and  each line through Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It preserves the conic γ, which is the unique regular  conic tangent to S at the three indeterminacy points of the integral R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Indeed, if there were two such distinct conics, then their total intersection  index at the three latter points would be no less than 6, – a contradiction  to B´ezout Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, σQ,S is the γ-angular symmetry, and hence,  it coincides with σQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This implies that σQ(S) = S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' ✷  Claim 12 together with σQ-invariance of the zero locus γ of the rational  function R implies that R ◦ σQ = ±R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The restriction of the integral R to  the line ℓ through Q tangent to the conic γ at the point (1, 1) is holomorphic  and nonconstant at (1, 1), since its numerator being restricted to ℓ has order  6 zero at (1, 1), and so does its denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The point (1, 1) is fixed by  σQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, the above equality holds with sign ”+” near the point (1, 1),  and hence, everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Case iv) is treated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The proposition is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  56  Recall that in our case of a multibilliard containing a conic with an  exotic dual billiard structure, the involution associated to each vertex is  a projective angular symmetry (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' As it is shown below, this  together with the next proposition implies Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='15 Consider an exotic rationally integrable dual billiard on  a conic γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let R be its canonical integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let A ∈ CP2, and let σA :  CP2 → CP2 be a projective angular symmetry centered at A that preserves  the foliation R = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then σA is the γ-angular symmetry centered at A,  and A belongs to the list from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof Set d = deg R ∈ 2N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The (punctured) conic γ is the only conical leaf  of multiplicity d  2 of the foliation R = const, see the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Therefore, σA(γ) = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, A /∈ γ, since σA is a projective involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Thus, its restriction to each line ℓ through A permutes its intersection points  with γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, σA is the γ-angular symmetry centered at A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It preserves  the set of indeterminacy points of the integral R, whose number is either  two, or three, since it preserves the foliation R = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, it preserves  the union of lines tangent to γ at the indeterminacy points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Consider first Case 2a), where there are two intederminacy points: in  the affine chart (z, w) these are the origin O and the infinity point B of  the parabola γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let LO, LB denote the lines tangent to γ at O and at B  respectively, and let Q be their intersection point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us show that A = Q:  this is Case (i) from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The infinity line LB is a leaf of the  foliation R = const, while LO isn’t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, the lines LB and LO are  σA-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Suppose the contrary: A ̸= Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Say, A /∈ LO;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' the case A /∈ LB  is treated analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then the restriction of the involution σA to each line  ℓ ̸= LB, OB through A fixes A and its intersection points distinct from A  with the lines LO, LB and OB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The number of the latter points is at least  two, unless A is one of their pairwise intersection points Q, O, B ∈ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But  A ̸= O, B, since A /∈ γ, and A ̸= Q by assumption, – a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus,  A = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then there are three indeterminacy points, and we can name  them by X, Y , Z so that R|Y Z ≡ const ̸= ∞, while R|XY , R|XZ ≡ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In  Subcase 2b1) X = (1, 1), and Y , Z are the origin and the infinity point  of the parabola γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The involution σA should fix one of the indeterminacy  points and permute two other ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We claim that it fixes X and permutes  Y and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Indeed, suppose the contrary: σA fixes, say, Y and permutes X  and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then σA(XY ) = Y Z and σA(XZ) = XZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' On the other hand,  σA should send level sets of the integral R to its level sets, since this is  57  true for generic, irreducible level sets of degree deg R, and remains valid  after passing to limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, σA should preserve the infinity level set, since  σA(XZ) = XZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' On the other hand, it should permute it with a finite level  set containing the line Y Z, since σA(XY ) = Y Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The contradiction thus  obtained proves that σA(X) = X and σA(Y ) = Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This implies that A is  the intersection point of the line Y Z with the tangent line to γ at X, and  the corresponding involution σA permutes the intersection point of each line  through A with the lines XY and XZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, the pair (A, σA) is the same,  as in the Cases (ii) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' These cases are obtained one from the  other by complex projective transformation, as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case 2c1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' (Case 2c2) is obtained from it by projective transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=')  The integral R has three indeterminacy points lying on the conic γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let  us denote them by X, Y , Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then one of them, say, X, is fixed by the  γ-angular symmetry σA, and the two other ones are permuted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Indeed, if  all of them were fixed, then the three distinct tangent lines to γ at them  would intersect at the point Q, which is impossible: through every point  lying outside the conic γ there are only two tangent lines to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This implies  that A is the intersection point of the line tangent to γ at X and the line  Y Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, it is admissible in the sense of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='31, case (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  In Case 2c2) all the admissible vertices are real, since so are the inde-  terminacy points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In Case 2c1) X = (1, 1) is the unique real indeterminacy  point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' the other ones are Y = (ε, ¯ε) and Z = (¯ε, ε), where ε = e  2πi  3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The  intersection of the line {w = 2z − 1} tangent to γ at (1, 1) and the line  Y Z is the admissible vertex (0, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Each one of the complex lines XZ,  XY has non-real slope, and hence, X is its unique real point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore,  the admissible vertex lying there is not real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, in Case 2c1) the point  (0, −1) is the unique real admissible vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The corresponding integral R = Rd has three indeterminacy  points: the origin, the point (1, 1) and the infinity point of the conic γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The line {z = 1} through the two latter indeterminacy points lies in a level  curve of the integral R: namely, in its polar locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  On the other hand,  R is non-constant on the lines {z = 0} and {z = w} passing through the  origin and the other indeterminacy points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This implies that every projective  transformation preserving the foliation R = const should fix the origin, and  hence, the Oz-axis: the corresponding tangent line to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us show that it  cannot be a γ-angular symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Suppose the contrary: it is the γ-angular  symmetry σA centered at a point A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then σA has to fix the origin and to  permute the two other indeterminacy points, as in the case discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Therefore, A is the intersection point of the line {z = 1} through them  58  and the Oz-axis: thus, A = (1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, the involution σA fixes the line  {z = 1}, which lies in the polar locus of the integral R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, it preserves  the whole polar locus, as in the above Case 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The polar locus consists  of the above line, the regular conic α := {w = −8z2} and an irreducible  rational cubic, see [27, proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, σA fixes the conic α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence,  it permutes its infinite point (coinciding with that of γ) and its other, finite  intersection point (1, −8) with the line {z = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  On the other hand, it  should send the infinite point to the other point (1, 1) ∈ γ ∩ {z = 1}, since  σA is the γ-angular symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The contradiction thus obtained proves that  if a multibilliard contains a conic with exotic dual billiard structure of type  2d), then it contains no vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='31 is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' ✷  Proof  of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The complex projective equivalence of bil-  liards of type (ii) is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us prove the analogous second statement  of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='32 on billiards of type (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The dual billiard structure on  γ of type 2c1) admits the order 3 symmetry (z, w) �→ (εz, ¯εw) cyclically per-  muting the indeterminacy points of the integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, it also permutes  cyclically admissible vertices and hence, acts transitively on them and on  their unordered pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The same statement holds for type 2c2), since the  dual billiards 2c1) and 2c2) are complex-projectively isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This im-  plies the second statement of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In the case of type 2c2) the  indeterminacy points are real, and hence, so are the admissible vertices, and  the order 3 symmetry is a real projective transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This together with  the above discussion implies the third statement of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6  Admissible vertices of real pencils of conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof of  Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='24  Proof  of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The ambient projective plane CP2 is the  projectivization of a three-dimensional complex space C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The complex  conjugation involution acting on C3 induces its action on CP2, which will  be also called conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It sends projective lines to projective lines and  preserves the complexification of every real pencil of conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case a): pencil of real conics through four distinct (may be complex)  points A, B, C, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The conjugation permutes the points A, B, C, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Therefore, it permutes vertices M1, M2 and M3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Hence, at least one of  them is fixed (say, M1), or equivalently, real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The line through the other  points M2, M3 should be fixed, and thus, real, since the union {M2, M3} is  invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Finally, the involution σM1 is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let now the point KAB be real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then the ambient line AB is invariant  59  under the conjugation, since the collection of complex lines through pairs of  permuted basic points of the pencil is invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, it is a real line,  the union {A, B} is conjugation invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, so is {C, D}, and the line  CD is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, the involution σKAB is also real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Note that the case, when C, D are real and A, B aren’t is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In  this case A and B are permuted by the conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, the points M2  and M3 are also permuted, and thus, they are not real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Similarly, , KBC  and KAC are permuted, KBD and KAD are permuted, and hence, they are  not real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case b): real pencil of conics through 3 points A, B, C tangent at the  point C to the same line L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The point C and the line L  should be obviously fixed by conjugation, and hence, real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, the  points A, B are either both fixed, or permuted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, the line AB is real,  and so is the intersection point M = AB ∩ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The point KAB ∈ AB is  also real, since complex conjugation acting on complex projective line sends  harmonic quadruples to harmonic quadruples and the harmonicity property  of a quadruple of points is invariant under two transpositions: one permuting  its two first points;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' the other one permuting its two last points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore,  the line CKAB is real, and so is σM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The global projective involutions σC  and σKAB are both real, since so are the lines AB and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In the case, when  the dual billiard structure at C is quasi-global and is defined by a real conic,  the corresponding involution σC is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case c): real pencil of conics through two distinct points A, C tangent at  them to two given lines LA and LC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' A priori, the points A and  C need not be real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' For example, a pencil of concentric circles satisfies the  above statements with A = [1 : i : 0], C = [1 : −i : 0]: the so-called isotropic  points at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The line AC is real, and so is M = LA ∩ LC, since the  complex conjugation either permutes A and C (and hence, the lines LA and  LC), or fixes them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, the involution σM is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The point M′,  which is an arbitrary point of the complex line AC needs not be real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But  if it is real, then so is the involution σM′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Indeed, σM′ can be equivalently  defined to fix M′ and the line through M and the point K ∈ AC for which  the quadruple M′, K, A, C is harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' If M′ is real, then so is K, as in the  above case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, the line MK is real and so is σM′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Cases d) and e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Reality of the vertex A, is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Equivalence of  reality of the involution σA,S and reality of the conic S, follows from reality  of the vertex A and from the fact that a projective involition of a complex  line having at least one real fixed point is real, if and only if its other fixed  point is also real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='24 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  60  3  Rationally 0-homogeneously integrable piecewise  smooth projective billiards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proof of Theorems  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='38, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='39, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='45  The first step of the above-mentioned classification is the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1 Let a planar projective billiard with piecewise C4-smooth bound-  ary containing a nonlinear arc be rationally 0-homogeneously integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Then the C4-smooth pieces of its boundary are conical arcs and straight-  line segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The projective billiard structure of each conical arc is either  of dual pencil type, or an exotic one from Statement 2) of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof  A rational 0-homogeneous integral of the billiard is automatically  such an integral for the projective billiard on each nonlinear arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, by  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='16, the nonlinear arcs are conics, and each of them is equipped  with a projective billiard structure either of dual pencil type, or exotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Below we describe the possible combinations of conical arcs and straight-  line segments equipped with projective billiard structures that yield alto-  gether a rationally integrable projective billiard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We reduce this description  to the classification of rationally integrable dual multibilliards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' To do this,  we use the projective duality given by orthogonal polarity, see Subsection  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1, which transforms a projective billiard to a dual multibilliard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We show  that the former is rationally 0-homogeneously integrable, if and only if the  latter is rationally integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We present a one-to-one correspondence be-  tween rational 0-homogeneous integrals of the former and rational integrals  of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Afterwards the main results on classification of rationally 0-  homogeneously integrable projective billiards follow immediately by duality  from those on dual multibilliards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1  Duality between projective billiards and dual multibil-  liards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Correspondence between integrals  Consider the orthogonal polarity, which sends each two-dimensional sub-  space in the Euclidean space R3  x1,x2,x3 to its orthogonal one-dimensional  subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The projectivization of this correspondence is a projective dual-  ity that sends each projective line in RP2  [x1:x2:x3] to a point in RP2, called  its dual point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It sends an immersed smooth2 strictly convex curve α to its  2Note that in general, the straightforward parametrization of the dual curve α∗ induced  by a Cm-smooth parametrization of the curve α is not Cm-smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But one can choose  the parameter of the dual curve to make it Cm-smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  61  dual: the immersed smooth strictly convex curve α∗, whose points are dual  to the projective lines tangent to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Consider a projective billiard in R2  x1,x2 with piecewise C4-smooth bound-  ary such that each its C4-smooth arc is either strictly convex, or a straight-  line segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This holds automatically, if all the nonlinear boundary arcs  are conical, as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Consider the ambient plane R2  x1,x2 as the  horizontal plane {x3 = 1} ⊂ R3  x1,x2,x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' We identify it with the standard  affine chart {x3 = 1} ∈ RP2  [x1:x2:x3] by tautological projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The above  duality sends each nonlinear C4-smooth boundary arc α to the dual curve  α∗ ⊂ RP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' For every point Q ∈ α let LQ denote the projective line tangent  to α at Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let Q∗ denote the line dual to Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It is tangent to the dual curve  α∗ at the point P = L∗  Q dual to LQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The projective billiard reflection at Q is  the nontrivial affine involution acting on TQR2, which fixes the points of the  line LQ and fixes the line N(Q) of the transversal line field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Its projectiviza-  tion acts as an involution RP1 → RP1 on the space RP1 of lines through Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The duality conjugates the latter involution acting on lines to a projective  involution σP acting on the projective line Q∗ = LP tangent to α∗ at P and  fixing the points P and N ∗(Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The projective involution family (σP )P ∈α∗  thus obtained is a dual billiard structure on the curve α∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It will be called  the dual billiard structure dual to the projective billiard on α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2 Consider a projective billiard as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Its dual multibil-  liard is the collection of curves α∗ in RP2 dual to its C4-smooth nonlinear  boundary arcs α, equipped with the dual billiard structure defined above,  and the points A (called vertices) dual to the ambient lines a of the straight-  line billiard boundary segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Each vertex A is equipped with the fol-  lowing dual billiard structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let U denote the union of all the billiard  boundary intervals lying in the line a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Each point Q ∈ U is dual to a line q  through A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The projective billiard reflection involution acting on the space  RP1 of lines through Q is conjugated by duality to a projective involution  σA,q : q → q fixing A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The family of involutions σA,q, Q ∈ U, yields the  prescribed dual billiard structure at the point A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  For every (x1, x2) ∈ R2 and (v1, v2) ∈ T(x1,x2)R2 set  r := (x1, x2, 1),  v = (v1, v2, 0) ∈ R3,  M = M(r, v) := [r, v] = (−v2, v1, ∆),  ∆ = ∆(x1, x2, v) = x1v2 − x2v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1)  For every fixed r the map M is a linear operator sending the space T(x1,x2)R2  isomorphically onto the orthogonal complement r⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Its projectivization  62  sends the space of lines in R2 through r onto the projective line dual to  the point (x1, x2) = [x1 : x2 : 1] ∈ R2 ⊂ RP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Each line through r is  sent onto its dual point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, the projectivized map M yields a well-  defined map from the space of projective lines to RP2 that coincides with the  above projective duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In particular, it conjugates the projective billiard  reflections at billiard boundary points to the corresponding dual multibil-  liard involutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, we can consider that the dual multibilliard lies  in the projective plane RP2 with homogeneous coordinates [M1 : M2 : M3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3 1) A projective billiard is rationally 0-homogeneously in-  tegrable, if and only if its dual multibilliard is rationally integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  2) Each rational 0-homogeneous integral of degree n (if any) of the pro-  jective billiard is a rational 0-homogeneous function of the moment vector  M = (M1, M2, M3), see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1), of the same degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  3) Let R be a rational integral of the dual multibilliard written in ho-  mogeneous coordinates [M1 : M2 : M3] as a ratio of two homogeneous  polynomials of degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then R[r, v] is a rational 0-homogeneous integral  of the projective billiard of the same degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof  The statements of the proposition extend [27, propositions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='23,  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='24] (formulated for a projective billiard on a connected curve) to projec-  tive billiards with piecewise C4-smooth boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The proofs given in [27,  subsections 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2] remain valid in this more general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4 The minimal degree of rational 0-homogeneous integral of a  projective billiard is equal to the minimal degree of rational integral of its  dual multibilliard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2  Case of dual pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proof of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='38, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='39, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='40  We use the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='5 Let P be a pencil of conics, P∗ be its dual pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  1) Let α ⊂ R2 ⊂ RP2 be a conical arc whose ambient conic lies in P∗,  equipped with the projective billiard structure defined by P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then its dual is  the dual conical arc α∗ equipped by the dual billiard structure of pencil type,  defined by the pencil P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The converse statement also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  2) A planar projective billiard is of dual pencil type, defined by the dual  pencil P∗, if and only if its dual multibilliard is of pencil type, defined by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  63  Proof  Statement 1) of the proposition follows from definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The def-  initions of dual multibilliard of pencil type and projective billiard of dual  pencil type are dual to each other: the standard (skew) admissible lines for  the dual pencil P∗ are dual to the standard (skew) admissible vertices for  the pencil P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This implies Statement 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Proof  of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let a projective billiard with piecewise C4-  smooth boundary containing a nonlinear arc be rationally integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then  its dual multibilliard is rationally integrable (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, all  its curves are conics, and the nonlinear arcs of projective billiard boundary  are conical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let there be at least two arcs of two distinct conics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Then  the multibilliard contains their dual conics, which are also distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence,  they are equipped with the dual billiard structure defined by the pencil P  containing them, each conic of the multibilliard lies in the same pencil P and  is equipped with the dual billiard structure defined by P (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Thus, all the conical arcs of the projective billiard boundary lie in the dual  pencil P∗ and are equipped with the projective billiard structures defined  by P∗, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='5, Statement 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='38 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Proof  of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let in a projective billiard all the nonlinear  boundary arcs be conics lying in the same dual pencil P∗, equipped with  the projective billiard structure defined by P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then the curves of the dual  multibilliard are conics lying in the pencil P, equipped with the dual billiard  structure defined by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The projective billiard is rationally 0-homogeneously  integrable, if and only if the dual multibilliard is rationally integrable, by  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3, Statement 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The latter holds, if and only if the multibil-  liard is of pencil type (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Or equivalently, if and only if the  projective billiard is of dual pencil type, see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='5, Statement 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='39 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='40 follows immediately from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='27 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4  by duality, since (neighbor) skew admissible lines for a dual pencil P∗ are  dual to (neighbor) skew admissible vertices for the pencil P and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3  Exotic projective billiards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='45  Let a projective billiard has boundary that consists of conical arcs of one  and the same conic equipped with an exotic dual billiard structure from  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='16, Case 2), and maybe some straightline segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It is ra-  tionally 0-homogeneously integrable, if and only if the corresponding dual  multibilliard is rationally integrable, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3, Statement 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In  appropriate coordinates the dual multibilliard consists of one conic γ =  64  {w = z2} ⊂ R2  z,w = {t = 1} ⊂ RP2  [z:w:t] equipped with the corresponding  exotic dual billiard structure from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11 and maybe some vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  It is rationally integrable, if and only if either it has no vertices, or each its  vertex is admissible in the sense of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This holds if and only  if the ambient lines of the projective billiard boundary segments are dual  to the admissible vertices, and their corresponding projective billiard struc-  tures are dual to the dual billiard structures at the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The lines dual  to the admissible vertices, equipped with the corresponding dual projective  billiard structures, will be called admissible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us find the admissible lines  case by case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' To do this, we use the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6 Consider the above parabola γ equipped with an exotic  dual billiard structure from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11, Case 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let C ⊂ RP2  [z:w:t] denote  the conic orthogonal-polar-dual to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  1) The projectivization [F] : RP2  [z:w:t] → RP2  [x1:x2:x3] of the linear map  F : (z, w, t) �→ (x1, x2, x3) := (z  2, t, w)  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2)  sends C to the parabola {x2x3 = x2  1}, which will be now referred to, as C,  C ∩ {x3 = 1} = {x2 = x2  1},  equipped with the corresponding projective billiard structure from Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='16, Case 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  2) For every point (z0, z2  0) ∈ γ the corresponding point of the dual curve  C has [x1 : x2 : x3]-coordinates [−z0 : z2  0 : 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  3) The points in C corresponding to O = (0, 0), (1, 1), ∞ ∈ γ are respec-  tively [0 : 0 : 1] = (0, 0), [−1 : 1 : 1] = (−1, 1), [0 : 1 : 0] = ∞ in the  coordinates [x1 : x2 : x3] and in the coordinates (x1, x2) in the affine chart  R2  x1,x2 = {x3 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof Statements 1) and 2) follow from [27, claim 14, subsection 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4] and  the discussion after it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Statement 3) follows from Statement 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Case 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The only admissible vertex of the dual billiard on γ is the  intersection point Q = [1 : 0 : 0] of the tangent lines to γ at the origin and  the infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It is equipped with the projective involution [z : w : t] �→ [−z :  w : t] fixing the points of the line Ow through the origin and the infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The duality sends the above tangent lines to the origin and to the infinity  respectively in the coordinates (x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, the dual line Q∗ is the line  through the origin and the infinity, equipped with the field of lines through  the point [1 : 0 : 0]: the horizontal line field orthogonal to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  65  Case 2b1) (Case 2b2) is treated analogously).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The only admissible vertex  Q = (0, −1) is the intersection point of the tangent line to γ at the point  (1, 1) and the line Ow through the origin and the infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The dual point  to the above tangent line is (−1, 1), by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6, Statement 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The  dual to the Ow-axis is the infinity point [1 : 0 : 0]: the intersection point  of the tangent lines at the origin and at the infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, the line Q∗  dual to Q is the line {x2 = 1} through the points (−1, 1) and [1 : 0 : 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let  us find the corresponding dual projective billiard structure on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The fixed  point line of the involution σQ is the line L = {w = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Indeed, σQ fixes  γ, and hence, the tangency points of the lines through Q tangent to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The  latter tangency points are (±1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, the fixed point line is the line L  through them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The line L intersects γ at the points (±1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Their dual lines  are tangent to C at the points (±1, 1), by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6, Statement 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Therefore, the dual point L∗ is the intersection point (0, −1) of the latter  tangent lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Finally, the admissible line Q∗ = {x2 = 1} is equipped with  the field of lines through the point (0, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case 2c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The dual billiard structure on the conic γ has three base  (indeterminacy) points: (0, 0), (1, 1), ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Each admissible vertex is the  intersection point of a line tangent to γ at one of them and the line through  two other ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The admissible vertices are (1, 0), (0, −1) and [1 : 1 : 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let  us find their dual lines and the projective billiard structures on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The  point Q = (1, 0) is the intersection point of the Oz-axis (which is tangent  to γ at (0, 0)) and the line {z = 1} (which is the line through the points  (1, 1) and ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The dual point to the Oz-axis is the origin (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The dual  point to the line {z = 1} is the point [−1 : 2 : 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Indeed, it is the point of  intersection of the lines tangent to C at the points (−1, 1) and infinity, by  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6, Statement 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The latter intersection point is [−1 : 2 : 0],  since the line tangent to C at (−1, 1) has slope −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Finally, the admissible  line Q∗ dual to Q is the line {x2 = −2x1} through the origin and the point  [−1 : 2 : 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us find its projective billiard structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The lines through Q  tangent to γ are the Oz-axis and the line {x2 = 2(x1 − 1)}, with tangency  points (0, 0) and (2, 4) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, the fixed point line of the  involution σQ is the line L = {w = 2z} through them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Its dual point L∗  is the intersection of the lines dual to (0, 0), (2, 4) ∈ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The latter lines are  tangent to C at the points (0, 0) and (−2, 4), by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6, Statement  3), and they intersect at (−1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, L∗ = (−1, 0), and the admissible line  Q∗ = {x2 = −2x1} is equipped with the field of lines through L∗ = (−1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The line dual to the admissible vertex (0, −1) is the line {x2 = 1}  equipped with the field of lines through the point (0, −1), as in Case 2b1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The admissible vertex [1 : 1 : 0] is the intersection point of the line  66  tangent to γ at infinity and the line through the points (0, 0) and (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Therefore, its dual line passes through infinity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=', is parallel to the Ox2-  axis) and the intersection point of the tangent lines to C at the corresponding  points (0, 0) and (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The latter intersection point is (− 1  2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence,  the line dual to [1 : 1 : 0] is {x1 = − 1  2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It intersects the conic C at two  points: the infinity and the point (− 1  2, 1  4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Its projective billiard structure  is the field of lines through the intersection point of the tangent lines to C  at the two latter points, as in the above cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Their intersection point is  [−1 : 1 : 0], since the slope of the tangent line to C at (− 1  2, 1  4) is equal to  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Finally, the admissible line [1 : 1 : 0]∗ = {x1 = − 1  2} is equipped with  the line field parallel to the vector (−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case 2c1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Then (0, −1) is the unique admissible vertex for the dual  billiard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The only admissible line is its dual line {x2 = 1} equipped with  the field of lines through the point (0, −1), as in Cases 2c2) and 2b1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' There are no admissible lines, since the dual billiard has no  admissible vertices (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='45 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  4  Integrals of dual pencil type billiards: examples  of degrees 4 and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='28,  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='41 and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='42  First we prove Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='28, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='41 and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then we provide ex-  amples of dual pencil type projective billiards with integrals of degrees 4 and  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Afterwards we discuss their realization by the so-called semi-(pseudo-)  Euclidean billiards, with nonlinear part of boundary being equipped with  normal line field for the standard (pseudo-) Euclidean form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1  Multibilliards of pencil type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='28  For every admissible vertex V from Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='21, Case a) (Mj or KEL)  equipped with the corresponding projective involution σV , let �V : R3 → R3  denote the linear involution whose projectivization is σV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  We normalize  it to fix the points of the two-dimensional subspace projected to the fixed  point line of σV and to act as the central symmetry α �→ −α on the one-  dimensional subspace projected to V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let �V ∗ denote its conjugate, acting on  the space R3∗ of linear functionals on R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The symmetric square Sym2(R3∗)  is identified with the space of homogeneous quadratic polynomials on R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The operators �V ∗ lifted to Sym2(R3∗) will be also denoted by �V ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In the  proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='28 we use the two following propositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  67  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1 Let a pencil have type 2a): conics through four different  points A, B, C, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' One has  ( �KEL �KLF )3 = Id  for every three distinct E, L, F ∈ {A, B, C, D}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1)  Proof  Let N ∈ {A, B, C, D} be the point distinct from E, L, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The  involutions σKEL, σKLF fix N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' σKEL fixes F and permutes E, L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' σKLF fixes  E and permutes L, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence, their product fixes N and makes an order three  cyclic permutation of the points E, L, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, Π := (σKEL ◦ σKLF )3 fixes  all the four points A, B, C, D ∈ RP2, hence Π = Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, ( �KEL �KLF )3 = Id  up to constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The latter constant factor should be equal to one,  since the operator in question has unit determinant, being a product of six  involutions �KST , each with determinant −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This proves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Recall that for every line X ⊂ RP2 by ξX ∈ R3∗ we denote a linear  functional vanishing on the two-dimensional subspace in R3 projected to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2 1) The subspace W ⊂ Sym2(R3∗) generated by the prod-  ucts ξEL(Y )ξ(EL)′(Y ) with (EL)′ being the line through the pair of points  {E′, L′} := {A, B, C, D} \\ {E, L}, is two-dimensional and �V ∗-invariant for  every admissible vertex V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Each operator �V ∗ corresponding to a standard  admissible vertex acts on W as the identity up to constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  2) The above functionals ξEL can be normalized so that  �K∗  AB(ξABξCD) = −ξABξCD, �K∗  AB(ξBCξAD) = −ξACξBD,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2)  and so that analogous formulas hold for the other operators �K∗  EL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof The zero conics of the polynomials ξEL(Y )ξ(EL)′(Y ) are the singular  conics AB ∪ CD, BC ∪ AD, AC ∪ BD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' They lie in the pencil of conics  through A, B, C, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence the space W spanned by these polynomials is  two-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Its �V ∗-invariance follows from σV -invariance of the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  For every V ∈ {M1, M2, M3} the involution σV fixes the three above conics,  and hence, each conic of the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus �V ∗|W = Id up to constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let us prove the first formula in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2) for arbitrary normalization of the  functionals ξAB and ξCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Every vector v ∈ R3 \\{0} with π(v) /∈ AB ∪CD is  sent by �KAB to the opposite side from the hyperplane π−1(CD) and to the  same side from the hyperplane π−1(AB), by definition: �KAB fixes the points  of the former hyperplane and acts as central symmetry on its complementary  invariant subspace π−1(KAB), which lies in the latter hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore,  it keeps the sign of the functional ξAB and changes the sign of ξCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hence,  it multiplies their product by −1, being an involution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  68  The operator �KAB permutes the conics BC ∪AD and AC ∪BD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' There-  fore, the functionals ξBC, ξAD, ξAC, ξBD can be normalized so that the  corresponding products ξBCξAD and ξACξBD be permuted by �K∗  AB with  change of sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2) is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us normalize ξAB and ξCD by  constant factors (this does not change formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2)) so that the analogue  of the second formula in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2) holds for �K∗  BC:  �K∗  BC(ξACξBD) = −ξABξCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3)  This together with the second formula in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2) and involutivity of the oper-  ators �V ∗ imply that  �K∗  BC �K∗  AB(ξACξBD) = − �K∗  BC(ξBCξAD) = ξBCξAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4)  Replacing the right-hand side in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3) by �K∗  AB(ξABξCD), applying �K∗  BC to  both sides, denoting H := �K∗  BC �K∗  AB, together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4) yield  H(ξABξCD) = ξACξBD, H(ξACξBD) = ξBCξAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='5)  One also has  H(ξBCξAD) = ξABξCD,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6)  since H3 = Id, by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore,  ξACξBD + ξBCξAD + ξABξCD = 0  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7)  since the terms in the latter sum form an orbit of order three two-dimensional  operator acting on W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us now prove the analogues of formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2) for  the other KEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' To this end, let us show that  �K∗  CD = �K∗  AB on the space W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='8)  Indeed, the composition σKCD ◦ σKAB fixes the three singular conics (and  hence, each conic of the pencil), by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, �K∗  CD = �K∗  AB on  W, up to constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The latter constant factor is equal to one, since  the operators in question take equal value at ξABξCD, by the first formula in  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2) (which holds for KAB replaced by KCD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This proves (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Formula  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='8) together with the other similar formulas, and already proved formula  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2) for the operators �K∗  AB, �K∗  BC imply the analogues of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2) for �K∗  CD,  �K∗  DA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us prove its analogue for �K∗  AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' One has  �K∗  BC �K∗  AC(ξACξBD) = − �K∗  BC(ξACξBD) = ξABξCD,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='9)  69  by formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2) for KBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Therefore, ξACξBD, ξABξCD together with a  third vector �K∗  BC �K∗  AC(ξABξCD) form the orbit of order three linear operator  �K∗  BC �K∗  AC, see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The sum of the vectors in the orbit should be equal to  zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7) implies that  �K∗  BC �K∗  AC(ξABξCD) = ξBCξAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Applying �K∗  BC to this equality yields �K∗  AC(ξABξCD) = −ξBCξAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Formula  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2) for KAC is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' For KBD if follows from its version for KAC as  above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2) is proved for all KEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  For the normalization chosen as in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2 formula  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='21), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=', (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Conversely, if (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7) holds, then the statements  of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2 also hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Relation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7) determines the collection of  products ξELξF N uniquely up to common constant factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof The first statement of the claim is already proved above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Its third,  uniqueness statement follows from two-dimensionality of the space W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' These  two statements together imply the second statement of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Proof  of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let the linear functionals ξEL be normalized  to satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='21), which is possible by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2 and the above claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Then they satisfy (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2), by the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Projective transformations of RP2  act on rational functions on RP2 (which can be represented as rational 0-  homogeneous functions of Y = (y1, y2, y3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The ratio ξABξCD  ξBCξAD is sent by  σKAB to ξABξCD  ξACξBD etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=', by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This implies invariance of the degree 12 ra-  tional function (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='22) under all the involutions σKEL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Its invariance under  the involutions corresponding to the standard vertices follows from Proposi-  tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2, Statement 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Thus, the integral in question is invariant under the  involutions of all the admissible vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It is invariant under the involution  of tangent line to a conic of the multibilliard, since so are its factors, which  are constant on the conics of the pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='28 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2  Dual pencil type projective billiards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proof of Theorem  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='41 and Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='42  Proof  of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Consider a dual pencil type projective billiard  given by a dual pencil of conics tangent to four distinct lines a, b, c, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Consider the corresponding dual multibilliard in RP2  [M1:M2:M3] obtained by  orthogonal polarity duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It is of pencil type, defined by the pencil of  conics through the points A, B, C, D dual to the latter lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Expression  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='25) considered as a rational 0-homogeneous function of M is a well-defined  70  function on RP2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It is a rational integral of the multibilliard, provided that  relation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='24) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' There indeed exist χem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='fn satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='24), by Theo-  rem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='28 and since each scalar product < r(en), M > is a linear functional  vanishing on the two-dimensional subspace π−1(EN) ⊂ R3 corresponding to  the line EN: r(en) ⊥ π−1(EN), by orthogonal polarity duality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore,  substituting M = [r, v] to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='25) yields an integral of the projective billiard,  by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3, Statement 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='41 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us find χem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='fn from linear equations implied  by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Substituting M = (0, 0, 1) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='24) yields  �  χem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='fn = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='10)  For every intersection point em set  x(em) := (x1(em), x2(em)) ∈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let now M ⊥ r(ab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then there exists a v = (v1, v2) = (v1, v2, 0) such that  M = [r(ab), v] = (−v2, v1, ∆ab), ∆ab := [x(ab), v] = x1(ab)v2 − x2(ab)v1,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11)  since v �→ [r, v] is a linear isomorphism R2 → r⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Substituting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='11) to  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='24) yields  χbc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='ad[x(bc) − x(ab), v][x(ad) − x(ab), v]  + χac;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='bd[x(ac) − x(ab), v][x(bd) − x(ab), v] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='12)  The vector differences (x(bc)−x(ab), x(bd)−x(ab) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='12) are proportional  (being parallel to the line b), and so are the other vector differences (parallel  to a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Therefore, equality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='12) is equivalent to the relation  χbc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='adρτ + χac;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='bdst = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='13)  Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='13) with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='10) and normalizing the χem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='fn so that χac;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='bd = 1  yields (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='42 is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3 The χem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='fn satisfying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='24) can be also found by all the pos-  sible substitutions M ⊥ r(em) for em = ab, bc, cd, ac, bd, ad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  This yields  a system of linear equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It appears that their matrix has rank two,  so that there exists a unique common non-zero solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Namely, all the  3x3-minors vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This follows from two-dimensionality of the subspace W  generated by the three quadratic forms < r(em), M >< r(fn), M >, which  in its turn follows from the fact that the three singular conics AB ∪ CD,  71  Figure 19:  AC ∪ BD, AD ∪ BC formed by the lines EM dual to the points em lie in  the same pencil of conics through the points A, B, C, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' On the other  hand, a direct calculation of 3x3-minors of the matrix of linear equations  and equating these minors to zero yields relations on the (oriented) lengths  s, τ, ρ, t, p = |cd − bc|, u = |bc − ac|, q = |cd − ad|, h = |ad − bd|, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' These relations are given by the following geometric theorem, which can  also be deduced from Sine Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The author believes that this theorem  is well-known, but he did not found a reference to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4 In a triangle XZT, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 19, let us take arbitrary points  Y , I on its sides ZT and XZ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let V denote the intersection  point of the lines XY and TI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Set  ρ := |Y V |, t := |V X|, s := |TV |, τ := |V I|,  u := |TY |, p := |Y Z|, q := |IZ|, h := |XI|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Then  pq  (p + u)(q + h) = ρτ  st ,  tp  (ρ + t)(p + u) =  τ  s + τ ,  sq  (s + τ)(q + h) =  ρ  ρ + t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='3  Generic dual pencil type projective billiards with inte-  grals of degrees 4 and 12  Let us construct explicit examples of dual pencil type projective billiards  with minimal degree of integral being equal to 4 and 12, with non-degenerate  dual pencil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Consider a dual pencil of conics tangent to four given distinct  lines: a, b, c, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Fix some its conic γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  We consider that it is a closed  curve in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Let us equip it with the projective billiard structure defined  by the pencil: the conics of the pencil are its complex caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Let us  construct the corresponding admissible lines m1, m2, m3 and kef, e, f ∈  {a, b, c, d}, e ̸= f, equipped with their central projective billiard structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  We consider that the intersection points ab, bc, cd, da of the tangent lines  72  Sform a convex quadrilateral in which γ is inscribed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Then the lines kac and  kbd both intersect the convex domain bounded by γ, see Figures 20 and 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='5 of projective billiards with integral of degree 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The  line kac cuts the domain bounded by γ into two pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Each of them is a  projective billiard bounded by an arc of the curve γ and a segment of the  line kac, both equipped with the corresponding projective billiard structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Both these projective billiards are rationally integrable with minimal degree  of integral equal to four (Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='39 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' However  the tangency points (marked in bold) of the curve γ with the lines a, b, c,  d are indeterminacy points of the projective billiard structure on γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' But  in each piece we can split its boundary arc lying in γ into open subarcs  separated by the tangency points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The projective billiard structure is well-  defined on the latter subarcs, and we can consider them as smaller smooth  pieces of the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' A way to exclude the indeterminacy points from the  boundary is to cut by the line m2 and to consider a smaller domain, bounded  by segments of the lines kac, m2 and an arc of the curve γ, now without  indeterminacies on smooth boundary arcs (except for the ”corners”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This  yields a curvilinear triangle equipped with a projective billiard structure,  also admitting a rational integral of minimal degree 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Figure 20: Three projective billiards (with boundaries marked in bold) with  rational integral of minimal degree 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The indeterminacies of the projective  billiard structure on γ are marked in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='6 with integral of degree 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Consider the two curvilinear  quadrilaterals with boundaries marked in bold at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  21 as projective  73  m2  d  m1  Y  m3  a  Cbilliards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The first one is bounded by segments of the lines kad, kbd, m2 and  an arc of the conic γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The second one is bounded by segments of the lines  kbd, kab, kac and an arc of the conic γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' (We need to note that the boundary  arc in γ in at least some of them contains (at least one) tangency point,  which is an indeterminacy point of the projective billiard structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=') Both  quadrilaterals considered as projective billiards are rationally integrable with  minimal degree of integral being equal to 12, by Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='39 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Figure 21: Two projective billiards (with boundaries marked in bold) with  integral of minimal degree 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='4  Semi-(pseudo-) Euclidean billiards with integrals of dif-  ferent degrees  Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='7 A projective billiard in R2  x1,x2 with piecewise smooth bound-  ary is called semi-Euclidean (semi-pseudo-Euclidean), if the nonlinear part  of the boundary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=', its complement to the union of straightline intervals  contained there, is equipped with normal line field for the standard Eu-  clidean metric dx2  1 + dx2  2 (respectively, for the standard pseudo-Euclidean  metric dx2  1 − dx3  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='8 A semi-Euclidean billiard is rationally 0-homogeneously in-  tegrable, if and only if the nonlinear part of its boundary is a finite union  of confocal conical arcs and segments of some of the admissible real lines  (listed below) for the corresponding confocal pencil of conics:  Case 1), pencil of confocal ellipses and hyperbolas:  the two symmetry axes of the ellipses, equipped with normal line field;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  74  Kab  Kbd  Kad  d  m  Y  m3  Kac  c- the lines L1, L2 through the foci F1, F2, orthogonal to the line F1F2,  each Lj is equipped with the field of lines through the other focus F2−j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  The billiard has quadratic integral, if and only if its boundary contains no  segments of lines L1,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' otherwise the minimal degree of integral is four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Case 2), pencil of confocal parabolas:  the common axis of the parabolas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  the line L through the focus that is orthogonal to the axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Both lines are equipped with the normal line field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The billiard has quadratic  integral, if and only if its boundary contains no segments of the line L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  otherwise the minimal degree of integral is four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof  The other admissible lines from Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='34 are not finite real  lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' For example, in Case 1) the dual pencil of confocal conics consists of  conics tangent to two given pairs of lines through the two isotropic points  [1 : ±i : 0] at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' In this case the only real skew admissible lines are  the lines L1 and L2, and they are opposite as skew admissible lines: they  correspond to two opposite intersection points of the above tangent lines,  namely, the foci F1 and F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Similarly in Case 2) the only real skew admissible  line is L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This together with Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='40 proves Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='9 Consider an ellipse and a line L1 through its left focus F1  that is orthogonal to the foci line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  22, the left part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Consider  the dashed domain bounded by the intersection segment of the line L1 with  the ellipse interior and the left elliptic arc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' the latter arc is equipped with  normal line field, and the segment with the field of lines through the other  focus F2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' This projective billiard admits a rational 0-homogeneous integral  of minimal degree four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' As the second focus F2 tends to infinity so that  the ellipse tends to a parabola with the focus F = F1, the above billiard  converges to a usual billiard (with normal line field) bounded by a segment  of the line L through F orthogonal to the parabola axis and by a parabola  arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' See the right part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The latter parabolic billiard is known  to have a polynomial integral of minimal degree four (and hence, a rational  0-homogeneous integral of the same degree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' It was first discovered in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='10 The projective billiards with rational 0-homogeneous in-  tegral of minimal degrees 4 and 12 presented at Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 20 and 21 respectively  are realized by semi-pseudo-Euclidean billiards in (R2, dx2  1 − dx2  2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Proof Take the points bd = b ∩ d and ac = a ∩ c to be the isotropic points  at infinity [1 : ±1 : 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  ✷  75  Figure 22: Billiards (dashed) with degree 4 integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' On the left: the semi-  Euclidean billiard bounded by a segment of the line L1 and an elliptic arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  As the ellipse degenerates to a horizontal parabola, it tends to the Euclidean  billiard on the right discovered in [34], with degree 4 polynomial integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  References  [1] Avila, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' De Simoi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Kaloshin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' An integrable deformation of an  ellipse of small eccentricity is an ellipse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' (2) 184 (2016),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 2, 527–558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  [2] Amiran, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Caustics and evolutes for convex planar domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Diff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Geometry, 28 (1988), 345–357.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  [3] Bialy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Convex billiards and a theorem by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hopf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=', 214(1)  (1993), 147–154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  [4] Bialy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' On totally integrable magnetic billiards on constant curvature  surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Announc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 19 (2012), 112–119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  [5] Bialy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Hopf rigidity for convex billiards on the hemisphere and hyper-  bolic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Discrete Contin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 33 (2013), No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 9, 3903–3913.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  [6] Bialy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Mironov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' On fourth-degree polynomial integrals of the  Birkhoff billiard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Steklov Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=', 295 (2016), No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1, 27–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  [7] Bialy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Mironov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Algebraic non-integrability of magnetic bil-  liards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' A 49 (2016), No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 45, 455101, 18 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  76  FF[8] Bialy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Mironov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Angular billiard and algebraic Birkhoff conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' in Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 313 (2017), 102–126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  [9] Bialy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Mironov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Algebraic Birkhoff conjecture for billiards on  Sphere and Hyperbolic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=', 115 (2017), 150–156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  [10] Bialy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Mironov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Polynomial non-integrability of magnetic bil-  liards on the sphere and the hyperbolic plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Russ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Surveys, 74  (2018), 187–209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  [11] Bialy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Mironov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' A survey on polynomial in momenta integrals  for billiard problems, Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=', 336 (2018), Issue 2131,  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='1098/rsta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='0418  [12] Bialy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Mironov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' The Birkhoff-Poritsky conjecture for centrally-  symmetric billiard tables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' (2) 196 (1) (2022), 389–413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='  [13] Bolotin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Integrable Birkhoff billiards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content='  [14] Bolotin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content=' Surveys 65 (2010), No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content=' (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Dyn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' 8 (2014), No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content='  [18] Dragovi´c, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content='  Arnold Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content='  [20] Fierobe, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content=' PhD thesis, ENS de Lyon, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content='  To appear in Israel J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
+page_content=' Preprint https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content=' Examples of reflective projective billiards and outer ghost  billiards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content=' On two-dimensional polynomially integrable billiards on  surfaces of constant curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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+page_content=' Preprint https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdAyT4oBgHgl3EQfl_jn/content/2301.00464v1.pdf'}
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diff --git a/qdE4T4oBgHgl3EQfvw0Q/content/tmp_files/2301.05244v1.pdf.txt b/qdE4T4oBgHgl3EQfvw0Q/content/tmp_files/2301.05244v1.pdf.txt
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@@ -0,0 +1,2027 @@
+MNRAS 000, 1–14 (2023)
+Preprint 16 January 2023
+Compiled using MNRAS LATEX style file v3.0
+Strong gravitational lensing’s ‘external shear’ is not shear
+Amy Etherington1,2 , James W. Nightingale1,2⋆ , Richard Massey1,2 , Sut-Ieng
+Tam3 , XiaoYue Cao4,5 , Anna Niemiec1, Qiuhan He2 , Andrew Robertson6 ,
+Ran Li5,4, Aristeidis Amvrosiadis2, Shaun Cole2, Jose M. Diego7, Carlos S. Frenk2,
+Brenda L. Frye8, David Harvey9, Mathilde Jauzac1,2,10,11, Anton M. Koekemoer10 ,
+David J. Lagattuta1, Marceau Limousin11, Guillaume Mahler1, Ellen Sirks12 &
+Charles L. Steinhardt13
+1Department of Physics, Centre for Extragalactic Astronomy, Durham University, South Rd, Durham, DH1 3LE, UK
+2Department of Physics, Institute for Computational Cosmology, Durham University, South Road, Durham DH1 3LE, UK
+3Academia Sinica Institute of Astronomy and Astrophysics (ASIAA), No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan
+4School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, China
+5National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100012, China
+6Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA
+7Instituto de F´ısica de Cantabria (CSIC-UC), Avda. Los Castros s/n, 39005 Santander, Spain
+8Department of Astronomy/Steward Observatory, University of Arizona, 933 N Cherry Ave., Tucson, AZ 85721, USA
+9Laboratoire d’Astrophysique, EPFL, Observatoire de Sauverny, 1290 Versoix, Switzerland
+10Space Telescope Science Institute, 3700 San Martin Dr., Baltimore, MD 21218, USA
+11Aix Marseille Univ, CNRS, CNES, LAM, F-13388 Marseille, France
+12Sydney Consortium for Particle Physics and Cosmology, School of Physics, The University of Sydney, NSW 2006, Australia
+13Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, København Ø 2100, Denmark
+ABSTRACT
+The distribution of mass in galaxy-scale strong gravitational lenses is often modelled as
+an elliptical power law plus ‘external shear’, which notionally accounts for neighbouring
+galaxies and cosmic shear. We show that it does not. Except in a handful of rare systems,
+the best-fit values of external shear do not correlate with independent measurements of
+shear: from weak lensing in 45 Hubble Space Telescope images, or in 50 mock images
+of lenses with complex distributions of mass. Instead, the best-fit shear is aligned with
+the major or minor axis of 88% of lens galaxies; and the amplitude of the external shear
+increases if that galaxy is disky. We conclude that ‘external shear’ attached to a power
+law model is not physically meaningful, but a fudge to compensate for lack of model
+complexity. Since it biases other model parameters that are interpreted as physically
+meaningful in several science analyses (e.g. measuring galaxy evolution, dark matter
+physics or cosmological parameters), we recommend that future studies of galaxy-scale
+strong lensing should employ more flexible mass models.
+Key words: gravitational lensing: strong — galaxies: structure
+1
+INTRODUCTION
+Gravitational lensing is the deflection of light rays by nearby
+concentrations of matter and their associated gravitational
+fields. If the light ray should pass straight through an object
+as massive as a galaxy, it can be deflected along multiple
+routes around the galaxy, and appear distorted into arcs or an
+‘Einstein ring’. Such galaxy-scale strong lensing has been used
+to infer the distribution of mass in massive elliptical galaxies
+⋆ Contact e-mail: james.w.nightingale@durham.ac.uk
+(Gavazzi et al. 2007; Koopmans et al. 2009; Auger et al.
+2010; Sonnenfeld et al. 2013; Bolton et al. 2012; Etherington
+et al. 2022a), to infer their dark matter content, stellar
+mass-to-light ratios, and inner structure (Massey et al. 2010;
+Sonnenfeld et al. 2012; Oldham & Auger 2018; Nightingale
+et al. 2019; Shu et al. 2015, 2016a). If the background source
+is variable, measurements of time delays between multiple
+images can be used to measure cosmological parameters
+(Kilbinger 2015) or the Hubble constant (Suyu et al. 2017;
+Wong et al. 2019; Harvey et al. 2020; Gomer & Williams
+2021). If the lens galaxy contains small substructures, which
+© 2023 The Authors
+arXiv:2301.05244v1  [astro-ph.CO]  12 Jan 2023
+
+2
+Etherington et al.
+would be a smoking gun of the ‘clumpy’ Cold Dark Matter
+(CDM) model, they would also perturb the multiple images
+(Natarajan & Springel 2004; Vegetti et al. 2010; Vegetti et al.
+2014; Li et al. 2016, 2017; Hezaveh et al. 2016; Ritondale
+et al. 2019; Despali et al. 2019; He et al. 2021; Amorisco et al.
+2022; Nightingale et al. 2022).
+When modelling the distribution of mass to fit strong
+lensing data, two additional free parameters are frequently
+included. The (amplitude and angle of) ‘external shear’ is
+intended to represent the cumulative deflection of light by all
+other gravitational potentials along the line of sight. Indeed,
+the best-fit value of external shear matched a model of the
+line of sight for three of six galaxy lenses studied by Wong
+et al. (2011). Subsequent papers have even proposed using
+external shear as relatively high signal-to-noise measurements
+of the ‘cosmic’ shear along individual lines of sight (Birrer
+et al. 2017; Desprez et al. 2018; Kuhn et al. 2020; Fleury
+et al. 2021; Hogg et al. 2022). However, the best-fit values
+of external shear did not match the lines of sight to the
+three other galaxies studied by Wong et al. (2011) — and,
+in general, best-fit values are much larger than expected for
+both galaxy lenses (Keeton et al. 1997; Witt & Mao 1997;
+Hilbert et al. 2007; Suyu et al. 2010; Barrera et al. 2021) and
+cluster lenses (Robertson et al. 2020; Limousin et al. 2022).
+Using mock observations, Cao et al. (2022, hereafter C22)
+demonstrated that the best-fit shear can be incorrect if the
+model of the mass distribution is missing complexity.
+In this paper we compare the external shear measured
+from Hubble Space Telescope (HST) imaging — of strong
+lensing galaxies from the SLACS survey (Bolton et al. 2008a),
+GALLERY survey (Shu et al. 2016b), and four lenses in
+clusters — against independent measurements of the shear
+along the same line-of-sight, observed as weak lensing of
+adjacent galaxies. To gain understanding, we also analyse
+C22’s mock HST imaging, generated without external shear,
+but fitted with shear as a free parameter.
+Comparing independent measurements of shear will test
+the hypothesis that strong lensing external shear is a real,
+physical quantity. Strong and weak lensing measurements
+average over different spatial scales and are obtained at differ-
+ent redshifts, so they might not be identical; but they should
+be strongly correlated. Analysing three lenses in the COS-
+MOS field (Faure et al. 2008) with sophisticated statistics,
+Kuhn et al. (2020) measured a smaller covariance between
+strong and weak lensing shears than the difference between
+individual systems, indicating that more data were required.
+In this work, with a much larger sample of galaxies, we simply
+aim to detect a correlation between the two probes.
+We define relevant concepts from lensing theory in Sec-
+tion 2. We then describe our observed and mock data in
+Section 3 and our analysis methods in Section 4. We present
+our results in Section 5 and investigate possible causes in
+Section 6. We interpret these in a wider context and conclude
+in Section 7. Throughout this paper, we assume a Planck
+2015 cosmological model (Ade et al. 2016).
+2
+GRAVITATIONAL LENSING THEORY
+Gravitational lensing describes the deflection of light rays
+from distant sources, by matter along its path to an observer,
+through angle α. This maps 2D coordinates of light on the
+distant source plane β to coordinates where they are observed
+on the foreground image plane θ = (θ1, θ2), via the lens
+equation
+β = θ − α(θ).
+(1)
+If the gravitational lens is much thinner than its angular
+diameter distance from the observer Dl, its distribution of
+mass can be treated as a 2D surface density projected along
+the line-of-sight Σ(θ) =
+�
+ρ(Dlθ1, Dlθ2, z) dz. The deflection
+angle is then the vector gradient α = ∇ψ of a 2D lensing
+potential
+ψ(θ) ≡ 2 Dls
+DlDs
+2G
+c2
+�
+Σ(θ′) ln |θ − θ′| d2θ′
+(2)
+= 2 Dls
+DlDs
+�
+Φ(Dlθ, z) dz ,
+(3)
+where Φ is the 3D Newtonian potential, and the prefactor
+(which involves angular diameter distances to the lens, to
+the source, and from the lens to the source) reflects the
+geometrical efficiency of a lens: peaking if it is half-way
+between the source and the observer.
+The Jacobian of the lens equation (1) is thus
+Aij ≡ ∂βi
+∂θj = δij − ∂αi
+∂θj = δij − ∂2ψ
+∂θiθj .
+(4)
+2.1
+Weak lensing
+If the source is much smaller than the scale of local variations
+in the gravitational field, the Jacobian can be approximated
+as constant
+Aij ≈
+�1 − κ − γ1
+−γ2
+−γ2
+1 − κ + γ1
+�
+(5)
+= (1 − κ)
+�1
+0
+0
+1
+�
+− γ
+�cos2φ
+sin2φ
+sin2φ
+cos2φ
+�
+,
+(6)
+where the convergence
+κ = 1
+2∇2ψ ,
+(7)
+and where the two components of shear
+�
+γ1, γ2
+�
+=
+�1
+2
+�∂2ψ
+∂θ2
+1
+− ∂2ψ
+∂θ2
+2
+�
+,
+∂2ψ
+∂θ1∂θ2
+�
+(8)
+can also be expressed in terms of shear amplitude γ2 = γ2
+1+γ2
+2
+and angle φ = ½ arctan (γ2/γ1). The convergence magnifies
+a source, and the shear changes its shape. Strictly, these
+quantities are only observable in combination as ‘reduced
+shear’ gi = γi/(1 − κ). However, in the weak lensing regime,
+κ ≪ 1, so gi ≈ γi.
+2.2
+Strong lensing
+If light from one side of a source is deflected differently to
+light from the other side, it can appear distorted in the image
+plane as an arc; it is also possible to see multiple images of a
+single source, if more than one solution exists with different
+α and θ. To reconstruct α(θ) it is usual to note that
+κ(θ) = Σ(θ)
+Σcrit
+with constant
+Σcrit = c2
+4πG
+Ds
+DlDls ,
+(9)
+which is equal to the mean surface mass density within the
+Einstein radius, REin. For axisymmetric lenses the Einstein
+MNRAS 000, 1–14 (2023)
+
+‘External shear’ is not shear
+3
+radius is uniquely defined by the radius of the circular tangen-
+tial critical curve that is produced where the magnification
+diverges in the lens plane. This occurs where the tangen-
+tial eigenvalue of the Jacobian (equation 4) λt = 1 − κ − γ
+is equal to zero. For asymmetric lenses, the definition of
+Einstein radius must be generalised; we choose to use the
+effective Einstein radius
+REin,eff =
+�
+A
+π ,
+(10)
+where A is the area enclosed by the tangential critical curve.
+When considering (typically Early-type) galaxy-scale
+lenses, it is common practise to parameterise the surface
+mass distribution as an elliptical power law (Suyu 2012)
+Σ(θ1, θ2; b, q, γ) = 3 − γ
+1 + q
+�
+b
+�
+θ2
+1 + θ2
+2/q2
+�γ−1
+,
+(11)
+where b ⩾ 0 is the angular scale length (referred to in some
+papers as the Einstein radius, but distinct from the more
+robust effective Einstein radius in Equation 10), 0 < q ⩽ 1
+is the projected minor to major axis ratio of the ellipti-
+cal isodensity contours, and (confusingly denoted) γ is the
+logarithmic slope of the mass distribution in 3D (for an
+‘isothermal’ distribution, γ = 2). If we also allow the mass to
+be translated to central coordinates (θc
+1, θc
+2) and reoriented
+to position angle φPL, which we measure counterclockwise
+from the positive θ1-axis, the model has six free parameters.
+The primary lens may not be the only source of shear.
+Any ‘external’ component due to other galaxies or clusters
+near the lens or along the ray path, and constant on scales
+larger than b (rather than the size of the source) is modelled
+as two more free parameters
+�
+γext
+1
+, γext
+2
+�
+= γext�
+cos (2φext), sin (2φext)
+�
+,
+(12)
+where γext is the amplitude and φext is the angle of the shear
+measured counterclockwise from the θ1-axis. This is applied
+as an additional component of α(θ). It does not change κ(θ).
+3
+DATA
+3.1
+Mock lens galaxies
+We analyse a set of 50 mock lens images, representative of
+data from the SLACS survey. They were generated by C22
+for an investigation into the systematic errors induced by
+the elliptical power-law model. We summarise the simulation
+procedure below; a detailed description can be found in
+Section 2.4 of that paper.
+The surface mass density of the lens galaxy comprises
+two components: a dark matter halo, parameterised by the
+spherical generalised Navarro, Frenk & White (gNFW) profile
+(Navarro et al. 1997; Zhao 1996; Cappellari et al. 2013), plus
+visible stellar matter, parameterised by a Multiple Gaussian
+Expansion (MGE; Cappellari 2002). The model parameters
+of the gNFW and MGE profiles of each lens galaxy are set
+to the best-fit parameters from fits of these distributions to
+SDSS-MaNGA stellar dynamics data, derived by Li et al.
+(2019) using the Jeans anisotropic model (JAM) method.
+The position angle of each Gaussian component in the MGE
+is fixed, however their axis ratios are free to vary, allowing
+for elliptical gradients in the mass distribution.
+The light distribution of the source galaxy is modelled by
+a single S´ersic profile (Graham & Driver 2005) with effective
+radius Reff = 0.15′′, S´ersic index n = 1, and axis ratio q = 0.7.
+The position in the source plane (xs, ys) is drawn from a
+Gaussian distribution with mean 0′′ and standard deviation
+0.1′′, and the position angle is uniformly selected between
+0◦ −180◦. The light from the source galaxy is ray-traced from
+the source plane at z = 0.6 to the image plane through the
+lens equation (equation 1), to simulate its lensed appearance.
+Further, to mimic observational effects, the image is con-
+volved with a Gaussian PSF with 0.05′′ standard deviation,
+and sampled by 0.05′′ square pixels. A flat background sky of
+0.1 electrons per second is assumed, and an exposure time of
+840 seconds is used to add Poisson noise from the source and
+background sky. The signal to noise ratio of the brightest
+pixel in the synthetic images is set to ∼50, by adjusting the
+intensity of the S´ersic source accordingly. No external shear
+was simulated in the mock data.
+3.2
+Observed lens galaxies
+We analyse three sets of galaxy-galaxy strong lenses. These
+include 42 lenses from the SLACS survey (Bolton et al. 2008a)
+that were fitted without significant residuals by Etherington
+et al. (2022b, hereafter E22)’s automated pipeline. Most are
+isolated field galaxies. They were found by searching for
+high redshift emission lines in the spectra of low-redshift
+galaxies obtained through a 3′′ fibre. They were then imaged
+by the HST Advanced Camera for Surveys (ACS) in the
+F814W band, and processed into stacked images with 0.05′′
+pixels. We also reprocessed these to measure weak lensing,
+following the procedure described by Tam et al. (2020), which
+supersamples the pixels to 0.03′′. We exclude lenses J1143-
+0144 and J1420+6019, for which only one exposure was
+obtained.
+We analyse 15 lenses from the GALLERY survey (Shu
+et al. 2016b) that were modelled by E22. These are also field
+galaxies, found by searching for compact Lyman-α-emitting
+source galaxies in spectra with a 2′′ fibre. They were imaged
+with the HST Wide Field Camera 3 (WFC3) in the F606W
+band and processed, following Shu et al. (2016a), into 0.04′′
+pixels. We do not attempt to measure weak lensing shear in
+these data.
+We analyse 4 galaxy-galaxy strong lenses in the outskirts
+of galaxy clusters, where we expect a 5–15% true external
+shear. Before beginning any analysis, we searched archival
+HST F814W imaging, and selected lenses with multiple imag-
+ing of sources that are extended a similar amount as the field
+lenses. Positions and redshifts of the selected lenses are given
+in Table 1. For MACS1149-GGL18 no source redshift has
+been recorded, where necessary we assume a source redshift
+of zs = 1.5 and test that the results do not change signifi-
+cantly when we change this assumption over a range of source
+redshifts from 0.5 to 2. Two of our selected lenses had been
+previously modelled by Desprez et al. (2018), although con-
+strained using only the positions of multiple images, rather
+than all the pixels. We analysed the HST data similarly to
+the SLACS lenses, except for the ‘cosmic snail’. For that
+lens alone, we do not measure weak lensing, but use the
+independent estimate of shear from Desprez et al. (2018)’s
+model IV of the galaxy cluster, constrained by cluster-scale
+MNRAS 000, 1–14 (2023)
+
+4
+Etherington et al.
+Lens name
+RA
+Dec
+zl
+zs
+MACS1149-GGL18
+177.410247
+22.352017
+0.544
+-
+Abell370-GGL19
+39.963013
+−1.534783
+0.375
+2.371
+MACS1149-GGL20
+177.402816
+22.436607
+0.544
+1.806
+RX J2129-GGL1 (snail)
+322.428780
+0.108071
+0.235
+1.610
+Table 1. Table of parameters for the 4 galaxy-galaxy lenses in the
+outskirts of clusters, the lens name refers to the cluster in which
+the lens resides.
+strong lensing, and shown by Desprez et al. (2018) to be
+consistent with ground-based measurements of weak lensing.
+4
+METHODS
+4.1
+Weak lensing analysis
+We identified galaxies on lines of sight adjacent to strong
+lenses using SExtractor (Bertin & Arnouts 1996), and mea-
+sured their shapes using the PyRRG (Harvey et al. 2019)
+implementation of the Rhodes et al. (2000) shear measure-
+ment method. This estimates the mean reduced shear in a
+patch of sky, by averaging galaxies’ apparent shapes
+ϵi = ϵint
+i
++ Gγi,
+(13)
+which have been transformed by weak lensing (Section 2.1)
+from an unknown intrinsic shape ϵint. The ‘shear responsivity’
+G varies as a function of galaxy flux, and its overall scaling
+has been calibrated on simulated images with known shear
+(Leauthaud et al. 2007). Under the assumption that galaxies’
+intrinisic shapes are randomly oriented, i.e. ⟨ϵint
+i ⟩ ≈ 0,
+⟨γi⟩ =
+�ϵi − ϵint
+i
+G
+�
+≈
+� ϵi
+G
+�
+−
+�ϵint
+i
+G
+�
+≈ ⟨ϵi⟩
+⟨G⟩.
+(14)
+Following Massey et al. (2007), we assume that the median
+redshift of the lensed galaxies is z ∼ 1.26. Thereafter, follow-
+ing Smail et al. (1994), we treat them all as being at this
+effective redshift. None of our results change significantly if
+we adjust this value.
+We average weak lensing shear measurements from the
+∼140 galaxies within 60′′ of the strong lens galaxy (no weights
+are applied to the galaxies that are averaged). The precision
+of this measurement is limited by the randomness in the
+distribution of the intrinsic shapes
+σ2
+int =
+��ϵint
+i
+G
+�2�
+.
+(15)
+We measure σint ∼ 0.3, consistent with Leauthaud et al.
+(2007), and hence uncertainty σint/
+√
+140 ≈ 0.02 on each
+component of mean shear. This is similar to uncertainty on
+our strong lensing measurements. None of our results change
+significantly if we use a 45′′ or 90′′ aperture instead.
+Although the line-of-sight directly through each galaxy-
+scale lens is not in the weak lensing regime, we assume that
+⟨gi⟩ ≈ ⟨γi⟩ still holds since the vast majority of adjacent lines
+of sights will be only weakly lensed. Nor do we attempt to
+model and subtract the weak shear due to the galaxy-scale
+lens itself. Doing so would mix the weak lensing and strong
+lensing analyses; and it is unnecessary at our achieved level of
+precision because the near-circular symmetry of most lenses
+−1.5
+−1.0
+−0.5
+0.0
+0.5
+1.0
+1.5
+−1.0
+−0.5
+0.0
+0.5
+1.0
+boxy (a4 = −0.1)
+ellipse
+disky (a4 = 0.1)
+Figure 1. Examples of boxy (a4 = −0.1, blue dashed curve) and
+disky (a4 = +0.1, pink dashed curve) perturbations to an ellipse
+(orange curve). The perturbations shown are ∼10 times larger
+than those typically observed. In both cases the perturbation at
+45◦, b4 = 0 (see equation 17).
+means that the lens contributes negligibly to ⟨γi⟩ inside a
+60′′ circular aperture.
+4.2
+Strong lensing analysis
+We analysed all data using the automated strong lens mod-
+elling software PyAutoLens1 (Nightingale et al. 2018, 2021b).
+This fits parameters of the lens model using all of the pixels
+in an image (not just e.g. locations of the centre of light, as
+in previous works).
+The pipelines used to fit the mock and observed data
+are described fully in C22 and E22 respectively. Briefly, we
+model the distribution of mass in both mock and real data
+using an elliptical power law (equation 11) plus external
+shear (equation 12). We then repeat the fit, fixing external
+shear γext = 0. We model the distribution of light in in real
+lens galaxies using a double S´ersic profile with a centre that
+is free to vary independently to that of the mass distribution,
+and for the source galaxy using an adaptive Voronoi mesh of
+pixels. For the mock data, we use C22’s fit in which the lens
+light is perfectly subtracted and the source light is modelled
+as an elliptical S´ersic. C22 also performed fits using a Voronoi
+mesh for the source light. However, since the mock data were
+created assuming a S´ersic source, the model we chose can
+perfectly describe the source, so any systematics we observe
+will be solely due to mismatch between the model and truth
+of the mass distribution, which is the point of interest in this
+study.
+4.3
+Multipole perturbations of an ellipse
+In Section 6.1 we shall investigate whether the strong lensing
+external shears depend on deviations from an elliptically
+symmetric distribution of mass. Specifically, we shall quan-
+tify the multipole deviations of two types of contour: the
+iso-convergence contour at κ = 1 of the gNFW+MGE dis-
+tributions used to create mock data, and the critical curves
+1 The PyAutoLens software is open source and available from
+https://github.com/Jammy2211/PyAutoLens
+MNRAS 000, 1–14 (2023)
+
+‘External shear’ is not shear
+5
+of both the mock and the observed galaxies. These con-
+tours are stored as a 2D array of points in polar coordinates
+[φcontour, Rcontour]. We calculate perpendicular deviations of
+each point from the true ellipse
+Rel(φ) =
+a
+√
+1 − ϵ2
+�
+1 − ϵ2 cos2(φ − φel)
+,
+(16)
+where a is the major axis, φel is the major axis orientation,
+and ϵ is the eccentricity (defined as ϵ2 ≡ 1 − b2/a2 where b
+is the minor axis). The deviations are then parameterised
+using multipoles
+δRm(φ; am, bm) =
+�
+am cos(m(φ−φel))+bm sin(m(φ−φel)) ,
+(17)
+where m is the order of the multipole, and am and bm are the
+magnitude of the deviations with symmetry along or at 45◦
+to the major and minor axes, respectively. We then perform
+a non-linear search to fit the model
+R(φ; a4, b4) = Rel(φ) + δR4(φ; a4, b4)
+(18)
+to the radial values of the contour. We assume uniform priors
+on the free parameters in the fit over a reasonable range and
+fit for them using the nested sampling algorithm dynesty.
+We assume the residual errors can be described by a Gaussian
+distribution and maximise the likelihood
+L(R|Ri, σ) =
+�
+i
+�
+1
+√
+2πσ2 exp
+�
+−(R(φi) − Ri)2
+2σ2
+��
+,
+(19)
+where Ri are the radial values of the contour and R(φi; a4, b4)
+are the model predicted values from equation (18) at each
+angular coordinate in the contour φi. Curves with best-fit
+values of a4 > 0 are ‘disky’; those with a4 < 0 are ‘boxy’ (see
+Figure 1).
+5
+RESULTS
+5.1
+SL shears do not correlate with WL shears
+Strong lensing measurements of shear γSL (obtained as the
+best-fit external shear γext) typically have amplitudes up to
+an order of magnitude larger than weak lensing measurements
+γWL along the same line of sight. The mock lenses have mean
+best-fit |⟨γSL⟩| = 0.019 ± 0.002, despite the true values all
+being γext = 0. Our measurement using a PL+ext mass
+model is consistent with C22’s value |⟨γSL⟩| = 0.015 using
+pixel-based source reconstructions. The real lenses have mean
+best-fit shear |⟨γSL⟩| = 0.098 ± 0.011, which is much larger
+than both the measured weak lensing shear |⟨γWL⟩| = 0.028±
+0.002 and the typical ∼ 1 − 3% shear expected along random
+lines of sight through the universe (Keeton et al. 1997), even
+accounting for the different scales on which they are averaged
+(Valageas et al. 2004; Wong et al. 2011).
+Strong lensing measurements of shear do not correlate
+with weak lensing measurements (Figure 2). To make this
+comparison (for the real lenses only), we first define rotated
+coordinate systems such that γSL
+1
+= γSL ⩾ 0 and γSL
+2
+= 0.
+Thus we need plot only three of the four components of
+shear. Second, we compensate for the different redshifts of
+the strongly lensed and weakly lensed sources by rescaling
+values of γSL
+1
+by (Ds/Dls)z′s=zs(Dls/Ds)z′s=1.26, i.e. the ef-
+fective value at the redshift of the weakly lensed galaxies
+−0.05
+0.00
+0.05
+0.10
+γWL
+1
+0.0
+0.1
+0.2
+0.3
+0.4
+γSL
+1
+�
+DS
+DLS
+�
+zs
+�
+DLS
+DS
+�
+1.26
+−0.05
+0.00
+0.05
+0.10
+γWL
+2
+Lens Name
+J0008-0004
+J0029-0055
+J0157-0056
+J0216-0813
+J0252+0039
+J0330-0020
+J0728+3835
+J0737+3216
+J0822+2652
+J0841+3824
+J0903+4116
+J0912+0029
+J0936+0913
+J0946+1006
+J0956+5100
+J0959+0410
+J0959+4416
+J1016+3859
+J1020+1122
+J1023+4230
+J1029+0420
+J1032+5322
+J1103+5322
+J1142+1001
+J1153+4612
+J1205+4910
+J1213+6708
+J1218+0830
+J1250+0523
+J1402+6321
+J1416+5136
+J1430+4105
+J1432+6317
+J1451-0239
+J1525+3327
+J1627-0053
+J1630+4520
+J2238-0754
+J2300+0022
+J2303+1422
+J2341+0000
+snail
+GGL20
+GGL19
+GGL18
+Figure 2. Values of the shear along the lines of sight to 39 galaxy-
+galaxy lenses, independently measured using strong lensing ‘exter-
+nal’ shear γSL and weak lensing γWL. Shears are oriented such
+that γSL
+2
+= 0, and rescaled to be at the same effective redshift. If
+strong and weak lensing shears were identical, all points would lie
+on the dashed lines. We instead find that external shears inferred
+from strong lensing are consistently larger than those measured
+by weak lensing, and not aligned.
+(see eqn 2). This scaling is exact only if the external shear is
+both real and dominated by neighbouring structures at the
+same redshift as the lens (Wong et al. 2011 found that 5/8 of
+the shear is from neighbouring structures). In any case, the
+rescaling is by a factor with mean of only 1.26 and rms 1.06,
+and our conclusions do not change if the rescaling is omit-
+ted or normalised to a different redshift. If strong and weak
+lensing measure the same quantity, we then expect γWL
+1
+to
+correlate with γSL
+1 , and γWL
+2
+to scatter around zero. We find
+that ⟨γWL
+2
+⟩ = −0.004 ± 0.003 is on average below zero, and
+MNRAS 000, 1–14 (2023)
+
+6
+Etherington et al.
+0.00
+0.02
+0.04
+0.06
+γext
+0
+20
+40
+60
+80
+|φPL+ext
+mass
+−φPL+ext
+ext
+|
+‘aligned’
+‘anti-aligned’
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+q
+0.0
+0.1
+0.2
+0.3
+0.4
+γext
+0
+20
+40
+60
+80
+|φPL+ext
+mass
+−φPL+ext
+ext
+|
+‘aligned’
+‘anti-aligned’
+0.4
+0.6
+0.8
+q
+Figure 3. Relative orientation of the strong lensing ‘external’ shear
+and the major axis of the lens mass, in mock (top panel) and real
+HST (bottom panel) data. In both cases, most of the shears are
+suspiciously aligned (φPL+ext
+mass
+− φPL+ext
+ext
+⩽ 30◦) or anti-aligned
+(φPL+ext
+mass
+− φPL+ext
+ext
+⩾ 60◦) with the lens mass distribution. If the
+γext parameter were measuring true external perturbations, the
+orientations would be random. Points are coloured by the best-fit
+axis ratio of the lens mass distribution. Highly elliptical lenses
+often lead to high values of γext.
+its scatter (0.02) is consistent with uncertainties calculated
+from the distribution of weak lensing shears. The best-fit
+slope γWL
+1
+= (−0.06 ± 0.04)γSL
+1
+actually infers a negative
+correlation, however this does not take into account the un-
+certainty on the strong lensing shears and so the uncertainty
+is likely underestimated. The Pearson correlation coefficient
+−0.19 ± 0.22 implies that, if there is a correlation, there is
+also a large amount of scatter.
+There are eight lenses for which γext ≈ γWL, including
+two of the four lenses which reside in the outskirts of clusters
+(see Section 5.3 for further discussion). However, there does
+not appear to be anything unique about these lenses that
+would make the shear possible to measure in these cases.
+5.2
+SL shears are (suspiciously) aligned with the mass
+For both mock and real data, the best-fit ‘external’ shear
+is usually aligned with the major axis of the lens mass
+(φPL+ext
+mass
+− φPL+ext
+ext
+⩽ 30◦) or with its minor axis (φPL+ext
+mass
+−
+φPL+ext
+ext
+⩾ 60◦): see figure 3. If external shears were truly
+measuring external perturbations, their orientations would
+be random (modulo intrinsic alignments between the shape of
+a galaxy and its surrounding tidal field, but these are much
+smaller than our achieved measurement precision; Zhang
+et al. 2022). The preference for aligning with the mass distri-
+bution again suggests an ‘internal’ shear that compensates
+for the inability of a power law model to represent the more
+complex true distribution of mass. Furthermore, the highest
+values of γext are also usually found in the most elliptical
+lenses.
+In mock data, 84% of external shears are aligned with
+the mass distribution: their mean offset is 3◦ with an rms
+scatter of 5◦. 14% of external shears are anti-aligned with
+the mass distribution, with a mean of 85◦ and scatter of 6◦.
+Only one lens has a best-fit external shear that is neither
+aligned nor anti-aligned, but this also has the lowest shear
+amplitude (γext = 0.0003), so the angle φext is ill-defined.
+The Pearson correlation coefficient between the best-fit axis
+ratios and external shears is −0.63.
+Real HST data produce a similar pattern, but with more
+scatter. Best-fit ‘external’ shears are aligned with the mass
+distribution in 68% of the lenses, and anti-aligned in 20%. All
+the remaining 12% have γext < 0.04, so the angles φPL+ext
+ext
+are noisy. The Pearson correlation coefficient between the
+best-fit axis ratios and external shears is −0.60. Despite
+the inferred external shears being an order of magnitude
+larger in the observations than in the mock data, the mass
+distributions are similarly elliptical: ⟨q⟩ = 0.77 ± 0.02 in the
+mocks, and 0.69 ± 0.02 for HST data.
+5.3
+Lenses in clusters
+5.3.1
+RX J2129-GGL1 (snail)
+Of the galaxy-galaxy lenses in clusters the snail measures
+shears that agree most closely between the two methods. No-
+tably, it measures the largest shear independent of the strong
+lensing method. Although this measurement was constrained
+by cluster scale strong lensing, Desprez et al. (2018) demon-
+strated that this value of shear is in agreement with that
+derived from weak lensing analysis of CLASH data (Figure
+13 of that paper). The strong lensing external shear is anti-
+aligned with it’s mass distribution, although this is expected
+since the mass distribution is coincidentally orientated with
+it’s major axis pointing towards the cluster centre (Table 2).
+5.3.2
+MACS1149-GGL20
+The shears measured using the independent probes for
+MACS1149-GGL20 are also in agreement, although the shear
+magnitude is much lower than is measured for the snail. In
+fact, this is one of the few lenses that measures a lower best-
+fit value of shear magnitude with strong lensing γext = 0.01
+than it does with the weak lensing method γWL = 0.04.
+Desprez et al. (2018) found that measurements of external
+shear from modelling the GGL alone were underestimated
+compared to the shears constrained using a full scale model of
+the galaxy cluster. However, both of these measurements are
+larger than either of the shears measured in this work. The au-
+thors measure an external shear magnitude γext = 0.13+0.08
+−0.06
+when modelling the potential of the lens as a double Pseudo-
+Isothermal Elliptical profile (dPIE), significantly larger than
+that measured in this work γext = 0.01+0.02
+−0.01. However, we
+measure a more elliptical mass distribution q = 0.51+0.02
+−0.01
+MNRAS 000, 1–14 (2023)
+
+‘External shear’ is not shear
+7
+Lens name
+SIS neighbour
+no neighbour
+γWL
+1
+γWL
+2
+γWL
+φWL
+φBCG
+φneighbour
+γext
+φext
+qPL
+φPL
+γext
+φext
+qPL
+φPL
+MACS1149-GGL18
+0.13+0.04
+-0.02
+40+5
+-5
+0.48+0.04
+-0.02
+147+9
+-8
+0.24+0.02
+-0.02
+26+2
+-2
+0.82+0.02
+-0.02
+174+19
+-22
+0.01+0.01
+-0.01
+−0.04+0.01
+-0.01
+0.04
+-40
+85
+95
+Abell370-GGL19
+0.06+0.01
+-0.01
+40+3
+-6
+0.92+0.01
+-0.01
+23+1
+-1
+0.07+0.03
+-0.02
+34+7
+-14
+0.72+0.03
+-0.02
+26+5
+-6
+0.05+0.02
+-0.02
+−0.01+0.02
+-0.02
+0.05
+-8
+-100
+35
+MACS1149-GGL20
+-
+-
+-
+-
+0.01+0.02
+-0.01
+13+47
+-49
+0.51+0.02
+-0.01
+106+1
+-2
+0.01+0.02
+-0.02
+0.04+0.02
+-0.02
+0.04
+40
+-83
+-
+RX J2129-GGL1 (snail)
+-
+-
+-
+-
+0.11+0.01
+-0.01
+35+2
+-2
+0.93+0.01
+-0.01
+−61+10
+-12
+0.08+0.08
+-0.08
+−0.02+-0.02
+–0.02
+0.08
+28
+-57
+-
+Table 2. Summary of strong and weak lensing parameters for the 4 galaxy-galaxy lenses that reside in clusters. All angles are in degrees
+anticlockwise from West.
+than was constrained by (Desprez et al. 2018) q = 0.81.
+The degeneracy between shear and axis ratio may therefore
+explain this discrepancy. As with the snail, the mass dis-
+tribution coincidentally points towards the cluster centre,
+therefore the anti-alignment of the external shear with the
+lens’ mass distribution that we infer is to be expected.
+5.3.3
+Abell370-GGL19
+We measure a similar shear magnitude with strong lensing
+for Abell370-GGL19 as we do with weak lensing, however
+the strong lensing external shear is suspiciously orientated
+towards a nearby neighbour galaxy and is aligned with the
+mass distribution (see Table 2). We therefore repeat the fit
+including free parameters for a singular isothermal sphere
+(SIS; γ = 2 and q = 1 in equation 11) fixed at the centre of
+the neighbour galaxy. The results including the mass of the
+neighbour galaxy do not change significantly (see the SIS
+neighbour column of Table 2 compared to the no neighbour
+column), although the power-law mass distribution does
+become less elliptical.
+5.3.4
+MACS1149-GGL18
+There is also a neighbour galaxy in close proximity to
+MACS1149-GGL18. We, therefore repeat the fit for includ-
+ing an SIS as was done for Abell370-GGL19. The shear is
+significantly overestimated as compared with weak lensing
+when the neighbour is not included in the fit. Including the
+neighbour galaxy halves the inference of strong lensing exter-
+nal shear, however this does not bring it into agreement with
+the weak lensing inference. The shear is anti-aligned with
+the power-law mass distribution which, given that the mass
+distribution is not aligned with the cluster galaxy in this
+case, suggests the external shear may be acting internally as
+discussed in the previous section.
+6
+ANALYSIS
+6.1
+External shear may compensate for boxiness/diskiness
+Our measurements in Section 5 suggest that γext mostly
+just compensates for the inability of a power law model to
+capture the complex distributions of mass (gNFW+MGE for
+the mocks, and likely more complex for real galaxies). This
+is consistent with the conclusion of Keeton et al. (1997), who
+inferred that the ⟨γext⟩ ∼ 10–15% required to fit point-source
+quad lenses, must reflect an inability of the lens model to
+capture a complex distribution of mass: perhaps misalignment
+between light and dark matter. Witt & Mao (1997) reached
+a similar conclusion, and derived an analytical prediction
+of the shear required by an elliptical potential to fit quad
+lenses.
+If the external shears result from a lack of complexity in
+the power law model to describe the underlying distribution of
+mass, one can ask what type of complexity the data requires.
+One possible deviation from the symmetry of an elliptical
+power law is boxiness and diskiness (see Section 4.3). We
+shall now investigate whether spurious external shear could
+arise to compensate for boxy/disky lens galaxies.
+6.1.1
+External shear creates boxy/disky critical curves
+An isothermal elliptical mass distribution has an elliptical
+critical curve (oriented in the same direction as the mass
+distribution but at 90◦ to the elongation of light from the
+source galaxy; Kochanek et al. 2004). However, changing the
+power law slope, γ ̸= 2, or adding an external shear, γext ̸= 0,
+perturbs the critical curves (left panel of Figure 4). These
+perturbations include significant a4/a moments (right panels
+of Figure 4), although they visually appear to be more than
+pure m = 4 modes (c.f. Figure 1).
+6.1.2
+Disky critical curves come from disky mass
+distributions
+The distributions of mass in our mock lenses happen to be
+almost all disky. We could measure any isodensity contour,
+but the κ=1 contour will be near the most sensitive region
+for lens fitting. These iso-convergence contours have mean
+⟨|a4/a|⟩ = 0.01 and ⟨|b4/a|⟩ = 0.0005. Only three lenses are
+boxy, but not usefully so, with ⟨a4/a⟩ = −0.0003.
+Critical curves of the best-fit PL+ext models to our
+mock data show a4/a moments highly correlated with those
+of the density contours (Figure 5). Again, ⟨|b4/a|⟩ = 0.0001 is
+an order of magnitude lower. Studying the same mocks, C22
+also noted that ‘external’ shear allowed the best-fit critical
+curves to better match the true critical curves. We find two
+systems with boxy critical curves a4/a < −0.01. Subject
+to some scatter, however, we conclude that the diskiness of
+isodensity contours and critical curves are highly correlated.
+Notably, all mock lenses whose best-fit external shear is
+aligned with the mass distribution have very disky critical
+curves (red points in Figure 6), and the three mock lenses
+with boxy critical curves have anti-aligned shear. Further-
+more, a4 typically increases with the external shear (Pearson
+correlation coefficient 0.45) and with the axis ratio of the lens
+mass (Pearson correlation coefficient −0.73). This may be ten-
+tative evidence that (some of) the dichotomy of aligned and
+anti-aligned shears may be caused by diskiness or boxiness.
+MNRAS 000, 1–14 (2023)
+
+8
+Etherington et al.
+−3
+−2
+−1
+0
+1
+2
+3
+−2.0
+−1.5
+−1.0
+−0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+γ = 1.6
+γ = 1.8
+SIE
+γ = 2.2
+γ = 2.4
+1.50
+1.75
+2.00
+2.25
+2.50
+2.75
+3.00
+γ
+−0.125
+−0.100
+−0.075
+−0.050
+−0.025
+0.000
+a4/a
+−3
+−2
+−1
+0
+1
+2
+3
+−2.0
+−1.5
+−1.0
+−0.5
+0.0
+0.5
+1.0
+1.5
+2.0
+γext
+1
+= 0.2
+γext
+1
+= 0.1
+SIE
+γext
+1
+= −0.1
+γext
+1
+= −0.2
+−0.3
+−0.2
+−0.1
+0.0
+0.1
+0.2
+0.3
+γext
+−0.15
+−0.10
+−0.05
+0.00
+a4/a
+Figure 4. What causes boxiness or diskiness? A Singular Isothermal Elliptical (SIE) mass distribution with a horizontal major axis has
+critical curves that are also elliptical with a horizontal major axis (orange). The critical curves are perturbed if the slope of the density
+profile γ ̸= 2 (top left panel) or the external shear γext ̸= 0 (bottom left panel). In particular, an aligned shear (γext
+1
+> 0) stretches the
+critical curves vertically (and the image horizontally); an anti-aligned shear (γext
+1
+< 0) does the opposite. Multipole measurements a4/a of
+the critical curve are shown as a function of slope (top right panel) and external shear (bottom right panel), where a4/a > 0 is “disky”
+and a4/a < 0 is “boxy”.
+−2.0
+−1.8
+−1.6
+−1.4
+−1.2
+−1.0
+log[ a4
+a +0.01](κ = 1)
+−2.50
+−2.25
+−2.00
+−1.75
+−1.50
+−1.25
+−1.00
+log[ a4
+a +0.01] (PL+ext critical curves)
+a4/a = 0
+a4/a = 0
+Figure 5. In mock lenses, disky (a4/a > 0) perturbations of the
+κ = 1 isodensity contour correlate with disky perturbations of the
+critical curves – which can also be measured for real lenses. To
+better visualise the correlation, values have been transformed by
+log[a4/a+0.01], with dashed lines indicating a4 = 0. Unfortunately,
+the mock data do not include any lenses with significantly boxy
+(a4/a < 0) distributions of mass. These points come from fits with
+Sersic sources, so the uncertainties are not comparable to those
+from analyses with pixellated sources.
+6.1.3
+Does boxiness/diskiness cause ‘external’ shear?
+Scatter in real data is larger than in the mocks. However,
+for SLACS and GALLERY lenses, the best-fit critical curves
+have mean ⟨|a4/a|⟩ = 0.016, similar to the mocks. Most (79%
+of) lenses with best-fit ‘external’ shear that is aligned with the
+mass distribution have disky critical curves; and most (70%)
+with anti-aligned shear have boxy critical curves (Figure 7).
+Moreover, lenses with the largest amplitude of external shear
+also have critical curves with the largest deviations from
+elliptical (Pearson correlation coefficient of 0.48 with a4 and
+0.65 with b4).
+This provides tentative evidence that ‘external’ shear
+in typical lensing analyses is really caused by the inabil-
+ity of parametric mass models to capture the complex dis-
+tribution of mass in a lens. A substantial portion of that
+complexity may be diskiness or boxiness of the mass dis-
+tribution. This creates diskiness or boxiness in the critical
+curves (Section 6.1.2), which leads to a spurious external
+shear (Section 6.1.1). We have not been able to quantify
+the relative contributions to external shear from true shear,
+diskiness/boxiness, or other sources.
+MNRAS 000, 1–14 (2023)
+
+‘External shear’ is not shear
+9
+0.00
+0.02
+0.04
+0.06
+γext
+0
+20
+40
+60
+80
+|φPL+ext
+mass
+−φPL+ext
+ext
+|
+‘aligned’
+‘anti-aligned’
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+a4/a
+Figure 6. For mock lenses that were simulated without external
+shear. Angle between the best-fit values of external shear and the
+major axis of the lens mass distribution, |φPL+ext
+mass
+− φPL+ext
+ext
+| in
+degrees, as a function of the amplitude of the best-fit external
+shear, γext. Points are coloured by the magnitude of the inferred
+critical curves deviation from elliptical symmetry a4/a, values of
+a4/a < 0 correspond to boxy critical curves and a4/a > 0 to disky
+ones. Note the red and blue colour points are orders of magnitude
+different, the blue range reaches a maximum of 0.001.
+0.0
+0.1
+0.2
+0.3
+0.4
+γext
+0
+20
+40
+60
+80
+|φPL+ext
+mass
+−φPL+ext
+ext
+|
+‘aligned’
+‘anti-aligned’
+0.00
+0.05
+0.10
+0.15
+0.20
+a4/a
+Figure 7. Same as for Figure 6 but for the observed SLACS and
+GALLERY lenses. The inferred external shears have a similar
+distribution of aligned and anti-aligned shears as the mock data
+sample, indicating they too may be acting internally. Note the in-
+crease in the scale of shear magnitude γext and elliptical deviations
+a4/a compared to the mock data sample.
+6.1.4
+More things probably cause ‘external’ shear too
+There are likely more sources of complexity in real mass
+distributions, which cause (or are compensated by) external
+shear. These confounding factors would explain the looser
+correlations and larger scatter than in the mocks. Just the
+observation that real lenses have external shears with am-
+plitudes six times greater than mocks implies that their
+distribution of mass deviates more from a power law.
+We speculate that the isodensity contours of a lens
+might be twisted (misaligned as a function of radius), like
+their isophotoes. Indeed, the critical curves of SLACS and
+GALLERY lenses have a handedness, with ⟨|b4/a|⟩ = 0.012
+two orders of magnitude larger than the mocks. If the critical
+curves do tell us something about the distribution of mass,
+this may indicate twisted isodensity contours. Twisting is
+also suggested by the inconsistently measured position angle
+when real data are fitted with and without external shear (see
+Figure 9); this is not present in the mocks. Van De Vyvere
+et al. (2022a) also found that twists in the underlying mass
+distribution are typically absorbed by changes in orientation
+of the mass distribution and shear in a PL+ext model.
+6.2
+External shear biases other SL model parameters
+Since best-fit values of ‘external’ shear may not (entirely)
+represent the physical quantity they are imagined to, we now
+test which other parameters in the mass model are biased
+by their inclusion, and which are still robustly measured. As
+an extreme alternative, we repeat all measurements but fix
+γext
+1
+= γext
+2
+= 0 when we refit the mass distribution.
+6.2.1
+Mock lenses
+The orientation of the mass distribution is robustly measured,
+with mean difference ⟨|φPL+ext
+mass
+− φPL
+mass|⟩ ∼ 1◦ and only ∼ 1◦
+of scatter when shear is excluded (figure 8). The axis ratio
+is also hardly changed, with ⟨qPL+ext − qPL⟩ = 0.01 ± 0.05.
+These values are so robust because all the MGE components
+share a common axis, which is therefore well-defined.
+However, the Einstein radii and slopes of the power-
+law mass model are systematically biased, by an amount
+that depends on the relative orientation of the shear and
+the mass (Einstein radii and power-law slopes are known
+to be degenerate parameters: see e.g. figure 5 of E22). For
+lenses whose shears were aligned with the mass distribution,
+removing γext increases the mean best-fit power-law slope
+by 0.20% ± 0.04 (blue points in figure 8; their mean best-fit
+Einstein radii decrease by 0.2 ± 0.03%). For lenses whose
+shears were anti-aligned, the power-law slopes decrease by
+0.07±0.03 (red points in figure 8; their mean best-fit Einstein
+radii increase by 0.08 ± 0.04). Across the entire sample, Ein-
+stein radii had been correctly measured when including γext
+(within 0.05 ± 0.17%; Cao et al. 2022), but are systematically
+underestimated by 0.2% ± 0.05% if shear is excluded.
+Our measurements might be caused by a bias described
+by Kochanek (2020) in the radial structure of a lens, when a
+model has too few azimuthal degrees of freedom. Kochanek
+provides the example of fitting a power-law model to a lens
+whose ellipticty increases with radius: the density slope is
+forced to spuriously shallow values to balance the shear inside
+the Einstein radius relative to the shear outside it. Further-
+more, Van De Vyvere et al. (2022a) found that decreasing
+ellipticity outside the Einstein radius spuriously increases
+γext for a power-law model, while ellipticity gradients inside
+or at the Einstein radius mainly bias the power-law slope.
+Our mocks whose shears align with the major axis may have
+a distribution of mass whose ellipticity increases with radius,
+and vice versa. Unfortunately, despite investigating a num-
+ber of quantities derived from the axis ratios of the MGE
+components, we were unable to confirm whether this is the
+underlying cause of the shear alignments.
+6.2.2
+Real lenses
+For SLACS and GALLERY lenses, the orientation of the
+mass distribution changes considerably when external shear
+MNRAS 000, 1–14 (2023)
+
+10
+Etherington et al.
+0
+2
+4
+6
+8
+|φPL+ext
+mass
+−φPL
+mass|
+0
+20
+40
+60
+80
+|φPL+ext
+mass
+−φPL+ext
+ext
+|
+‘aligned’
+‘anti-aligned’
+−0.7
+−0.6
+−0.5
+−0.4
+−0.3
+−0.2
+−0.1
+0.0
+0.1
+γPL+ext −γPL
+Figure 8. For mock lenses that were simulated without external
+shear. Angle between the best-fit external shear and the major axis
+of the lens mass distribution, |φPL+ext
+mass
+− φPL+ext
+ext
+| in degrees, as a
+function of the change in orientation of the major axis when fitting
+with and without an external shear, |φPL+ext
+mass
+− φPL
+mass|. Points are
+coloured by the difference in power-law slope inferred between the
+models fitted with and without an external shear (γPL+ext −γPL).
+Systems with best-fit shear aligned to the mass systematically
+decrease in power-law slope when the external shear is removed
+from the model (blue points). Anti-aligned shears exhibit the
+opposite behaviour (red points).
+0
+20
+40
+60
+80
+|φPL+ext
+mass
+−φPL
+mass|
+0
+20
+40
+60
+80
+|φPL+ext
+mass
+−φPL+ext
+ext
+|
+−0.6
+−0.4
+−0.2
+0.0
+0.2
+qPL+ext −qPL
+Figure 9. Orientation angle offset of the external shear from the
+PL+ext mass distribution (φPL+ext
+mass
+− φPL+ext
+ext
+) as a function of
+the difference in orientation angle when the mass distribution is
+fitted with and without an external shear (φPL+ext
+mass
+− φPL
+mass) for
+the observed SLACS and GALLERY samples. Scatter points are
+coloured by the difference in axis ratio of the models fitted with a
+PL+ext and a PL (qPL+ext − qPL).
+is removed, with ⟨|φPL+ext
+mass
+− φPL
+mass|⟩ ∼ 27◦ and ∼ 27◦ scatter
+(figure 9; note the different scale on the horizontal axis to
+figure 8). The best-fit axis ratio (indicated by colour in
+figure 9) also increases by 0.18 ± 0.016 for aligned systems,
+which become more spherical; decreases by 0.08 ± 0.10 for
+anti-aligned systems, which become more elliptical; and is
+inconsistent for the rest, with mean change ⟨qPL+ext −qPL⟩ =
+−0.02 ± 0.04.
+Neither the Einstein radii nor slopes of the power-law
+mass model are systematically biased when external shear
+is removed. We find ⟨∆REin/REin⟩ = −0.013+0.030
+−0.043, where
+∆REin ≡ RPL+ext
+Ein
+−RPL
+Ein. However, the best fit values scatter
+about the same mean, such that ⟨∆R2
+Ein/REin⟩ = 0.0452,
+which is two orders of magnitude larger than the equivalent
+0.00032 value for the mocks (figure 10). Furthermore, the
+scatter increases with increasing external shear magnitude
+(figure 11).
+The best-fit centre of the mass distribution moves on
+average by −0.031 ± 0.061′′ when external shear parameters
+are introduced, and on average remains 0.05 ± 0.06′′ from
+the centre of light. With or without shear, this offset is an
+order of magnitude greater than in the mock data.
+We suspect this indicates that the distribution of mass
+in real galaxies is more complex than that in the mocks:
+for example with multiple components that are rotationally
+offset from one another and no single, well-defined axis of
+ellipticity.
+7
+INTERPRETATION
+7.1
+Strong lensing external shears are not measuring shear
+We find that external shear, as measured by galaxy-galaxy
+strong lensing, does not correlate well with the true shear
+along a line-of-sight (as measured independently by weak
+lensing; Section 5.1 and Figure 2). Best-fit values of external
+shear are also frequently several times higher than that along
+typical lines of sight through the Universe, so only a small
+fraction of it could be due to true shear. Rather, external
+shear tends to align with either the major axis or minor axis
+of the lens mass distribution (Section 5.2). It appears to
+be compensating for the inflexibility of typical mass models
+(here an elliptical power-law) to represent the complex dis-
+tributions of mass. A substantial portion of that complexity
+appears to be diskiness and boxiness (Section 6.1), especially
+in mock data. In real data, isophotal twists, elliptical gradi-
+ents, and offsets in the centres and alignments of dark and
+stellar matter all increase uncertainty on model parameters.
+These have all been seen in the stellar mass of SLACS lenses
+(Nightingale et al. 2019). We have not been able to quantify
+how much each source contributes to the scatter or bias in a
+real measurement of external shear.
+Hogg et al. (2022) suggest a different, ‘minimal line-of-
+sight’ way of parameterising the shear that is less degenerate
+with lens model parameters, but this is still subject to biases
+in the shear parameters when simplifying assumptions are
+made for the lens model. For complex distributions of lens
+mass, the false inference of external shear will pose a challenge
+for efforts to use strong lensing to measure cosmic shear (e.g.
+Birrer et al. 2017; Fleury et al. 2021).
+7.2
+Implications for strong lensing science goals
+Including external shear alters the best-fit values of other
+parameters (Section 6.2). In mock data, some parameters do
+move closer to the known truth: the mean error in power law
+slopes is −0.02 ± 0.10 with shear, or −0.14 ± 0.20 without it.
+A ∼ 0.2% bias in Einstein radii also disappears with shear.
+However, the shear is not physically real and, contribut-
+ing zero convergence, does not correspond to a physically
+meaningful distribution of mass. For studies that rely on
+accurate reconstructions of the mass distribution (e.g. galaxy
+evolution, dark matter physics, and the Hubble constant),
+this degeneracy with key parameters will eventually limit
+MNRAS 000, 1–14 (2023)
+
+‘External shear’ is not shear
+11
+−0.1
+0.0
+∆REin/REin
+0
+5
+10
+15
+Number
+mean: -0.013
+16th(84th): -0.0566 (0.0166)
+mean: 0.002
+16th(84th): -0.0001 (0.0039)
+−0.5
+0.0
+∆γ
+mean: -0.092
+16th(84th): -0.3858 (0.1808)
+mean: -0.16
+16th(84th): -0.303 (-0.006)
+0.0
+0.5
+∆q
+mean: 0.06
+16th(84th): -0.099 (0.2198)
+mean: 0.027
+16th(84th): -0.011 (0.069)
+−0.2
+−0.1
+0.0
+0.1
+∆ centre
+mean: -0.031
+16th(84th): -0.0836 (0.0067)
+mean: -0.002
+16th(84th): -0.0031 (0.0)
+Figure 10. Histograms of the difference between inferred PL mass distribution model parameters with external shear as free parameters,
+minus those without shear, for the observed (pink) and mock data samples (orange). From left to right panels the parameters are:
+fractional Einstein radius, logarithmic slope of density profile, axis ratio of mass distribution, and radial distance of the centre of the mass
+distribution, in arcseconds.
+0.0
+0.1
+0.2
+0.3
+0.4
+γext
+−0.15
+−0.10
+−0.05
+0.00
+0.05
+RPL+ext
+Ein
+−RPL
+Ein
+RPL
+Ein
+Lens Name
+J0959+4416
+J1016+3859
+J1153+4612
+J1416+5136
+J0008-0004
+J0029-0055
+J0157-0056
+J0216-0813
+J0252+0039
+J0330-0020
+J0728+3835
+J0737+3216
+J0822+2652
+J0841+3824
+J0903+4116
+J0912+0029
+J0936+0913
+J0946+1006
+J0959+0410
+J1020+1122
+J1023+4230
+J1029+0420
+J1032+5322
+J1142+1001
+J1143-0144
+J1205+4910
+J1213+6708
+J1218+0830
+J1250+0523
+J1402+6321
+J1420+6019
+J1430+4105
+J1432+6317
+J1451-0239
+J1525+3327
+J1627-0053
+J1630+4520
+J2238-0754
+J2300+0022
+J2303+1422
+J2341+0000
+J0029+2544
+J0201+3228
+J0237-0641
+J0742+3341
+J0755+3445
+J0918+5105
+J1110+2808
+J1110+3649
+J1116+0915
+J1141+2216
+J1201+4743
+J1226+5457
+J2228+1205
+J2342-0120
+Figure 11. Fractional difference between Einstein radii inferred for
+models with and without the free parameters for external shear,
+as a function of the external shear amplitude.
+statistical precision. For example, a key result of the SLACS
+survey was that elliptical galaxies have isothermal (γ=2)
+mass profiles (Gavazzi et al. 2007; Vegetti & Koopmans 2009;
+Auger et al. 2010). Including external shear, E22 measured
+⟨γ⟩ = 2.0756+0.023
+−0.024 for SLACS galaxies; here we measure
+⟨γ⟩ = 2.0159 =+0.027
+−0.032 (Figure 10), highlighting the system-
+atic uncertainty that will remain fixed even if the sample
+size increases. Furthermore, a PL+ext model leaves false de-
+tections of subhalos in a mock disky galaxy (He et al. 2022)
+or real HST data (Nightingale et al. 2022).
+The Hubble constant H0 can be measured from the
+time delay between multiple images (Suyu et al. 2017; Wong
+et al. 2019; Birrer et al. 2019). However, assuming specific
+functional forms for the mass model can artificially break
+the mass-sheet transformation. Our tests on mock data (Sec-
+tion 6.2.1) demonstrate a coupling, predicted by Kochanek
+(2020), between angular and radial structure. Oversimple
+models bias the slope, and hence bias H0. C22 estimated
+∼9% bias in H0 when using a PL+ext model to interpret time
+delays generated from gNFW+MGE lenses. The angular de-
+grees of freedom added by ‘external’ shear are insufficient to
+compensate for even this complexity. Given that our analysis
+of SLACS and GALLERY lenses indicate even more angular
+degrees of freedom, such as twists in the mass distribution,
+biases for real lensing systems will likely be closer to the
+20–50% suggested by other studies (Schneider & Sluse 2013;
+Xu et al. 2016; Kochanek 2019; Gomer & Williams 2019,
+2020, 2021). More flexible models, such as adding an internal
+mass sheet to the PL+ext model which is contstrained via
+stellar dynamics, have been introduced to mitigate
+Combining H0 measurements from a large popula-
+tion of lenses might average away individual biases up to
+10kms−1Mpc−1, on the assumption that boxy and disky mass
+distributions are equally well represented (Van De Vyvere
+et al. 2022a). However, the diskiness of the mass distribution
+correlates with the diskiness of the (observable) PL+ext crit-
+MNRAS 000, 1–14 (2023)
+
+12
+Etherington et al.
+ical curves (Section 6.1.2), and our observationally-selected
+sample contains an overpopulation of 71% disky galaxies
+(Section 6.1.3). More flexible mass models (e.g. a PL with an
+internal mass sheet) have also been introduced (Birrer et al.
+2020), which infer unbiased H0 values on mock lens samples
+(Ding et al. 2021). Moreover, Van De Vyvere et al. (2022b)
+did not investigate the effect of b4 ‘twisting’ perturbations
+(Section 6.1.4) or mis-centering (Section 6.2.2), both of which
+we observe in the critical curves of real galaxies, and whose
+effects may not average away. For example, the centre of
+mass in the H0LiCOW model of lens WFI 2033-4723 is offset
+from the centre of light by ∼10× astrometric uncertainty
+(Suyu et al. 2010; Barrera et al. 2021), and a similar offset
+in iPTF16geu increases asymmetry (Diego et al. 2022). Such
+offsets are unphysical (Schaller et al. 2015), so it is not clear
+that complexities in the mass distributions of real lenses can
+be safely ignored by averaging over a population.
+7.3
+Future work: more complex mass models are needed
+External shear does not appear to be sufficient as (the sole)
+parameter to encode all the complexity in real lenses. Al-
+though Einstein radii are expected to be ‘model independent’
+within ∼2% uncertainty Bolton et al. (2008b); Sonnenfeld
+et al. (2013), the 4.5% RMS fractional difference we mea-
+sure with and without shear, suggests an unknown unknown.
+A single power-law model leads to mass discrepancies with
+stellar dynamics (Etherington et al. 2022a), and spurious
+false-positives in searches for dark matter subhalos (Nightin-
+gale et al. 2022). Further work (e.g. Cao et al. 2022; Van
+De Vyvere et al. 2020, 2022a) to understand the types of
+asymmetries that must be accounted for in the lens mod-
+elling, and the possibility of constraining such models, will
+be invaluable.
+What parametric forms allow sufficient complexity —
+in a minimum number of parameters that need to be con-
+strained? Even our gNFW+MGE mocks still do not capture
+the full complexity of real lenses, but their MGEs were forced
+to be aligned. Perhaps the model could have both a4 and b4
+perturbations, varying as a function of radius. Cluster-scale
+models frequently use a soft core inside some scale radius, and
+(Limousin et al. 2022) add B-spline functions. The number
+of free parameters must be balanced with the information
+available: pixellated mass models are generally undercon-
+strained (and although they might be able to fit observations
+without external shear; Valls-Gabaud et al. 2006, some line-
+of-sight shear is expected for galaxy-scale lenses). To instead
+increase the available information, the total mass could be
+decomposed into dark matter and stellar components, with
+the latter informed by the lens light (which is otherwise
+a nuisance). Whatever the parametric form, the azimuthal
+degrees of freedom must be defined carefully to avoid the
+bias described by Kochanek (2020) on the inference of the
+radial mass distribution. We suggest more studies (e.g. Van
+De Vyvere et al. 2020, 2022a, C22), embedded deeply in each
+specific science case, to quantify the impact of simplifying
+assumptions.
+A silver lining is that the sensitivity of galaxy-galaxy
+strong lensing data may be a new opportunity. Both our
+results and those of Van De Vyvere et al. (2022b) suggest
+that it might be possible to measure the diskiness or boxiness
+of galaxies’ dark matter halos. Our results are even more
+optimistic: if measurable multipole perturbations in critical
+curves traces those in the distributions of both mass and
+light, then one could study the dark morphology of galaxies
+at high redshift with relative ease.
+SOFTWARE CITATIONS
+This work uses the following software packages:
+• Astropy (Astropy Collaboration 2013; Price-Whelan et al.
+2018)
+• Corner.py (Foreman-Mackey 2016)
+• Dynesty (Speagle 2020)
+• Matplotlib (Hunter 2007)
+• Numba (Lam et al. 2015)
+• NumPy (van der Walt et al. 2011)
+• PyAutoFit (Nightingale et al. 2021a)
+• PyAutoLens (Nightingale & Dye 2015; Nightingale et al.
+2018, 2021b)
+• Scikit-image (Van der Walt et al. 2014)
+• Scikit-learn (Pedregosa et al. 2011)
+• Scipy (Virtanen et al. 2020)
+• SQLite (Hipp 2020)
+ACKNOWLEDGEMENTS
+We thank Yiping Shu, Guillaume Desprez, Johan Richard,
+Mathilde Jauzac, and Keichi Umetsu for providing data from
+previously published papers, and Liliya Williams and Adi
+Zitrin for stimulating discussions.
+AE is supported by UK STFC via grants ST/R504725/1
+and ST/T506047/1. JN and RM are supported by STFC
+via grant ST/T002565/1, and the UK Space Agency via
+grant ST/W002612/1. XYC and RL are supported by the
+National Nature Science Foundation of China (via grants
+11988101, 11773032, 12022306), the China Manned Space
+Project (science research grants CMS-CSST-2021-B01, CMS-
+CSST-2021-A01) and by the K.C.Wong Education Founda-
+tion. QH, AA, SMC, and CSF are supported by the ERC
+via grant GA 786910. MJ is supported by UKRI via grant
+MR/S017216/1. This work used both the Cambridge Ser-
+vice for Data Driven Discovery (CSD3) and the DiRAC
+Data-Centric system, which are operated by the Univer-
+sity of Cambridge and Durham University on behalf of the
+STFC DiRAC HPC Facility (www.dirac.ac.uk). These were
+funded by BIS capital grant ST/K00042X/1, STFC capi-
+tal grants ST/P002307/1, ST/R002452/1, ST/H008519/1,
+ST/K00087X/1, STFC Operations grants ST/K003267/1,
+ST/K003267/1, and Durham University. DiRAC is part of
+the UK National E-Infrastructure.
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+page_content=' Steinhardt13  1Department of Physics, Centre for Extragalactic Astronomy, Durham University, South Rd, Durham, DH1 3LE, UK  2Department of Physics, Institute for Computational Cosmology, Durham University, South Road, Durham DH1 3LE, UK  3Academia Sinica Institute of Astronomy and Astrophysics (ASIAA), No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
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+page_content=' 4, Roosevelt Road, Taipei 10617, Taiwan  4School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, China  5National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100012, China  6Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA  7Instituto de F´ısica de Cantabria (CSIC-UC), Avda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Los Castros s/n, 39005 Santander, Spain  8Department of Astronomy/Steward Observatory, University of Arizona, 933 N Cherry Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=', Tucson, AZ 85721, USA  9Laboratoire d’Astrophysique, EPFL, Observatoire de Sauverny, 1290 Versoix, Switzerland  10Space Telescope Science Institute, 3700 San Martin Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Baltimore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' MD 21218,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' USA  11Aix Marseille Univ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' CNES,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' LAM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' F-13388 Marseille,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' France  12Sydney Consortium for Particle Physics and Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' School of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The University of Sydney,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' NSW 2006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Australia  13Niels Bohr Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' University of Copenhagen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Lyngbyvej 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' København Ø 2100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Denmark  ABSTRACT  The distribution of mass in galaxy-scale strong gravitational lenses is often modelled as  an elliptical power law plus ‘external shear’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' which notionally accounts for neighbouring  galaxies and cosmic shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We show that it does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Except in a handful of rare systems,  the best-fit values of external shear do not correlate with independent measurements of  shear: from weak lensing in 45 Hubble Space Telescope images, or in 50 mock images  of lenses with complex distributions of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Instead, the best-fit shear is aligned with  the major or minor axis of 88% of lens galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' and the amplitude of the external shear  increases if that galaxy is disky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We conclude that ‘external shear’ attached to a power  law model is not physically meaningful, but a fudge to compensate for lack of model  complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Since it biases other model parameters that are interpreted as physically  meaningful in several science analyses (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' measuring galaxy evolution, dark matter  physics or cosmological parameters), we recommend that future studies of galaxy-scale  strong lensing should employ more flexible mass models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Key words: gravitational lensing: strong — galaxies: structure  1  INTRODUCTION  Gravitational lensing is the deflection of light rays by nearby  concentrations of matter and their associated gravitational  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' If the light ray should pass straight through an object  as massive as a galaxy, it can be deflected along multiple  routes around the galaxy, and appear distorted into arcs or an  ‘Einstein ring’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Such galaxy-scale strong lensing has been used  to infer the distribution of mass in massive elliptical galaxies  ⋆ Contact e-mail: james.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='nightingale@durham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='uk  (Gavazzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Koopmans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Auger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Bolton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Etherington  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2022a), to infer their dark matter content, stellar  mass-to-light ratios, and inner structure (Massey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Oldham & Auger 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Nightingale  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2015, 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' If the background source  is variable, measurements of time delays between multiple  images can be used to measure cosmological parameters  (Kilbinger 2015) or the Hubble constant (Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Harvey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Gomer & Williams  2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' If the lens galaxy contains small substructures, which  © 2023 The Authors  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='05244v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='CO]  12 Jan 2023  2  Etherington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  would be a smoking gun of the ‘clumpy’ Cold Dark Matter  (CDM) model, they would also perturb the multiple images  (Natarajan & Springel 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Vegetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Vegetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2016, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Hezaveh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Ritondale  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Despali et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Amorisco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Nightingale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  When modelling the distribution of mass to fit strong  lensing data, two additional free parameters are frequently  included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The (amplitude and angle of) ‘external shear’ is  intended to represent the cumulative deflection of light by all  other gravitational potentials along the line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Indeed,  the best-fit value of external shear matched a model of the  line of sight for three of six galaxy lenses studied by Wong  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Subsequent papers have even proposed using  external shear as relatively high signal-to-noise measurements  of the ‘cosmic’ shear along individual lines of sight (Birrer  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Desprez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Kuhn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Fleury  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Hogg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' However, the best-fit values  of external shear did not match the lines of sight to the  three other galaxies studied by Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2011) — and,  in general, best-fit values are much larger than expected for  both galaxy lenses (Keeton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Witt & Mao 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Hilbert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Barrera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2021) and  cluster lenses (Robertson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Limousin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Using mock observations, Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2022, hereafter C22)  demonstrated that the best-fit shear can be incorrect if the  model of the mass distribution is missing complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  In this paper we compare the external shear measured  from Hubble Space Telescope (HST) imaging — of strong  lensing galaxies from the SLACS survey (Bolton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2008a),  GALLERY survey (Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2016b), and four lenses in  clusters — against independent measurements of the shear  along the same line-of-sight, observed as weak lensing of  adjacent galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' To gain understanding, we also analyse  C22’s mock HST imaging, generated without external shear,  but fitted with shear as a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Comparing independent measurements of shear will test  the hypothesis that strong lensing external shear is a real,  physical quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Strong and weak lensing measurements  average over different spatial scales and are obtained at differ-  ent redshifts, so they might not be identical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' but they should  be strongly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Analysing three lenses in the COS-  MOS field (Faure et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2008) with sophisticated statistics,  Kuhn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2020) measured a smaller covariance between  strong and weak lensing shears than the difference between  individual systems, indicating that more data were required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  In this work, with a much larger sample of galaxies, we simply  aim to detect a correlation between the two probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  We define relevant concepts from lensing theory in Sec-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We then describe our observed and mock data in  Section 3 and our analysis methods in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We present  our results in Section 5 and investigate possible causes in  Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We interpret these in a wider context and conclude  in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Throughout this paper, we assume a Planck  2015 cosmological model (Ade et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  2  GRAVITATIONAL LENSING THEORY  Gravitational lensing describes the deflection of light rays  from distant sources, by matter along its path to an observer,  through angle α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' This maps 2D coordinates of light on the  distant source plane β to coordinates where they are observed  on the foreground image plane θ = (θ1, θ2), via the lens  equation  β = θ − α(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  (1)  If the gravitational lens is much thinner than its angular  diameter distance from the observer Dl, its distribution of  mass can be treated as a 2D surface density projected along  the line-of-sight Σ(θ) =  �  ρ(Dlθ1, Dlθ2, z) dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The deflection  angle is then the vector gradient α = ∇ψ of a 2D lensing  potential  ψ(θ) ≡ 2 Dls  DlDs  2G  c2  �  Σ(θ′) ln |θ − θ′| d2θ′  (2)  = 2 Dls  DlDs  �  Φ(Dlθ, z) dz ,  (3)  where Φ is the 3D Newtonian potential, and the prefactor  (which involves angular diameter distances to the lens, to  the source, and from the lens to the source) reflects the  geometrical efficiency of a lens: peaking if it is half-way  between the source and the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  The Jacobian of the lens equation (1) is thus  Aij ≡ ∂βi  ∂θj = δij − ∂αi  ∂θj = δij − ∂2ψ  ∂θiθj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  (4)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  Weak lensing  If the source is much smaller than the scale of local variations  in the gravitational field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' the Jacobian can be approximated  as constant  Aij ≈  �1 − κ − γ1  −γ2  −γ2  1 − κ + γ1  �  (5)  = (1 − κ)  �1  0  0  1  �  − γ  �cos2φ  sin2φ  sin2φ  cos2φ  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  (6)  where the convergence  κ = 1  2∇2ψ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  (7)  and where the two components of shear  �  γ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' γ2  �  =  �1  2  �∂2ψ  ∂θ2  1  − ∂2ψ  ∂θ2  2  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  ∂2ψ  ∂θ1∂θ2  �  (8)  can also be expressed in terms of shear amplitude γ2 = γ2  1+γ2  2  and angle φ = ½ arctan (γ2/γ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The convergence magnifies  a source, and the shear changes its shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Strictly, these  quantities are only observable in combination as ‘reduced  shear’ gi = γi/(1 − κ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' However, in the weak lensing regime,  κ ≪ 1, so gi ≈ γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  Strong lensing  If light from one side of a source is deflected differently to  light from the other side, it can appear distorted in the image  plane as an arc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' it is also possible to see multiple images of a  single source, if more than one solution exists with different  α and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' To reconstruct α(θ) it is usual to note that  κ(θ) = Σ(θ)  Σcrit  with constant  Σcrit = c2  4πG  Ds  DlDls ,  (9)  which is equal to the mean surface mass density within the  Einstein radius, REin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' For axisymmetric lenses the Einstein  MNRAS 000, 1–14 (2023)  ‘External shear’ is not shear  3  radius is uniquely defined by the radius of the circular tangen-  tial critical curve that is produced where the magnification  diverges in the lens plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' This occurs where the tangen-  tial eigenvalue of the Jacobian (equation 4) λt = 1 − κ − γ  is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' For asymmetric lenses, the definition of  Einstein radius must be generalised;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' we choose to use the  effective Einstein radius  REin,eff =  �  A  π ,  (10)  where A is the area enclosed by the tangential critical curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  When considering (typically Early-type) galaxy-scale  lenses, it is common practise to parameterise the surface  mass distribution as an elliptical power law (Suyu 2012)  Σ(θ1, θ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' b, q, γ) = 3 − γ  1 + q  �  b  �  θ2  1 + θ2  2/q2  �γ−1  ,  (11)  where b ⩾ 0 is the angular scale length (referred to in some  papers as the Einstein radius, but distinct from the more  robust effective Einstein radius in Equation 10), 0 < q ⩽ 1  is the projected minor to major axis ratio of the ellipti-  cal isodensity contours, and (confusingly denoted) γ is the  logarithmic slope of the mass distribution in 3D (for an  ‘isothermal’ distribution, γ = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' If we also allow the mass to  be translated to central coordinates (θc  1, θc  2) and reoriented  to position angle φPL, which we measure counterclockwise  from the positive θ1-axis, the model has six free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  The primary lens may not be the only source of shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Any ‘external’ component due to other galaxies or clusters  near the lens or along the ray path, and constant on scales  larger than b (rather than the size of the source) is modelled  as two more free parameters  �  γext  1  , γext  2  �  = γext�  cos (2φext), sin (2φext)  �  ,  (12)  where γext is the amplitude and φext is the angle of the shear  measured counterclockwise from the θ1-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' This is applied  as an additional component of α(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' It does not change κ(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  3  DATA  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  Mock lens galaxies  We analyse a set of 50 mock lens images, representative of  data from the SLACS survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' They were generated by C22  for an investigation into the systematic errors induced by  the elliptical power-law model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We summarise the simulation  procedure below;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' a detailed description can be found in  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='4 of that paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  The surface mass density of the lens galaxy comprises  two components: a dark matter halo, parameterised by the  spherical generalised Navarro, Frenk & White (gNFW) profile  (Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Zhao 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Cappellari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2013), plus  visible stellar matter, parameterised by a Multiple Gaussian  Expansion (MGE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Cappellari 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The model parameters  of the gNFW and MGE profiles of each lens galaxy are set  to the best-fit parameters from fits of these distributions to  SDSS-MaNGA stellar dynamics data, derived by Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  (2019) using the Jeans anisotropic model (JAM) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  The position angle of each Gaussian component in the MGE  is fixed, however their axis ratios are free to vary, allowing  for elliptical gradients in the mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  The light distribution of the source galaxy is modelled by  a single S´ersic profile (Graham & Driver 2005) with effective  radius Reff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='15′′, S´ersic index n = 1, and axis ratio q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  The position in the source plane (xs, ys) is drawn from a  Gaussian distribution with mean 0′′ and standard deviation  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1′′, and the position angle is uniformly selected between  0◦ −180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The light from the source galaxy is ray-traced from  the source plane at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='6 to the image plane through the  lens equation (equation 1), to simulate its lensed appearance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Further, to mimic observational effects, the image is con-  volved with a Gaussian PSF with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='05′′ standard deviation,  and sampled by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='05′′ square pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' A flat background sky of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1 electrons per second is assumed, and an exposure time of  840 seconds is used to add Poisson noise from the source and  background sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The signal to noise ratio of the brightest  pixel in the synthetic images is set to ∼50, by adjusting the  intensity of the S´ersic source accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' No external shear  was simulated in the mock data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  Observed lens galaxies  We analyse three sets of galaxy-galaxy strong lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' These  include 42 lenses from the SLACS survey (Bolton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2008a)  that were fitted without significant residuals by Etherington  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2022b, hereafter E22)’s automated pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Most are  isolated field galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' They were found by searching for  high redshift emission lines in the spectra of low-redshift  galaxies obtained through a 3′′ fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' They were then imaged  by the HST Advanced Camera for Surveys (ACS) in the  F814W band, and processed into stacked images with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='05′′  pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We also reprocessed these to measure weak lensing,  following the procedure described by Tam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2020), which  supersamples the pixels to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='03′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We exclude lenses J1143-  0144 and J1420+6019, for which only one exposure was  obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  We analyse 15 lenses from the GALLERY survey (Shu  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2016b) that were modelled by E22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' These are also field  galaxies, found by searching for compact Lyman-α-emitting  source galaxies in spectra with a 2′′ fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' They were imaged  with the HST Wide Field Camera 3 (WFC3) in the F606W  band and processed, following Shu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2016a), into 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='04′′  pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We do not attempt to measure weak lensing shear in  these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  We analyse 4 galaxy-galaxy strong lenses in the outskirts  of galaxy clusters, where we expect a 5–15% true external  shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Before beginning any analysis, we searched archival  HST F814W imaging, and selected lenses with multiple imag-  ing of sources that are extended a similar amount as the field  lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Positions and redshifts of the selected lenses are given  in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' For MACS1149-GGL18 no source redshift has  been recorded, where necessary we assume a source redshift  of zs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5 and test that the results do not change signifi-  cantly when we change this assumption over a range of source  redshifts from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Two of our selected lenses had been  previously modelled by Desprez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2018), although con-  strained using only the positions of multiple images, rather  than all the pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We analysed the HST data similarly to  the SLACS lenses, except for the ‘cosmic snail’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' For that  lens alone, we do not measure weak lensing, but use the  independent estimate of shear from Desprez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2018)’s  model IV of the galaxy cluster, constrained by cluster-scale  MNRAS 000, 1–14 (2023)  4  Etherington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Lens name  RA  Dec  zl  zs  MACS1149-GGL18  177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='410247  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='352017  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='544 Abell370-GGL19  39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='963013  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='534783  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='375  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='371  MACS1149-GGL20  177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='402816  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='436607  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='544  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='806  RX J2129-GGL1 (snail)  322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='428780  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='108071  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='235  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='610  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Table of parameters for the 4 galaxy-galaxy lenses in the  outskirts of clusters, the lens name refers to the cluster in which  the lens resides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  strong lensing, and shown by Desprez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2018) to be  consistent with ground-based measurements of weak lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  4  METHODS  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  Weak lensing analysis  We identified galaxies on lines of sight adjacent to strong  lenses using SExtractor (Bertin & Arnouts 1996), and mea-  sured their shapes using the PyRRG (Harvey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2019)  implementation of the Rhodes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2000) shear measure-  ment method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' This estimates the mean reduced shear in a  patch of sky, by averaging galaxies’ apparent shapes  ϵi = ϵint  i  + Gγi,  (13)  which have been transformed by weak lensing (Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1)  from an unknown intrinsic shape ϵint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The ‘shear responsivity’  G varies as a function of galaxy flux, and its overall scaling  has been calibrated on simulated images with known shear  (Leauthaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Under the assumption that galaxies’  intrinisic shapes are randomly oriented, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' ⟨ϵint  i ⟩ ≈ 0,  ⟨γi⟩ =  �ϵi − ϵint  i  G  �  ≈  � ϵi  G  �  −  �ϵint  i  G  �  ≈ ⟨ϵi⟩  ⟨G⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  (14)  Following Massey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2007), we assume that the median  redshift of the lensed galaxies is z ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Thereafter, follow-  ing Smail et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (1994), we treat them all as being at this  effective redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' None of our results change significantly if  we adjust this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  We average weak lensing shear measurements from the  ∼140 galaxies within 60′′ of the strong lens galaxy (no weights  are applied to the galaxies that are averaged).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The precision  of this measurement is limited by the randomness in the  distribution of the intrinsic shapes  σ2  int =  ��ϵint  i  G  �2�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  (15)  We measure σint ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3, consistent with Leauthaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  (2007), and hence uncertainty σint/  √  140 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='02 on each  component of mean shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' This is similar to uncertainty on  our strong lensing measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' None of our results change  significantly if we use a 45′′ or 90′′ aperture instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Although the line-of-sight directly through each galaxy-  scale lens is not in the weak lensing regime, we assume that  ⟨gi⟩ ≈ ⟨γi⟩ still holds since the vast majority of adjacent lines  of sights will be only weakly lensed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Nor do we attempt to  model and subtract the weak shear due to the galaxy-scale  lens itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Doing so would mix the weak lensing and strong  lensing analyses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' and it is unnecessary at our achieved level of  precision because the near-circular symmetry of most lenses  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  boxy (a4 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1)  ellipse  disky (a4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1)  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Examples of boxy (a4 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1, blue dashed curve) and  disky (a4 = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1, pink dashed curve) perturbations to an ellipse  (orange curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The perturbations shown are ∼10 times larger  than those typically observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' In both cases the perturbation at  45◦, b4 = 0 (see equation 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  means that the lens contributes negligibly to ⟨γi⟩ inside a  60′′ circular aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  Strong lensing analysis  We analysed all data using the automated strong lens mod-  elling software PyAutoLens1 (Nightingale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2018, 2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  This fits parameters of the lens model using all of the pixels  in an image (not just e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' locations of the centre of light, as  in previous works).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  The pipelines used to fit the mock and observed data  are described fully in C22 and E22 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Briefly, we  model the distribution of mass in both mock and real data  using an elliptical power law (equation 11) plus external  shear (equation 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We then repeat the fit, fixing external  shear γext = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We model the distribution of light in in real  lens galaxies using a double S´ersic profile with a centre that  is free to vary independently to that of the mass distribution,  and for the source galaxy using an adaptive Voronoi mesh of  pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' For the mock data, we use C22’s fit in which the lens  light is perfectly subtracted and the source light is modelled  as an elliptical S´ersic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' C22 also performed fits using a Voronoi  mesh for the source light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' However, since the mock data were  created assuming a S´ersic source, the model we chose can  perfectly describe the source, so any systematics we observe  will be solely due to mismatch between the model and truth  of the mass distribution, which is the point of interest in this  study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3  Multipole perturbations of an ellipse  In Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1 we shall investigate whether the strong lensing  external shears depend on deviations from an elliptically  symmetric distribution of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Specifically, we shall quan-  tify the multipole deviations of two types of contour: the  iso-convergence contour at κ = 1 of the gNFW+MGE dis-  tributions used to create mock data, and the critical curves  1 The PyAutoLens software is open source and available from  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='com/Jammy2211/PyAutoLens  MNRAS 000, 1–14 (2023)  ‘External shear’ is not shear  5  of both the mock and the observed galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' These con-  tours are stored as a 2D array of points in polar coordinates  [φcontour, Rcontour].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We calculate perpendicular deviations of  each point from the true ellipse  Rel(φ) =  a  √  1 − ϵ2  �  1 − ϵ2 cos2(φ − φel)  ,  (16)  where a is the major axis, φel is the major axis orientation,  and ϵ is the eccentricity (defined as ϵ2 ≡ 1 − b2/a2 where b  is the minor axis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The deviations are then parameterised  using multipoles  δRm(φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' am, bm) =  �  am cos(m(φ−φel))+bm sin(m(φ−φel)) ,  (17)  where m is the order of the multipole, and am and bm are the  magnitude of the deviations with symmetry along or at 45◦  to the major and minor axes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We then perform  a non-linear search to fit the model  R(φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' a4, b4) = Rel(φ) + δR4(φ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' a4, b4)  (18)  to the radial values of the contour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We assume uniform priors  on the free parameters in the fit over a reasonable range and  fit for them using the nested sampling algorithm dynesty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  We assume the residual errors can be described by a Gaussian  distribution and maximise the likelihood  L(R|Ri, σ) =  �  i  �  1  √  2πσ2 exp  �  −(R(φi) − Ri)2  2σ2  ��  ,  (19)  where Ri are the radial values of the contour and R(φi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' a4, b4)  are the model predicted values from equation (18) at each  angular coordinate in the contour φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Curves with best-fit  values of a4 > 0 are ‘disky’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' those with a4 < 0 are ‘boxy’ (see  Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  5  RESULTS  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  SL shears do not correlate with WL shears  Strong lensing measurements of shear γSL (obtained as the  best-fit external shear γext) typically have amplitudes up to  an order of magnitude larger than weak lensing measurements  γWL along the same line of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The mock lenses have mean  best-fit |⟨γSL⟩| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='019 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='002, despite the true values all  being γext = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Our measurement using a PL+ext mass  model is consistent with C22’s value |⟨γSL⟩| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='015 using  pixel-based source reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The real lenses have mean  best-fit shear |⟨γSL⟩| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='098 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='011, which is much larger  than both the measured weak lensing shear |⟨γWL⟩| = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='028±  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='002 and the typical ∼ 1 − 3% shear expected along random  lines of sight through the universe (Keeton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 1997), even  accounting for the different scales on which they are averaged  (Valageas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Strong lensing measurements of shear do not correlate  with weak lensing measurements (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' To make this  comparison (for the real lenses only), we first define rotated  coordinate systems such that γSL  1  = γSL ⩾ 0 and γSL  2  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Thus we need plot only three of the four components of  shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Second, we compensate for the different redshifts of  the strongly lensed and weakly lensed sources by rescaling  values of γSL  1  by (Ds/Dls)z′s=zs(Dls/Ds)z′s=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='26, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' the ef-  fective value at the redshift of the weakly lensed galaxies  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='10  γWL  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='4  γSL  1  �  DS  DLS  �  zs  �  DLS  DS  �  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='26  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='10  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='γWL  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='Lens Name  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='J0008-0004  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='J0029-0055  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
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+page_content='J0216-0813  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='J0252+0039  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='J0330-0020  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
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+page_content='snail  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='GGL20  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
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+page_content='Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Values of the shear along the lines of sight to 39 galaxy-  galaxy lenses, independently measured using strong lensing ‘exter-  nal’ shear γSL and weak lensing γWL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Shears are oriented such  that γSL  2  = 0, and rescaled to be at the same effective redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' If  strong and weak lensing shears were identical, all points would lie  on the dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We instead find that external shears inferred  from strong lensing are consistently larger than those measured  by weak lensing, and not aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  (see eqn 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' This scaling is exact only if the external shear is  both real and dominated by neighbouring structures at the  same redshift as the lens (Wong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2011 found that 5/8 of  the shear is from neighbouring structures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' In any case, the  rescaling is by a factor with mean of only 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='26 and rms 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='06,  and our conclusions do not change if the rescaling is omit-  ted or normalised to a different redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' If strong and weak  lensing measure the same quantity, we then expect γWL  1  to  correlate with γSL  1 , and γWL  2  to scatter around zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We find  that ⟨γWL  2  ⟩ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='004 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='003 is on average below zero, and  MNRAS 000, 1–14 (2023)  6  Etherington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='06  γext  0  20  40  60  80  |φPL+ext  mass  −φPL+ext  ext  |  ‘aligned’  ‘anti-aligned’  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='9  q  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='4  γext  0  20  40  60  80  |φPL+ext  mass  −φPL+ext  ext  |  ‘aligned’  ‘anti-aligned’  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='8  q  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Relative orientation of the strong lensing ‘external’ shear  and the major axis of the lens mass, in mock (top panel) and real  HST (bottom panel) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' In both cases, most of the shears are  suspiciously aligned (φPL+ext  mass  − φPL+ext  ext  ⩽ 30◦) or anti-aligned  (φPL+ext  mass  − φPL+ext  ext  ⩾ 60◦) with the lens mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' If the  γext parameter were measuring true external perturbations, the  orientations would be random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Points are coloured by the best-fit  axis ratio of the lens mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Highly elliptical lenses  often lead to high values of γext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  its scatter (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='02) is consistent with uncertainties calculated  from the distribution of weak lensing shears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The best-fit  slope γWL  1  = (−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='04)γSL  1  actually infers a negative  correlation, however this does not take into account the un-  certainty on the strong lensing shears and so the uncertainty  is likely underestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The Pearson correlation coefficient  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='22 implies that, if there is a correlation, there is  also a large amount of scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  There are eight lenses for which γext ≈ γWL, including  two of the four lenses which reside in the outskirts of clusters  (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3 for further discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' However, there does  not appear to be anything unique about these lenses that  would make the shear possible to measure in these cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  SL shears are (suspiciously) aligned with the mass  For both mock and real data, the best-fit ‘external’ shear  is usually aligned with the major axis of the lens mass  (φPL+ext  mass  − φPL+ext  ext  ⩽ 30◦) or with its minor axis (φPL+ext  mass  −  φPL+ext  ext  ⩾ 60◦): see figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' If external shears were truly  measuring external perturbations, their orientations would  be random (modulo intrinsic alignments between the shape of  a galaxy and its surrounding tidal field, but these are much  smaller than our achieved measurement precision;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Zhang  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The preference for aligning with the mass distri-  bution again suggests an ‘internal’ shear that compensates  for the inability of a power law model to represent the more  complex true distribution of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Furthermore, the highest  values of γext are also usually found in the most elliptical  lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  In mock data, 84% of external shears are aligned with  the mass distribution: their mean offset is 3◦ with an rms  scatter of 5◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 14% of external shears are anti-aligned with  the mass distribution, with a mean of 85◦ and scatter of 6◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Only one lens has a best-fit external shear that is neither  aligned nor anti-aligned, but this also has the lowest shear  amplitude (γext = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0003), so the angle φext is ill-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  The Pearson correlation coefficient between the best-fit axis  ratios and external shears is −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Real HST data produce a similar pattern, but with more  scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Best-fit ‘external’ shears are aligned with the mass  distribution in 68% of the lenses, and anti-aligned in 20%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' All  the remaining 12% have γext < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='04, so the angles φPL+ext  ext  are noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The Pearson correlation coefficient between the  best-fit axis ratios and external shears is −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Despite  the inferred external shears being an order of magnitude  larger in the observations than in the mock data, the mass  distributions are similarly elliptical: ⟨q⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='02 in the  mocks, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='02 for HST data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3  Lenses in clusters  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  RX J2129-GGL1 (snail)  Of the galaxy-galaxy lenses in clusters the snail measures  shears that agree most closely between the two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' No-  tably, it measures the largest shear independent of the strong  lensing method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Although this measurement was constrained  by cluster scale strong lensing, Desprez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2018) demon-  strated that this value of shear is in agreement with that  derived from weak lensing analysis of CLASH data (Figure  13 of that paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The strong lensing external shear is anti-  aligned with it’s mass distribution, although this is expected  since the mass distribution is coincidentally orientated with  it’s major axis pointing towards the cluster centre (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  MACS1149-GGL20  The shears measured using the independent probes for  MACS1149-GGL20 are also in agreement, although the shear  magnitude is much lower than is measured for the snail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' In  fact, this is one of the few lenses that measures a lower best-  fit value of shear magnitude with strong lensing γext = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='01  than it does with the weak lensing method γWL = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Desprez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2018) found that measurements of external  shear from modelling the GGL alone were underestimated  compared to the shears constrained using a full scale model of  the galaxy cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' However, both of these measurements are  larger than either of the shears measured in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The au-  thors measure an external shear magnitude γext = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='13+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='08  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='06  when modelling the potential of the lens as a double Pseudo-  Isothermal Elliptical profile (dPIE), significantly larger than  that measured in this work γext = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='01+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' However, we  measure a more elliptical mass distribution q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='51+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='02  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='01  MNRAS 000, 1–14 (2023)  ‘External shear’ is not shear  7  Lens name  SIS neighbour  no neighbour  γWL  1  γWL  2  γWL  φWL  φBCG  φneighbour  γext  φext  qPL  φPL  γext  φext  qPL  φPL  MACS1149-GGL18  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='13+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='02  40+5  5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
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+page_content='02  –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
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+page_content='08  28  57 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Summary of strong and weak lensing parameters for the 4 galaxy-galaxy lenses that reside in clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' All angles are in degrees  anticlockwise from West.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  than was constrained by (Desprez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2018) q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  The degeneracy between shear and axis ratio may therefore  explain this discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' As with the snail, the mass dis-  tribution coincidentally points towards the cluster centre,  therefore the anti-alignment of the external shear with the  lens’ mass distribution that we infer is to be expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3  Abell370-GGL19  We measure a similar shear magnitude with strong lensing  for Abell370-GGL19 as we do with weak lensing, however  the strong lensing external shear is suspiciously orientated  towards a nearby neighbour galaxy and is aligned with the  mass distribution (see Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We therefore repeat the fit  including free parameters for a singular isothermal sphere  (SIS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' γ = 2 and q = 1 in equation 11) fixed at the centre of  the neighbour galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The results including the mass of the  neighbour galaxy do not change significantly (see the SIS  neighbour column of Table 2 compared to the no neighbour  column), although the power-law mass distribution does  become less elliptical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='4  MACS1149-GGL18  There is also a neighbour galaxy in close proximity to  MACS1149-GGL18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We, therefore repeat the fit for includ-  ing an SIS as was done for Abell370-GGL19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The shear is  significantly overestimated as compared with weak lensing  when the neighbour is not included in the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Including the  neighbour galaxy halves the inference of strong lensing exter-  nal shear, however this does not bring it into agreement with  the weak lensing inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The shear is anti-aligned with  the power-law mass distribution which, given that the mass  distribution is not aligned with the cluster galaxy in this  case, suggests the external shear may be acting internally as  discussed in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  6  ANALYSIS  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  External shear may compensate for boxiness/diskiness  Our measurements in Section 5 suggest that γext mostly  just compensates for the inability of a power law model to  capture the complex distributions of mass (gNFW+MGE for  the mocks, and likely more complex for real galaxies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' This  is consistent with the conclusion of Keeton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (1997), who  inferred that the ⟨γext⟩ ∼ 10–15% required to fit point-source  quad lenses, must reflect an inability of the lens model to  capture a complex distribution of mass: perhaps misalignment  between light and dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Witt & Mao (1997) reached  a similar conclusion, and derived an analytical prediction  of the shear required by an elliptical potential to fit quad  lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  If the external shears result from a lack of complexity in  the power law model to describe the underlying distribution of  mass, one can ask what type of complexity the data requires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  One possible deviation from the symmetry of an elliptical  power law is boxiness and diskiness (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We  shall now investigate whether spurious external shear could  arise to compensate for boxy/disky lens galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  External shear creates boxy/disky critical curves  An isothermal elliptical mass distribution has an elliptical  critical curve (oriented in the same direction as the mass  distribution but at 90◦ to the elongation of light from the  source galaxy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Kochanek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' However, changing the  power law slope, γ ̸= 2, or adding an external shear, γext ̸= 0,  perturbs the critical curves (left panel of Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' These  perturbations include significant a4/a moments (right panels  of Figure 4), although they visually appear to be more than  pure m = 4 modes (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  Disky critical curves come from disky mass  distributions  The distributions of mass in our mock lenses happen to be  almost all disky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We could measure any isodensity contour,  but the κ=1 contour will be near the most sensitive region  for lens fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' These iso-convergence contours have mean  ⟨|a4/a|⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='01 and ⟨|b4/a|⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Only three lenses are  boxy, but not usefully so, with ⟨a4/a⟩ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Critical curves of the best-fit PL+ext models to our  mock data show a4/a moments highly correlated with those  of the density contours (Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Again, ⟨|b4/a|⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0001 is  an order of magnitude lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Studying the same mocks, C22  also noted that ‘external’ shear allowed the best-fit critical  curves to better match the true critical curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We find two  systems with boxy critical curves a4/a < −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Subject  to some scatter, however, we conclude that the diskiness of  isodensity contours and critical curves are highly correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Notably, all mock lenses whose best-fit external shear is  aligned with the mass distribution have very disky critical  curves (red points in Figure 6), and the three mock lenses  with boxy critical curves have anti-aligned shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Further-  more, a4 typically increases with the external shear (Pearson  correlation coefficient 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='45) and with the axis ratio of the lens  mass (Pearson correlation coefficient −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' This may be ten-  tative evidence that (some of) the dichotomy of aligned and  anti-aligned shears may be caused by diskiness or boxiness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  MNRAS 000, 1–14 (2023)  8  Etherington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  −3  −2  −1  0  1  2  3  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='6  γ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='8  SIE  γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  γ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='75  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='25  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='50  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='75  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='00  γ  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='125  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='100  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='075  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='050  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='000  a4/a  −3  −2  −1  0  1  2  3  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  γext  1  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  γext  1  = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  SIE  γext  1  = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  γext  1  = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3  γext  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='15  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='10  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='00  a4/a  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' What causes boxiness or diskiness?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' A Singular Isothermal Elliptical (SIE) mass distribution with a horizontal major axis has  critical curves that are also elliptical with a horizontal major axis (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The critical curves are perturbed if the slope of the density  profile γ ̸= 2 (top left panel) or the external shear γext ̸= 0 (bottom left panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' In particular, an aligned shear (γext  1  > 0) stretches the  critical curves vertically (and the image horizontally);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' an anti-aligned shear (γext  1  < 0) does the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Multipole measurements a4/a of  the critical curve are shown as a function of slope (top right panel) and external shear (bottom right panel), where a4/a > 0 is “disky”  and a4/a < 0 is “boxy”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='8  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='6  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='4  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  log[ a4  a +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='01](κ = 1)  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='50  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='25  −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='00  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='75  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='50  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='25  −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='00  log[ a4  a +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='01] (PL+ext critical curves)  a4/a = 0  a4/a = 0  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' In mock lenses, disky (a4/a > 0) perturbations of the  κ = 1 isodensity contour correlate with disky perturbations of the  critical curves – which can also be measured for real lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' To  better visualise the correlation, values have been transformed by  log[a4/a+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='01], with dashed lines indicating a4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Unfortunately,  the mock data do not include any lenses with significantly boxy  (a4/a < 0) distributions of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' These points come from fits with  Sersic sources, so the uncertainties are not comparable to those  from analyses with pixellated sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3  Does boxiness/diskiness cause ‘external’ shear?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Scatter in real data is larger than in the mocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' However,  for SLACS and GALLERY lenses, the best-fit critical curves  have mean ⟨|a4/a|⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='016, similar to the mocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Most (79%  of) lenses with best-fit ‘external’ shear that is aligned with the  mass distribution have disky critical curves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' and most (70%)  with anti-aligned shear have boxy critical curves (Figure 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Moreover, lenses with the largest amplitude of external shear  also have critical curves with the largest deviations from  elliptical (Pearson correlation coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='48 with a4 and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='65 with b4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  This provides tentative evidence that ‘external’ shear  in typical lensing analyses is really caused by the inabil-  ity of parametric mass models to capture the complex dis-  tribution of mass in a lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' A substantial portion of that  complexity may be diskiness or boxiness of the mass dis-  tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' This creates diskiness or boxiness in the critical  curves (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2), which leads to a spurious external  shear (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We have not been able to quantify  the relative contributions to external shear from true shear,  diskiness/boxiness, or other sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  MNRAS 000, 1–14 (2023)  ‘External shear’ is not shear  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='06  γext  0  20  40  60  80  |φPL+ext  mass  −φPL+ext  ext  |  ‘aligned’  ‘anti-aligned’  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='10  a4/a  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' For mock lenses that were simulated without external  shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Angle between the best-fit values of external shear and the  major axis of the lens mass distribution, |φPL+ext  mass  − φPL+ext  ext  | in  degrees, as a function of the amplitude of the best-fit external  shear, γext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Points are coloured by the magnitude of the inferred  critical curves deviation from elliptical symmetry a4/a, values of  a4/a < 0 correspond to boxy critical curves and a4/a > 0 to disky  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Note the red and blue colour points are orders of magnitude  different, the blue range reaches a maximum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='4  γext  0  20  40  60  80  |φPL+ext  mass  −φPL+ext  ext  |  ‘aligned’  ‘anti-aligned’  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='20  a4/a  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Same as for Figure 6 but for the observed SLACS and  GALLERY lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The inferred external shears have a similar  distribution of aligned and anti-aligned shears as the mock data  sample, indicating they too may be acting internally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Note the in-  crease in the scale of shear magnitude γext and elliptical deviations  a4/a compared to the mock data sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='4  More things probably cause ‘external’ shear too  There are likely more sources of complexity in real mass  distributions, which cause (or are compensated by) external  shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' These confounding factors would explain the looser  correlations and larger scatter than in the mocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Just the  observation that real lenses have external shears with am-  plitudes six times greater than mocks implies that their  distribution of mass deviates more from a power law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  We speculate that the isodensity contours of a lens  might be twisted (misaligned as a function of radius), like  their isophotoes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Indeed, the critical curves of SLACS and  GALLERY lenses have a handedness, with ⟨|b4/a|⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='012  two orders of magnitude larger than the mocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' If the critical  curves do tell us something about the distribution of mass,  this may indicate twisted isodensity contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Twisting is  also suggested by the inconsistently measured position angle  when real data are fitted with and without external shear (see  Figure 9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' this is not present in the mocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Van De Vyvere  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2022a) also found that twists in the underlying mass  distribution are typically absorbed by changes in orientation  of the mass distribution and shear in a PL+ext model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  External shear biases other SL model parameters  Since best-fit values of ‘external’ shear may not (entirely)  represent the physical quantity they are imagined to, we now  test which other parameters in the mass model are biased  by their inclusion, and which are still robustly measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' As  an extreme alternative, we repeat all measurements but fix  γext  1  = γext  2  = 0 when we refit the mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  Mock lenses  The orientation of the mass distribution is robustly measured,  with mean difference ⟨|φPL+ext  mass  − φPL  mass|⟩ ∼ 1◦ and only ∼ 1◦  of scatter when shear is excluded (figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The axis ratio  is also hardly changed, with ⟨qPL+ext − qPL⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  These values are so robust because all the MGE components  share a common axis, which is therefore well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  However, the Einstein radii and slopes of the power-  law mass model are systematically biased, by an amount  that depends on the relative orientation of the shear and  the mass (Einstein radii and power-law slopes are known  to be degenerate parameters: see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' figure 5 of E22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' For  lenses whose shears were aligned with the mass distribution,  removing γext increases the mean best-fit power-law slope  by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='20% ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='04 (blue points in figure 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' their mean best-fit  Einstein radii decrease by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='03%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' For lenses whose  shears were anti-aligned, the power-law slopes decrease by  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='07±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='03 (red points in figure 8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' their mean best-fit Einstein  radii increase by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='04).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Across the entire sample, Ein-  stein radii had been correctly measured when including γext  (within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='17%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2022), but are systematically  underestimated by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2% ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='05% if shear is excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Our measurements might be caused by a bias described  by Kochanek (2020) in the radial structure of a lens, when a  model has too few azimuthal degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Kochanek  provides the example of fitting a power-law model to a lens  whose ellipticty increases with radius: the density slope is  forced to spuriously shallow values to balance the shear inside  the Einstein radius relative to the shear outside it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Further-  more, Van De Vyvere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2022a) found that decreasing  ellipticity outside the Einstein radius spuriously increases  γext for a power-law model, while ellipticity gradients inside  or at the Einstein radius mainly bias the power-law slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Our mocks whose shears align with the major axis may have  a distribution of mass whose ellipticity increases with radius,  and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Unfortunately, despite investigating a num-  ber of quantities derived from the axis ratios of the MGE  components, we were unable to confirm whether this is the  underlying cause of the shear alignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  Real lenses  For SLACS and GALLERY lenses, the orientation of the  mass distribution changes considerably when external shear  MNRAS 000, 1–14 (2023)  10  Etherington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  0  2  4  6  8  |φPL+ext  mass  −φPL  mass|  0  20  40  60  80  |φPL+ext  mass  −φPL+ext  ext  |  ‘aligned’  ‘anti-aligned’  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='7  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='6  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  γPL+ext −γPL  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' For mock lenses that were simulated without external  shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Angle between the best-fit external shear and the major axis  of the lens mass distribution, |φPL+ext  mass  − φPL+ext  ext  | in degrees, as a  function of the change in orientation of the major axis when fitting  with and without an external shear, |φPL+ext  mass  − φPL  mass|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Points are  coloured by the difference in power-law slope inferred between the  models fitted with and without an external shear (γPL+ext −γPL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Systems with best-fit shear aligned to the mass systematically  decrease in power-law slope when the external shear is removed  from the model (blue points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Anti-aligned shears exhibit the  opposite behaviour (red points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  0  20  40  60  80  |φPL+ext  mass  −φPL  mass|  0  20  40  60  80  |φPL+ext  mass  −φPL+ext  ext  |  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='6  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  qPL+ext −qPL  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Orientation angle offset of the external shear from the  PL+ext mass distribution (φPL+ext  mass  − φPL+ext  ext  ) as a function of  the difference in orientation angle when the mass distribution is  fitted with and without an external shear (φPL+ext  mass  − φPL  mass) for  the observed SLACS and GALLERY samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Scatter points are  coloured by the difference in axis ratio of the models fitted with a  PL+ext and a PL (qPL+ext − qPL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  is removed, with ⟨|φPL+ext  mass  − φPL  mass|⟩ ∼ 27◦ and ∼ 27◦ scatter  (figure 9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' note the different scale on the horizontal axis to  figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The best-fit axis ratio (indicated by colour in  figure 9) also increases by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='016 for aligned systems,  which become more spherical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' decreases by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='10 for  anti-aligned systems, which become more elliptical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' and is  inconsistent for the rest, with mean change ⟨qPL+ext −qPL⟩ =  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Neither the Einstein radii nor slopes of the power-law  mass model are systematically biased when external shear  is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We find ⟨∆REin/REin⟩ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='013+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='030  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='043, where  ∆REin ≡ RPL+ext  Ein  −RPL  Ein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' However, the best fit values scatter  about the same mean, such that ⟨∆R2  Ein/REin⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0452,  which is two orders of magnitude larger than the equivalent  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='00032 value for the mocks (figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Furthermore, the  scatter increases with increasing external shear magnitude  (figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  The best-fit centre of the mass distribution moves on  average by −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='031 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='061′′ when external shear parameters  are introduced, and on average remains 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='05 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='06′′ from  the centre of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' With or without shear, this offset is an  order of magnitude greater than in the mock data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  We suspect this indicates that the distribution of mass  in real galaxies is more complex than that in the mocks:  for example with multiple components that are rotationally  offset from one another and no single, well-defined axis of  ellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  7  INTERPRETATION  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  Strong lensing external shears are not measuring shear  We find that external shear, as measured by galaxy-galaxy  strong lensing, does not correlate well with the true shear  along a line-of-sight (as measured independently by weak  lensing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1 and Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Best-fit values of external  shear are also frequently several times higher than that along  typical lines of sight through the Universe, so only a small  fraction of it could be due to true shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Rather, external  shear tends to align with either the major axis or minor axis  of the lens mass distribution (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' It appears to  be compensating for the inflexibility of typical mass models  (here an elliptical power-law) to represent the complex dis-  tributions of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' A substantial portion of that complexity  appears to be diskiness and boxiness (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1), especially  in mock data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' In real data, isophotal twists, elliptical gradi-  ents, and offsets in the centres and alignments of dark and  stellar matter all increase uncertainty on model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  These have all been seen in the stellar mass of SLACS lenses  (Nightingale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We have not been able to quantify  how much each source contributes to the scatter or bias in a  real measurement of external shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Hogg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2022) suggest a different, ‘minimal line-of-  sight’ way of parameterising the shear that is less degenerate  with lens model parameters, but this is still subject to biases  in the shear parameters when simplifying assumptions are  made for the lens model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' For complex distributions of lens  mass, the false inference of external shear will pose a challenge  for efforts to use strong lensing to measure cosmic shear (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Fleury et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  Implications for strong lensing science goals  Including external shear alters the best-fit values of other  parameters (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' In mock data, some parameters do  move closer to the known truth: the mean error in power law  slopes is −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='10 with shear, or −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='20 without it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  A ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2% bias in Einstein radii also disappears with shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  However, the shear is not physically real and, contribut-  ing zero convergence, does not correspond to a physically  meaningful distribution of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' For studies that rely on  accurate reconstructions of the mass distribution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' galaxy  evolution, dark matter physics, and the Hubble constant),  this degeneracy with key parameters will eventually limit  MNRAS 000, 1–14 (2023)  ‘External shear’ is not shear  11  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  ∆REin/REin  0  5  10  15  Number  mean: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='013  16th(84th): -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0566 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0166)  mean: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='002  16th(84th): -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0001 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0039)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  ∆γ  mean: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='092  16th(84th): -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3858 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1808)  mean: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='16  16th(84th): -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='303 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='006)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5  ∆q  mean: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='06  16th(84th): -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='099 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2198)  mean: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='027  16th(84th): -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='011 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='069)  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1  ∆ centre  mean: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='031  16th(84th): -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0836 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0067)  mean: -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='002  16th(84th): -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0031 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0)  Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Histograms of the difference between inferred PL mass distribution model parameters with external shear as free parameters,  minus those without shear, for the observed (pink) and mock data samples (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' From left to right panels the parameters are:  fractional Einstein radius, logarithmic slope of density profile, axis ratio of mass distribution, and radial distance of the centre of the mass  distribution, in arcseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
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+page_content='Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Fractional difference between Einstein radii inferred for  models with and without the free parameters for external shear,  as a function of the external shear amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  statistical precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' For example, a key result of the SLACS  survey was that elliptical galaxies have isothermal (γ=2)  mass profiles (Gavazzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Vegetti & Koopmans 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Auger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Including external shear, E22 measured  ⟨γ⟩ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0756+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='023  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='024 for SLACS galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' here we measure  ⟨γ⟩ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='0159 =+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='027  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='032 (Figure 10), highlighting the system-  atic uncertainty that will remain fixed even if the sample  size increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Furthermore, a PL+ext model leaves false de-  tections of subhalos in a mock disky galaxy (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2022)  or real HST data (Nightingale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  The Hubble constant H0 can be measured from the  time delay between multiple images (Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Wong  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' However, assuming specific  functional forms for the mass model can artificially break  the mass-sheet transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Our tests on mock data (Sec-  tion 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1) demonstrate a coupling, predicted by Kochanek  (2020), between angular and radial structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Oversimple  models bias the slope, and hence bias H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' C22 estimated  ∼9% bias in H0 when using a PL+ext model to interpret time  delays generated from gNFW+MGE lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The angular de-  grees of freedom added by ‘external’ shear are insufficient to  compensate for even this complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Given that our analysis  of SLACS and GALLERY lenses indicate even more angular  degrees of freedom, such as twists in the mass distribution,  biases for real lensing systems will likely be closer to the  20–50% suggested by other studies (Schneider & Sluse 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Kochanek 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Gomer & Williams 2019,  2020, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' More flexible models, such as adding an internal  mass sheet to the PL+ext model which is contstrained via  stellar dynamics, have been introduced to mitigate  Combining H0 measurements from a large popula-  tion of lenses might average away individual biases up to  10kms−1Mpc−1, on the assumption that boxy and disky mass  distributions are equally well represented (Van De Vyvere  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' However, the diskiness of the mass distribution  correlates with the diskiness of the (observable) PL+ext crit-  MNRAS 000, 1–14 (2023)  12  Etherington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  ical curves (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2), and our observationally-selected  sample contains an overpopulation of 71% disky galaxies  (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' More flexible mass models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' a PL with an  internal mass sheet) have also been introduced (Birrer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  2020), which infer unbiased H0 values on mock lens samples  (Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Moreover, Van De Vyvere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2022b)  did not investigate the effect of b4 ‘twisting’ perturbations  (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='4) or mis-centering (Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='2), both of which  we observe in the critical curves of real galaxies, and whose  effects may not average away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' For example, the centre of  mass in the H0LiCOW model of lens WFI 2033-4723 is offset  from the centre of light by ∼10× astrometric uncertainty  (Suyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Barrera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2021), and a similar offset  in iPTF16geu increases asymmetry (Diego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Such  offsets are unphysical (Schaller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2015), so it is not clear  that complexities in the mass distributions of real lenses can  be safely ignored by averaging over a population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='3  Future work: more complex mass models are needed  External shear does not appear to be sufficient as (the sole)  parameter to encode all the complexity in real lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Al-  though Einstein radii are expected to be ‘model independent’  within ∼2% uncertainty Bolton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2008b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Sonnenfeld  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2013), the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='5% RMS fractional difference we mea-  sure with and without shear, suggests an unknown unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  A single power-law model leads to mass discrepancies with  stellar dynamics (Etherington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2022a), and spurious  false-positives in searches for dark matter subhalos (Nightin-  gale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Further work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Van  De Vyvere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2020, 2022a) to understand the types of  asymmetries that must be accounted for in the lens mod-  elling, and the possibility of constraining such models, will  be invaluable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  What parametric forms allow sufficient complexity —  in a minimum number of parameters that need to be con-  strained?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Even our gNFW+MGE mocks still do not capture  the full complexity of real lenses, but their MGEs were forced  to be aligned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Perhaps the model could have both a4 and b4  perturbations, varying as a function of radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Cluster-scale  models frequently use a soft core inside some scale radius, and  (Limousin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2022) add B-spline functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' The number  of free parameters must be balanced with the information  available: pixellated mass models are generally undercon-  strained (and although they might be able to fit observations  without external shear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Valls-Gabaud et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2006, some line-  of-sight shear is expected for galaxy-scale lenses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' To instead  increase the available information, the total mass could be  decomposed into dark matter and stellar components, with  the latter informed by the lens light (which is otherwise  a nuisance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Whatever the parametric form, the azimuthal  degrees of freedom must be defined carefully to avoid the  bias described by Kochanek (2020) on the inference of the  radial mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' We suggest more studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Van  De Vyvere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2020, 2022a, C22), embedded deeply in each  specific science case, to quantify the impact of simplifying  assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  A silver lining is that the sensitivity of galaxy-galaxy  strong lensing data may be a new opportunity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Both our  results and those of Van De Vyvere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' (2022b) suggest  that it might be possible to measure the diskiness or boxiness  of galaxies’ dark matter halos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Our results are even more  optimistic: if measurable multipole perturbations in critical  curves traces those in the distributions of both mass and  light, then one could study the dark morphology of galaxies  at high redshift with relative ease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  SOFTWARE CITATIONS  This work uses the following software packages:  Astropy (Astropy Collaboration 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Price-Whelan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  2018)  Corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='py (Foreman-Mackey 2016)  Dynesty (Speagle 2020)  Matplotlib (Hunter 2007)  Numba (Lam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2015)  NumPy (van der Walt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2011)  PyAutoFit (Nightingale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2021a)  PyAutoLens (Nightingale & Dye 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' Nightingale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  2018, 2021b)  Scikit-image (Van der Walt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2014)  Scikit-learn (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2011)  Scipy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' 2020)  SQLite (Hipp 2020)  ACKNOWLEDGEMENTS  We thank Yiping Shu, Guillaume Desprez, Johan Richard,  Mathilde Jauzac, and Keichi Umetsu for providing data from  previously published papers, and Liliya Williams and Adi  Zitrin for stimulating discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='  AE is supported by UK STFC via grants ST/R504725/1  and ST/T506047/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' JN and RM are supported by STFC  via grant ST/T002565/1, and the UK Space Agency via  grant ST/W002612/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' XYC and RL are supported by the  National Nature Science Foundation of China (via grants  11988101, 11773032, 12022306), the China Manned Space  Project (science research grants CMS-CSST-2021-B01, CMS-  CSST-2021-A01) and by the K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='Wong Education Founda-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' QH, AA, SMC, and CSF are supported by the ERC  via grant GA 786910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' MJ is supported by UKRI via grant  MR/S017216/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' This work used both the Cambridge Ser-  vice for Data Driven Discovery (CSD3) and the DiRAC  Data-Centric system, which are operated by the Univer-  sity of Cambridge and Durham University on behalf of the  STFC DiRAC HPC Facility (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='dirac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' These were  funded by BIS capital grant ST/K00042X/1, STFC capi-  tal grants ST/P002307/1, ST/R002452/1, ST/H008519/1,  ST/K00087X/1, STFC Operations grants ST/K003267/1,  ST/K003267/1, and Durham University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
+page_content=' DiRAC is part of  the UK National E-Infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
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+page_content=', 2011, Comput Sci  Eng, 13, 22  MNRAS 000, 1–14 (2023)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/qdE4T4oBgHgl3EQfvw0Q/content/2301.05244v1.pdf'}
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+Fast and accurate determination of the curvature-corrected field emission
+current
+Debabrata Biswas1, 2, a) and Rajasree Ramachandran1, 2
+1)Bhabha Atomic Research Centre, Mumbai 400 085, INDIA
+2)Homi Bhabha National Institute, Mumbai 400 094, INDIA
+The curvature-corrected field emission current density, obtained by linearizing at or below the Fermi energy,
+is investigated. Two special cases, corresponding to the peak of the normal energy distribution and the mean
+normal energy, are considered. It is found that the current density evaluated using the mean normal energy
+results in errors in the net emission current below 3% for apex radius of curvature, Ra ≥ 5nm and for apex
+fields Ea in the range 3 − 10 V/nm for an emitter having work-function φ = 4.5eV. An analytical expression
+for the net field emission current is also obtained for locally parabolic tips using the generalized cosine law.
+The errors are found to be below 6% for Ra ≥ 5nm over an identical range of apex field strengths. The
+benchmark current is obtained by numerically integrating the current density over the emitter surface and
+the current density itself computed by integrating over the energy states using the exact Gamow factor and
+the Kemble form for the WKB transmission coefficient. The analytical expression results in a remarkable
+speed-up in the computation of the net emission current and is especially useful for large area field emitters
+having tens of thousands of emission sites.
+I.
+INTRODUCTION
+Recent studies have shown that field emitters with tip
+radius in the nanometer range can be best modelled ac-
+curately by taking into account the variation in local field
+in the tunneling region, which is roughly 1-2nm from
+the emitter surface depending on the field strength1–5.
+When the apex radius of curvature (Ra) of the emitter
+is large (Ra > 100nm), the local field is roughly constant
+in this region even though the field enhancement factor
+itself may be large3.
+Thus, the Murphy-Good current
+density6–14 is quite likely adequate3 for Ra > 100nm
+while for emitters with Ra < 100nm, errors first start
+building up at smaller field strengths and for Ra ≤ 10nm,
+the errors become large over a wide range of fields3,5.
+The necessity for curvature-corrections was illustrated
+recently1 using the experimental results for a single
+Molybdenum emitter tip2 with a FESEM-estimated end-
+cap apex radius of curvature in the 5-10nm range with
+the square-shaped pyramidal base having a side-length
+Lb ∈ [1.25, 1.35]µm.
+Interestingly, even on using the
+Fowler-Nordheim16 current density that ignores image-
+charge contribution and seriously under-predicts the cur-
+rent density, the fit was good1,17 but required an emission
+area of 130000nm2. In contrast, the area of a hemisphere
+of radius 10nm is only about 628nm2.
+On the other
+hand1, the Murphy-Good current density (that takes
+into account image-charge contribution to the tunneling
+potential15), used with the generalized cosine law18,21 of
+local field variation around the emitter tip, had a best
+fit to the experimental data with Ra = 9.79nm which is
+within the estimated range of Ra. However, the value
+of field enhancement required the base-length Lb to be
+0.65µm which is clearly outside the 1.25− 1.35µm range.
+a)Electronic mail: dbiswas.hbni@gmail.com
+Thus, while the Fowler-Nordheim current density has
+gross non-conformity with the physical dimensions, the
+Murphy-Good current density seems to be in need of a
+correction. Indeed, on using a curvature-corrected (CC)
+expression for emission current3, the best fit to experi-
+mental data required1 Ra ≈ 5.41nm and Lb ≈ 1.275µm,
+both of which are within the range of their respective
+estimated values. This one-off validation could be a co-
+incidence and more such experiments, observation and
+data analysis are required to explore and put on a firm
+footing, the limits of validity of each model19,20.
+The evidence so far seems to suggest that a curvature-
+corrected field emission theory is necessary for nano-
+tipped emitters. An elementary form of this3 was used
+in Ref. [1], based on a tunneling potential having a single
+correction term. Since then, an approximately univer-
+sal tunneling potential having an additional curvature
+correction term has been established22 using the nonlin-
+ear line charge model22–26 and tested against the finite-
+element software COMSOL4. A curvature-corrected ana-
+lytical current density has also been determined5 by suit-
+ably algebraic approximation of the exact Gamow factor
+and its linearization at the Fermi energy. While the re-
+sults are promising, there is a scope for improving its
+accuracy by choosing a different linearization-energy. It
+is also desirable to have an analytical expression for the
+net field emission current applicable for Ra ≥ 5nm over
+a wide range of fields. The present communication seeks
+to establish accurate analytical expressions for both, the
+curvature-corrected local current density, as well as the
+net emission current for smooth locally parabolic emit-
+ters.
+The issue of accuracy in analytical expressions for cur-
+rent density has recently been investigated in Ref. [14]
+for emitters where curvature corrections are unimportant
+(Ra > 100nm).
+The three major factors investigated
+were: (a) the form in which the Gamow factor, G, is
+cast (b) the use of e−G to determine the transmission
+arXiv:2301.11545v1  [physics.app-ph]  27 Jan 2023
+
+2
+coefficient and (c) the energy at which the Gamow factor
+should be linearized in order to obtain an approximate
+analytical form for the current density. It was found14
+that if an analytical form of the current density is used
+to determine the net emission current, only the second
+and third factors are important. For instance, the use
+of e−G to determine the transmission coefficient leads to
+errors at larger local fields where the tunneling barrier
+transitions from ‘strong’ to ‘weak’. A better way of de-
+termining the transmission coefficient within the WKB
+approximation is the Kemble28,29 formula (1 + eG)−1.
+Another significant cause of error can be ascribed to the
+energy at which the Gamow factor is linearized in order
+to obtain an approximate analytical form for the current
+density. In the traditional approach to cold field emis-
+sion, the Gamow factor is linearized at the Fermi energy.
+While this holds at smaller values of the local field, it
+leads to large errors at higher fields due to the shift in the
+normal energy distribution away from the Fermi energy.
+In the following, we shall continue to use the traditional
+representation of the Gamow factor in term of the Forbes
+approximation7 for the WKB integral, and add curvature
+corrections to it.
+In section II, we shall make use of a curvature-
+corrected current density that makes use of a Kemble
+correction and a shifted point of linearization. We shall
+compare the results by choosing the energy correspond-
+ing to the peak of the normal energy distribution as well
+as the mean normal energy. While both results are en-
+couraging, the mean normal energy is more accurate es-
+pecially at lower field strengths. Finally, an evaluation
+of the net field-emission current is carried out using the
+generalized cosine law in section III and compared with
+the exact WKB result. Summary and discussions form
+the concluding section.
+II.
+AN ACCURATE CURVATURE-CORRECTED
+CURRENT DENSITY
+A widely adopted method to obtain an analytical ex-
+pression for the current density is to Taylor expand the
+Gamow factor about the Fermi energy EF in order to
+carry out the energy integration. Recent studies14 show
+that this is adequate at smaller local field strengths
+for which the electrons closer to the Fermi energy pre-
+dominantly tunnel through.
+As the field strength in-
+creases, the height and width of the tunneling barrier
+decreases and the electrons well below the Fermi energy
+start contributing to the net emitted current.
+This is
+evident from the shift in the peak of the normal energy
+distribution13 of the emitted electrons as the local field
+increases. Hence, for cold field emission, an expansion of
+the Gamow factor around the peak of the normal energy
+distribution or the mean normal energy seems preferable.
+This is likely to yield a better approximation for field
+emission current density applicable over a wide range of
+fields.
+The use of e−G is also a factor that contributes to the
+errors at higher fields where the barrier becomes weak.
+The transmission coefficient in the Kemble form28 can be
+approximated as14
+T(E) =
+1
+1 + eG ≈ e−G �
+1 − e−G�
+.
+(1)
+Used alongside the linearization of the Gamow factor,
+this is likely to provide a simple yet reasonably accurate
+expression for the field emission current density.
+A.
+Expansion of the Gamow factor and the curvature
+corrected current density
+The Gamow factor is expressed as
+G = g
+� s2
+s1
+�
+VT (s) − Eds.
+(2)
+Here, g = 2
+√
+2m/ℏ, m the mass of the electron and ℏ the
+reduced Planck’s constant h/(2π). In Eq. (2), VT is the
+tunneling potential energy, E is the normal component of
+electron energy at the emission surface and s1, s2 are the
+zeroes of the integrand. The curvature-corrected form of
+the tunneling potential energy is4,30
+VT (s) ≈ EF + φ + Vext(s) −
+B
+s(1 + s/2R)
+(3)
+where, φ is the work function, EF the Fermi energy while
+the external potential energy Vext takes the form,
+Vext(s) ≈ −qEls
+�
+1 − s
+R + 4
+3
+� s
+R
+�2�
+(4)
+with q
+the magnitude of electronic charge,
+B
+=
+q2/(16πϵ0), El the local electric field, and s denoting the
+normal distance from the surface of the emitter.
+The
+quantity R−1 is the mean curvature31,32 so that R is
+the harmonic mean of the principle radii of curvature
+R1 and R2 at the emission site i.e. R = 2/(R−1
+1
++ R−1
+2 ).
+The curvature-corrected external potential of Eq. 4 fol-
+lows directly from Eq. (35) of Ref. [4] which holds in the
+region close to the apex for all axially symmetric emit-
+ters in a parallel plate diode configuration. For a more
+detailed exposition, the reader may refer to appendix A
+on the tunneling potential33.
+Using the curvature-corrected tunneling potential en-
+ergy of Eq. (3), an approximate form for the Gamow
+factor can be found numerically to be5
+G = 2
+3g ϕ3/2
+qEl
+[ν(y) + xw1(y) + x2w2(y) + x3w3(y)] (5)
+= 2
+3g ϕ3/2
+qEl
+νc(y).
+(6)
+
+3
+Here, ϕ = EF +φ−E, x = ϕ/(qElR), y = 2√qBEl/ϕ and
+the curvature-corrected barrier function νc(y) = ν(y) +
+xw1(y) + x2w2(y) + x3w3(y), where4
+ν(y) = 1 − y2 + 1
+3y2 ln y
+(7)
+w1(y) = 10
+13 − 2
+11y2 + 1
+80y4 +
+1
+200y2 ln y
+(8)
+w2(y) = 10
+11 + 2
+11y2 − 1
+6y4 +
+1
+200y2 ln y
+(9)
+w3(y) = −41
+10 + 39
+20y2 + 1
+3y4 −
+1
+150y2 ln y.
+(10)
+Note that xw1(y), x2w2(y), x3w3(y) are the curvature
+corrections that arise due to R dependent terms in the
+external as well as image charge potential. As R → ∞
+in the planar limit, x → 0, so that νc(y) reduces to ν(y)
+which corresponds to the use of the Schottky-Nordheim
+barrier.
+We shall hereafter denote the linearization energy by
+Em.
+On expansion of the curvature-corrected Gamow
+factor and retaining the linear term, we obtain
+G(E) ≈ G(Em) − (E − Em)tcm
+dm
+(11)
+where tcm = tc(Em) with
+tcm = t(ym) + xmt1(ym) + x2
+mt2(ym) + x3
+mt3(ym)
+=
+�
+1 + y2
+m
+9 − 1
+9y2
+m ln ym
+�
++
+xm
+�25
+13 − 237
+1100y2
+m −
+1
+480y4
+m −
+7
+1200y2
+m ln ym
+�
++
+x2
+m
+�70
+33 + 589
+3300y2
+m + 1
+18y4
+m +
+1
+200y2
+m ln ym
+�
++
+x3
+m
+�
+−123
+10 + 2929
+900 y2
+m + 1
+9y4
+m − 1
+90y2
+m ln ym
+�
+(12)
+where
+ym
+=
+cs
+√El/ϕm,
+d−1
+m
+=
+g ϕ1/2
+m
+El ,
+cs
+=
+1.199985 eV (V/nm)−1/2, ϕm = EF + φ − Em and
+xm = ϕm/(qElR).
+The Gamow factor at Em can be
+expressed as
+G(Em) = BFNϕ3/2
+m
+νcm
+El
+(13)
+where νcm = ν(ym)+xmw1(ym)+x2
+mw2(ym)+x3
+mw3(ym).
+The field emission current density
+J =
+2mq
+(2π)2ℏ3
+� EF
+0
+(EF − E)
+1
+1 + eG(E) dE
+(14)
+≈
+2mq
+(2π)2ℏ3
+� EF
+0
+(EF − E)e−G(E) �
+1 − e−G(E) + . . .
+�
+dE
+can be expressed on completing the integration over en-
+ergy states as
+Jm
+cc ≈ AFN
+1
+ϕm
+E2
+l
+t2cm
+e−Bcc
+�
+1 − e−Bcc
+4
+�
+(15)
+Bcc = BFNϕ3/2
+m
+νcm
+El
+− tcm
+dm
+(EF − Em)
+(16)
+where AFN
+≃
+1.541434 µA eV V−2,
+BFN
+≃
+6.830890 eV−3/2 V nm−1 are the usual Fowler-Nordheim
+constants. The curvature-corrected current density Jm
+cc
+(Eq. (15)), with the incorporation of the first Kemble cor-
+rection and linearization of the Gamow factor at Em pro-
+vides an analytical expression that can be used to evalu-
+ate the net field emission current from a curved emitter,
+either by numerically integrating over the surface or by
+using the local field variation over the emitter surface to
+obtain an approximate analytical expression for the net
+field emission current.
+B.
+Numerical verification
+The exact WKB result (referred to hereafter as the
+benchmark) obtained by (i) finding the Gamow factor
+exactly by numerical integration (ii) use of the Kemble
+form of transmission coefficient and (iii) numerical inte-
+gration over energy to obtain the current density, can be
+used to validate Eq. (15). Since we shall be comparing
+net emission currents rather than current-densities, the
+local current density is integrated over the surface near
+the apex to obtain the net current numerically.
+The geometrical entity we are focusing on is an axially-
+symmetric emitter having an apex radius of curvature Ra
+and height h = 300Ra. It is mounted on a parallel plate
+diode where the generalized cosine law18,21 of local field
+variation holds:
+El = Ea
+z/h
+�
+(z/h)2 + (ρ/Ra)2 = Ea cos ˜θ.
+(17)
+In the above, h is the height of the emitter, Ra is the apex
+radius of curvature and Ea the apex field. Eq. (17) holds
+for all axially symmetric emitters where the tips are lo-
+cally approximated well by a parabola z ≈ h − ρ2/(2Ra)
+upto ρ ≈ Ra.
+Thus the only parameters required are
+h, Ra and the apex field34–37 Ea, since the generalized
+cosine law18,21 for local fields holds for such emitter-tips.
+Note that the benchmark also uses the parabolic approx-
+imation and the generalized cosine law for determining
+the net emission current38.
+In the following, we shall
+consider EF = 8.5eV and φ = 4.5eV. The apex fields con-
+sidered are in the range [3,10] V/nm which correspond
+to scaled barrier fields39 Ea/Eφ in the range 0.21333 -
+0.71109 where Eφ = (0.6944617 eV−2Vnm−1)φ2.
+
+4
+ 0
+ 3
+ 6
+ 9
+ 12
+ 15
+ 18
+ 21
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+ 10
+Relative Error (%)
+Ea (V/nm)
+Mean NE, Ra = 5nm
+Peak NED, Ra = 5nm
+εF - 2 dF/tF, Ra = 5nm
+εF - dF/tF, Ra = 5nm
+εF, Ra = 5nm
+FIG. 1. The absolute relative error in the net emission current
+with respect to the exact WKB result. Five cases are shown
+with various linearization energy Em. ‘Mean NE’ refers to the
+exact mean normal energy, ‘Peak NED’ refers to the exact
+energy at which the normal energy distribution peaks, ‘EF ’
+refers to Em = EF , ‘EF − 2dF /tF ’ is the approximate mean
+normal energy, while ‘EF −dF /tF ’ is the approximate peak of
+the normal energy distribution.
+ 0
+ 2
+ 4
+ 6
+ 8
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+ 10
+Relative Error (%)
+Ea (V/nm)
+Mean NE, Ra = 50nm
+Peak NED, Ra = 50nm
+εF - 2 dF/tF, Ra = 50nm
+εF - dF/tF, Ra = 50nm
+εF, Ra = 50nm
+FIG. 2. As in case of Fig. 1 with Ra = 50nm. Note that the
+error for Em = EF increases at higher values of Ea.
+It is clear that there are severals factors at play when
+comparing the error with respect to the exact WKB re-
+sult. We shall discuss two of these from the broad pic-
+ture available to us.
+The first is the effect of curva-
+ture correction which reflects in the approximate Gamow
+factor in Eq. (5).
+Since the expansion is in powers of
+x = ϕ/(qElR), the approximate Gamow factor is prone
+to errors at smaller values of El and R. Thus, irrespec-
+tive of the energy at which the linearization is carried
+out, lower fields and radius of curvature are prone to er-
+rors. In general, at higher R and El, the curvature errors
+are expected to reduce. The second important consider-
+ation is the energy at which the linearization is carried
+out. Since the peak of the normal energy distribution
+moves away from EF at higher fields for a given Ra, lin-
+earization at Em = EF should in general lead to larger
+errors at higher fields strengths. Apart from these two,
+there are other subtle effects that decide the magnitude
+of relative error at a given field strength as we shall see.
+Note that on the surface of an emitter, El reduces away
+from the apex while R increases and this leads to a mild
+decrease in the expansion parameter x.
+ 0
+ 1
+ 2
+ 3
+ 4
+ 7.6
+ 7.7
+ 7.8
+ 7.9
+ 8
+ 8.1
+ 8.2
+ 8.3
+ 8.4
+ 8.5
+Normal energy distribution
+εn (eV)
+Ra = 5nm
+Ra = 20nm
+Ra = 50nm
+FIG. 3. The normal energy distribution at Ea = 3V/nm for
+Ra = 5, 20 and 50nm. Note the shift in the distribution away
+from the EF (= 8.5eV here) for larger values of Ra.
+With this perspective, we shall compare the absolute
+relative errors at Ra = 5nm and Ra = 50nm shown in
+Figs. 1 and 2 respectively, for various values of Em dis-
+played in the legends. Clearly ‘Mean NE’, which refers
+to the exact mean normal energy determined numerically
+(see appendix C), performs well at Ra = 5nm at all field
+strengths while Em = EF shows large errors especially at
+lower fields.
+Even at Ra = 50nm where curvature er-
+rors are expected to be smaller, ‘Mean NE’ as well as the
+approximate mean normal energy (Em ≈ EF − 2dF /tF )
+perform well while in case of E = EF , the linearization
+error dominates leading to larger errors at higher field
+strengths. The energy value corresponding to the peak of
+the normal energy distribution (‘Peak NED’) also gives
+good results though the errors are somewhat high for
+smaller apex fields at Ra = 5nm.
+Some of the trends in Figs. 1 and 2 are easy to un-
+derstand. For instance, at Ra = 5nm, the errors fall as
+expected with an increase in Ea in all cases (except for a
+mild increase at Em = EF for Ea > 9V/nm). The larger
+than expected error (approximately 21%) for Em = EF at
+Ea = 3V/nm however seems intriguing. To understand
+this better, the normal energy distribution (see Fig. 3)
+for different values of Ra at Ea = 3V/nm is quiet in-
+structive.
+The peak of the normal energy distribution
+
+5
+ 16
+ 18
+ 20
+ 22
+ 24
+ 26
+ 28
+ 30
+ 32
+ 7.6
+ 7.7
+ 7.8
+ 7.9
+ 8
+ 8.1
+ 8.2
+ 8.3
+ 8.4
+ 8.5
+Gamow factor
+εn (eV)
+Ra = 5nm; Exact
+Ra = 5nm; εm = εF
+Ra = 5nm; εm = εF - 2dF/tF
+Ra = 50nm; Exact
+Ra = 50nm; εm = εF
+Ra = 5nm; εm = εF - 2dF/tF
+FIG. 4. The exact Gamow factor at Ea = 3V/nm is com-
+pared with the linearized Gamow factors with the point of
+linearization at Em = EF and Em = EF − 2dF /tF . The upper
+set of curves correspond to Ra = 5nm while the lower set is
+for Ra = 50nm. The two linearized versions are nearly equiv-
+alent at Ra = 50nm while, for Ra = 5nm linearizing at the
+approximate mean energy yields results closer to the exact
+Gamow factor over the energy range of interest.
+shifts slightly away from EF as Ra increases. Note also
+that the distributions have a long tail. The linearized
+Gamow factor in the corresponding normal energy range
+is shown in Fig. 4. For Ra = 5nm, linearization at EF
+results in larger deviations from the exact Gamow fac-
+tor compared to linearization at EF − 2dF /tF . Not sur-
+prisingly, the relative error in net emission current drops
+from about 21% to about 13% in moving from Em = EF
+to Em = EF − 2dF /tF .
+At Ra = 50nm, curvature effects are smaller and the
+linearized Gamow factor does not noticeably deviate from
+the exact Gamow factor (see Fig. 4 for Ea = 3V/nm).
+Thus, the errors remain more or less similar at all lin-
+earization energies. The magnitude of the error at a par-
+ticular Ea depends on how closely the linearized Gamow
+factor approximates the exact Gamow factor over the
+relevant range of normal energies. For Ea > 5V/nm at
+Em = EF , the increase in error is expected due to the
+shift in normal energy distribution away from EF and the
+corresponding deviation of the linearized Gamow factor
+from the exact Gamow factor.
+In order to verify that the trend observed in moving
+from Ra = 5nm to Ra = 50nm is gradual, we show the
+results for Ra = 10nm and Ra = 20nm in Figs. 5 and 6.
+It is apparent from these results that linearization at the
+exact mean normal energy (‘Mean NE’) is optimum for
+all values of Ra and Ea with errors generally below 3%.
+The approximate mean normal energy Em ≈ EF −2dF /tF
+is only marginally worse with errors exceeding 6% only
+at Ra = 5nm.
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+ 10
+Relative Error (%)
+Ea (V/nm)
+Mean NE, Ra = 10nm
+Peak NED, Ra = 10nm
+εF - 2 dF/tF, Ra = 10nm
+εF - dF/tF, Ra = 10nm
+εF, Ra = 10nm
+FIG. 5. As in case of Fig. 1 with Ra = 10nm.
+ 0
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+ 10
+Relative Error (%)
+Ea (V/nm)
+Mean NE, Ra = 20nm
+Peak NED, Ra = 20nm
+εF - 2 dF/tF, Ra = 20nm
+εF - dF/tF, Ra = 20nm
+εF, Ra = 20nm
+FIG. 6. As in case of Fig. 1 with Ra = 20nm.
+III.
+THE NET CURVATURE-CORRECTED EMISSION
+CURRENT
+The curvature-corrected expression for the current
+density, with linearization at the mean normal energy,
+can be used to arrive at an analytical expression for the
+net emission current on using the generalized cosine law
+of local field variation El = Ea cos ˜θ (Eq. (17)). Assum-
+ing a sharp locally parabolic emitter tip, the total emitted
+current can be evaluated using the expression13
+I ≈ 2πR2
+a
+�
+Jcc(˜θ) sin ˜θ
+cos4 ˜θ
+× C(˜θ) d˜θ
+(18)
+where C(˜θ) is a correction factor which, for a sharp emit-
+ter (h/Ra >> 1), is approximately unity. In the follow-
+ing we shall assume the emitter to be reasonably sharp
+so that C ≈ 1.
+The basic idea is to express Jcc in terms of ˜θ by
+
+6
+replacing all the local fields using El = Ea cos ˜θ.
+A
+further simplification can be made by the substitution
+1/ cos ˜θ = 1 + u and retaining only terms upto O(u2) in
+Bcc and t−2
+cm. The approximation is expected to be good
+at lower apex fields since the emission is limited to an
+area closer to the apex, while at higher fields, where the
+emission area is larger, this might lead to larger errors.
+Writing Bcc ≈ D0 + D1u and t−2
+cm ≈ F0 + F1u, the
+integration can be carried out easily. Note that, it gener-
+ally suffices to integrate upto ρ = Ra which, for a sharp
+emitter, corresponds to ˜θ = π/4 or u =
+√
+2 − 1 = u0.
+Thus,
+I ≈ 2πR2
+aGAFN
+1
+ϕma
+E2
+aF0e−D0
+(19)
+where Em is the mean normal energy, while
+G ≈ 1
+D1
++ F1
+F0
+1
+D2
+1
+− e−D0
+4
+� 1
+2D1
++ F1
+F0
+1
+4D2
+1
+�
+−
+e−D1u0
+� 1
+D1
++ F1
+F0
+1 + D1u0
+D2
+1
+− e−D0−D1u0
+4
+×
+� 1
+2D1
++ F1
+F0
+1 + 2D1u0
+4D2
+1
+��
+.
+(20)
+Expressions for D0, D1, F0 and F1 can be found in ap-
+pendix B.
+ 0
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+ 10
+εm = exact mean NE
+Relative Error (%)
+Ea (V/nm)
+Ra = 5nm
+Ra = 10nm
+Ra = 20nm
+Ra = 50nm
+FIG. 7. The magnitude of the relative error in the analytical
+expression for the curvature-corrected current (Eq. 19) com-
+pared to the exact WKB result. Here Em is the exact mean
+normal energy.
+In Fig. 7, we compare the magnitude of the relative
+error in the net current as given by Eq. (19) and (20)
+with respect to the exact WKB result which has been
+used as the benchmark throughout this study with Em
+as the exact mean normal energy. Clearly, the analytical
+expression is adequate for a wide range of fields and apex
+ 0
+ 3
+ 6
+ 9
+ 12
+ 15
+ 18
+ 21
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+ 10
+εm = εF
+Relative Error (%)
+Ea (V/nm)
+Ra = 5nm
+Ra = 10nm
+Ra = 20nm
+Ra = 50nm
+FIG. 8. The magnitude of the relative error in the analyt-
+ical expression for the curvature-corrected current (Eq. 19)
+compared to the exact WKB result for Em = EF .
+radius of curvature. The increase in error at higher fields
+is due to the linearization of Bcc and t−2
+cm in the variable
+u which is a measure of the distance from the apex. This
+is however a small price to pay for a compact analytical
+expression for the net emission current.
+For the sake of comparison, we also show the relative
+errors in the net current obtained using the analytical
+expressions in Eq. (19) and (20) with Em = EF .
+The
+trends are similar to those shown in section II B where the
+linearized current density is integrated numerically over
+the emitter end-cap. The errors are more pronounced at
+smaller apex radius of curvature and apex field strengths.
+Clearly, linearization at the mean normal energy ensures
+smaller errors over a wide range of fields and radius of
+curvature.
+ 0
+ 2
+ 4
+ 6
+ 8
+ 10
+ 12
+ 14
+ 16
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+ 10
+εm = εF - 2dF/tF
+Relative Error (%)
+Ea (V/nm)
+Ra = 5nm
+Ra = 10nm
+Ra = 20nm
+Ra = 50nm
+FIG. 9. The magnitude of the relative error in the analytical
+expression for the curvature-corrected current (Eq. 19) com-
+pared to the exact WKB result. Here, Em = EF − 2dF /tF .
+
+7
+While the errors are reasonably small when the exact
+mean normal energy is used, it require the computation of
+integrals that marginally offsets the use of an analytical
+expression for the net current.
+In Fig. 9, we provide
+a comparison of the magnitude of relative errors with
+respect to the exact WKB result, using the approximate
+Em = EF − 2dF /tF .
+While the errors for Ra = 5nm
+are somewhat large, the approximate value of the mean
+normal energy may be used profitably for Ra ≥ 10nm.
+TABLE I. Comparison of time required for 10000 evaluations
+of the net emission current for Ea ∈ [3, 10]V/nm and Ra ∈
+[5, 50]nm. The ‘Scale Factor’ is the ratio of the time taken by
+‘WKB Exact’ and the time taken by a given method. It gives
+a rough indication of the speed-up achieved. Here Em is the
+mean normal energy. Also shown is the average relative error
+with respect to ‘WKB exact’.
+Method
+Time (s) Scale Factor Average Error
+WKB Exact
+315.1 s
+1
+—
+WKB Fit
+23.8 s
+13.24
+1.67%
+Eq. (19) with
+56.7 s
+5.55
+2.06%
+Em exact
+Eq. (19) with
+0.0003 s
+106
+3.14%
+Em approximate
+Finally, Table I provides a comparison of the CPU
+time (in seconds) required on a standard desktop to se-
+rially compute the net emission current for 104 combina-
+tions of Ea and Ra in the range of apex fields and radius
+of curvature considered in this paper. Thus, there are
+100 values of Ra spaced uniformly in the range [5,50]nm
+and 100 values of Ea spaced uniformly in the [3,10]V/nm
+range. In the table, ‘WKB Fit’ refers to the use of Eq. (5)
+for the Gamow factor and numerical integration over en-
+ergy while ‘WKB exact’ refers to the ‘exact’ numerical
+evaluation of the Gamow factor followed by numerical
+integration over energy. The last two rows refer to the
+analytical formula for the net current of Eq. (19) with ‘Em
+exact’ evaluated as outlined in appendix C and ‘Em ap-
+proximate’ as Em ≈ EF − 2dF /tF . Clearly, linearization
+at the approximate mean normal energy results in fast
+computation of the net emission current using Eq. (19)
+by a factor ≈ 80000 compared to ‘WKB fit’ and about
+106 compared to the ‘WKB exact’ result. This is only
+marginally offset by a larger error for Ra = 5nm as seen
+in Fig. 9.
+The average relative error in the 5 − 50nm
+range is however small as shown in Table I.
+IV.
+CONCLUSIONS
+We have presented an expression for the curvature-
+corrected current density obtained by linearization at an
+energy Em ≤ EF and insertion of a correction term to ac-
+count for the Kemble transmission coefficient. Numerical
+results show that the mean normal energy is a suitable
+candidate for the linearization energy Em and predicts the
+net emission current to within 3% accuracy compared to
+the exact WKB result for Ra ≥ 5nm and over a wide
+range of field.
+We have also obtained an analytical expression for the
+net emission current using the generalized cosine law of
+local field variation. It requires only the apex radius of
+curvature Ra and the apex electric field Ea and is able
+to calculate the net field-emission current to within 6%
+accuracy compared to the current obtained by explic-
+itly integrating the exact WKB current density over the
+emitter tip for Ra ≥ 5nm and a wide range of apex fields.
+Both of these results are expected to be useful in deal-
+ing with sharp emitters having tip radius Ra ≥ 5nm.
+The expression for current density can be used in all sit-
+uations including those where the emitter does not have
+any special symmetry. On the other hand, the expres-
+sion for the net emission current is extremely useful for
+axially symmetric emitters with smooth locally parabolic
+tips mounted in a parallel plate configuration, consider-
+ing that the speed-up achieved in current computation
+is enormous.
+The accuracies obtained in all cases are
+good, given that even minor experimental uncertainties
+can lead to far larger changes in the net emission current.
+Finally, the analytical expression for emission current
+is especially useful for a fast determination of net emis-
+sion current from a large area field emitter have thou-
+sands of axially symmetric emitters40–42.
+V.
+AUTHOR DECLARATIONS
+A.
+Conflict of interest
+There is no conflict of interest to disclose.
+B.
+Data Availability
+The data that supports the findings of this study are
+available within the article.
+C.
+Author Contributions
+Debabrata Biswas Conceptualization (lead), data
+curation (equal), formal analysis (equal), methodology
+(lead), software (equal), validation (supporting), visual-
+ization (equal), original draft (lead), review and editing
+(supporting).
+Rajasree Ramachandran Conceptualization (sup-
+porting), data curation (equal), formal analysis (equal),
+methodology (supporting), software (equal), validation
+(lead), visualization (equal), original draft (supporting),
+review and editing (lead).
+
+8
+VI.
+REFERENCE
+1D. Biswas and R. Kumar, J. Vac. Sci. Technol. B 37, 040603
+(2019).
+2C. Lee, S. Tsujino and R. J. Dwayne Miller, Appl. Phys. Lett.
+113, 013505 (2018).
+3D. Biswas and R. Ramachandran, J. Vac. Sci. Technol. B 37,
+021801 (2019).
+4R. Ramachandran and D. Biswas, J. Appl. Phys. 129, 184301
+(2021).
+5D. Biswas and R. Ramachandran, J. Appl. Phys. 129, 194303
+(2021).
+6E. L. Murphy and R. H. Good, Phys. Rev. 102, 1464 (1956).
+7R. G. Forbes, App. Phys. Lett. 89, 113122 (2006).
+8R. G. Forbes and J. H. B. Deane, Proc. R. Soc. A 463, 2907
+(2007).
+9J. H. B. Deane and R. G. Forbes, J. Phys. A: Math. Theor. 41,
+395301 (2008).
+10K. L. Jensen, Introduction to the physics of electron emission,
+Chichester, U.K., Wiley, 2018.
+11K. L. Jensen, J. Appl. Phys. 126, 065302 (2019).
+12R. G. Forbes, J. Appl. Phys. 126, 210901 (2019).
+13D. Biswas, Physics of Plasmas 25, 043105 (2018).
+14D. Biswas, J. Appl. Phys. 131, 154301 (2022).
+15L. W. Nordheim, Proc. R. Soc. London, Ser. A 121, 626 (1928).
+16R. H. Fowler and L. W. Nordheim, Proc. Roy. Soc. Ser. A 119,
+173 (1928).
+17The emission area was assumed to be independent of the local
+field.
+18D. Biswas, G. Singh, S. G. Sarkar and R. Kumar, Ultrami-
+croscopy 185, 1 (2018).
+19For a recent analysis on a single emitter not requiring curvature
+correction, see [20].
+20E. O. Popov, S. V. Filippov and A. G. Kolosko, J. Vac. Sci.
+Technol. B 41, 012801 (2023).
+21D. Biswas, G. Singh and R. Ramachandran, Physica E 109, 179
+(2019).
+22D. Biswas, G. Singh and R. Kumar, J. App. Phys. 120, 124307
+(2016).
+23E. Mesa, E. Dubado-Fuentes, and J. J. Saenz, J. Appl. Phys. 79,
+39 (1996).
+24E. G. Pogorelov, A. I. Zhbanov, and Y.-C. Chang, Ultrami-
+croscopy 109, 373 (2009).
+25J. R. Harris, K. L. Jensen, and D. A. Shiffler, J. Phys. D 48,
+385203(2015).
+26J. R. Harris, K. L. Jensen, W. Tang, and D. A. Schiffler, J. Vac.
+Sci. Technol. B 34, 041215 (2016).
+27K. L. Jensen, Journal of Applied Physics 111, 054916 (2012).
+28E. C. Kemble, Phys. Rev. 48, 549 (1935).
+29R. G. Forbes, Journal of Applied Physics 103, 114911 (2008).
+30D. Biswas, R. Ramachandran and G. Singh, Phys. Plasmas 25,
+013113 (2018); ibid. 29, 129901 (2022).
+31In Ref. [32], an effective spherical approximation was used to
+generate the potential with R−1 as the mean of the two principle
+curvature R−1
+1
+and R−1
+2 . The authors state32 “This approach re-
+turns a high precision result comparable to the approach reported
+by Biswas and Ramachandran”, referring to the results in Ref. [4
+and 5] that used the first and second corrections to the external
+potential with R = R2. In Ref. [4] the external potential was de-
+rived for general axially symmetric emitters using the nonlinear
+line charge model.
+32J. Ludwick, M. Cahay, N. Hernandez, H. Hall, J. O’Mara,
+K. L. Jensen, J. H. B. Deane, R. G. Forbes, T. C. Back, Journal
+of Applied Physics 130, 144302 (2021).
+33In previous publications4,30, R was approximated as R2 and the
+results were found to be close to numerically determined external
+potentials using COMSOL for various shapes. It is shown in the
+appendix using the results of Refs. [4 and 30], that in the first cor-
+rection s/R, R is the harmonic mean Rm = 2/(R−1
+1
++R−1
+2 ). The
+second correction 4s2/(3R2) with R = Rm is an approximation,
+albeit a marginally improved one compared to the identification
+R ≈ R2.
+34The local field at the emitter-apex, Ea, is related to the applied or
+macroscopic field E0 through the apex field enhancement factor
+γa. See for instance Refs. [35–37].
+35D. Biswas, Physics of Plasmas 25, 043113 (2018).
+36D. Biswas, Physics of Plasmas, 26, 073106 (2019).
+37T. A. de Assis, F. F. Dall’Agnol and R. G. Forbes, J. Phys:
+Condens. Matter 34, 493001 (2022).
+38At high fields, contributions beyond ρ ≈ Ra cannot be altogether
+neglected. While the validity of the parabolic approximation and
+the cosine law (except in hemi-ellipsoids) start breaking down
+for ρ > Ra, the curvature corrected current density of Eq. (15)
+continues to hold and can be used to determine the net emission
+current.
+39R. G. Forbes, J. Vac. Sci. Technol. B26, 209 (2008).
+40D. Biswas and R. Rudra, Physics of Plasmas 25, 083105 (2018).
+41D. Biswas and R. Rudra, J. Vac. Sci. Technol. B38, 023207
+(2020).
+42D. Biswas, J. Vac. Sci. Technol. B38, 063201 (2020).
+Appendix A: The tunneling potential
+The electric field, El, close to the emitter surface is
+assumed to be a constant so that the corresponding po-
+tential can be expressed as Vext(s) = −Els where s is the
+normal distance from the surface of the emitter and El is
+the magnitude of the local electric field. The assumption
+holds good when the radius of curvature at the emission
+site is large (typically R > 100nm).
+As R decreases, corrections become important and
+these can be expressed as powers of s/R. Thus,
+Vext(s) = −Els
+�
+1 + c1
+s
+R + . . . + cn
+� s
+R
+�n
++ . . .
+�
+.
+(A1)
+The
+effective
+spherical
+approximation
+used
+in
+Ref. [32] leads to cn
+=
+(−1)n so that Vext(s)
+=
+−ElR [1 − 1/(1 + s/R)] with R−1 = 2/(R−1
+1
++ R−1
+2 ).
+In Ref. [30], following the analysis of the hemiellipsoid,
+the hyperboloid and the hemisphere, it was concluded
+that c1 = −1, c2 = 4/3 and R ≈ R2 where R2 is the
+second (smaller) principle radius of curvature.
+With
+these identifications, the external potential was found
+to approximate the numerically determined external
+potentials for other emitter shapes as well30. Ref. [4] uses
+the nonlinear line charge model22 for axially symmetric
+emitters to arrive at an approximate form close to the
+apex. In the following, we shall show that the results
+of both Ref. [4 and 30] can be recast in the form where
+{c1 = −1, R = Rm} exactly while {c2 = 4/3, R2 = R2
+m}
+is approximate but fairly accurate close to the apex.
+In addition to the approximate results in section II30,
+Ref. [30] also provides in the appendix, a systematic ex-
+pansion of the external potential in powers of s for the
+hemi-ellipsoid.
+In terms of the prolate spheroidal co-
+ordinates (η, ξ, ϕ)
+
+9
+x = L
+�
+(η2 − 1)(1 − ξ2) cos ϕ
+(A2)
+y = L
+�
+(η2 − 1)(1 − ξ2) sin ϕ
+(A3)
+z = Lηξ
+(A4)
+it was found that
+Vext(s) = V (s) =
+�
+d1s + d2s2 + d3s3�
+(A5)
+where
+d1 = Vηa1
+(A6)
+d2 = Vηa2 + 1
+2Vηηa2
+1
+(A7)
+d3 = Vηa3 + Vηηa1a2 + Vξηa1b2 + 1
+6Vηηηa3
+1.
+(A8)
+The derivatives of the potential at a point (η0, ξ0) on the
+surface of the hemiellipsoid are
+Vη = Elhη
+(A9)
+Vηη = −Elhη 2η0
+η2
+0 − 1
+(A10)
+Vηξ = Elhη
+1
+ξ0
+(A11)
+Vηηη = Elhη
+8η2
+0
+(η2
+0 − 1)2
+(A12)
+where
+hη = L
+�
+(η2
+0 − ξ2
+0)/(η2
+0 − 1)
+(A13)
+hξ = L
+�
+(η2
+0 − ξ2
+0)/(1 − ξ2
+0).
+(A14)
+The coefficients
+a1 = 1
+hη
+(A15)
+a2 =
+1
+2h2
+ξ
+η0
+η2
+0 − ξ2
+0
+(A16)
+a3 = −
+1
+2hηh2
+ξ
+η2
+0 + ξ2
+0
+(η2
+0 − ξ2
+0)2
+(A17)
+b2 = − 1
+2h2
+ξ
+ξ0
+η2
+0 − ξ2
+0
+(A18)
+while the principle radii of curvature are
+R1 = Ra
+(η2
+0 − ξ2
+0)3/2
+(η2
+0 − 1)3/2
+(A19)
+R2 = Ra
+(η2
+0 − ξ2
+0)1/2
+(η2
+0 − 1)1/2
+(A20)
+where Ra is the apex radius of curvature. On putting
+together these results, the values of d1, d2 and d3 are
+d1 = −El
+(A21)
+d2 = El
+2Ra
+(η2
+0 − 1)1/2
+(η2
+0 − ξ2
+0)3/2
+�
+2η2
+0 − 1 − ξ2
+0
+�
+= El
+Rm
+(A22)
+d3 = −4
+3
+El
+R2m
+�
+3 + 4η4
+0 + ξ4
+0 − 2η2
+0(3 + ξ2
+0)
+�
+(1 + ξ2
+0 − 2η2
+0)2
+(A23)
+= −4
+3
+El
+R2m
+�
+1 − 2(η2
+0 − 1)(1 − ξ2
+0)
+(1 + ξ2
+0 − 2η2
+0)2
+�
+(A24)
+= −4
+3
+El
+R2m
+[1 − C]
+(A25)
+where
+Rm =
+2
+(1/R1 + 1/R2)
+= 2Ra
+η2
+0 − ξ2
+0)3/2
+(η2
+0 − 1)1/2(2η2
+0 − 1 − ξ2
+0).
+(A26)
+The external potential thus takes the form
+Vext(s) = −Els
+�
+1 −
+� s
+Rm
+�
++ 4
+3
+� s
+Rm
+�2
+(1 − C)
+�
+(A27)
+for the hemiellipsoid emitter. In terms of ρ2
+0 = x2
+0 + y2
+0
+where x0, y0 are on the surface of the hemiellipsoid, the
+correction term C = ρ2
+0/(2R2
+a). Thus,
+Vext(s) = −Els
+�
+1 −
+s
+Rm
++ 4
+3
+s2
+R2m
+�
+1 − 1
+2
+ρ2
+0
+R2a
+��
+.
+(A28)
+Note that close to the apex, ρ/Ra << 1 while Rm ≈ R2.
+A more general result, valid for all axially symmetric
+emitters in a parallel plate geometry, was arrived at us-
+ing the nonlinear line charge model4. In such cases, the
+external potential can be expressed as (see Eq. (35) of
+Ref [4]),
+Vext(s) ≈ −Els
+�
+1 − s
+Ra
+(1 − ρ2
+0
+R2a
+) + 4
+3
+s2
+R2a
+(1 − 5
+2
+ρ2
+0
+R2a
+)
+�
+.
+(A29)
+Close to the apex 1/Rm ≈ (1/Ra)(1 − ρ2
+0/R2
+a) while
+1/R2
+m ≈ (1/R2
+a)(1 − 2ρ2
+0/R2
+a). Thus, Eq. (A29) can be
+expressed as
+Vext(s) ≈ −Els
+�
+1 −
+s
+Rm
++ 4
+3
+s2
+R2m
+(1 − 1
+2
+ρ2
+0
+R2a
+)
+�
+.
+(A30)
+
+10
+This is identical to the result obtained for the hemiellip-
+soid (see Eq. A28) but applicable generally for all axially
+symmetric emitters. Approximating (1 − ρ2
+0/(2R2
+a)) ≈ 1
+leads us to an approximate universal form for the exter-
+nal potential (see Eq. (4)) close to the emitter surface.
+Note that Eq. (A29) can also be expressed as
+Vext(s) ≈ −Els
+�
+1 −
+s
+Rm
++ 4
+3
+s2
+R1Rm
+�
+(A31)
+The correction terms,
+s
+Rm and 4
+3
+s2
+R1Rm are exact for any
+point on the hemiellipsoid surface.
+For other emitter
+shapes4, the two correction terms may have extra fac-
+tors that can be ascribed to the non-linear line charge
+distribution.
+Since these have been ignored as an ap-
+proximation, we choose to adopt the form in Eq. (4) with
+R = Rm as an approximate but accurate representation
+of the external potential in the tunneling region.
+Finally, while the change from R2 to Rm reduces the
+need for approximations, its impact on the net field emis-
+sion current is small compared to a neglect of the second
+correction term 4s2/(3R2
+m), especially at smaller values
+of Ra and Ea. For instance, at Ra = 5nm and work-
+function φ = 4.5eV, the error in net emission current on
+using R2 in Eq. (4) is about 13% at Ea = 5V/nm, while
+it is around 62% on ignoring 4s2/(3R2
+m) altogether. At
+Ea = 4V/nm, the error in net emission current on using
+R2 in Eq. (4) remains roughly the same while the error
+grows to around 92% on ignoring 4s2/(3R2
+m). In each
+of these cases, the exact WKB method is used and the
+benchmark current is obtained using Rm in Eq. (4).
+Appendix B: The coefficients D0, D1, F0 and F1
+We shall briefly outline the derivation of the coeffi-
+cients D0, D1, F0 and F1 and state the results. The de-
+pendence on u in Bcc and tcm arise from the variation in
+El and xm over the surface of the emitter. Thus,
+Bcc = Bcc(El, xm) = Bcc( Ea
+1 + u, xma
+1 + u) ≈ D0 + D1u
+(B1)
+so that D0
+= Bcc(Ea, xma).
+In the above, xm
+=
+ϕm/(qElRm) ≈ xma/(1 + u) where xma = ϕm/(qEaRa).
+The approximation holds for tall emitters where El =
+Ea(z/h)/
+�
+(z/h)2 + (ρ/Ra)2 ≈ Ea/
+�
+1 + ρ2/R2a.
+The
+coefficient D1 can be written as
+D1 =
+�dBcc
+du
+�
+u=0
+=
+�dEl
+du
+�
+u=0
+×
+�dBcc
+dEl
+�
+El=Ea
++
+�dxm
+du
+�
+u=0
+×
+�dBcc
+dxm
+�
+xm=xma
+.
+(B2)
+Since El = Ea/(1 + u), (dEl/du)u=0 = −Ea. Similarly,
+since xm = xma/(1 + u), (dxm/du)u=0 = −xma. Thus,
+D1 = −Ea(dBcc/dEl)El=Ea − xma(dBcc/dxm)xm=xma
+(B3)
+This can be further expressed as
+D1 = Bcc(Ea) − BFNϕ3/2
+m
+Ea
+�
+Ea
+dνcm
+dEl
+���
+Ea + xma
+dνcm
+dxm
+���
+xma
+�
++ gϕ1/2
+m (EF − Em)
+Ea
+�
+Ea
+dtcm
+dEl
+���
+Ea + xma
+dtcm
+dxm
+���
+xma
+�
+.
+(B4)
+It is simpler to express dνcm/dEl and dtcm/dEl as
+dνcm
+dEl
+���
+Ea = dνcm
+d(y2)
+���
+yma
+d(y2)
+dEl
+���
+Ea
+(B5)
+dtcm
+dEl
+���
+Ea = dtcm
+d(y2)
+���
+yma
+d(y2)
+dEl
+���
+Ea
+(B6)
+and use the fact that d(y2)/dEl = 4qB/ϕ2
+m. The quanti-
+ties dνcm
+dxm
+���
+xma and dtcm
+dEl
+���
+xma can be obtained directly and
+expressed as
+dνcm
+dxm
+���
+xma
+= w1(yma) + 2xmaw2(yma) + 3x2
+maw3(yma)
+dtcm
+dxm
+���
+xma = t1(yma) + 2xmat2(yma) + 3x2
+mat3(yma).
+These results can be combined to obtain
+D1 = Bcc(Ea)
+− BFN
+4qB
+ϕ1/2
+m
+dνcm
+d(y2)
+���
+y2ma
++ 4Bqg(EF − Em)
+ϕ3/2
+m
+dtcm
+d(y2)
+���
+y2ma
+− BFNϕ3/2
+m
+Ea
+xma
+dνcm
+dxm
+���
+xma
++ (EF − Em)
+dma
+xma
+dtcm
+dxm
+���
+xma
+(B7)
+where y2
+ma = 4qBEa/ϕ2
+m. Finally,
+� dνcm
+d(y2)
+�
+y2=y2ma
+=
+u0(yma) + xmau1(yma) + x2
+mau2(yma) + x3
+mau3(yma)
+(B8)
+with
+u0(yma) = −1 + 1
+6(1 + ln y2
+ma) + 1
+6
+(B9)
+u1(yma) = − 2
+11 + 2
+80y2
+ma +
+1
+400(1 + ln y2
+ma) (B10)
+u2(yma) = 2
+11 + 2
+3y2
+ma +
+1
+400(1 + ln y2
+ma)
+(B11)
+u3(yma) = 39
+20 + 2
+3y2
+ma −
+1
+300(1 + ln y2
+ma).
+(B12)
+
+11
+Similarly,
+� dtcm
+d(y2)
+�
+y2=y2ma
+=
+p0(yma) + xmap1(yma) + x2
+map2(yma) + x3
+map3(yma)
+(B13)
+with
+p0(y) = 1
+9 − 1
+18(1 + ln y2) + 1
+6
+(B14)
+p1(y) = − 237
+1100 −
+1
+240y2 −
+7
+2400(1 + ln y2) (B15)
+p2(y) = 589
+3300 + 1
+9y2 +
+1
+400(1 + ln y2)
+(B16)
+p3(y) = 2929
+900 + 2
+9y2 − 1
+45(1 + ln y2).
+(B17)
+This completes the evaluation of D1 in Eq. (B7).
+The coefficients F0 and F1 are defined as
+1
+t2cm(El, xm) =
+1
+t2cm( Ea
+1+u, xma
+1+u) ≈ F0 + F1u
+(B18)
+so that F0 = 1/t2
+cm(Ea, xma). The coefficient F1 is
+F1 = dt−2
+cm
+du
+���
+u=0 =
+2
+t3cm
+�
+Ea
+dtcm
+dEl
+���
+Ea + xma
+dtcm
+dxm
+���
+xma
+�
+(B19)
+which can be finally expressed as
+F1 =
+2
+t3cm
+�4qBEa
+ϕ2m
+dtcm
+d(y2)
+���
+y2ma
++ xma
+dtcm
+dxm
+���
+xma
+�
+.
+(B20)
+Appendix C: The ‘exact’ mean normal energy
+The exact mean normal energy can be determined
+starting with the joint distribution f(EN, ˜θ) or equiva-
+lently f(EN, ρ)13. In terms of ρ, it can be expressed as
+⟨EN⟩ = S1/S2 where
+S1 =
+� �
+dρdEN ρ
+�
+1 + ρ2/R2a(EF − EN)ENT(EN, ρ)
+S2 =
+� �
+dρdEN ρ
+�
+1 + ρ2/R2a(EF − EN)T(EN, ρ).
+In the above T(EN, ρ) ≈ 1/(1+eG(EN,ρ)) is the transmis-
+sion coefficient for an electron having normal energy EN
+at a point ρ on the emitter-tip z ≈ h − ρ2/(2Ra), having
+a local field El = Ea(z/h)/
+�
+z2/h2 + ρ2/R2a. It can be
+determined using Eqns. (5) - (10) for the Gamow factor
+G.
+
diff --git a/s9FJT4oBgHgl3EQfcyxN/content/tmp_files/load_file.txt b/s9FJT4oBgHgl3EQfcyxN/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..8ed699aad9de4a66394ba437180650e7de5b4d07
--- /dev/null
+++ b/s9FJT4oBgHgl3EQfcyxN/content/tmp_files/load_file.txt
@@ -0,0 +1,627 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf,len=626
+page_content='Fast and accurate determination of the curvature-corrected field emission  current  Debabrata Biswas1, 2, a) and Rajasree Ramachandran1, 2  1)Bhabha Atomic Research Centre, Mumbai 400 085, INDIA  2)Homi Bhabha National Institute, Mumbai 400 094, INDIA  The curvature-corrected field emission current density, obtained by linearizing at or below the Fermi energy,  is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Two special cases, corresponding to the peak of the normal energy distribution and the mean  normal energy, are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' It is found that the current density evaluated using the mean normal energy  results in errors in the net emission current below 3% for apex radius of curvature, Ra ≥ 5nm and for apex  fields Ea in the range 3 − 10 V/nm for an emitter having work-function φ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='5eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' An analytical expression  for the net field emission current is also obtained for locally parabolic tips using the generalized cosine law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The errors are found to be below 6% for Ra ≥ 5nm over an identical range of apex field strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The  benchmark current is obtained by numerically integrating the current density over the emitter surface and  the current density itself computed by integrating over the energy states using the exact Gamow factor and  the Kemble form for the WKB transmission coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The analytical expression results in a remarkable  speed-up in the computation of the net emission current and is especially useful for large area field emitters  having tens of thousands of emission sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  INTRODUCTION  Recent studies have shown that field emitters with tip  radius in the nanometer range can be best modelled ac-  curately by taking into account the variation in local field  in the tunneling region, which is roughly 1-2nm from  the emitter surface depending on the field strength1–5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  When the apex radius of curvature (Ra) of the emitter  is large (Ra > 100nm), the local field is roughly constant  in this region even though the field enhancement factor  itself may be large3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Thus, the Murphy-Good current  density6–14 is quite likely adequate3 for Ra > 100nm  while for emitters with Ra < 100nm, errors first start  building up at smaller field strengths and for Ra ≤ 10nm,  the errors become large over a wide range of fields3,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The necessity for curvature-corrections was illustrated  recently1 using the experimental results for a single  Molybdenum emitter tip2 with a FESEM-estimated end-  cap apex radius of curvature in the 5-10nm range with  the square-shaped pyramidal base having a side-length  Lb ∈ [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='25, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='35]µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Interestingly, even on using the  Fowler-Nordheim16 current density that ignores image-  charge contribution and seriously under-predicts the cur-  rent density, the fit was good1,17 but required an emission  area of 130000nm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' In contrast, the area of a hemisphere  of radius 10nm is only about 628nm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  On the other  hand1, the Murphy-Good current density (that takes  into account image-charge contribution to the tunneling  potential15), used with the generalized cosine law18,21 of  local field variation around the emitter tip, had a best  fit to the experimental data with Ra = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='79nm which is  within the estimated range of Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' However, the value  of field enhancement required the base-length Lb to be  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='65µm which is clearly outside the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='25− 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='35µm range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  a)Electronic mail: dbiswas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='hbni@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='com  Thus, while the Fowler-Nordheim current density has  gross non-conformity with the physical dimensions, the  Murphy-Good current density seems to be in need of a  correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Indeed, on using a curvature-corrected (CC)  expression for emission current3, the best fit to experi-  mental data required1 Ra ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='41nm and Lb ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='275µm,  both of which are within the range of their respective  estimated values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' This one-off validation could be a co-  incidence and more such experiments, observation and  data analysis are required to explore and put on a firm  footing, the limits of validity of each model19,20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The evidence so far seems to suggest that a curvature-  corrected field emission theory is necessary for nano-  tipped emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' An elementary form of this3 was used  in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' [1], based on a tunneling potential having a single  correction term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Since then, an approximately univer-  sal tunneling potential having an additional curvature  correction term has been established22 using the nonlin-  ear line charge model22–26 and tested against the finite-  element software COMSOL4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' A curvature-corrected ana-  lytical current density has also been determined5 by suit-  ably algebraic approximation of the exact Gamow factor  and its linearization at the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' While the re-  sults are promising, there is a scope for improving its  accuracy by choosing a different linearization-energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' It  is also desirable to have an analytical expression for the  net field emission current applicable for Ra ≥ 5nm over  a wide range of fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The present communication seeks  to establish accurate analytical expressions for both, the  curvature-corrected local current density, as well as the  net emission current for smooth locally parabolic emit-  ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The issue of accuracy in analytical expressions for cur-  rent density has recently been investigated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' [14]  for emitters where curvature corrections are unimportant  (Ra > 100nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The three major factors investigated  were: (a) the form in which the Gamow factor, G, is  cast (b) the use of e−G to determine the transmission  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='11545v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='app-ph]  27 Jan 2023  2  coefficient and (c) the energy at which the Gamow factor  should be linearized in order to obtain an approximate  analytical form for the current density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' It was found14  that if an analytical form of the current density is used  to determine the net emission current, only the second  and third factors are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' For instance, the use  of e−G to determine the transmission coefficient leads to  errors at larger local fields where the tunneling barrier  transitions from ‘strong’ to ‘weak’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' A better way of de-  termining the transmission coefficient within the WKB  approximation is the Kemble28,29 formula (1 + eG)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Another significant cause of error can be ascribed to the  energy at which the Gamow factor is linearized in order  to obtain an approximate analytical form for the current  density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' In the traditional approach to cold field emis-  sion, the Gamow factor is linearized at the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  While this holds at smaller values of the local field, it  leads to large errors at higher fields due to the shift in the  normal energy distribution away from the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  In the following, we shall continue to use the traditional  representation of the Gamow factor in term of the Forbes  approximation7 for the WKB integral, and add curvature  corrections to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  In section II, we shall make use of a curvature-  corrected current density that makes use of a Kemble  correction and a shifted point of linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' We shall  compare the results by choosing the energy correspond-  ing to the peak of the normal energy distribution as well  as the mean normal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' While both results are en-  couraging, the mean normal energy is more accurate es-  pecially at lower field strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Finally, an evaluation  of the net field-emission current is carried out using the  generalized cosine law in section III and compared with  the exact WKB result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Summary and discussions form  the concluding section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  AN ACCURATE CURVATURE-CORRECTED  CURRENT DENSITY  A widely adopted method to obtain an analytical ex-  pression for the current density is to Taylor expand the  Gamow factor about the Fermi energy EF in order to  carry out the energy integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Recent studies14 show  that this is adequate at smaller local field strengths  for which the electrons closer to the Fermi energy pre-  dominantly tunnel through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  As the field strength in-  creases, the height and width of the tunneling barrier  decreases and the electrons well below the Fermi energy  start contributing to the net emitted current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  This is  evident from the shift in the peak of the normal energy  distribution13 of the emitted electrons as the local field  increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Hence, for cold field emission, an expansion of  the Gamow factor around the peak of the normal energy  distribution or the mean normal energy seems preferable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  This is likely to yield a better approximation for field  emission current density applicable over a wide range of  fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The use of e−G is also a factor that contributes to the  errors at higher fields where the barrier becomes weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The transmission coefficient in the Kemble form28 can be  approximated as14  T(E) =  1  1 + eG ≈ e−G �  1 − e−G�  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  (1)  Used alongside the linearization of the Gamow factor,  this is likely to provide a simple yet reasonably accurate  expression for the field emission current density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Expansion of the Gamow factor and the curvature  corrected current density  The Gamow factor is expressed as  G = g  � s2  s1  �  VT (s) − Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  (2)  Here, g = 2  √  2m/ℏ, m the mass of the electron and ℏ the  reduced Planck’s constant h/(2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (2), VT is the  tunneling potential energy, E is the normal component of  electron energy at the emission surface and s1, s2 are the  zeroes of the integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The curvature-corrected form of  the tunneling potential energy is4,30  VT (s) ≈ EF + φ + Vext(s) −  B  s(1 + s/2R)  (3)  where, φ is the work function, EF the Fermi energy while  the external potential energy Vext takes the form,  Vext(s) ≈ −qEls  �  1 − s  R + 4  3  � s  R  �2�  (4)  with q  the magnitude of electronic charge,  B  =  q2/(16πϵ0), El the local electric field, and s denoting the  normal distance from the surface of the emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The  quantity R−1 is the mean curvature31,32 so that R is  the harmonic mean of the principle radii of curvature  R1 and R2 at the emission site i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' R = 2/(R−1  1  + R−1  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The curvature-corrected external potential of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 4 fol-  lows directly from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (35) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' [4] which holds in the  region close to the apex for all axially symmetric emit-  ters in a parallel plate diode configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' For a more  detailed exposition, the reader may refer to appendix A  on the tunneling potential33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Using the curvature-corrected tunneling potential en-  ergy of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (3), an approximate form for the Gamow  factor can be found numerically to be5  G = 2  3g ϕ3/2  qEl  [ν(y) + xw1(y) + x2w2(y) + x3w3(y)] (5)  = 2  3g ϕ3/2  qEl  νc(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  (6)  3  Here, ϕ = EF +φ−E, x = ϕ/(qElR), y = 2√qBEl/ϕ and  the curvature-corrected barrier function νc(y) = ν(y) +  xw1(y) + x2w2(y) + x3w3(y), where4  ν(y) = 1 − y2 + 1  3y2 ln y  (7)  w1(y) = 10  13 − 2  11y2 + 1  80y4 +  1  200y2 ln y  (8)  w2(y) = 10  11 + 2  11y2 − 1  6y4 +  1  200y2 ln y  (9)  w3(y) = −41  10 + 39  20y2 + 1  3y4 −  1  150y2 ln y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  (10)  Note that xw1(y), x2w2(y), x3w3(y) are the curvature  corrections that arise due to R dependent terms in the  external as well as image charge potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' As R → ∞  in the planar limit, x → 0, so that νc(y) reduces to ν(y)  which corresponds to the use of the Schottky-Nordheim  barrier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  We shall hereafter denote the linearization energy by  Em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  On expansion of the curvature-corrected Gamow  factor and retaining the linear term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' we obtain  G(E) ≈ G(Em) − (E − Em)tcm  dm  (11)  where tcm = tc(Em) with  tcm = t(ym) + xmt1(ym) + x2  mt2(ym) + x3  mt3(ym)  =  �  1 + y2  m  9 − 1  9y2  m ln ym  �  +  xm  �25  13 − 237  1100y2  m −  1  480y4  m −  7  1200y2  m ln ym  �  +  x2  m  �70  33 + 589  3300y2  m + 1  18y4  m +  1  200y2  m ln ym  �  +  x3  m  �  −123  10 + 2929  900 y2  m + 1  9y4  m − 1  90y2  m ln ym  �  (12)  where  ym  =  cs  √El/ϕm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  d−1  m  =  g ϕ1/2  m  El ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  cs  =  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='199985 eV (V/nm)−1/2, ϕm = EF + φ − Em and  xm = ϕm/(qElR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The Gamow factor at Em can be  expressed as  G(Em) = BFNϕ3/2  m  νcm  El  (13)  where νcm = ν(ym)+xmw1(ym)+x2  mw2(ym)+x3  mw3(ym).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The field emission current density  J =  2mq  (2π)2ℏ3  � EF  0  (EF − E)  1  1 + eG(E) dE  (14)  ≈  2mq  (2π)2ℏ3  � EF  0  (EF − E)e−G(E) �  1 − e−G(E) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  �  dE  can be expressed on completing the integration over en-  ergy states as  Jm  cc ≈ AFN  1  ϕm  E2  l  t2cm  e−Bcc  �  1 − e−Bcc  4  �  (15)  Bcc = BFNϕ3/2  m  νcm  El  − tcm  dm  (EF − Em)  (16)  where AFN  ≃  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='541434 µA eV V−2,  BFN  ≃  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='830890 eV−3/2 V nm−1 are the usual Fowler-Nordheim  constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The curvature-corrected current density Jm  cc  (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (15)), with the incorporation of the first Kemble cor-  rection and linearization of the Gamow factor at Em pro-  vides an analytical expression that can be used to evalu-  ate the net field emission current from a curved emitter,  either by numerically integrating over the surface or by  using the local field variation over the emitter surface to  obtain an approximate analytical expression for the net  field emission current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Numerical verification  The exact WKB result (referred to hereafter as the  benchmark) obtained by (i) finding the Gamow factor  exactly by numerical integration (ii) use of the Kemble  form of transmission coefficient and (iii) numerical inte-  gration over energy to obtain the current density, can be  used to validate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Since we shall be comparing  net emission currents rather than current-densities, the  local current density is integrated over the surface near  the apex to obtain the net current numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The geometrical entity we are focusing on is an axially-  symmetric emitter having an apex radius of curvature Ra  and height h = 300Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' It is mounted on a parallel plate  diode where the generalized cosine law18,21 of local field  variation holds:  El = Ea  z/h  �  (z/h)2 + (ρ/Ra)2 = Ea cos ˜θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  (17)  In the above, h is the height of the emitter, Ra is the apex  radius of curvature and Ea the apex field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (17) holds  for all axially symmetric emitters where the tips are lo-  cally approximated well by a parabola z ≈ h − ρ2/(2Ra)  upto ρ ≈ Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Thus the only parameters required are  h, Ra and the apex field34–37 Ea, since the generalized  cosine law18,21 for local fields holds for such emitter-tips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Note that the benchmark also uses the parabolic approx-  imation and the generalized cosine law for determining  the net emission current38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  In the following, we shall  consider EF = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='5eV and φ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='5eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The apex fields con-  sidered are in the range [3,10] V/nm which correspond  to scaled barrier fields39 Ea/Eφ in the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='21333 -  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='71109 where Eφ = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='6944617 eV−2Vnm−1)φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  4  0  3  6  9  12  15  18  21  3  4  5  6  7  8  9  10  Relative Error (%)  Ea (V/nm)  Mean NE, Ra = 5nm  Peak NED, Ra = 5nm  εF - 2 dF/tF, Ra = 5nm  εF - dF/tF, Ra = 5nm  εF, Ra = 5nm  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The absolute relative error in the net emission current  with respect to the exact WKB result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Five cases are shown  with various linearization energy Em.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' ‘Mean NE’ refers to the  exact mean normal energy, ‘Peak NED’ refers to the exact  energy at which the normal energy distribution peaks, ‘EF ’  refers to Em = EF , ‘EF − 2dF /tF ’ is the approximate mean  normal energy, while ‘EF −dF /tF ’ is the approximate peak of  the normal energy distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  0  2  4  6  8  3  4  5  6  7  8  9  10  Relative Error (%)  Ea (V/nm)  Mean NE, Ra = 50nm  Peak NED, Ra = 50nm  εF - 2 dF/tF, Ra = 50nm  εF - dF/tF, Ra = 50nm  εF, Ra = 50nm  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' As in case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 1 with Ra = 50nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Note that the  error for Em = EF increases at higher values of Ea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  It is clear that there are severals factors at play when  comparing the error with respect to the exact WKB re-  sult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' We shall discuss two of these from the broad pic-  ture available to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The first is the effect of curva-  ture correction which reflects in the approximate Gamow  factor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Since the expansion is in powers of  x = ϕ/(qElR), the approximate Gamow factor is prone  to errors at smaller values of El and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Thus, irrespec-  tive of the energy at which the linearization is carried  out, lower fields and radius of curvature are prone to er-  rors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' In general, at higher R and El, the curvature errors  are expected to reduce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The second important consider-  ation is the energy at which the linearization is carried  out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Since the peak of the normal energy distribution  moves away from EF at higher fields for a given Ra, lin-  earization at Em = EF should in general lead to larger  errors at higher fields strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Apart from these two,  there are other subtle effects that decide the magnitude  of relative error at a given field strength as we shall see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Note that on the surface of an emitter, El reduces away  from the apex while R increases and this leads to a mild  decrease in the expansion parameter x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  0  1  2  3  4  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='6  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='7  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='8  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='9  8  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='1  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='2  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='3  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='4  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='5  Normal energy distribution  εn (eV)  Ra = 5nm  Ra = 20nm  Ra = 50nm  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The normal energy distribution at Ea = 3V/nm for  Ra = 5, 20 and 50nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Note the shift in the distribution away  from the EF (= 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='5eV here) for larger values of Ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  With this perspective, we shall compare the absolute  relative errors at Ra = 5nm and Ra = 50nm shown in  Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 1 and 2 respectively, for various values of Em dis-  played in the legends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Clearly ‘Mean NE’, which refers  to the exact mean normal energy determined numerically  (see appendix C), performs well at Ra = 5nm at all field  strengths while Em = EF shows large errors especially at  lower fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Even at Ra = 50nm where curvature er-  rors are expected to be smaller, ‘Mean NE’ as well as the  approximate mean normal energy (Em ≈ EF − 2dF /tF )  perform well while in case of E = EF , the linearization  error dominates leading to larger errors at higher field  strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The energy value corresponding to the peak of  the normal energy distribution (‘Peak NED’) also gives  good results though the errors are somewhat high for  smaller apex fields at Ra = 5nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Some of the trends in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 1 and 2 are easy to un-  derstand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' For instance, at Ra = 5nm, the errors fall as  expected with an increase in Ea in all cases (except for a  mild increase at Em = EF for Ea > 9V/nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The larger  than expected error (approximately 21%) for Em = EF at  Ea = 3V/nm however seems intriguing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' To understand  this better, the normal energy distribution (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 3)  for different values of Ra at Ea = 3V/nm is quiet in-  structive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The peak of the normal energy distribution  5  16  18  20  22  24  26  28  30  32  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='6  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='7  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='8  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='9  8  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='1  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='2  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='3  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='4  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='5  Gamow factor  εn (eV)  Ra = 5nm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Exact  Ra = 5nm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' εm = εF  Ra = 5nm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' εm = εF - 2dF/tF  Ra = 50nm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Exact  Ra = 50nm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' εm = εF  Ra = 5nm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' εm = εF - 2dF/tF  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The exact Gamow factor at Ea = 3V/nm is com-  pared with the linearized Gamow factors with the point of  linearization at Em = EF and Em = EF − 2dF /tF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The upper  set of curves correspond to Ra = 5nm while the lower set is  for Ra = 50nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The two linearized versions are nearly equiv-  alent at Ra = 50nm while, for Ra = 5nm linearizing at the  approximate mean energy yields results closer to the exact  Gamow factor over the energy range of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  shifts slightly away from EF as Ra increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Note also  that the distributions have a long tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The linearized  Gamow factor in the corresponding normal energy range  is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' For Ra = 5nm, linearization at EF  results in larger deviations from the exact Gamow fac-  tor compared to linearization at EF − 2dF /tF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Not sur-  prisingly, the relative error in net emission current drops  from about 21% to about 13% in moving from Em = EF  to Em = EF − 2dF /tF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  At Ra = 50nm, curvature effects are smaller and the  linearized Gamow factor does not noticeably deviate from  the exact Gamow factor (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 4 for Ea = 3V/nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Thus, the errors remain more or less similar at all lin-  earization energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The magnitude of the error at a par-  ticular Ea depends on how closely the linearized Gamow  factor approximates the exact Gamow factor over the  relevant range of normal energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' For Ea > 5V/nm at  Em = EF , the increase in error is expected due to the  shift in normal energy distribution away from EF and the  corresponding deviation of the linearized Gamow factor  from the exact Gamow factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  In order to verify that the trend observed in moving  from Ra = 5nm to Ra = 50nm is gradual, we show the  results for Ra = 10nm and Ra = 20nm in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  It is apparent from these results that linearization at the  exact mean normal energy (‘Mean NE’) is optimum for  all values of Ra and Ea with errors generally below 3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The approximate mean normal energy Em ≈ EF −2dF /tF  is only marginally worse with errors exceeding 6% only  at Ra = 5nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  0  2  4  6  8  10  12  3  4  5  6  7  8  9  10  Relative Error (%)  Ea (V/nm)  Mean NE, Ra = 10nm  Peak NED, Ra = 10nm  εF - 2 dF/tF, Ra = 10nm  εF - dF/tF, Ra = 10nm  εF, Ra = 10nm  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' As in case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 1 with Ra = 10nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  0  1  2  3  4  5  6  7  8  3  4  5  6  7  8  9  10  Relative Error (%)  Ea (V/nm)  Mean NE, Ra = 20nm  Peak NED, Ra = 20nm  εF - 2 dF/tF, Ra = 20nm  εF - dF/tF, Ra = 20nm  εF, Ra = 20nm  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' As in case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 1 with Ra = 20nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  THE NET CURVATURE-CORRECTED EMISSION  CURRENT  The curvature-corrected expression for the current  density, with linearization at the mean normal energy,  can be used to arrive at an analytical expression for the  net emission current on using the generalized cosine law  of local field variation El = Ea cos ˜θ (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (17)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Assum-  ing a sharp locally parabolic emitter tip, the total emitted  current can be evaluated using the expression13  I ≈ 2πR2  a  �  Jcc(˜θ) sin ˜θ  cos4 ˜θ  × C(˜θ) d˜θ  (18)  where C(˜θ) is a correction factor which, for a sharp emit-  ter (h/Ra >> 1), is approximately unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' In the follow-  ing we shall assume the emitter to be reasonably sharp  so that C ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The basic idea is to express Jcc in terms of ˜θ by  6  replacing all the local fields using El = Ea cos ˜θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  A  further simplification can be made by the substitution  1/ cos ˜θ = 1 + u and retaining only terms upto O(u2) in  Bcc and t−2  cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The approximation is expected to be good  at lower apex fields since the emission is limited to an  area closer to the apex, while at higher fields, where the  emission area is larger, this might lead to larger errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Writing Bcc ≈ D0 + D1u and t−2  cm ≈ F0 + F1u, the  integration can be carried out easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Note that, it gener-  ally suffices to integrate upto ρ = Ra which, for a sharp  emitter, corresponds to ˜θ = π/4 or u =  √  2 − 1 = u0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Thus,  I ≈ 2πR2  aGAFN  1  ϕma  E2  aF0e−D0  (19)  where Em is the mean normal energy, while  G ≈ 1  D1  + F1  F0  1  D2  1  − e−D0  4  � 1  2D1  + F1  F0  1  4D2  1  �  −  e−D1u0  � 1  D1  + F1  F0  1 + D1u0  D2  1  − e−D0−D1u0  4  ×  � 1  2D1  + F1  F0  1 + 2D1u0  4D2  1  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  (20)  Expressions for D0, D1, F0 and F1 can be found in ap-  pendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  0  1  2  3  4  5  6  3  4  5  6  7  8  9  10  εm = exact mean NE  Relative Error (%)  Ea (V/nm)  Ra = 5nm  Ra = 10nm  Ra = 20nm  Ra = 50nm  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The magnitude of the relative error in the analytical  expression for the curvature-corrected current (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 19) com-  pared to the exact WKB result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Here Em is the exact mean  normal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 7, we compare the magnitude of the relative  error in the net current as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (19) and (20)  with respect to the exact WKB result which has been  used as the benchmark throughout this study with Em  as the exact mean normal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Clearly, the analytical  expression is adequate for a wide range of fields and apex  0  3  6  9  12  15  18  21  3  4  5  6  7  8  9  10  εm = εF  Relative Error (%)  Ea (V/nm)  Ra = 5nm  Ra = 10nm  Ra = 20nm  Ra = 50nm  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The magnitude of the relative error in the analyt-  ical expression for the curvature-corrected current (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 19)  compared to the exact WKB result for Em = EF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  radius of curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The increase in error at higher fields  is due to the linearization of Bcc and t−2  cm in the variable  u which is a measure of the distance from the apex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' This  is however a small price to pay for a compact analytical  expression for the net emission current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  For the sake of comparison, we also show the relative  errors in the net current obtained using the analytical  expressions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (19) and (20) with Em = EF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The  trends are similar to those shown in section II B where the  linearized current density is integrated numerically over  the emitter end-cap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The errors are more pronounced at  smaller apex radius of curvature and apex field strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Clearly, linearization at the mean normal energy ensures  smaller errors over a wide range of fields and radius of  curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  0  2  4  6  8  10  12  14  16  3  4  5  6  7  8  9  10  εm = εF - 2dF/tF  Relative Error (%)  Ea (V/nm)  Ra = 5nm  Ra = 10nm  Ra = 20nm  Ra = 50nm  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The magnitude of the relative error in the analytical  expression for the curvature-corrected current (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 19) com-  pared to the exact WKB result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Here, Em = EF − 2dF /tF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  7  While the errors are reasonably small when the exact  mean normal energy is used, it require the computation of  integrals that marginally offsets the use of an analytical  expression for the net current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 9, we provide  a comparison of the magnitude of relative errors with  respect to the exact WKB result, using the approximate  Em = EF − 2dF /tF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  While the errors for Ra = 5nm  are somewhat large, the approximate value of the mean  normal energy may be used profitably for Ra ≥ 10nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Comparison of time required for 10000 evaluations  of the net emission current for Ea ∈ [3, 10]V/nm and Ra ∈  [5, 50]nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The ‘Scale Factor’ is the ratio of the time taken by  ‘WKB Exact’ and the time taken by a given method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' It gives  a rough indication of the speed-up achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Here Em is the  mean normal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Also shown is the average relative error  with respect to ‘WKB exact’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Method  Time (s) Scale Factor Average Error  WKB Exact  315.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='1 s  1  —  WKB Fit  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='8 s  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='24  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='67%  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (19) with  56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='7 s  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='55  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='06%  Em exact  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (19) with  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='0003 s  106  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='14%  Em approximate  Finally, Table I provides a comparison of the CPU  time (in seconds) required on a standard desktop to se-  rially compute the net emission current for 104 combina-  tions of Ea and Ra in the range of apex fields and radius  of curvature considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Thus, there are  100 values of Ra spaced uniformly in the range [5,50]nm  and 100 values of Ea spaced uniformly in the [3,10]V/nm  range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' In the table, ‘WKB Fit’ refers to the use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (5)  for the Gamow factor and numerical integration over en-  ergy while ‘WKB exact’ refers to the ‘exact’ numerical  evaluation of the Gamow factor followed by numerical  integration over energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The last two rows refer to the  analytical formula for the net current of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (19) with ‘Em  exact’ evaluated as outlined in appendix C and ‘Em ap-  proximate’ as Em ≈ EF − 2dF /tF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Clearly, linearization  at the approximate mean normal energy results in fast  computation of the net emission current using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (19)  by a factor ≈ 80000 compared to ‘WKB fit’ and about  106 compared to the ‘WKB exact’ result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' This is only  marginally offset by a larger error for Ra = 5nm as seen  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The average relative error in the 5 − 50nm  range is however small as shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  CONCLUSIONS  We have presented an expression for the curvature-  corrected current density obtained by linearization at an  energy Em ≤ EF and insertion of a correction term to ac-  count for the Kemble transmission coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Numerical  results show that the mean normal energy is a suitable  candidate for the linearization energy Em and predicts the  net emission current to within 3% accuracy compared to  the exact WKB result for Ra ≥ 5nm and over a wide  range of field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  We have also obtained an analytical expression for the  net emission current using the generalized cosine law of  local field variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' It requires only the apex radius of  curvature Ra and the apex electric field Ea and is able  to calculate the net field-emission current to within 6%  accuracy compared to the current obtained by explic-  itly integrating the exact WKB current density over the  emitter tip for Ra ≥ 5nm and a wide range of apex fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Both of these results are expected to be useful in deal-  ing with sharp emitters having tip radius Ra ≥ 5nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The expression for current density can be used in all sit-  uations including those where the emitter does not have  any special symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' On the other hand, the expres-  sion for the net emission current is extremely useful for  axially symmetric emitters with smooth locally parabolic  tips mounted in a parallel plate configuration, consider-  ing that the speed-up achieved in current computation  is enormous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The accuracies obtained in all cases are  good, given that even minor experimental uncertainties  can lead to far larger changes in the net emission current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Finally, the analytical expression for emission current  is especially useful for a fast determination of net emis-  sion current from a large area field emitter have thou-  sands of axially symmetric emitters40–42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  AUTHOR DECLARATIONS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Conflict of interest  There is no conflict of interest to disclose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Data Availability  The data that supports the findings of this study are  available within the article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Author Contributions  Debabrata Biswas Conceptualization (lead), data  curation (equal), formal analysis (equal), methodology  (lead), software (equal), validation (supporting), visual-  ization (equal), original draft (lead), review and editing  (supporting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Rajasree Ramachandran Conceptualization (sup-  porting), data curation (equal), formal analysis (equal),  methodology (supporting), software (equal), validation  (lead), visualization (equal), original draft (supporting),  review and editing (lead).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  8  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  REFERENCE  1D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Biswas and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Kumar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Vac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' B 37, 040603  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Tsujino and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Dwayne Miller, Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  113, 013505 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Biswas and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Ramachandran, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Vac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' B 37,  021801 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  4R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Ramachandran and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Biswas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 129, 184301  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  5D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Biswas and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Ramachandran, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 129, 194303  (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  6E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Murphy and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Good, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 102, 1464 (1956).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  7R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Forbes, App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 89, 113122 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  8R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Forbes and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Deane, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' A 463, 2907  (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  9J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Deane and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Forbes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' A: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 41,  395301 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  10K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Jensen, Introduction to the physics of electron emission,  Chichester, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=', Wiley, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  11K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Jensen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 126, 065302 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  12R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Forbes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 126, 210901 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  13D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Biswas, Physics of Plasmas 25, 043105 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  14D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Biswas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 131, 154301 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  15L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Nordheim, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' London, Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' A 121, 626 (1928).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  16R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Fowler and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Nordheim, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' A 119,  173 (1928).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  17The emission area was assumed to be independent of the local  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  18D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Biswas, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Singh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Sarkar and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Kumar, Ultrami-  croscopy 185, 1 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  19For a recent analysis on a single emitter not requiring curvature  correction, see [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  20E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Popov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Filippov and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Kolosko, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Vac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' B 41, 012801 (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  21D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Biswas, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Singh and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Ramachandran, Physica E 109, 179  (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  22D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Biswas, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Singh and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Kumar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 120, 124307  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  23E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Mesa, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Dubado-Fuentes, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Saenz, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 79,  39 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  24E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Pogorelov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Zhbanov, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Chang, Ultrami-  croscopy 109, 373 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  25J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Harris, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Jensen, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Shiffler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' D 48,  385203(2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  26J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Harris, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Jensen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Tang, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Schiffler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Vac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' B 34, 041215 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  27K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Jensen, Journal of Applied Physics 111, 054916 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  28E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Kemble, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 48, 549 (1935).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  29R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Forbes, Journal of Applied Physics 103, 114911 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  30D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Biswas, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Ramachandran and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Singh, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Plasmas 25,  013113 (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' ibid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' 29, 129901 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  31In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' [32], an effective spherical approximation was used to  generate the potential with R−1 as the mean of the two principle  curvature R−1  1  and R−1  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The authors state32 “This approach re-  turns a high precision result comparable to the approach reported  by Biswas and Ramachandran”, referring to the results in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' [4  and 5] that used the first and second corrections to the external  potential with R = R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' [4] the external potential was de-  rived for general axially symmetric emitters using the nonlinear  line charge model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  32J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Ludwick, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Cahay, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Hernandez, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Hall, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' O’Mara,  K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Jensen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Deane, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Forbes, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Back, Journal  of Applied Physics 130, 144302 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  33In previous publications4,30, R was approximated as R2 and the  results were found to be close to numerically determined external  potentials using COMSOL for various shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' It is shown in the  appendix using the results of Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' [4 and 30], that in the first cor-  rection s/R, R is the harmonic mean Rm = 2/(R−1  1  +R−1  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The  second correction 4s2/(3R2) with R = Rm is an approximation,  albeit a marginally improved one compared to the identification  R ≈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  34The local field at the emitter-apex, Ea, is related to the applied or  macroscopic field E0 through the apex field enhancement factor  γa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' See for instance Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' [35–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  35D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Biswas, Physics of Plasmas 25, 043113 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  36D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Biswas, Physics of Plasmas, 26, 073106 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  37T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' de Assis, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Dall’Agnol and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Forbes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Phys:  Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Matter 34, 493001 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  38At high fields, contributions beyond ρ ≈ Ra cannot be altogether  neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' While the validity of the parabolic approximation and  the cosine law (except in hemi-ellipsoids) start breaking down  for ρ > Ra, the curvature corrected current density of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (15)  continues to hold and can be used to determine the net emission  current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  39R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Forbes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Vac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' B26, 209 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  40D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Biswas and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Rudra, Physics of Plasmas 25, 083105 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  41D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Biswas and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Rudra, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Vac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' B38, 023207  (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  42D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Biswas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Vac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' B38, 063201 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Appendix A: The tunneling potential  The electric field, El, close to the emitter surface is  assumed to be a constant so that the corresponding po-  tential can be expressed as Vext(s) = −Els where s is the  normal distance from the surface of the emitter and El is  the magnitude of the local electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The assumption  holds good when the radius of curvature at the emission  site is large (typically R > 100nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  As R decreases, corrections become important and  these can be expressed as powers of s/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Thus,  Vext(s) = −Els  �  1 + c1  s  R + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' + cn  � s  R  �n  + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  (A1)  The  effective  spherical  approximation  used  in  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' [32] leads to cn  =  (−1)n so that Vext(s)  =  −ElR [1 − 1/(1 + s/R)] with R−1 = 2/(R−1  1  + R−1  2 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' [30], following the analysis of the hemiellipsoid,  the hyperboloid and the hemisphere, it was concluded  that c1 = −1, c2 = 4/3 and R ≈ R2 where R2 is the  second (smaller) principle radius of curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  With  these identifications, the external potential was found  to approximate the numerically determined external  potentials for other emitter shapes as well30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' [4] uses  the nonlinear line charge model22 for axially symmetric  emitters to arrive at an approximate form close to the  apex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' In the following, we shall show that the results  of both Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' [4 and 30] can be recast in the form where  {c1 = −1, R = Rm} exactly while {c2 = 4/3, R2 = R2  m}  is approximate but fairly accurate close to the apex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  In addition to the approximate results in section II30,  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' [30] also provides in the appendix, a systematic ex-  pansion of the external potential in powers of s for the  hemi-ellipsoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  In terms of the prolate spheroidal co-  ordinates (η, ξ, ϕ)  9  x = L  �  (η2 − 1)(1 − ξ2) cos ϕ  (A2)  y = L  �  (η2 − 1)(1 − ξ2) sin ϕ  (A3)  z = Lηξ  (A4)  it was found that  Vext(s) = V (s) =  �  d1s + d2s2 + d3s3�  (A5)  where  d1 = Vηa1  (A6)  d2 = Vηa2 + 1  2Vηηa2  1  (A7)  d3 = Vηa3 + Vηηa1a2 + Vξηa1b2 + 1  6Vηηηa3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  (A8)  The derivatives of the potential at a point (η0, ξ0) on the  surface of the hemiellipsoid are  Vη = Elhη  (A9)  Vηη = −Elhη 2η0  η2  0 − 1  (A10)  Vηξ = Elhη  1  ξ0  (A11)  Vηηη = Elhη  8η2  0  (η2  0 − 1)2  (A12)  where  hη = L  �  (η2  0 − ξ2  0)/(η2  0 − 1)  (A13)  hξ = L  �  (η2  0 − ξ2  0)/(1 − ξ2  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  (A14)  The coefficients  a1 = 1  hη  (A15)  a2 =  1  2h2  ξ  η0  η2  0 − ξ2  0  (A16)  a3 = −  1  2hηh2  ξ  η2  0 + ξ2  0  (η2  0 − ξ2  0)2  (A17)  b2 = − 1  2h2  ξ  ξ0  η2  0 − ξ2  0  (A18)  while the principle radii of curvature are  R1 = Ra  (η2  0 − ξ2  0)3/2  (η2  0 − 1)3/2  (A19)  R2 = Ra  (η2  0 − ξ2  0)1/2  (η2  0 − 1)1/2  (A20)  where Ra is the apex radius of curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' On putting  together these results, the values of d1, d2 and d3 are  d1 = −El  (A21)  d2 = El  2Ra  (η2  0 − 1)1/2  (η2  0 − ξ2  0)3/2  �  2η2  0 − 1 − ξ2  0  �  = El  Rm  (A22)  d3 = −4  3  El  R2m  �  3 + 4η4  0 + ξ4  0 − 2η2  0(3 + ξ2  0)  �  (1 + ξ2  0 − 2η2  0)2  (A23)  = −4  3  El  R2m  �  1 − 2(η2  0 − 1)(1 − ξ2  0)  (1 + ξ2  0 − 2η2  0)2  �  (A24)  = −4  3  El  R2m  [1 − C]  (A25)  where  Rm =  2  (1/R1 + 1/R2)  = 2Ra  η2  0 − ξ2  0)3/2  (η2  0 − 1)1/2(2η2  0 − 1 − ξ2  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  (A26)  The external potential thus takes the form  Vext(s) = −Els  �  1 −  � s  Rm  �  + 4  3  � s  Rm  �2  (1 − C)  �  (A27)  for the hemiellipsoid emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' In terms of ρ2  0 = x2  0 + y2  0  where x0, y0 are on the surface of the hemiellipsoid, the  correction term C = ρ2  0/(2R2  a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Thus,  Vext(s) = −Els  �  1 −  s  Rm  + 4  3  s2  R2m  �  1 − 1  2  ρ2  0  R2a  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  (A28)  Note that close to the apex, ρ/Ra << 1 while Rm ≈ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  A more general result, valid for all axially symmetric  emitters in a parallel plate geometry, was arrived at us-  ing the nonlinear line charge model4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' In such cases, the  external potential can be expressed as (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (35) of  Ref [4]),  Vext(s) ≈ −Els  �  1 − s  Ra  (1 − ρ2  0  R2a  ) + 4  3  s2  R2a  (1 − 5  2  ρ2  0  R2a  )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  (A29)  Close to the apex 1/Rm ≈ (1/Ra)(1 − ρ2  0/R2  a) while  1/R2  m ≈ (1/R2  a)(1 − 2ρ2  0/R2  a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Thus, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (A29) can be  expressed as  Vext(s) ≈ −Els  �  1 −  s  Rm  + 4  3  s2  R2m  (1 − 1  2  ρ2  0  R2a  )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  (A30)  10  This is identical to the result obtained for the hemiellip-  soid (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' A28) but applicable generally for all axially  symmetric emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Approximating (1 − ρ2  0/(2R2  a)) ≈ 1  leads us to an approximate universal form for the exter-  nal potential (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (4)) close to the emitter surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (A29) can also be expressed as  Vext(s) ≈ −Els  �  1 −  s  Rm  + 4  3  s2  R1Rm  �  (A31)  The correction terms,  s  Rm and 4  3  s2  R1Rm are exact for any  point on the hemiellipsoid surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  For other emitter  shapes4, the two correction terms may have extra fac-  tors that can be ascribed to the non-linear line charge  distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Since these have been ignored as an ap-  proximation, we choose to adopt the form in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (4) with  R = Rm as an approximate but accurate representation  of the external potential in the tunneling region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Finally, while the change from R2 to Rm reduces the  need for approximations, its impact on the net field emis-  sion current is small compared to a neglect of the second  correction term 4s2/(3R2  m), especially at smaller values  of Ra and Ea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' For instance, at Ra = 5nm and work-  function φ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='5eV, the error in net emission current on  using R2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (4) is about 13% at Ea = 5V/nm, while  it is around 62% on ignoring 4s2/(3R2  m) altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' At  Ea = 4V/nm, the error in net emission current on using  R2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (4) remains roughly the same while the error  grows to around 92% on ignoring 4s2/(3R2  m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' In each  of these cases, the exact WKB method is used and the  benchmark current is obtained using Rm in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  Appendix B: The coefficients D0, D1, F0 and F1  We shall briefly outline the derivation of the coeffi-  cients D0, D1, F0 and F1 and state the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The de-  pendence on u in Bcc and tcm arise from the variation in  El and xm over the surface of the emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Thus,  Bcc = Bcc(El, xm) = Bcc( Ea  1 + u, xma  1 + u) ≈ D0 + D1u  (B1)  so that D0  = Bcc(Ea, xma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  In the above, xm  =  ϕm/(qElRm) ≈ xma/(1 + u) where xma = ϕm/(qEaRa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The approximation holds for tall emitters where El =  Ea(z/h)/  �  (z/h)2 + (ρ/Ra)2 ≈ Ea/  �  1 + ρ2/R2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The  coefficient D1 can be written as  D1 =  �dBcc  du  �  u=0  =  �dEl  du  �  u=0  ×  �dBcc  dEl  �  El=Ea  +  �dxm  du  �  u=0  ×  �dBcc  dxm  �  xm=xma  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  (B2)  Since El = Ea/(1 + u), (dEl/du)u=0 = −Ea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Similarly,  since xm = xma/(1 + u), (dxm/du)u=0 = −xma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Thus,  D1 = −Ea(dBcc/dEl)El=Ea − xma(dBcc/dxm)xm=xma  (B3)  This can be further expressed as  D1 = Bcc(Ea) − BFNϕ3/2  m  Ea  �  Ea  dνcm  dEl  ���  Ea + xma  dνcm  dxm  ���  xma  �  + gϕ1/2  m (EF − Em)  Ea  �  Ea  dtcm  dEl  ���  Ea + xma  dtcm  dxm  ���  xma  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  (B4)  It is simpler to express dνcm/dEl and dtcm/dEl as  dνcm  dEl  ���  Ea = dνcm  d(y2)  ���  yma  d(y2)  dEl  ���  Ea  (B5)  dtcm  dEl  ���  Ea = dtcm  d(y2)  ���  yma  d(y2)  dEl  ���  Ea  (B6)  and use the fact that d(y2)/dEl = 4qB/ϕ2  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The quanti-  ties dνcm  dxm  ���  xma and dtcm  dEl  ���  xma can be obtained directly and  expressed as  dνcm  dxm  ���  xma  = w1(yma) + 2xmaw2(yma) + 3x2  maw3(yma)  dtcm  dxm  ���  xma = t1(yma) + 2xmat2(yma) + 3x2  mat3(yma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  These results can be combined to obtain  D1 = Bcc(Ea)  − BFN  4qB  ϕ1/2  m  dνcm  d(y2)  ���  y2ma  + 4Bqg(EF − Em)  ϕ3/2  m  dtcm  d(y2)  ���  y2ma  − BFNϕ3/2  m  Ea  xma  dνcm  dxm  ���  xma  + (EF − Em)  dma  xma  dtcm  dxm  ���  xma  (B7)  where y2  ma = 4qBEa/ϕ2  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' Finally,  � dνcm  d(y2)  �  y2=y2ma  =  u0(yma) + xmau1(yma) + x2  mau2(yma) + x3  mau3(yma)  (B8)  with  u0(yma) = −1 + 1  6(1 + ln y2  ma) + 1  6  (B9)  u1(yma) = − 2  11 + 2  80y2  ma +  1  400(1 + ln y2  ma) (B10)  u2(yma) = 2  11 + 2  3y2  ma +  1  400(1 + ln y2  ma)  (B11)  u3(yma) = 39  20 + 2  3y2  ma −  1  300(1 + ln y2  ma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  (B12)  11  Similarly,  � dtcm  d(y2)  �  y2=y2ma  =  p0(yma) + xmap1(yma) + x2  map2(yma) + x3  map3(yma)  (B13)  with  p0(y) = 1  9 − 1  18(1 + ln y2) + 1  6  (B14)  p1(y) = − 237  1100 −  1  240y2 −  7  2400(1 + ln y2) (B15)  p2(y) = 589  3300 + 1  9y2 +  1  400(1 + ln y2)  (B16)  p3(y) = 2929  900 + 2  9y2 − 1  45(1 + ln y2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  (B17)  This completes the evaluation of D1 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (B7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  The coefficients F0 and F1 are defined as  1  t2cm(El, xm) =  1  t2cm( Ea  1+u, xma  1+u) ≈ F0 + F1u  (B18)  so that F0 = 1/t2  cm(Ea, xma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' The coefficient F1 is  F1 = dt−2  cm  du  ���  u=0 =  2  t3cm  �  Ea  dtcm  dEl  ���  Ea + xma  dtcm  dxm  ���  xma  �  (B19)  which can be finally expressed as  F1 =  2  t3cm  �4qBEa  ϕ2m  dtcm  d(y2)  ���  y2ma  + xma  dtcm  dxm  ���  xma  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  (B20)  Appendix C: The ‘exact’ mean normal energy  The exact mean normal energy can be determined  starting with the joint distribution f(EN, ˜θ) or equiva-  lently f(EN, ρ)13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' In terms of ρ, it can be expressed as  ⟨EN⟩ = S1/S2 where  S1 =  � �  dρdEN ρ  �  1 + ρ2/R2a(EF − EN)ENT(EN, ρ)  S2 =  � �  dρdEN ρ  �  1 + ρ2/R2a(EF − EN)T(EN, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content='  In the above T(EN, ρ) ≈ 1/(1+eG(EN,ρ)) is the transmis-  sion coefficient for an electron having normal energy EN  at a point ρ on the emitter-tip z ≈ h − ρ2/(2Ra), having  a local field El = Ea(z/h)/  �  z2/h2 + ρ2/R2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' It can be  determined using Eqns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
+page_content=' (5) - (10) for the Gamow factor  G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/s9FJT4oBgHgl3EQfcyxN/content/2301.11545v1.pdf'}
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+This article is a part of the special collection  
+Cognitive Innovation for Social Change: Multiple Disciplinary Perspectives towards Sustainable Development 
+1 
+ AVANT, Vol. XIII, No. 2 
+ISSN: 2082-6710 avant.edu.pl/en 
+DOI: 10.26913/avant.202208 
+ 
+Piloting Virtual Reality Photo-Based Tours 
+among Students of a Filipino Language Class: 
+A Case of Emergency Remote Teaching in Japan 
+ 
+Roberto Bacani Figueroa Jr* 
+ 
+University of the Philippines Open University, Philippines 
+*corresponding author: rbfigueroa1@up.edu.ph 
+ 
+Florinda Amparo Palma Gil 
+Tokyo University of Foreign Studies, Japan 
+ 
+Hiroshi Taniguchi 
+University of the Philippines Open University, Philippines 
+ 
+Received 6 September 2020; accepted 28 January 2021; published 24 August 2022. 
+ 
+Abstract 
+The State of Emergency declaration in Japan due to the COVID-19 pandemic affected 
+many aspects of society in the country, much like the rest of the world. One sector that 
+felt its disruptive impact was education. As educational institutions raced to implement 
+emergency remote teaching (ERT) to continue providing the learning needs of students, 
+some have opened to innovative interventions. This paper describes a case of ERT where 
+Filipino vocabulary was taught to a class of Japanese students taking Philippine Studies 
+in a Japanese university using a cognitive innovation based on virtual reality, an immer-
+sive technology often researched for immersion and presence. Students were divided 
+into three groups to experience six lessons designed around virtual reality photo-based 
+tours at different immersion levels. While the effect of immersion on satisfaction was not 
+found to be statistically significant, presence and satisfaction were found to be corre-
+lated. Despite challenges that were encountered, benefits like enjoyment, increased en-
+gagement, and perceived learning were reported by the students. Our findings exemplify 
+how emerging multisensory technologies can be used to enhance affective and cognitive 
+dimensions of human experience while responding to gaps created by the spatial limita-
+tions of remote learning. 
+Keywords: emergency remote teaching in Japan; Filipino language teaching; language 
+learning; virtual reality photo-based tour; virtual reality in education 
+ 
+
+OPEN
+ACCESSRoberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi 
+ 
+ 
+2 
+1. Introduction 
+The COVID-19 pandemic has greatly changed societies in almost every aspect of life since 
+the end of 2019. One area that has been heavily affected by this global crisis is education. 
+The year 2020 served as a turning point for educators across the globe, causing them to 
+reevaluate their pedagogy and look for innovative ways to overcome the challenge of 
+limiting in-person meetings for teaching and learning. The effects varied depending on 
+the subject being taught, the cultural setting, and the technologies available. This paper 
+hopes to contribute to this fast-growing body of knowledge on context-specific innova-
+tions by presenting a study that describes the process of prototyping a technology-based 
+innovation in the context of what is known in the educational literature as emergency 
+remote teaching (ERT), which was defined as the sudden and temporary shift of delivery 
+to a distance or remote mode of instruction as a result of extremely disruptive phenom-
+ena (Hodges et al., 2020; Mohmmed et al., 2020). Results from this study may be of in-
+terest not only to researchers in the field of education, but also in other fields like 
+psychology, information technology, and cognitive science as they have implications on 
+the dynamics of immersion, presence, and satisfaction in multi-sensory experiences 
+brought about by immersive technologies.  
+ 
+1.1. Foreign Language Teaching in Japan 
+Popular foreign language learning methods in Japan include in-person lessons at private 
+language schools, online one-on-one lessons, and the use of language applications on mo-
+bile phones or personal computers. However, what is referred to as foreign language 
+learning, mostly refers to learning the English language. While the English language is 
+taught from the elementary level to the university level, with grammar and translation 
+methods dominating over others, the teaching of foreign languages other than English is 
+mostly unheard of (Butler, 2007). Among languages that were rarely taught in Japan is 
+Filipino, the national language of the Philippines. 
+Among Japanese universities, Osaka University and Tokyo University of Foreign Studies 
+are the only ones that offer a full major course in the Filipino language as of this writing. 
+With this, comes the scarcity of teaching and learning materials that are content-based 
+(Laranjo, 2020), especially those targeting Japanese learners. These universities have fa-
+cilitated Filipino language education using traditional in-person classroom-based or 
+blended pedagogy using a learning management system (LMS). Learning activities that 
+would provide a more immersive experience for learners and establish relevance and 
+context of the target language were offered in the form of study abroad programs or 
+study tours embedded within the four-year curriculum.  
+1.2. The Situation of Education in Japan at the Dawn of the Pandemic 
+Just like the rest of the world, university teachers and students in Japan did not expect 
+the effect that the pandemic would bring to the academic year 2020. When the COVID-
+19 outbreak occurred, the Japanese government called a nationwide state of emergency 
+
+Piloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  
+ 
+3 
+and restricted people’s mobility entering and leaving Japan. As a result, foreign students 
+who returned to their home countries during the spring break in 2020 were included in 
+the reentry ban (Osumi, 2020). At the same time, 80 to 90% of Japanese universities re-
+called their students studying abroad, giving no guarantee of returning to their programs 
+(Wortley, 2021). According to the result of a nationwide survey released by Japan’s Min-
+istry of Education, Culture, Sports, Science, and Technology (MEXT) in May of 2020, as 
+a countermeasure to the COVID-19 outbreak, 930 universities out of 1046 that re-
+sponded (86.9%), delayed the start of classes and more than 96.6% of the said universi-
+ties either decided or were considering conducting distance learning using different 
+forms of media. Since the most popular form of education in Japanese universities before 
+the pandemic was face-to-face classes, teachers were forced to hurriedly prepare for 
+switching to online classes during the short delay (Inoue, 2020). On the other hand, in-
+coming first-year students, expecting to start a new active life in their new school, had to 
+attend online classes without being able to set foot on campus and before experiencing 
+face-to-face lectures with their professors (Hirabayashi, 2020).  
+Although Japanese universities had already been introduced to the use of LMS for lan-
+guage education even before the pandemic, actual usage by teachers, staff, and students 
+was still minimal (Murakami, 2016). Teachers had to make significant adjustments in 
+shifting to online classes. The Mainichi Shinbun, one of the major newspapers in Japan, 
+surveyed 66 institutions and showed one teacher describing how his preparation time 
+has gone up fivefold while some teachers thought of doing online classes as becoming 
+Youtubers or that they should be like radio broadcasters (Mainichi Japan, 2020). As for 
+the students, it was reported that “…while 59.2 percent were positive about online clas-
+ses, 21 percent said they did not wish to take part, reflecting concerns about the quality 
+of education that remote learning provides and finding the right environment for partic-
+ipating” (Kyodo News, 2020, para 2). In fact, in a survey from an organization petitioning 
+the return of classes to face-to-face, it was reported that the quality of lessons was lower 
+than face-to-face was the second highest answer given by 845 out of 1,500 respondents, 
+regarding the perceived disadvantages of an online class (DaigakuIkitai, 2020). Even at 
+the University of Tokyo, which was the fastest to adapt to online classes, 70% of the stu-
+dents who were surveyed said that online classes were mere approximations of regular 
+classes (Shoji, 2020). However, an exploratory study on first-year students’ perception 
+of university online lessons reported that the efficiency of technological innovation 
+brought about a positive response to online learning as well as changed the feeling of 
+loneliness of in-home study to being able to concentrate on individual learning efficiently 
+through online lessons (Hirabayashi, 2020). 
+Teachers’ lack of knowledge in developing efficient online classes seemingly presents 
+a challenge in achieving the fourth sustainable development goal, which is about en-
+suring equitable and quality education for everyone. However, it also presents an op-
+portunity to explore, exploit, and explain innovative ideas for revolutionizing education 
+that would address the gaps and restrictions presented by the current situation (Ki-
+takka, 2020).  
+
+Roberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi 
+ 
+ 
+4 
+1.3. Virtual Reality Technology in Emergency Remote Teaching (ERT) 
+Virtual reality (VR) is a system of specific hardware and software that brings about 
+a real-time, computer-generated three-dimensional environment that users, having 
+a perceived self-location, can navigate and interact with (Hayward, 1993). Despite ha-
+ving many features, two were found to be common among those investigated by VR re-
+searchers: Presence and Immersion. Presence or telepresence is the feeling of being there 
+in a technology-mediated environment (Heeter, 1992), while immersion is defined as the 
+aspects of hardware and software systems, such as display quality and stereoscopy, 
+which could facilitate presence (Slater & Wilbur, 1977).  
+VR has gained increasing popularity in technology-integrated classrooms as reported in 
+studies done by Chang et al. (2016), Coyne et al. (2018), and Kim et al. (2001). The ability 
+of VR to immerse learners in a virtual environment or simulate a geographical location 
+without having to go there is potentially useful in addressing the gap caused by limited 
+mobility among learners due to the lockdowns caused by the pandemic around the 
+world. One instance of a VR application that provides opportunities to teachers without 
+a programming background is the virtual reality photo-based tour or simply VR tour. VR 
+tours are interactive tours based on 360 photos of real-world locations with hot spots 
+around the environment that provide information about an area or objects. Equipment 
+and devices for 360 photos have become more affordable in recent years.  
+Furthermore, affordable and easy-to-use platforms for creating interactive tours using 
+these media have become widely available. The lockdowns that caused the lack of access 
+to location-based educational activities have increased the potential use of VR tours in 
+education and other learning contexts, especially among courses and subjects that would 
+have involved field trips in normal cases. However, reports were scarce during the time 
+that this study was conceptualized.  
+Even rarer were studies on remote teaching and learning methodologies for the Filipino 
+language. The few papers found on teaching Filipino as a foreign language mainly focused 
+on the details of the curriculum and the background of students. Only some of these pa-
+pers mentioned the methodology of teaching and the challenges of teaching Filipino - 
+Quirolgico-Pottier (1997), Luquin (2016), Pambid-Domingo (2010), Barrios-Le Blanc 
+(2010), and Laranjo (2020). Pambid-Domingo (2010) mentioned the use of authentic 
+materials, videos, and songs, while Barrios-Le Blanc (2010) explained teachers' tech-
+niques in using poetry for teaching. However, none of them investigated using infor-
+mation and communication technology (ICT) in teaching the subject matter among 
+Japanese students. This could be brought about by the general apprehension among 
+teachers in Japan about using ICT-based novel interventions (Teaching and Learning In-
+ternational Survey [TALIS], 2018).  
+According to the 2018 Teaching and Learning International Survey (TALIS) survey by 
+the Organization for Economic Co-operation and Development, Japan ranked 45 out of 
+50 on teachers’ preparedness in using ICT for teaching. Fewer teachers felt "well pre-
+
+Piloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  
+ 
+5 
+pared" or "very well prepared" in ICT for teaching in Japan compared to other participat-
+ing countries. While Japanese universities have learning management systems, they are 
+used for less than 20% of all courses. Moreover, due to a lack of staff for creating digital 
+content and maintaining ICT systems, insufficient ICT skills for both academic and gen-
+eral staff, and a limited budget, ICT tools were mostly used for syllabus systems (89%), 
+student information systems (63%), and campus WiFi networks (79%), impeding the 
+introduction and promotion of ICT tools for educational use (Funamori, 2017). 
+A few months into the pandemic, news articles have shared how ICT was used by various 
+learning institutions in Japan to fill the gap created by the state of emergency caused by 
+the COVID-19 outbreak (O’Donoghue, 2020). However, the authors have not yet found 
+any study that outlined how a particular ICT-based intervention was designed and how 
+it affected the students’ learning experience and outcome. This is also true even in for-
+eign language teaching and learning, much more so among less popular languages like 
+Filipino. Aside from a recent study reporting some motivational effects of VR tours in 
+a university in Japan (Figueroa et al., 2020), no article regarding VR tours for teaching 
+the Filipino language in Japan in the context of remote teaching has been found as of this 
+writing. Thus, when the state of emergency was first announced in the country, and 
+a shift to online classes was decided by the university administration, an exploratory 
+study on utilizing VR tours in teaching vocabulary to zero-beginner learners of the Fili-
+pino language was carried out. 
+ 
+1.4. Aims of The Study 
+This study aimed to provide insights as to how an intervention using VR tours could have 
+an impact on Japanese students of a Filipino Language class in the context of ERT during 
+the time of the COVID-19 outbreak in Japan. More specifically, it aims to answer three 
+research questions (RQ): 
+RQ1: How different are the levels of satisfaction between students who experienced 
+VR tours with varying levels of immersion?  
+RQ2: How are the levels of satisfaction among students related to presence in the 
+VR tours? 
+  
+RQ3: What are the other benefits of incorporating VR tours in an online Basic Fili-
+pino Language Class designed for Japanese students? 
+ 
+
+Roberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi 
+ 
+ 
+6 
+2. Materials and Methodology 
+This study investigated the impact of VR tours on Japanese students of a Filipino Lan-
+guage class in the context of ERT during the time of the COVID-19 outbreak in Japan using 
+both quantitative and qualitative analysis. In the quantitative analysis, descriptive statis-
+tics and non-parametric tests were used, while in the qualitative analysis, reflexive the-
+matic analysis was used.  
+ 
+2.1. Virtual Reality Photo-based Tours (VR Tours) 
+The researchers adopted six VR tours into a Filipino Language Course for beginners at 
+a Tokyo-based university over six lessons from May 22 to June 26 of 2020. Each VR tour 
+was inserted into a 90-minute online synchronous lesson. The aims of each tour were 
+fourfold: (1) to increase students’ vocabulary, (2) to develop students’ ears for recogni-
+zing Filipino sounds and rhythm, (3) to let them experience the Philippines albeit virtu-
+ally, and (4) to let them experience a new way of learning a language. By the end of the 
+six VR tours, the students were expected to be able to introduce popular Japanese tourist 
+spots in a VR tour of their own using the vocabulary that they had learned from the tours 
+and to be able to introduce the places using the correct pronunciation and rhythm. 
+Each tour introduced a place in the Philippines while featuring seven new Filipino words. 
+The tours were given titles based on their main location of interest, and the seven words 
+introduced in each tour were related to the things, people, places, and events that could 
+be found in these locations. Figure 1 shows the tour used for Lesson 6, which was entitled 
+Plasa. A hyperlink to the interactive online tour is also provided as a note for those in-
+terested in experiencing it. 
+Figure 1 
+Lesson 6: Plasa Virtual Tour 
+ 
+Note. URL: (https://figtreeacademy.org/vr/plaza/) 
+
+VRPiloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  
+ 
+7 
+Table 1 shows a detailed list of the tours for reference. The first, second, fourth, and sixth 
+tours had audio guidance in Japanese and Filipino by two narrators (a Japanese native 
+speaker and a Filipino native speaker). In contrast, the third and fifth tours were nar-
+rated purely in Filipino by a Filipino native speaker. 
+ 
+Table 1 
+The VR Tours 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+2.2. Participants 
+All six lessons were conducted among a class of 15 first-year Japanese students majoring 
+in Philippine Studies. All students in the class volunteered to participate in the study.  
+ 
+2.3. Data Collection 
+Data collected were the following: pre-VR Tour survey, post-semester survey, six after-
+class surveys, two focus group discussions, and observation notes. Survey questions can 
+be found in Appendix A1 and A2; focus group discussion questions are in Appendix B; 
+and observation notes are in Appendix C. 
+The pre-VR tour survey was used to decide the groupings. Random selection was carried 
+out among learners who had compatible phones. The participants were divided into 
+three groups - high immersion group, moderate immersion group, and low immersion 
+group. Quantitative items from five out of six after-class surveys were used for answering 
+RQ1 and RQ2. Open-ended questions from six after-class surveys were used for answe-
+ring RQ3 and the results were compared to the data collected from the post-semester 
+survey, two focus group discussions, and observation notes. 
+ 
+
+Number
+Title
+Date
+Language
+1
+Rizal Park (name of a park) /
+May 22
+Japanese and Filipino
+Pilot Tour
+2
+Aurora (name of a province)
+May 29
+Japanese and Filipino
+3
+Quiapo Church (name of
+June 5
+Filipino
+church)
+4
+LRT (light railway
+June 12
+Japanese and Filipino
+transportation)
+5
+Museo (museum)
+June 19
+Filipino
+6
+Plasa (plaza)
+June 26
+Japanese and FilipinoRoberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi 
+ 
+ 
+8 
+2.4. Procedure 
+As shown in the procedural diagram in Figure 2 below, the study was implemented in 
+several phases. As part of the preparation phase, students were asked to accomplish 
+a pre-VR tour survey. Students were assigned to three groups based on their phone’s 
+specifications revealed by their answers to the pre-VR tour survey.  
+ 
+Figure 2 
+Procedural Diagram of the Study 
+ 
+ 
+ 
+ 
+Six lessons were conducted with groups following the same lesson structure. At the be-
+ginning of the lesson, students were asked to do a pre-test of target vocabulary 
+words. Each virtual tour served as one of the main activities of each lesson. Lessons were 
+structured into four phases [Lesson Introduction → Virtual Tour → Focus on Grammar 
+→ Application].  
+In the Lesson Introduction phase, the theme and the objectives of the lesson were intro-
+duced. In the Virtual Tour phase, a new set of vocabulary related to the lesson's theme 
+was introduced using a VR tour. Students assigned to the high immersion group partici-
+pated in all the VR tours using their phones and VR goggles delivered to their homes. 
+Those assigned to the moderate immersion group participated in all the VR tours using 
+their computers or smartphones, but without using VR goggles. Those assigned to the 
+
+Preparation
+Pre-VRTourSurvey
+Pre-Test
+LESSON
+Implementation
+x 6 times
+(*1)
+Post-Test
+After-class Survey
+(*1)Aftereachlesson,theteacherand
+twootherresearchersdiscussedand
+wrotedowntheirobservations
+Post-semesterSurvey
+Reflection
+FocusGroup
+DiscussionsPiloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  
+ 
+9 
+low immersion group viewed and listened to PowerPoint-based tours that had the same 
+content as the VR tours but used 2D photos. During this phase, the teacher reminded the 
+students to experience the VR tours three times for vocabulary reinforcement.  
+ This was followed by the Focus on Grammar phase, where new grammar was taught. In 
+the Application phase, students were asked to use the presented vocabulary words and 
+new grammar to form sentences describing sample photos provided by the teacher. Fi-
+nally, a post-test containing items related to the presented vocabulary words was given, 
+followed by an after-class survey.  
+After conducting lessons 1, 2, and 3 (Rizal Park, Aurora, and Quiapo Church), the students 
+were divided into pairs. They were asked to make their own tour introducing a place in 
+Japan by using 2D photos and Google Slides. They were allowed to choose the language 
+they would use as long as they included seven Filipino words from the three tours. Like-
+wise, after lessons 4, 5, and 6 (LRT, Museo, and Plasa), the students were again divided 
+into pairs. They were asked to make their own VR tour using the platform Story Spheres, 
+and VR photos of places in Tokyo which were provided to them. They were given instruc-
+tion to use seven words from the last three tours and the vocabulary words learned 
+within the semester.  
+Another survey was conducted at the end of the semester, and from the results, six stu-
+dents were invited for two focus group discussions. Two students from each group were 
+chosen to join the FGDs. The students who were interviewed were the ones who men-
+tioned something related to the VR tours in their answers in the post-semester survey. 
+After each lesson, the teacher and two other researchers discussed and wrote down their 
+observation notes.       
+ 
+2.5. Iterative Improvement based on Challenges Encountered 
+Strategic integration of VR tours in class involved iterations of creative thinking, brain-
+storming, testing, evaluation, revision, and synthesis -activities that characterize a cog-
+nitive innovation. Like in most innovations, several challenges were encountered during 
+implementation. These challenges were addressed by adjusting the design and imple-
+mentation of succeeding lessons described briefly below. However, other challenges 
+were not addressed due to time constraints and contextual limitations.  
+Device Compatibility. The students were grouped according to the compatibility of 
+their mobile devices with VR Tours and their willingness to try VR Tours. However, two 
+participants in the high immersion group had incompatible phones. Hence, the groupings 
+had to be adjusted after the first lesson resulting in the decision to collect quantitative 
+data from lessons 2 to 6 only.  
+Device and Tour Usability. An orientation was conducted since it was the first time for 
+many of the participants to use VR tours. However, high immersion group participants 
+
+Roberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi 
+ 
+ 
+10 
+continued to have difficulty using the device until the second tour. They started becom-
+ing comfortable only from the third tour with careful guidance and reminders in operat-
+ing the tours. 
+Tour Technical Problems. Participants from all three groups experienced technical dif-
+ficulty in loading the audio of the tours, most especially the high immersion group par-
+ticipants. Hotspots were supposed to activate certain audio explanations when gazed on 
+or clicked. Some of the audio files were slightly larger, which caused them to be loaded 
+rather slowly, especially when the student’s internet connection was intermittent. This 
+interrupted the flow of the lessons significantly. However, these were resolved by re-
+starting the application or refreshing the page, or by using a different mobile device.  
+Tour Language. Since the participants were Japanese speakers who generally had no 
+exposure to the Filipino language, it was decided to have the first tour be explained in 
+Japanese with highlights on Filipino terms that were the target words for that lesson. 
+Labels corresponding to the target words were also placed on the hotspots. However, 
+this removed any challenge in the task and lessened the impact of the tour in terms of 
+student engagement. Thus, the succeeding tours did not have textual labels, which 
+turned out to be useful in training the students’ listening skills. On the other hand, the 
+two tours that had audio in all Filipino increased the difficulty of the task immensely. As 
+a response, the teacher translated and explained the script's meaning after the third lis-
+tening part so students could consult the script as often as they wanted to make sense of 
+what the tour guide was saying in each hotspot.  
+ 
+3. Data Analysis 
+Quantitative and qualitative analyses were performed to answer the research questions 
+of the study. In this section, we discuss the three research questions and the analyses 
+performed to answer them. All quantitative procedures were carried out using R, 
+an open-source software offering libraries for statistical analysis (R Core Team, 2012). 
+R-Studio, a commonly used integrated development environment for R, was utilized for 
+easy documentation and organization of data. 
+RQ 1: Levels of Immersion and Satisfaction Ratings 
+Participants’ satisfaction ratings collected from after-class surveys of Lessons 2 to 6 were 
+analyzed and grouped by immersion level. The omnibus test used to determine the effect 
+of immersion level on satisfaction was the Kruskal-Wallis test as a non-parametric alter-
+native to the analysis of variance for the following reasons: 1) There was a small number 
+of participants who were randomly allocated to the treatment groups (i.e., moderate and 
+high immersion groups); 2) Samples were mutually independent; and 3) Dependent va-
+riables were at least in the ordinal scale. Boxplot visualizations were generated using 
+the ggplot2 library (Wickham, 2016), while the Kruskal-Wallis function was performed 
+using the stats library (R Core Team, 2012). 
+ 
+
+Piloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  
+ 
+11 
+RQ 2: Presence and Satisfaction Ratings 
+This section describes the tests performed and plots that were visually inspected to an-
+swer research question 2. Participants’ satisfaction and presence ratings collected from 
+after-class surveys of Lessons 2 to 6 were analyzed. Scatterplots of satisfaction and pre-
+sence ratings were generated in five lessons to visualize the monotonic relationship be-
+tween the two variables using the ggplot2 library (Wickham, 2016). Kendall’s tau was 
+calculated to determine the correlation between satisfaction and presence ratings based 
+on the assumptions: 1) Data from the paired observations appeared to follow a mono-
+tonic relationship, and 2) variables were measured at least in the ordinal scale. Further-
+more, Kendall’s tau was chosen over Spearman’s rho because the p-values of the former 
+were found to be more accurate with smaller sample sizes (Marshall & Boggis, 2016), 
+which was the case in the current study.  
+RQ3: Benefits of VR tours  
+The qualitative analysis employed in this study was reflexive thematic analysis. To exe-
+cute the analysis, data were collected from the six after-class surveys, the transcription of 
+the two FGDs, and the observation notes. These data were collected to answer RQ3: What 
+are the other benefits of incorporating VR tours in an online Basic Filipino Language Class 
+designed for Japanese students?  
+The data analyzed from the six after-class surveys were the reasons given by the partici-
+pants regarding the rating of their experience of trying each VR Tour. The reasons were 
+categorized into codes and sub-themes until two main themes emerged from the catego-
+rization. After this process, the codes and sub-themes were triangulated with the an-
+swers of the six participants invited to the FGDs, results from the post-semester survey, 
+and observation notes of the authors.  
+Categorizing the reasons given by the participants regarding the rating of their VR Tour 
+experience produced 20 codes which were further categorized into 7 sub-themes and 
+finally into two main themes – the Benefits of VR Tours and the Problems and Chal-
+lenges encountered while doing the VR Tours.  
+ 
+4. Results 
+Notable findings from analyses that were carried out are organized according to the 
+three research questions. 
+RQ 1: Levels of Immersion and Satisfaction Ratings 
+The boxplots in Figure 3 visually illustrate the median and interquartile range of satis-
+faction ratings given by the three groups. Outliers were found in the high immersion 
+group in the second lesson and the sixth lesson, and the moderate immersion group in the 
+fifth lesson.  
+ 
+
+Roberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi 
+ 
+ 
+12 
+Figure 3 
+Satisfaction Boxplots of Three Immersion Groups per Lesson 
+ 
+ 
+ 
+Figure 4 
+Scatterplots of Satisfaction and Presence Ratings in Five Lessons 
+ 
+ 
+The medians of satisfaction ratings by the low immersion, moderate immersion, and high 
+immersion groups in the second lesson were 8 (IQR = 5 - 8), 9 (IQR = 9 - 10), and 7 (IQR 
+= 7 - 8), respectively. The medians of satisfaction ratings by the low immersion, moderate 
+
+Lesson2:AuroraTour
+Lesson3:QuiapoTour
+Lesson4:LRTTour
+10-
+8 -
+6 -
+Satisfaction
+0
+0
+2
+Lesson 5: Museo Tour
+Lesson6:Plasatour
+10
+2
+8 -
+9
+4
+0
+1
+2
+0
+1
+2
+ImmersionLesson2:AuroraTour
+Lesson3:QuiapoTour
+Lesson4:LRTTour
+10
+8·
+6
+4
+Presence
+6
+8
+10
+Lesson5:MuseoTour
+Lesson6:Plasatour
+10
+8
+6
+6
+8
+10
+6
+8
+10
+SatisfactionPiloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  
+ 
+13 
+immersion, and high immersion groups in the third lesson were 8 (IQR = 7 - 10), 9 (IQR = 
+8 - 10), and 8 (IQR = 7 - 8), respectively. The medians of satisfaction ratings by the low 
+immersion, moderate immersion, and high immersion groups in the fourth lesson were 8 
+(IQR = 7 - 10), 9 (IQR = 9 - 10), and 9 (IQR = 7 - 9), respectively. The medians of satisfac-
+tion ratings by the low immersion, moderate immersion, and high immersion groups in the 
+fifth lesson were 6 (IQR = 5 - 8), 10 (IQR = 0), and 7 (IQR = 6 - 9), respectively. Finally, 
+the medians of satisfaction ratings by the low immersion, moderate immersion, and high 
+immersion groups in the sixth lesson were 8 (IQR = 7 - 10), 10 (IQR = 9 - 10), and 8 (IQR 
+= 8 – 9), respectively. The medians and interquartile ranges presented in the boxplots 
+also revealed that students in the moderate immersion group generally gave higher sa-
+tisfaction ratings compared to the low and high immersion groups. 
+However, no statistically significant differences were found among three groups who ex-
+perienced tours in the second, [H(2) = 4.52, p = .11]; third, [H(2)= 2.49, p = .39]; fourth, 
+[H(2)= 1.63, p = .44]; fifth, [H(2)= 4.99, p = .08]; and sixth, [H(2)= 3.08, p = .22], lessons.  
+RQ 2: Presence and Satisfaction Ratings 
+The scatterplots shown in Figure 4 illustrate the monotonic relationship between the 
+two variables in five lessons. Results revealed that there was a statistically significant 
+strong correlation (N=15) between presence and satisfaction ratings in the second (τb = 
+.50, p=.02), third (τb = .51, p=.02), fourth (τb = .67, p=.002), fifth (τb = .65, p=.002), and 
+sixth (τb = .56, p=.01) lessons.  
+RQ3: Benefits of VR tours  
+The codes and categories collected in the thematic analysis are presented in Table 2. Two 
+main themes emerged: Benefits and Challenges. Since the challenges were already dis-
+cussed in the methodology section as part of the iterative improvement process, this sec-
+tion will discuss the subthemes related to benefits.  
+ Increased Engagement. Out of the 20 codes identified, most of the reasons given by the 
+participants for their VR Tour experience ratings alluded to the excitement of doing 
+something new or the feeling of being transported to a different place which made them 
+say that learning was enjoyable. These trends were also the first things that the teacher 
+observed among students who used the VR tours for the first time. Participants of the 
+high immersion group and moderate immersion group were observed to have eagerly and 
+diligently engaged in learning the new sets of vocabulary words and in fini-shing the ac-
+tivities of each tour. The novelty of the intervention naturally stirred students' interest 
+in participating in each class. Although a few chose to use the conventional PowerPoint 
+presentation-based tours because they said that they were not good with technology or 
+that their mobile phones and PC were not compatible, five out of the six participants of 
+the focus group discussions said that the VR tours were their most en-joyable activity in 
+class. The tours made them feel that they were in the Philippines. One of the participants 
+said, “I could feel the Philippines with the VR tour even if I’m not there.” Three out of the 
+five participants even said that the application activity of the course, which allowed them 
+
+Roberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi 
+ 
+ 
+14 
+to make their own tours, was extremely fun and made them think about the difficulties 
+encountered by the teacher and researchers in setting up the tours. 
+ 
+Table 2 
+Codes and Themes Created from the Categorization Process 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+Perceived Vocabulary Retention. The second prevailing theme in the observations and 
+survey results was that students had perceived ease of remembering the words because 
+of the context provided by the virtual tours in the lessons. They recalled that they quickly 
+remembered and understood how the words would be used in the Philippines as they 
+referenced objects in the tour, which made it easy to associate them with their meanings. 
+Many of them associated their ease of remembering the words with the feeling of being 
+in the actual place. This was further validated by the participants of the focus group dis-
+cussions. A participant commented, “We can see the real Philippines, and with the actual 
+
+Main Themes
+Sub-themes
+Codes
+Benefits
+Increased
+-Excitement towards a new experience (12)
+Engagement
+-Excitement in feeling as if in the actual place (10)
+-Enjoymentinlearning(7)
+Perceived
+-Ease of memorizing words (9)
+Vocabulary
+-Easeofunderstanding (6)
+Retention
+-Acquisition of actual usage of new words (6)
+-Acquisition of new word s (5)
+-Acquisition of correct pronunciation and rhythm (3)
+Interest
+-Admirationoftheview(16)
+-InterestingoingandseeingthePhilippines(3)
+Authentic
+-Acquisition ofinformation about actualplaces and the
+Information
+actualenvironment/atmosphereinthePhilippines(8)
+about the
+-Acquisition ofinformation aboutthedailylifeinthe
+Philippines
+Philippines likerush hour, crowded train,goingto
+church,fashionof people,etc.(8)
+Challenges
+Computer/Phone
+-Trouble in listeningto theaudio (5)
+Trouble
+-Trouble in playingthe tour smoothly and continuously
+(4)
+-Trouble in manipulating or operating the VR Tour (2)
+Obscureness of
+-Difficulty in understanding the photos (2)
+Photo
+-Monotonousnessofphotos(2)
+-Unrelatednessofphotostothethemeofthetour(2)
+Difficultywiththe
+-Disappointment with listening to only in Filipino
+language used for
+withoutJapanese(forsometours)(3)
+the script
+-Disappointmentin listening to more Japanesethan
+Filipino (forsometours)(2)Piloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  
+ 
+15 
+situation, I can learn the vocabulary.” Aside from comments about the benefit of VR Tours 
+for language learning, this study also found comments about the effi-ciency of the pro-
+gram design itself for language learning like, “the lesson was repeated learning. We 
+learned the vocabulary before VR and during VR. We encountered those vocabulary many 
+times,” and “I was able to listen to Filipino sentences. I was able to practice listening to 
+Filipino words and enjoyed learning words.” From comments like these, the authors 
+learned that the study was able to achieve the aims of each tour. 
+Interest in the Subject Matter. The third prevailing theme was that the VR Tours stirred 
+their interest not just in the activity, but also in the subject matter: the Philippines and 
+its national language, one of the goals of the whole program. As further explained in the 
+latter section, according to the post-semester survey, students with an initially slightly 
+negative view of the country changed their impression of it. The VR tours made them 
+want to visit it more. This was also supported by the frequency of the students’ answers 
+regarding the aesthetic value of the tours, which led them to be interested in the lessons 
+as reported in the after-class surveys. 
+Authentic Information about the Philippines. The fourth prevailing theme found from 
+after-class surveys was the authenticity of the information about the Philippines that the 
+participants were able to get from the VR Tours. Participants wrote that they could “feel 
+the atmosphere” of the places introduced in the tours, like the church or the train. Half of 
+the participants also mentioned that they were happy to be able to “learn about the his-
+tory of the Philippines”, or “see everyday life in the Philippines.” The authenticity of the 
+information they got from the tours made them feel as if they were walking quietly inside 
+a church full of people praying or as if they were also stuck inside a crowded train. Two 
+focus group discussion participants agreed that “All tours were fun.” It was followed by 
+the comment, “the image about the Philippines made me think I want to go there.” 
+Changes in perception of the Philippines. Aside from the after-class surveys, data 
+were gathered from the post-semester survey. The answers showed important points 
+about the benefits of using VR Tours in a Filipino language class. Answers revealed that 
+some of the students who experienced the VR tours considered the activity one of the 
+most useful activities during the period and that experiencing the tours changed their 
+perception of the Philippines. The six students who wrote about this also said during the 
+focus group discussions that their perception of the Philippines significantly changed. 
+Some of them said that before the tours, they were anxious about visiting the country 
+because they had only associated the Philippines with hot weather and poverty. How-
+ever, after experiencing the VR tours, they discovered something new. Answers from stu-
+dents like “I thought many people are poor, but once I saw Manila so developed, I wanted 
+to go” and “I am getting interested in the Philippines more and more. I want to visit it” set 
+the tone of the focus group discussions. They also mentioned that experiencing local and 
+rural life was one of the things that made the tours interesting because what the media 
+had shown them was only about tourist attractions, politics, or poverty. This made them 
+feel more excited to visit the country and learn more about the language to prepare them-
+selves for the actual visit.  
+
+Roberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi 
+ 
+ 
+16 
+5. Discussion 
+The study described in this paper primarily aimed to investigate how integrating VR 
+tours supported a class in Japan that shifted to synchronous online classes as an emer-
+gency remote teaching strategy due to the COVID-19 pandemic. The findings of the study 
+unearthed several themes that are worth discussing.  
+ 
+5.1. Emergency Remote Teaching as an enabler for Cognitive Innovation 
+As Gummerum and Denham (2014) explained, a cognitive innovation starts with an ex-
+ploratory phase of creatively looking for ideas and probing boundaries, followed by 
+choosing ideas that could be exploited, tested, and improved. It is then followed by re-
+flecting on the experience and synthesizing what has transpired to explain phenomena 
+and generate new questions and ideas. At first glance, one might think that cognitive in-
+novation can only thrive in a more relaxed, unconstrained, and open environment.  
+However, in this instance, the cognitive innovation of incorporating novel VR tours into 
+an online lesson that involved iteratively brainstorming lesson design and structure-re-
+lated ideas, selecting which of these ideas to exploit, and constantly reflecting on learning 
+points and things to improve each week, was enabled by limitations brought about by 
+the state of emergency in Japan during the COVID-19 outbreak.  
+While the term emergency was used in both emergency remote teaching and state of 
+emergency, its contextual meaning is not comparable to a dying patient being rushed to 
+a hospital or a burning house. Rather, it pertains to a state where certain interventions 
+that were not normally allowed would be considered because of an extremely disruptive 
+phenomenon. For example, in a normal situation, a teacher in Japan would not be allowed 
+to teach a class online in a traditional university. However, because of the state of emer-
+gency pronouncement by the government where universities must disallow in-person 
+activities to prevent the spread of COVID-19, conducting classes online through various 
+platforms has become permissible to provide a way for educational institutions to de-
+liver services to their learners. In the same way, because of the sudden closure of borders 
+that prevented learners from learning Filipino vocabulary in a more immersive environ-
+ment through field trips, VR was considered to be a possible means of allowing learners 
+to somehow experience certain places in the Philippines while learning the language 
+without having to go there.  
+Before the phenomenon, the idea of using virtual tours in a university setting was not 
+something that could be easily considered as students were able to receive ample stimuli 
+and realistic input from study abroad programs and in-person interactions with native 
+speakers in the physical classroom. The void created by the phenomenon made teachers, 
+learners, and school administrators open to new ideas that would not have been ideal in 
+normal cases, as revealed by reflection conversations held among the observers and the 
+teacher. In this way, temporal and spatial restrictions in education gave way to the open-
+ness to new ways of experiencing places while learning and actively participating in ma-
+
+Piloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  
+ 
+17 
+king the succeeding iterations better. It also made researchers more attentive to feed-
+back and less afraid to exploit alternatives. The phenomenon was so unique that no pre-
+vious guideline could have restricted them from trying new ideas.  
+While the challenges and difficulties encountered in the exploratory study supported re-
+cent findings (Graeske & Sjöberg, 2021; Phoon et al., 2021), specifically those involving 
+the use of VR goggles and internet connectivity, reflecting on them were useful in as-
+sessing how VR tours as an online learning supplementary intervention can be replicated 
+successfully in the future. As of this writing, the tours created have been used in two 
+subsequent offerings of the same course, where most of the exploration was presented 
+without VR goggles. However, VR is progressively recognized as a powerful tool for 
+learning and productivity by institutions and governments. Moreover, with some inter-
+national gatherings like conferences being held in virtual reality, technology companies 
+developing more affordable and portable VR goggles, and groups championing standar-
+dization of its use in the educational setting (European Committee for Standardization, 
+2022), the capability of having the tours viewed using VR goggles by all students will 
+most likely increase should another situation like this arise in the future.  
+ 
+5.2. VR tour: Novelty, Presence, and Satisfaction 
+Novelty helps with subsequent learning (Fenker et al., 2008; Schomaker et al., 2014). It 
+is also associated with a release of dopamine in the hippocampus (Biel & Bunzeck, 2019), 
+which is related to motivation and pleasure. However, novelty is expected to wane as 
+learners get more familiar with the intervention. The novelty of VR could have contrib-
+uted to the satisfaction ratings of learners in the first and second lessons. However, the 
+satisfaction ratings and sustained high interest of students in the four succeeding lessons 
+suggested that there was something more than the novelty of VR tours that made them 
+interested not only in the target vocabulary but also in the host country of the language. 
+An affective aspect of the learning experience made them want to learn more about the 
+country and the culture besides the language. Presence or the feeling of being in the vir-
+tual place was frequently found to be the reason for students enjoying the activity. The 
+results of the correlation analysis supported this observation by reporting a strong cor-
+relation between satisfaction and presence ratings in all five lessons.  
+ 
+5.3. Learning Experience more than Outcome 
+All fifteen students performed well in the course. Their learning ability as well as the 
+other interventions like the practice exercises, could have mediated actual vocabulary 
+acquisition in this study. Thus, post-test scores were not included. However, the impact 
+of VR tours, as revealed in their interviews, gave us a glimpse of how they would perform 
+in the succeeding courses. Studies have shown that enjoyment in the learning experience 
+is positively related to actual engagement and learning attitudes (Cybinski & Selvana-
+than, 2005), which lead to better learning outcomes in the long-term. 
+
+Roberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi 
+ 
+ 
+18 
+      5.4. Immersion, Presence, and Affective Outcomes 
+While the relationship between immersion and presence has been well established in 
+the literature (Gorini et al., 2011; Lessiter et al., 2001), the study’s findings were insight-
+ful because they brought about interesting realizations regarding the objective features 
+of technology (i.e., immersion) and presence being conduits for the affective dimension 
+of the human experience. While immersion was previously found in the literature to be 
+directly influential on presence, the study's findings revealed that the ensuing affective 
+outcomes were only associated with presence implying that the technological features 
+of the device did not solely predict desirable affective outcomes from the multisensorial 
+virtual reality experience. The explanation for this can only be gleaned from further stu-
+dies. However, based on personal observation, it could be investigated using the lenses 
+of user experience (UX) and ergonomics. Furthermore, the relationship between pres-
+ence and satisfaction led to the question, “What makes presence satisfying in a virtual ex-
+perience?” While it has been social presence that has been found to reduce the adverse 
+effects of physical distance and increase satisfaction among remote learners in past stu-
+dies (Moore, 2013; Richardson et al., 2017), results from this study suggest that presence 
+experienced in a virtual space regardless of interaction or social connection could in-
+crease satisfaction and perceived learning. This can be further investigated by looking 
+into the constructs of interest and sense of place. One could test whether their influence 
+on emotions could be moderated by this spatial form of presence. 
+ 
+6. Conclusion and Recommendations 
+We are ten years away from the deadline of achieving the UN’s Sustainable Development 
+Goals, a part of which is achieving quality education for all. In the context of language 
+education, it is our humble belief that achieving quality education cannot be proven by 
+having high scores in exams and tests as proven by the discrepancy of language test 
+scores and actual language learning as exemplified by the study of Nicholson (2015). In-
+stead, we believe that quality education can be achieved by providing enjoyable expe-
+riences that establish relevance and authenticity as positive emotions facilitate retention 
+of what students have learned (Dulay & Burt, 1977; Krashen, 1982). In this paper, we 
+presented a new way of learning a foreign language in a remote teaching context through 
+a cognitive innovation based on VR. However, our findings were not without limitations. 
+The study’s lack of generalizability was partly due to the small number of participants. 
+Therefore, an experimental study on testing the effect of immersion on satisfaction levels 
+with larger group sizes is suggested for future researchers. The random assignment of 
+students to groups was limited by the capabilities of their own mobile devices. While this 
+was expected from the study’s current context, future researchers may benefit from the 
+fast-paced evolution of technology and recruit participants whose mobile devices would 
+be compatible with VR applications. Furthermore, a face-to-face classroom setting would 
+ensure that everyone would have the chance to use VR goggles.  
+
+Piloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  
+ 
+19 
+In conclusion, we hope that the study's contributions, albeit small, would add to the va-
+rious means of responding to geographical and spatial limitations in education brought 
+about by the pandemic by creating cognitive innovations based on a novel technology 
+like VR. Moreover, the initial findings of the study could support and guide future re-
+searchers who are interested in presence, immersion, and other aspects of VR, as well as 
+their impact on learning outcomes. Furthermore, practitioners may benefit from the 
+study as it presented one of the many alternatives and supplements to location-based 
+learning provided by study abroad programs and field trips that would otherwise be in-
+accessible to more than half of the world’s population due to poverty, geopolitical rea-
+sons, or disruptions brought about by phenomena like the COVID-19 pandemic. The 
+realizations gleaned from the findings opened up questions regarding immersion’s rela-
+tionship with affective outcomes and suggested lenses for further study. Moreover, the 
+study suggested new hypotheses to be investigated regarding presence and the affective 
+dimension of the human experience. 
+ 
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+Lessiter, J., Freeman, J., Keogh, E., & Davidoff, J. (2001). A cross-media presence question-
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+ronments, 10(3), 282-297. 
+Luquin, E. (2016). Ang pagtuturo ng Filipino sa mga Pranses (Panayam). Daluyan (pp. 
+224-232). Sentro ng Wikang Filipino. 
+Mainichi Japan. (2020, May 9). Japan univ. lecturers worry over crumbling class quality 
+as virus forces courses online. The Mainichi. Retrieved from https://www.mainichi.jp 
+Marshall, E. & Boggis, E. (2016). The statistics tutor’s quick guide to commonly used statis-
+tical tests. The University of Sheffield.  
+Mohmmed, A. O., Khidhir, B. A., Nazeer, A., & Vijayan, V. J. (2020). Emergency remote 
+teaching during Coronavirus pandemic: The current trend and future directive at 
+Middle East College Oman. Innovative Infrastructure Solutions, 5(3), 1-11. 
+Moore, M. G. (2013). The theory of transactional distance. In Handbook of distance edu-
+cation (pp. 84-103). Routledge. 
+Murakami, C. (2016). Japanese University students and learning management systems - 
+日本人大学生と学習管理システム-.Learning (学習特別号),23(2). 
+Nicholson, S. J. (2015). Evaluating the TOEIC® in South Korea: Practicality, reliability and 
+validity. International Journal of Education, 7(1), 221-233. 
+O’Donoghue, J. J. (2020, April 21). In the era of COVID-19, a shift to digital forms of teach-
+ing in Japan. The Japan Times. Retrieved from https://www.japantimes.co.jp 
+OECD. (2019, June 19). TALIS 2018 results (volume I): Teachers and school leaders as life-
+long learners. OECD Publishing. 
+
+Roberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi 
+ 
+ 
+22 
+Osumi, M. (2020, June 19). Japan's re-entry ban threatens scholarships, admission and 
+graduation 
+for 
+foreign 
+students. 
+The 
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+Times. 
+Retrieved 
+from 
+https://www.japantimes.co.jp 
+Pambid-Domingo, N. (2010). Introductory Filipino at UCLA. In R. S. Mabanglo & R. G. Ga-
+lang (Eds.), Essays on Filipino language and literature (pp. 139-148). Anvil Publishing. 
+Phoon, G. C., Idris, M. Z., & Nugrahani, R. (2021). Virtual reality (VR) in 21st. century ed-
+ucation: The opportunities and challenges of digital learning in classroom. Asian Pen-
+didikan, 1(2), 105-110. 
+Quirolgico-Pottier, M. (1997). Ang pagtuturo ng Filipino sa mga Pranses. Daluyan (pp. 
+11-20). Sentro ng Wikang Filipino.  
+R Core Team. (2012). R: A language and environment for statistical computing. 2013 Vi-
+enna. Austria R Foundation for Statistical Computing. 
+Richardson, J. C., Maeda, Y., Lv, J., & Caskurlu, S. (2017). Social presence in relation to 
+students' satisfaction and learning in the online environment: A meta-analysis. Com-
+puters in Human Behavior, 71, 402-417. 
+Schomaker, J., van Bronkhorst, M. L. V., & Meeter, M. (2014). Exploring a novel environ-
+ment improves motivation and promotes recall of words. Frontiers in Psychology, 
+5(918). doi: 10.3389/fpsyg.2014.00918 
+Shoji, K. (2020, November 7). Japan's students struggle to embrace online learning amid 
+COVID-19. The Japan Times. Retrieved from https://www.japantimes.co.jp 
+Slater, M., & Wilbur, S. (1997). A framework for immersive virtual environments (FIVE): 
+Speculations on the role of presence in virtual environments. Presence: Teleopera-
+tors & Virtual Environments, 6(6), 603-616. 
+Wickham, H. (2016). Programming with ggplot2. ggplot2 (pp. 241-253). Springer. 
+Wortley, K. (2021, March 20). The pandemic left Japanese students studying abroad 
+scrambling. A year later, what’s happened to their academic dreams? The Japan 
+Times. Retrieved from https://www.japantimes.co.jp 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Piloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  
+ 
+23 
+Appendices 
+ 
+Appendix A1 
+Survey Forms Used for Analysis 
+One of the main data used in this study was the after-class survey. From all the questions, 
+the authors focused mainly on questions number two and three. Question number two 
+was mainly used for quantitative analysis and question number three was mainly used 
+for qualitative analysis. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+After-class Survey Questions
+8.少了一体上、今筱今日语巢使确率
+以龙思以？
+1.名前/二木一么(English:Name/Nickname
+English: How much do you see yourself using the Filipino words you
+learned today in the future after the tour?
+低 (Lowest) 高 (Highest)
+1 --- 2 --- 3 --- 4 --- 5 --- 6 --- 7 --- 8 --- 9--- 10
+2.今回の体俩龙
+9.VR岁一の中食感
+English: How would you rate your experience?
+English:Whatwerethepositivefeelingsyouhad duringtheVRtour?
+良<(Lowest)良(Highest)
+1 --- 2 --- 3 --- 4 --- 5 --- 6 --- 7 --- 8 --- 9 --- 10
+3.周2の伍の理由述～
+10.VR一の中良感
+English: What's the reason for your rating in number 2?
+English: What were the negative feelings you had during the VR tour?
+4.岁了一自体件机面百感龙？
+11.一遥人下<龙English:ChooseOne
+English: How interested were you in the actual experience?
+VR一龙の享真龙の感
+面白(Lowest)面白(Highest)
+English: During the VR Tour, I felt like I was just looking at a photo.
+1 --- 2 --- 3 --- 4 --- 5 --- 6 --- 7 --- 8 --- 9--- 10
+一本当体の感
+English: I felt like I was in an actual tour.
+12.龙の写真見龙、一本当体
+の感？
+L？
+English:Howmuchdidyoufeelthatyouwereinthetour and notjust
+English: How much were you interested in the lesson's content (new
+looking at a photo?
+words)?
+感(Lowest)感(Highest)
+1 --- 2 --- 3 --- 4 --- 5--- 6 --- 7 --- 8 --- 9--- 10
+1 --- 2 --- 3 --- 4 --- 5 --- 6 --- 7 --- 8 --- 9--- 10
+6.奥味持下部分法？当の忘人下遥人下龙去
+12.今俊の授の一体思
+主？世思心？
+English: What were the most interesting parts?
+English: Would you like to do more of these tours in future online
+classes? Why or why not?
+7.の少了一忘体膝上、自身将来今日誓个
+13.の他今回の体、提案、實周等
+の使の想像？の场面
+英语在使思？
+English: Please share other comments, suggestions, or questions
+English: Do you see yourself using the Filipino words you learned
+regarding the whole experience.
+today in the future after thetour?If yes, how?Roberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi 
+ 
+ 
+24 
+Appendix A2 
+Post-semester Survey 
+The result of the qualitative analysis using question number three of the after-class sur-
+vey was cross-referenced with the answers of the students in the post-semester survey. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Post-semester Survey
+About Taking Filipino as Major
+About the Content of the Class
+I.Why did you decide to study Filipino?
+2. What do you want to become in the future (what is your dream job,
+4.How much did you understand about the lessons?
+dream life.etc)?
+ONot atall
+Overymuch
+3.WhatwasyourimageofthePhilippinesandoftheFilipinosbefore
+taking the Filipino class?
+5.Which part of the class did you like? Check all that apply
+4. What was your image of the Philippines and of the Filipinos after
+OGrammar explanation
+taking the Filipino class?
+OVocabulary quizzes
+About Online Class
+OOral recitation (trying to make sentences)
+1.How did you like studying online? ONot at all
+OVery Much
+OShort class discussion about the theme of the lesson
+2. What are the advantages/merits of studying online?
+OReflection each week (Muni-muni)
+3. What are the disadvantages of studying online?
+OVirtual Tours,
+4.How was the length of each class? Overy short
+OVery long
+OActive Learning (Video about Learning a Foreign Language, Making
+5.If given a choice again, where would you like to have this class,
+Virtual Tours)
+online or in an actual classroom?
+Oonline
+OIn an actual classroom
+OWhichever is fine
+6.In general, which part of the class is most useful for you? Check all
+About the Content of theClass
+that apply.
+I.In general, how do you like the content of the class?
+OGrammar explanation
+ONot at all OVery much
+OVocabulary quizzes
+2.Do you think the themes/topics chosen for this class were appropriate
+OOral recitation (trying to make sentences)
+(self-introduction,family,favoritethings,birthday,themesofthevirtual
+OShort class discussion about the theme of the lesson
+tours like Rizal Park, Quiapa Church, etc.)?
+OReflection each week (Muni-muni)
+ONot at allOVery much
+OVirtual Tours,
+3. Which theme or topic did you like? Check all that apply.
+OActive Learning (Video about Learning a Foreign Language, Making
+OSelf- introduction, Family
+Virtual Tours)
+OBirthday
+OFavorite Things,
+7.Which Filipino words or expressions do you think you will remember
+OThings inside a room
+for a long time?
+ODescribing People
+ODescribing Place,
+8. What is the most important message or information did you learn in
+ORizal Park
+this class?
+OAurora
+9. Please give any suggestion on how to improve this class.
+OLRT
+OPlasa
+OMuseo,
+OUniversities in the PhilippinesPiloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  
+ 
+25 
+Appendix B 
+Focus Group Discussion (FGD) Questions 
+In addition to the answers of the students in the post-semester survey, the result of the 
+qualitative analysis using question number three of the after-class survey was also cross-
+referenced with the answers of selected students during the two FGDs conducted. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Focus Group Discussion (FGD) Questions:
+1．世、語の勉强遥捉の？
+11.思？
+、語积？
+What did you think about the Virtual tours?
+Why did you decide to study Filipino? Did you choose
+Filipino or not?
+12.役立？、YES
+の役思？
+2．語の学の味
+Were they useful? If yes, how?
+？11  の、答
+From a scale of 1 to 10, how interested are you in learning
+13.一番、好了一？何故
+Filipino?
+下加？
+What was your favorite tour? Why?
+3.
+語の学何期待？
+、の期待法美现？
+14. 一番、面白了一？、
+What were you expecting to learn? Did your expectations
+机法何故下？
+come true?
+What was your least favorite tour? Why?
+4.
+语の授知
+15．了～の心時間共变化？
+、の感？
+の、何故、化？
+How did you feel when you learned that the Filipino class
+Did your interest in the tours change through time? How and
+will be held online?
+why?
+5.
+語の才授の最初の1、２、
+16．、来、自分語姿
+の感？
+？
+寸？簡单下？
+Did you imagine yourself speaking Filipino in the future?
+How were your first few days of learning Filipino in an
+online class? Was it difficult? Was it easy?
+17.の强？
+effect？
+6.
+授の中下一番、楽活動何下？
+Did the virtual tours make your image stronger? Was it
+飞机法何故下？
+effective?
+What were themost enjoyable activities in the class? Why?
+18．困難感何
+7.
+、授前の考
+？
+L?
+What were the problems you encountered with the virtual
+How did you think of the Philippines before the class?
+tours?
+8.
+、、の考
+19．来年以降の授、一于了一
+办？
+加?
+How do you think of the Philippines now?
+Would you recommend using virtual tours in future classes?
+、の方·考方、授の
+20.の答、何故？？
+中方·考方变？
+Why/Why not?
+、？
+If your view changed, was there something in class that
+helped you change it? How?
+10.R以外面
+办?
+Is there any activity you had fun except VR tour?Roberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi 
+ 
+ 
+26 
+Appendix C 
+Observation Notes 
+The notes of the authors based on observations during the classes and from the discus-
+sion and brainstorming during the weekly researchers’ meetings were used to address 
+the challenges encountered during each tour session. The challenges were tackled by ad-
+justing the design and implementation of succeeding lessons. 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+Observation Notes
+TOUR 1: RIZAL
+TOUR 2: AURORA
+1.
+Started with an orientation on how to use VR Goggles.
+The new grouping suits the students. The students seem
+2.
+Students had a hard time using the goggles.
+happier.
+3. 
+They were excited and eager to do the VR Tours.
+2.
+Things went smoothly this time.
+4.
+I guessI wasnotabletoclearlyexplainthegroupings
+3.
+We could finish up to after-class survey plus we were able to
+because at the end of the class, some students say they didn't
+dothelesson ofthe day.
+knowwhattheyshoulddoorshould'vehavedoneduringthe
+4.
+I heard positive feedback from the students today. I think
+first VR Tour.
+they enjoyed the tour.
+5.
+Some students were lost.
+5.
+The pre-test and post-test seemed very easy for them
+6.
+Two to three students complained that they did not hear
+anything.
+7.
+Everything took a lot of time - couldn't do the lesson for the
+day,just theVR Tour.TheTour needmoretimeto explain-
+maybe next time a separate day for Orientation, Pre-VR Tour
+Survey, practice, etc.
+8.
+Around2studentsrequestforchangeofgroupingstoGroup
+3(PowerPoint-basedTour)becausetheirPCcouldn'ttake
+the tour's memory. One said she is not good with technology
+so she really didn't know what to do during the whole
+process.
+No time for after-class survey. We couldn't do it.
+10. The post-test seemed very easy for them so they all got a
+perfect score.I think this is because we listenedto the audio
+file several times and thetargetwords werewritten explicitly
+in thetour.Maybethelabels should betaken out fromthe
+nexttour
+TOUR 3: QUIAPO CHURCH
+TOUR 4: LRT
+1.Thepre-test and post-test seemed very easy for them. Or they
+1.They also enjoyed this tour because it seemed real.
+are very good in guessing? Especially for the pre-test or they
+They asked why there were only guys on the train. I
+already know the words from other classes? Some words may
+explained about “women's car"
+havealreadybeenintroducedinotherclasses-needto
+3.
+I'm not really sure if G1 students are still using the goggles.
+considerthis in next experiment.
+They have their videos turned off to save internet memory. I
+2.
+A lot of students -G1 and G2 said they liked the photo -
+couldn't force them to turn it on because of school
+very beautiful and it was their first time “to enter"a church
+regulations to respect students' feelings regarding turning on
+3.
+One student who was supposed to be in G2 used the VR Tour
+and off videos. What to do about this? For now, just keep on
+linkforGl.Myfault!Needtosendlinkexclusivelytoeach
+reminding them to use goggles.
+G! students not on the group chat.
+4.
+Theywere shocked tohear/listentoanall-Filipinotourbut
+some students said it was good because they want to listen to
+moreFilipino ifpossible.
+TOUR5:PLASA
+TOUR 6:MUSEO
+1.
+Alot of students enjoyed thephoto.They said thisand the
+1.
+The last tour.Some students who were part of Gl and G2felt
+Quiapa, Church are the two most beautiful photos among the
+sad about this.Actually,fromG3I didn'thearanyfeedback
+tours.
+all throughout the tour. Well, they said they really didn't
+2.
+TheActiveLearning-VRTourusingPPTbystudentswasa
+want to join because they are not good with pc and that if
+success. Out of the 8 pairs, 3 pairs made their tour using the
+they joined,theywould havehadalot of pcproblems
+Filipino language. This was not required, but they said they
+2.
+They said they learned about the fashion of Filipino people so
+wanted to try using the Filipino they have learned.
+it was very interesting.
+3.
+The tour is going smoothly. We could do lessons and could
+I discovered that some studentsdidn't answerthe survey soI
+finish the post-test and after-class survey.
+needtogivethemafollowup onthis.
\ No newline at end of file
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+page_content='This article is a part of the special collection Cognitive Innovation for Social Change: Multiple Disciplinary Perspectives towards Sustainable Development 1 AVANT, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
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+page_content=' 2 ISSN: 2082-6710 avant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='pl/en DOI: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='26913/avant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='202208  Piloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class: A Case of Emergency Remote Teaching in Japan  Roberto Bacani Figueroa Jr*  University of the Philippines Open University, Philippines *corresponding author: rbfigueroa1@up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='ph  Florinda Amparo Palma Gil Tokyo University of Foreign Studies, Japan  Hiroshi Taniguchi University of the Philippines Open University, Philippines  Received 6 September 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' accepted 28 January 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' published 24 August 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Abstract The State of Emergency declaration in Japan due to the COVID-19 pandemic affected many aspects of society in the country, much like the rest of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' One sector that felt its disruptive impact was education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' As educational institutions raced to implement emergency remote teaching (ERT) to continue providing the learning needs of students, some have opened to innovative interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' This paper describes a case of ERT where Filipino vocabulary was taught to a class of Japanese students taking Philippine Studies in a Japanese university using a cognitive innovation based on virtual reality, an immer- sive technology often researched for immersion and presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Students were divided into three groups to experience six lessons designed around virtual reality photo-based tours at different immersion levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' While the effect of immersion on satisfaction was not found to be statistically significant, presence and satisfaction were found to be corre- lated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Despite challenges that were encountered, benefits like enjoyment, increased en- gagement, and perceived learning were reported by the students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Our findings exemplify how emerging multisensory technologies can be used to enhance affective and cognitive dimensions of human experience while responding to gaps created by the spatial limita- tions of remote learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Keywords: emergency remote teaching in Japan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Filipino language teaching;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' language learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' virtual reality photo-based tour;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' virtual reality in education  OPEN ACCESSRoberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi  2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Introduction The COVID-19 pandemic has greatly changed societies in almost every aspect of life since the end of 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' One area that has been heavily affected by this global crisis is education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The year 2020 served as a turning point for educators across the globe, causing them to reevaluate their pedagogy and look for innovative ways to overcome the challenge of limiting in-person meetings for teaching and learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The effects varied depending on the subject being taught, the cultural setting, and the technologies available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' This paper hopes to contribute to this fast-growing body of knowledge on context-specific innova- tions by presenting a study that describes the process of prototyping a technology-based innovation in the context of what is known in the educational literature as emergency remote teaching (ERT), which was defined as the sudden and temporary shift of delivery to a distance or remote mode of instruction as a result of extremely disruptive phenom- ena (Hodges et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Mohmmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Results from this study may be of in- terest not only to researchers in the field of education, but also in other fields like psychology, information technology, and cognitive science as they have implications on the dynamics of immersion, presence, and satisfaction in multi-sensory experiences brought about by immersive technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Foreign Language Teaching in Japan Popular foreign language learning methods in Japan include in-person lessons at private language schools, online one-on-one lessons, and the use of language applications on mo- bile phones or personal computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' However, what is referred to as foreign language learning, mostly refers to learning the English language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' While the English language is taught from the elementary level to the university level, with grammar and translation methods dominating over others, the teaching of foreign languages other than English is mostly unheard of (Butler, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Among languages that were rarely taught in Japan is Filipino, the national language of the Philippines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Among Japanese universities, Osaka University and Tokyo University of Foreign Studies are the only ones that offer a full major course in the Filipino language as of this writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' With this, comes the scarcity of teaching and learning materials that are content-based (Laranjo, 2020), especially those targeting Japanese learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' These universities have fa- cilitated Filipino language education using traditional in-person classroom-based or blended pedagogy using a learning management system (LMS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Learning activities that would provide a more immersive experience for learners and establish relevance and context of the target language were offered in the form of study abroad programs or study tours embedded within the four-year curriculum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  The Situation of Education in Japan at the Dawn of the Pandemic Just like the rest of the world, university teachers and students in Japan did not expect the effect that the pandemic would bring to the academic year 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' When the COVID- 19 outbreak occurred, the Japanese government called a nationwide state of emergency  Piloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  3 and restricted people’s mobility entering and leaving Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' As a result, foreign students who returned to their home countries during the spring break in 2020 were included in the reentry ban (Osumi, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' At the same time, 80 to 90% of Japanese universities re- called their students studying abroad, giving no guarantee of returning to their programs (Wortley, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' According to the result of a nationwide survey released by Japan’s Min- istry of Education, Culture, Sports, Science, and Technology (MEXT) in May of 2020, as a countermeasure to the COVID-19 outbreak, 930 universities out of 1046 that re- sponded (86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='9%), delayed the start of classes and more than 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='6% of the said universi- ties either decided or were considering conducting distance learning using different forms of media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Since the most popular form of education in Japanese universities before the pandemic was face-to-face classes, teachers were forced to hurriedly prepare for switching to online classes during the short delay (Inoue, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' On the other hand, in- coming first-year students, expecting to start a new active life in their new school, had to attend online classes without being able to set foot on campus and before experiencing face-to-face lectures with their professors (Hirabayashi, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Although Japanese universities had already been introduced to the use of LMS for lan- guage education even before the pandemic, actual usage by teachers, staff, and students was still minimal (Murakami, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Teachers had to make significant adjustments in shifting to online classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The Mainichi Shinbun, one of the major newspapers in Japan, surveyed 66 institutions and showed one teacher describing how his preparation time has gone up fivefold while some teachers thought of doing online classes as becoming Youtubers or that they should be like radio broadcasters (Mainichi Japan, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' As for the students, it was reported that “…while 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='2 percent were positive about online clas- ses, 21 percent said they did not wish to take part, reflecting concerns about the quality of education that remote learning provides and finding the right environment for partic- ipating” (Kyodo News, 2020, para 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' In fact, in a survey from an organization petitioning the return of classes to face-to-face, it was reported that the quality of lessons was lower than face-to-face was the second highest answer given by 845 out of 1,500 respondents, regarding the perceived disadvantages of an online class (DaigakuIkitai, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Even at the University of Tokyo, which was the fastest to adapt to online classes, 70% of the stu- dents who were surveyed said that online classes were mere approximations of regular classes (Shoji, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  However, an exploratory study on first-year students’ perception of university online lessons reported that the efficiency of technological innovation brought about a positive response to online learning as well as changed the feeling of loneliness of in-home study to being able to concentrate on individual learning efficiently through online lessons (Hirabayashi, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Teachers’ lack of knowledge in developing efficient online classes seemingly presents a challenge in achieving the fourth sustainable development goal, which is about en- suring equitable and quality education for everyone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' However, it also presents an op- portunity to explore, exploit, and explain innovative ideas for revolutionizing education that would address the gaps and restrictions presented by the current situation (Ki- takka, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Roberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi  4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Virtual Reality Technology in Emergency Remote Teaching (ERT) Virtual reality (VR) is a system of specific hardware and software that brings about a real-time, computer-generated three-dimensional environment that users, having a perceived self-location, can navigate and interact with (Hayward, 1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Despite ha- ving many features, two were found to be common among those investigated by VR re- searchers: Presence and Immersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Presence or telepresence is the feeling of being there in a technology-mediated environment (Heeter, 1992), while immersion is defined as the aspects of hardware and software systems, such as display quality and stereoscopy, which could facilitate presence (Slater & Wilbur, 1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' VR has gained increasing popularity in technology-integrated classrooms as reported in studies done by Chang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2016), Coyne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2018), and Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The ability of VR to immerse learners in a virtual environment or simulate a geographical location without having to go there is potentially useful in addressing the gap caused by limited mobility among learners due to the lockdowns caused by the pandemic around the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' One instance of a VR application that provides opportunities to teachers without a programming background is the virtual reality photo-based tour or simply VR tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' VR tours are interactive tours based on 360 photos of real-world locations with hot spots around the environment that provide information about an area or objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Equipment and devices for 360 photos have become more affordable in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Furthermore, affordable and easy-to-use platforms for creating interactive tours using these media have become widely available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The lockdowns that caused the lack of access to location-based educational activities have increased the potential use of VR tours in education and other learning contexts, especially among courses and subjects that would have involved field trips in normal cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' However, reports were scarce during the time that this study was conceptualized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Even rarer were studies on remote teaching and learning methodologies for the Filipino language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The few papers found on teaching Filipino as a foreign language mainly focused on the details of the curriculum and the background of students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Only some of these pa- pers mentioned the methodology of teaching and the challenges of teaching Filipino - Quirolgico-Pottier (1997), Luquin (2016), Pambid-Domingo (2010), Barrios-Le Blanc (2010), and Laranjo (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=" Pambid-Domingo (2010) mentioned the use of authentic materials, videos, and songs, while Barrios-Le Blanc (2010) explained teachers' tech- niques in using poetry for teaching." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' However, none of them investigated using infor- mation and communication technology (ICT) in teaching the subject matter among Japanese students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  This could be brought about by the general apprehension among teachers in Japan about using ICT-based novel interventions (Teaching and Learning In- ternational Survey [TALIS], 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' According to the 2018 Teaching and Learning International Survey (TALIS) survey by the Organization for Economic Co-operation and Development, Japan ranked 45 out of 50 on teachers’ preparedness in using ICT for teaching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Fewer teachers felt "well pre-  Piloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  5 pared" or "very well prepared" in ICT for teaching in Japan compared to other participat- ing countries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' While Japanese universities have learning management systems, they are used for less than 20% of all courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Moreover, due to a lack of staff for creating digital content and maintaining ICT systems, insufficient ICT skills for both academic and gen- eral staff, and a limited budget, ICT tools were mostly used for syllabus systems (89%), student information systems (63%), and campus WiFi networks (79%), impeding the introduction and promotion of ICT tools for educational use (Funamori, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' A few months into the pandemic, news articles have shared how ICT was used by various learning institutions in Japan to fill the gap created by the state of emergency caused by the COVID-19 outbreak (O’Donoghue, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' However, the authors have not yet found any study that outlined how a particular ICT-based intervention was designed and how it affected the students’ learning experience and outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' This is also true even in for- eign language teaching and learning, much more so among less popular languages like Filipino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Aside from a recent study reporting some motivational effects of VR tours in a university in Japan (Figueroa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', 2020), no article regarding VR tours for teaching the Filipino language in Japan in the context of remote teaching has been found as of this writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Thus, when the state of emergency was first announced in the country, and a shift to online classes was decided by the university administration, an exploratory study on utilizing VR tours in teaching vocabulary to zero-beginner learners of the Fili- pino language was carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Aims of The Study This study aimed to provide insights as to how an intervention using VR tours could have an impact on Japanese students of a Filipino Language class in the context of ERT during the time of the COVID-19 outbreak in Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' More specifically, it aims to answer three research questions (RQ): RQ1: How different are the levels of satisfaction between students who experienced VR tours with varying levels of immersion?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' RQ2: How are the levels of satisfaction among students related to presence in the VR tours?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  RQ3: What are the other benefits of incorporating VR tours in an online Basic Fili- pino Language Class designed for Japanese students?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Roberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi  6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Materials and Methodology This study investigated the impact of VR tours on Japanese students of a Filipino Lan- guage class in the context of ERT during the time of the COVID-19 outbreak in Japan using both quantitative and qualitative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' In the quantitative analysis, descriptive statis- tics and non-parametric tests were used, while in the qualitative analysis, reflexive the- matic analysis was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Virtual Reality Photo-based Tours (VR Tours) The researchers adopted six VR tours into a Filipino Language Course for beginners at a Tokyo-based university over six lessons from May 22 to June 26 of 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Each VR tour was inserted into a 90-minute online synchronous lesson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The aims of each tour were fourfold: (1) to increase students’ vocabulary, (2) to develop students’ ears for recogni- zing Filipino sounds and rhythm, (3) to let them experience the Philippines albeit virtu- ally, and (4) to let them experience a new way of learning a language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' By the end of the six VR tours, the students were expected to be able to introduce popular Japanese tourist spots in a VR tour of their own using the vocabulary that they had learned from the tours and to be able to introduce the places using the correct pronunciation and rhythm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Each tour introduced a place in the Philippines while featuring seven new Filipino words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The tours were given titles based on their main location of interest, and the seven words introduced in each tour were related to the things, people, places, and events that could be found in these locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Figure 1 shows the tour used for Lesson 6, which was entitled Plasa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' A hyperlink to the interactive online tour is also provided as a note for those in- terested in experiencing it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Figure 1 Lesson 6: Plasa Virtual Tour  Note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' URL: (https://figtreeacademy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='org/vr/plaza/)  VRPiloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  7 Table 1 shows a detailed list of the tours for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The first, second, fourth, and sixth tours had audio guidance in Japanese and Filipino by two narrators (a Japanese native speaker and a Filipino native speaker).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' In contrast, the third and fifth tours were nar- rated purely in Filipino by a Filipino native speaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Table 1  The VR Tours  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Participants All six lessons were conducted among a class of 15 first-year Japanese students majoring in Philippine Studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' All students in the class volunteered to participate in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Data Collection Data collected were the following: pre-VR Tour survey, post-semester survey, six after- class surveys, two focus group discussions, and observation notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Survey questions can be found in Appendix A1 and A2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' focus group discussion questions are in Appendix B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' and observation notes are in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The pre-VR tour survey was used to decide the groupings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Random selection was carried out among learners who had compatible phones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The participants were divided into three groups - high immersion group, moderate immersion group, and low immersion group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Quantitative items from five out of six after-class surveys were used for answering RQ1 and RQ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Open-ended questions from six after-class surveys were used for answe- ring RQ3 and the results were compared to the data collected from the post-semester survey, two focus group discussions, and observation notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Number Title Date Language 1 Rizal Park (name of a park) / May 22 Japanese and Filipino Pilot Tour 2 Aurora (name of a province) May 29 Japanese and Filipino 3 Quiapo Church (name of June 5 Filipino church) 4 LRT (light railway June 12 Japanese and Filipino transportation) 5 Museo (museum) June 19 Filipino 6 Plasa (plaza) June 26 Japanese and FilipinoRoberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi  8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Procedure As shown in the procedural diagram in Figure 2 below, the study was implemented in several phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' As part of the preparation phase, students were asked to accomplish a pre-VR tour survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Students were assigned to three groups based on their phone’s specifications revealed by their answers to the pre-VR tour survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Figure 2 Procedural Diagram of the Study  Six lessons were conducted with groups following the same lesson structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' At the be- ginning of the lesson, students were asked to do a pre-test of target vocabulary words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Each virtual tour served as one of the main activities of each lesson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Lessons were structured into four phases [Lesson Introduction → Virtual Tour → Focus on Grammar → Application].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' In the Lesson Introduction phase, the theme and the objectives of the lesson were intro- duced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=" In the Virtual Tour phase, a new set of vocabulary related to the lesson's theme was introduced using a VR tour." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Students assigned to the high immersion group partici- pated in all the VR tours using their phones and VR goggles delivered to their homes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Those assigned to the moderate immersion group participated in all the VR tours using their computers or smartphones, but without using VR goggles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Those assigned to the  Preparation Pre-VRTourSurvey Pre-Test LESSON Implementation x 6 times (*1) Post-Test After-class Survey (*1)Aftereachlesson,theteacherand twootherresearchersdiscussedand wrotedowntheirobservations Post-semesterSurvey Reflection FocusGroup DiscussionsPiloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  9 low immersion group viewed and listened to PowerPoint-based tours that had the same content as the VR tours but used 2D photos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' During this phase, the teacher reminded the students to experience the VR tours three times for vocabulary reinforcement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' This was followed by the Focus on Grammar phase, where new grammar was taught.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' In the Application phase, students were asked to use the presented vocabulary words and new grammar to form sentences describing sample photos provided by the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Fi- nally, a post-test containing items related to the presented vocabulary words was given, followed by an after-class survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' After conducting lessons 1, 2, and 3 (Rizal Park, Aurora, and Quiapo Church), the students were divided into pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' They were asked to make their own tour introducing a place in Japan by using 2D photos and Google Slides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' They were allowed to choose the language they would use as long as they included seven Filipino words from the three tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Like- wise, after lessons 4, 5, and 6 (LRT, Museo, and Plasa), the students were again divided into pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' They were asked to make their own VR tour using the platform Story Spheres, and VR photos of places in Tokyo which were provided to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' They were given instruc- tion to use seven words from the last three tours and the vocabulary words learned within the semester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Another survey was conducted at the end of the semester, and from the results, six stu- dents were invited for two focus group discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Two students from each group were chosen to join the FGDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The students who were interviewed were the ones who men- tioned something related to the VR tours in their answers in the post-semester survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' After each lesson, the teacher and two other researchers discussed and wrote down their observation notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Iterative Improvement based on Challenges Encountered Strategic integration of VR tours in class involved iterations of creative thinking, brain- storming, testing, evaluation, revision, and synthesis -activities that characterize a cog- nitive innovation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Like in most innovations, several challenges were encountered during implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' These challenges were addressed by adjusting the design and imple- mentation of succeeding lessons described briefly below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' However, other challenges were not addressed due to time constraints and contextual limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Device Compatibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The students were grouped according to the compatibility of their mobile devices with VR Tours and their willingness to try VR Tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' However, two participants in the high immersion group had incompatible phones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Hence, the groupings had to be adjusted after the first lesson resulting in the decision to collect quantitative data from lessons 2 to 6 only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Device and Tour Usability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' An orientation was conducted since it was the first time for many of the participants to use VR tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' However, high immersion group participants  Roberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi  10 continued to have difficulty using the device until the second tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' They started becom- ing comfortable only from the third tour with careful guidance and reminders in operat- ing the tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Tour Technical Problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Participants from all three groups experienced technical dif- ficulty in loading the audio of the tours, most especially the high immersion group par- ticipants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Hotspots were supposed to activate certain audio explanations when gazed on or clicked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Some of the audio files were slightly larger, which caused them to be loaded rather slowly, especially when the student’s internet connection was intermittent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' This interrupted the flow of the lessons significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' However, these were resolved by re- starting the application or refreshing the page, or by using a different mobile device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Tour Language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Since the participants were Japanese speakers who generally had no exposure to the Filipino language, it was decided to have the first tour be explained in Japanese with highlights on Filipino terms that were the target words for that lesson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Labels corresponding to the target words were also placed on the hotspots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' However, this removed any challenge in the task and lessened the impact of the tour in terms of student engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Thus, the succeeding tours did not have textual labels, which turned out to be useful in training the students’ listening skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' On the other hand, the two tours that had audio in all Filipino increased the difficulty of the task immensely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content="  As a response, the teacher translated and explained the script's meaning after the third lis- tening part so students could consult the script as often as they wanted to make sense of what the tour guide was saying in each hotspot." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Data Analysis Quantitative and qualitative analyses were performed to answer the research questions of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' In this section, we discuss the three research questions and the analyses performed to answer them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' All quantitative procedures were carried out using R, an open-source software offering libraries for statistical analysis (R Core Team, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' R-Studio, a commonly used integrated development environment for R, was utilized for easy documentation and organization of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' RQ 1: Levels of Immersion and Satisfaction Ratings Participants’ satisfaction ratings collected from after-class surveys of Lessons 2 to 6 were analyzed and grouped by immersion level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The omnibus test used to determine the effect of immersion level on satisfaction was the Kruskal-Wallis test as a non-parametric alter- native to the analysis of variance for the following reasons: 1) There was a small number of participants who were randomly allocated to the treatment groups (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', moderate and high immersion groups);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 2) Samples were mutually independent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' and 3) Dependent va- riables were at least in the ordinal scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Boxplot visualizations were generated using the ggplot2 library (Wickham, 2016), while the Kruskal-Wallis function was performed using the stats library (R Core Team, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Piloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  11 RQ 2: Presence and Satisfaction Ratings This section describes the tests performed and plots that were visually inspected to an- swer research question 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Participants’ satisfaction and presence ratings collected from after-class surveys of Lessons 2 to 6 were analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Scatterplots of satisfaction and pre- sence ratings were generated in five lessons to visualize the monotonic relationship be- tween the two variables using the ggplot2 library (Wickham, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Kendall’s tau was calculated to determine the correlation between satisfaction and presence ratings based on the assumptions: 1) Data from the paired observations appeared to follow a mono- tonic relationship, and 2) variables were measured at least in the ordinal scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Further- more, Kendall’s tau was chosen over Spearman’s rho because the p-values of the former were found to be more accurate with smaller sample sizes (Marshall & Boggis, 2016), which was the case in the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' RQ3: Benefits of VR tours The qualitative analysis employed in this study was reflexive thematic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' To exe- cute the analysis, data were collected from the six after-class surveys, the transcription of the two FGDs, and the observation notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' These data were collected to answer RQ3: What are the other benefits of incorporating VR tours in an online Basic Filipino Language Class designed for Japanese students?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  The data analyzed from the six after-class surveys were the reasons given by the partici- pants regarding the rating of their experience of trying each VR Tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The reasons were categorized into codes and sub-themes until two main themes emerged from the catego- rization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' After this process, the codes and sub-themes were triangulated with the an- swers of the six participants invited to the FGDs, results from the post-semester survey, and observation notes of the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Categorizing the reasons given by the participants regarding the rating of their VR Tour experience produced 20 codes which were further categorized into 7 sub-themes and finally into two main themes – the Benefits of VR Tours and the Problems and Chal- lenges encountered while doing the VR Tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Results Notable findings from analyses that were carried out are organized according to the three research questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' RQ 1: Levels of Immersion and Satisfaction Ratings The boxplots in Figure 3 visually illustrate the median and interquartile range of satis- faction ratings given by the three groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Outliers were found in the high immersion group in the second lesson and the sixth lesson, and the moderate immersion group in the fifth lesson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Roberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi  12 Figure 3 Satisfaction Boxplots of Three Immersion Groups per Lesson  Figure 4 Scatterplots of Satisfaction and Presence Ratings in Five Lessons  The medians of satisfaction ratings by the low immersion, moderate immersion, and high immersion groups in the second lesson were 8 (IQR = 5 - 8), 9 (IQR = 9 - 10), and 7 (IQR = 7 - 8), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The medians of satisfaction ratings by the low immersion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' moderate  Lesson2:AuroraTour Lesson3:QuiapoTour Lesson4:LRTTour 10- 8 - 6 - Satisfaction 0 0 2 Lesson 5: Museo Tour Lesson6:Plasatour 10 2 8 - 9 4 0 1 2 0 1 2 ImmersionLesson2:AuroraTour Lesson3:QuiapoTour Lesson4:LRTTour 10 8· 6 4 Presence 6 8 10 Lesson5:MuseoTour Lesson6:Plasatour 10 8 6 6 8 10 6 8 10 SatisfactionPiloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  13 immersion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' and high immersion groups in the third lesson were 8 (IQR = 7 - 10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 9 (IQR = 8 - 10),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' and 8 (IQR = 7 - 8),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The medians of satisfaction ratings by the low immersion, moderate immersion, and high immersion groups in the fourth lesson were 8 (IQR = 7 - 10), 9 (IQR = 9 - 10), and 9 (IQR = 7 - 9), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The medians of satisfac- tion ratings by the low immersion, moderate immersion, and high immersion groups in the fifth lesson were 6 (IQR = 5 - 8), 10 (IQR = 0), and 7 (IQR = 6 - 9), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Finally, the medians of satisfaction ratings by the low immersion, moderate immersion, and high immersion groups in the sixth lesson were 8 (IQR = 7 - 10), 10 (IQR = 9 - 10), and 8 (IQR = 8 – 9), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The medians and interquartile ranges presented in the boxplots also revealed that students in the moderate immersion group generally gave higher sa- tisfaction ratings compared to the low and high immersion groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' However, no statistically significant differences were found among three groups who ex- perienced tours in the second, [H(2) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='52, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='11];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' third, [H(2)= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='49, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='39];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' fourth, [H(2)= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='63, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='44];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' fifth, [H(2)= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='99, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='08];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' and sixth, [H(2)= 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='08, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='22], lessons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' RQ 2: Presence and Satisfaction Ratings The scatterplots shown in Figure 4 illustrate the monotonic relationship between the two variables in five lessons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Results revealed that there was a statistically significant strong correlation (N=15) between presence and satisfaction ratings in the second (τb = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='50, p=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='02), third (τb = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='51, p=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='02), fourth (τb = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='67, p=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='002), fifth (τb = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='65, p=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='002), and sixth (τb = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='56, p=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='01) lessons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' RQ3: Benefits of VR tours The codes and categories collected in the thematic analysis are presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Two main themes emerged: Benefits and Challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Since the challenges were already dis- cussed in the methodology section as part of the iterative improvement process, this sec- tion will discuss the subthemes related to benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Increased Engagement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Out of the 20 codes identified, most of the reasons given by the participants for their VR Tour experience ratings alluded to the excitement of doing something new or the feeling of being transported to a different place which made them say that learning was enjoyable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' These trends were also the first things that the teacher observed among students who used the VR tours for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Participants of the high immersion group and moderate immersion group were observed to have eagerly and diligently engaged in learning the new sets of vocabulary words and in fini-shing the ac- tivities of each tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=" The novelty of the intervention naturally stirred students' interest in participating in each class." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Although a few chose to use the conventional PowerPoint presentation-based tours because they said that they were not good with technology or that their mobile phones and PC were not compatible, five out of the six participants of the focus group discussions said that the VR tours were their most en-joyable activity in class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The tours made them feel that they were in the Philippines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' One of the participants said, “I could feel the Philippines with the VR tour even if I’m not there.” Three out of the five participants even said that the application activity of the course, which allowed them  Roberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi  14 to make their own tours, was extremely fun and made them think about the difficulties encountered by the teacher and researchers in setting up the tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Table 2 Codes and Themes Created from the Categorization Process  Perceived Vocabulary Retention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The second prevailing theme in the observations and survey results was that students had perceived ease of remembering the words because of the context provided by the virtual tours in the lessons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' They recalled that they quickly remembered and understood how the words would be used in the Philippines as they referenced objects in the tour, which made it easy to associate them with their meanings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Many of them associated their ease of remembering the words with the feeling of being in the actual place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' This was further validated by the participants of the focus group dis- cussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' A participant commented,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' “We can see the real Philippines,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' and with the actual  ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Main ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Themes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Sub-themes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Codes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Benefits ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Increased ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='-Excitement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='towards ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='new ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='experience ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='(12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Engagement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='-Excitement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='feeling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='if ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='actual ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='place ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='(10) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='-Enjoymentinlearning(7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Perceived ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='-Ease ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='memorizing ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='words ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='(9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Vocabulary ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='-Easeofunderstanding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
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+page_content=' We encountered those vocabulary many times,” and “I was able to listen to Filipino sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' I was able to practice listening to Filipino words and enjoyed learning words.” From comments like these, the authors learned that the study was able to achieve the aims of each tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Interest in the Subject Matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The third prevailing theme was that the VR Tours stirred their interest not just in the activity, but also in the subject matter: the Philippines and its national language, one of the goals of the whole program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' As further explained in the latter section, according to the post-semester survey, students with an initially slightly negative view of the country changed their impression of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The VR tours made them want to visit it more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' This was also supported by the frequency of the students’ answers regarding the aesthetic value of the tours, which led them to be interested in the lessons as reported in the after-class surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Authentic Information about the Philippines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The fourth prevailing theme found from after-class surveys was the authenticity of the information about the Philippines that the participants were able to get from the VR Tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Participants wrote that they could “feel the atmosphere” of the places introduced in the tours, like the church or the train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Half of the participants also mentioned that they were happy to be able to “learn about the his- tory of the Philippines”, or “see everyday life in the Philippines.” The authenticity of the information they got from the tours made them feel as if they were walking quietly inside a church full of people praying or as if they were also stuck inside a crowded train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Two focus group discussion participants agreed that “All tours were fun.” It was followed by the comment, “the image about the Philippines made me think I want to go there.” Changes in perception of the Philippines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Aside from the after-class surveys, data were gathered from the post-semester survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The answers showed important points about the benefits of using VR Tours in a Filipino language class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Answers revealed that some of the students who experienced the VR tours considered the activity one of the most useful activities during the period and that experiencing the tours changed their perception of the Philippines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The six students who wrote about this also said during the focus group discussions that their perception of the Philippines significantly changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Some of them said that before the tours, they were anxious about visiting the country because they had only associated the Philippines with hot weather and poverty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  How- ever, after experiencing the VR tours, they discovered something new.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Answers from stu- dents like “I thought many people are poor, but once I saw Manila so developed, I wanted to go” and “I am getting interested in the Philippines more and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' I want to visit it” set the tone of the focus group discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' They also mentioned that experiencing local and rural life was one of the things that made the tours interesting because what the media had shown them was only about tourist attractions, politics, or poverty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' This made them feel more excited to visit the country and learn more about the language to prepare them- selves for the actual visit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Roberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi  16 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Discussion The study described in this paper primarily aimed to investigate how integrating VR tours supported a class in Japan that shifted to synchronous online classes as an emer- gency remote teaching strategy due to the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The findings of the study unearthed several themes that are worth discussing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Emergency Remote Teaching as an enabler for Cognitive Innovation As Gummerum and Denham (2014) explained, a cognitive innovation starts with an ex- ploratory phase of creatively looking for ideas and probing boundaries, followed by choosing ideas that could be exploited, tested, and improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' It is then followed by re- flecting on the experience and synthesizing what has transpired to explain phenomena and generate new questions and ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' At first glance, one might think that cognitive in- novation can only thrive in a more relaxed, unconstrained, and open environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' However, in this instance, the cognitive innovation of incorporating novel VR tours into an online lesson that involved iteratively brainstorming lesson design and structure-re- lated ideas, selecting which of these ideas to exploit, and constantly reflecting on learning points and things to improve each week, was enabled by limitations brought about by the state of emergency in Japan during the COVID-19 outbreak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' While the term emergency was used in both emergency remote teaching and state of emergency, its contextual meaning is not comparable to a dying patient being rushed to a hospital or a burning house.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Rather, it pertains to a state where certain interventions that were not normally allowed would be considered because of an extremely disruptive phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' For example, in a normal situation, a teacher in Japan would not be allowed to teach a class online in a traditional university.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  However, because of the state of emer- gency pronouncement by the government where universities must disallow in-person activities to prevent the spread of COVID-19, conducting classes online through various platforms has become permissible to provide a way for educational institutions to de- liver services to their learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' In the same way, because of the sudden closure of borders that prevented learners from learning Filipino vocabulary in a more immersive environ- ment through field trips, VR was considered to be a possible means of allowing learners to somehow experience certain places in the Philippines while learning the language without having to go there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Before the phenomenon, the idea of using virtual tours in a university setting was not something that could be easily considered as students were able to receive ample stimuli and realistic input from study abroad programs and in-person interactions with native speakers in the physical classroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The void created by the phenomenon made teachers, learners, and school administrators open to new ideas that would not have been ideal in normal cases, as revealed by reflection conversations held among the observers and the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' In this way, temporal and spatial restrictions in education gave way to the open- ness to new ways of experiencing places while learning and actively participating in ma-  Piloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  17 king the succeeding iterations better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' It also made researchers more attentive to feed- back and less afraid to exploit alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The phenomenon was so unique that no pre- vious guideline could have restricted them from trying new ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' While the challenges and difficulties encountered in the exploratory study supported re- cent findings (Graeske & Sjöberg, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Phoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', 2021), specifically those involving the use of VR goggles and internet connectivity, reflecting on them were useful in as- sessing how VR tours as an online learning supplementary intervention can be replicated successfully in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' As of this writing, the tours created have been used in two subsequent offerings of the same course, where most of the exploration was presented without VR goggles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' However, VR is progressively recognized as a powerful tool for learning and productivity by institutions and governments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Moreover, with some inter- national gatherings like conferences being held in virtual reality, technology companies developing more affordable and portable VR goggles, and groups championing standar- dization of its use in the educational setting (European Committee for Standardization, 2022), the capability of having the tours viewed using VR goggles by all students will most likely increase should another situation like this arise in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' VR tour: Novelty, Presence, and Satisfaction Novelty helps with subsequent learning (Fenker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Schomaker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' It is also associated with a release of dopamine in the hippocampus (Biel & Bunzeck, 2019), which is related to motivation and pleasure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' However, novelty is expected to wane as learners get more familiar with the intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The novelty of VR could have contrib- uted to the satisfaction ratings of learners in the first and second lessons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' However, the satisfaction ratings and sustained high interest of students in the four succeeding lessons suggested that there was something more than the novelty of VR tours that made them interested not only in the target vocabulary but also in the host country of the language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' An affective aspect of the learning experience made them want to learn more about the country and the culture besides the language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Presence or the feeling of being in the vir- tual place was frequently found to be the reason for students enjoying the activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The results of the correlation analysis supported this observation by reporting a strong cor- relation between satisfaction and presence ratings in all five lessons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Learning Experience more than Outcome All fifteen students performed well in the course.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Their learning ability as well as the other interventions like the practice exercises, could have mediated actual vocabulary acquisition in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Thus, post-test scores were not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' However, the impact of VR tours, as revealed in their interviews, gave us a glimpse of how they would perform in the succeeding courses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Studies have shown that enjoyment in the learning experience is positively related to actual engagement and learning attitudes (Cybinski & Selvana- than, 2005), which lead to better learning outcomes in the long-term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Roberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi  18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Immersion, Presence, and Affective Outcomes While the relationship between immersion and presence has been well established in the literature (Gorini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Lessiter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', 2001), the study’s findings were insight- ful because they brought about interesting realizations regarding the objective features of technology (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', immersion) and presence being conduits for the affective dimension of the human experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=" While immersion was previously found in the literature to be directly influential on presence, the study's findings revealed that the ensuing affective outcomes were only associated with presence implying that the technological features of the device did not solely predict desirable affective outcomes from the multisensorial virtual reality experience." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The explanation for this can only be gleaned from further stu- dies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' However, based on personal observation, it could be investigated using the lenses of user experience (UX) and ergonomics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Furthermore, the relationship between pres- ence and satisfaction led to the question, “What makes presence satisfying in a virtual ex- perience?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' While it has been social presence that has been found to reduce the adverse effects of physical distance and increase satisfaction among remote learners in past stu- dies (Moore, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Richardson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', 2017), results from this study suggest that presence experienced in a virtual space regardless of interaction or social connection could in- crease satisfaction and perceived learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' This can be further investigated by looking into the constructs of interest and sense of place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' One could test whether their influence on emotions could be moderated by this spatial form of presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Conclusion and Recommendations We are ten years away from the deadline of achieving the UN’s Sustainable Development Goals, a part of which is achieving quality education for all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' In the context of language education, it is our humble belief that achieving quality education cannot be proven by having high scores in exams and tests as proven by the discrepancy of language test scores and actual language learning as exemplified by the study of Nicholson (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' In- stead, we believe that quality education can be achieved by providing enjoyable expe- riences that establish relevance and authenticity as positive emotions facilitate retention of what students have learned (Dulay & Burt, 1977;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Krashen, 1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' In this paper, we presented a new way of learning a foreign language in a remote teaching context through a cognitive innovation based on VR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' However, our findings were not without limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The study’s lack of generalizability was partly due to the small number of participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Therefore, an experimental study on testing the effect of immersion on satisfaction levels with larger group sizes is suggested for future researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The random assignment of students to groups was limited by the capabilities of their own mobile devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' While this was expected from the study’s current context, future researchers may benefit from the fast-paced evolution of technology and recruit participants whose mobile devices would be compatible with VR applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Furthermore, a face-to-face classroom setting would ensure that everyone would have the chance to use VR goggles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content="  Piloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  19 In conclusion, we hope that the study's contributions, albeit small, would add to the va- rious means of responding to geographical and spatial limitations in education brought about by the pandemic by creating cognitive innovations based on a novel technology like VR." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Moreover, the initial findings of the study could support and guide future re- searchers who are interested in presence, immersion, and other aspects of VR, as well as their impact on learning outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Furthermore, practitioners may benefit from the study as it presented one of the many alternatives and supplements to location-based learning provided by study abroad programs and field trips that would otherwise be in- accessible to more than half of the world’s population due to poverty, geopolitical rea- sons, or disruptions brought about by phenomena like the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The realizations gleaned from the findings opened up questions regarding immersion’s rela- tionship with affective outcomes and suggested lenses for further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Moreover, the study suggested new hypotheses to be investigated regarding presence and the affective dimension of the human experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
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+page_content=' International Education Studies, 14(8), 76-83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Gummerum, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', & Denham, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Cognitive innovation: From cell to society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Europe’s Journal of Psychology, 10(4), 586-588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Hayward, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Situating cyberspace: The popularisation of virtual reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Future visions: New technologies of the screen (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 180-204).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' British Film Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Heeter, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Being there: The subjective experience of presence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Presence: Teleoper- ators & Virtual Environments, 1(2), 262-271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Hirabayashi, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' コロナ禍における大学のオンライン授業に対する新入生の認 識についての探索的研究[ Corona Exploratory research regarding the recognition of new students to online university classes in].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 共栄大学研究論集= The Journal of Kyoei University, (19), 55-66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Hodges, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', Moore, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', Lockee, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', Trust, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', & Bond, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The difference be- tween emergency remote teaching and online learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' EDUCAUSE Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' https://er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='educause.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='edu/articles/2020/3/the-difference-between-emergency-re- mote-teaching-and-online-learning Inoue, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 学生によるオンライン授業の点検・評価 [Inspection and evaluation of online lessons by students].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 環太平洋大学研究紀要= BULLETIN OF INTERNA- TIONAL PACIFIC UNIVERSITY, (17), 59-67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Piloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  21 JAPAN MEXT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 新型コロナウイルス感染症対策に関する大学等の対応状況につ いて[Response status of universities, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' regarding measures against new corona- virus infections].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='記者.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', Park, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', Yuk, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', & Lee, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Virtual reality simulations in physics education.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Interactive Multimedia Electronic Journal of Computer-Enhanced Learn- ing, 3(2), 1-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Kittaka, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2020, April 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Coronavirus crisis offers chance to update Japanese schools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The Japan Times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Retrieved from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='japantimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='jp Krashen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Acquiring a second language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' World Englishes, 1(3), 97-101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Laranjo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Mapping Philippine studies in Northeast Asia: A SWOT analysis of Southeast Asian Studies programs from China, Japan and Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' SUVANNABHUMI Multi-disciplinary Journal of Southeast Asian Studies, 111-130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Lessiter, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', Freeman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', Keogh, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', & Davidoff, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' A cross-media presence question- naire: The ITC-sense of presence inventory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Presence: Teleoperators & Virtual Envi- ronments, 10(3), 282-297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Luquin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Ang pagtuturo ng Filipino sa mga Pranses (Panayam).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Daluyan (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 224-232).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Sentro ng Wikang Filipino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Mainichi Japan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
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+page_content=' Japan univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' lecturers worry over crumbling class quality as virus forces courses online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The Mainichi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Retrieved from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='mainichi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='jp Marshall, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
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+page_content=' The statistics tutor’s quick guide to commonly used statis- tical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The University of Sheffield.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
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+page_content=' Emergency remote teaching during Coronavirus pandemic: The current trend and future directive at Middle East College Oman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Innovative Infrastructure Solutions, 5(3), 1-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Moore, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
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+page_content=' The theory of transactional distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' In Handbook of distance edu- cation (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
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+page_content=' Routledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Murakami, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
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+page_content=' Japanese University students and learning management systems - 日本人大学生と学習管理システム-.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Learning (学習特別号),23(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Nicholson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Evaluating the TOEIC® in South Korea: Practicality, reliability and validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' International Journal of Education, 7(1), 221-233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' O’Donoghue, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2020, April 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
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+page_content=' Retrieved from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
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+page_content=' (2019, June 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' TALIS 2018 results (volume I): Teachers and school leaders as life- long learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
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+page_content='  Roberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi  22 Osumi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2020, June 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=" Japan's re-entry ban threatens scholarships, admission and graduation for foreign students." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The Japan Times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Retrieved from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='japantimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='jp Pambid-Domingo, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Introductory Filipino at UCLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' In R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Mabanglo & R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Ga- lang (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' ), Essays on Filipino language and literature (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 139-148).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Anvil Publishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Phoon, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', Idris, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', & Nugrahani, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Virtual reality (VR) in 21st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' century ed- ucation: The opportunities and challenges of digital learning in classroom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Asian Pen- didikan, 1(2), 105-110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Quirolgico-Pottier, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Ang pagtuturo ng Filipino sa mga Pranses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Daluyan (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 11-20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Sentro ng Wikang Filipino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' R Core Team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' R: A language and environment for statistical computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 2013 Vi- enna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Austria R Foundation for Statistical Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Richardson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', Maeda, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', Lv, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', & Caskurlu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=" Social presence in relation to students' satisfaction and learning in the online environment: A meta-analysis." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Com- puters in Human Behavior, 71, 402-417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Schomaker, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', van Bronkhorst, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', & Meeter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Exploring a novel environ- ment improves motivation and promotes recall of words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Frontiers in Psychology, 5(918).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='3389/fpsyg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='00918 Shoji, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2020, November 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=" Japan's students struggle to embrace online learning amid COVID-19." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The Japan Times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Retrieved from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='japantimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='jp Slater, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=', & Wilbur, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  A framework for immersive virtual environments (FIVE): Speculations on the role of presence in virtual environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Presence: Teleopera- tors & Virtual Environments, 6(6), 603-616.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Wickham, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Programming with ggplot2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' ggplot2 (pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 241-253).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Wortley, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' (2021, March 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The pandemic left Japanese students studying abroad scrambling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' A year later, what’s happened to their academic dreams?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The Japan Times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Retrieved from https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='japantimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='jp  Piloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  23  Appendices  Appendix A1 Survey Forms Used for Analysis One of the main data used in this study was the after-class survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' From all the questions, the authors focused mainly on questions number two and three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Question number two was mainly used for quantitative analysis and question number three was mainly used for qualitative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  After-class Survey Questions 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='少了一体上、今筱今日语巢使确率 以龙思以？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='名前/二木一么(English:Name/Nickname English: How much do you see yourself using the Filipino words you learned today in the future after the tour?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 低 (Lowest) 高 (Highest) 1 --- 2 --- 3 --- 4 --- 5 --- 6 --- 7 --- 8 --- 9--- 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='今回の体俩龙 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='VR岁一の中食感 English: How would you rate your experience?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' English:Whatwerethepositivefeelingsyouhad duringtheVRtour?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 良<(Lowest)良(Highest) 1 --- 2 --- 3 --- 4 --- 5 --- 6 --- 7 --- 8 --- 9 --- 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='周2の伍の理由述～ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content="VR一の中良感 English: What's the reason for your rating in number 2?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' English: What were the negative feelings you had during the VR tour?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='岁了一自体件机面百感龙？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='一遥人下<龙English:ChooseOne English: How interested were you in the actual experience?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' VR一龙の享真龙の感 面白(Lowest)面白(Highest) English: During the VR Tour, I felt like I was just looking at a photo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 1 --- 2 --- 3 --- 4 --- 5 --- 6 --- 7 --- 8 --- 9--- 10 一本当体の感 English: I felt like I was in an actual tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='龙の写真見龙、一本当体 の感？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' L？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=" English:Howmuchdidyoufeelthatyouwereinthetour and notjust English: How much were you interested in the lesson's content (new looking at a photo?" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' words)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 感(Lowest)感(Highest) 1 --- 2 --- 3 --- 4 --- 5--- 6 --- 7 --- 8 --- 9--- 10 1 --- 2 --- 3 --- 4 --- 5 --- 6 --- 7 --- 8 --- 9--- 10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='奥味持下部分法？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='当の忘人下遥人下龙去 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='今俊の授の一体思 主？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='世思心？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' English: What were the most interesting parts?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' English: Would you like to do more of these tours in future online classes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Why or why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='の少了一忘体膝上、自身将来今日誓个 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='の他今回の体、提案、實周等 の使の想像？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='の场面 英语在使思？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' English: Please share other comments, suggestions, or questions English: Do you see yourself using the Filipino words you learned regarding the whole experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' today in the future after thetour?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='If yes, how?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Roberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi  24 Appendix A2 Post-semester Survey The result of the qualitative analysis using question number three of the after-class sur- vey was cross-referenced with the answers of the students in the post-semester survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Post-semester Survey About Taking Filipino as Major About the Content of the Class I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Why did you decide to study Filipino?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' What do you want to become in the future (what is your dream job, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='How much did you understand about the lessons?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' dream life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='etc)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' ONot atall Overymuch 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='WhatwasyourimageofthePhilippinesandoftheFilipinosbefore taking the Filipino class?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Which part of the class did you like?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Check all that apply 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' What was your image of the Philippines and of the Filipinos after OGrammar explanation taking the Filipino class?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' OVocabulary quizzes About Online Class OOral recitation (trying to make sentences) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='How did you like studying online?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' ONot at all OVery Much OShort class discussion about the theme of the lesson 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' What are the advantages/merits of studying online?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' OReflection each week (Muni-muni) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' What are the disadvantages of studying online?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' OVirtual Tours, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='How was the length of each class?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Overy short OVery long OActive Learning (Video about Learning a Foreign Language, Making 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='If given a choice again, where would you like to have this class, Virtual Tours) online or in an actual classroom?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Oonline OIn an actual classroom OWhichever is fine 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='In general, which part of the class is most useful for you?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Check all About the Content of theClass that apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='In general, how do you like the content of the class?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  OGrammar explanation ONot at all OVery much OVocabulary quizzes 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Do you think the themes/topics chosen for this class were appropriate OOral recitation (trying to make sentences) (self-introduction,family,favoritethings,birthday,themesofthevirtual OShort class discussion about the theme of the lesson tours like Rizal Park, Quiapa Church, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=')?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' OReflection each week (Muni-muni) ONot at allOVery much OVirtual Tours, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Which theme or topic did you like?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Check all that apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' OActive Learning (Video about Learning a Foreign Language, Making OSelf- introduction, Family Virtual Tours) OBirthday OFavorite Things, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Which Filipino words or expressions do you think you will remember OThings inside a room for a long time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' ODescribing People ODescribing Place, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' What is the most important message or information did you learn in ORizal Park this class?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' OAurora 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Please give any suggestion on how to improve this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' OLRT OPlasa OMuseo, OUniversities in the PhilippinesPiloting Virtual Reality Photo-Based Tours among Students of a Filipino Language Class …  25 Appendix B Focus Group Discussion (FGD) Questions In addition to the answers of the students in the post-semester survey, the result of the qualitative analysis using question number three of the after-class survey was also cross- referenced with the answers of selected students during the two FGDs conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Focus Group Discussion (FGD) Questions: 1．世、語の勉强遥捉の？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='思？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 、語积？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' What did you think about the Virtual tours?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Why did you decide to study Filipino?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Did you choose Filipino or not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='役立？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='、YES の役思？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 2．語の学の味 Were they useful?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' If yes, how?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' ？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='11  の、答 From a scale of 1 to 10, how interested are you in learning 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='一番、好了一？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='何故 Filipino?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 下加？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' What was your favorite tour?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Why?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 語の学何期待？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 、の期待法美现？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 一番、面白了一？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='、 What were you expecting to learn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Did your expectations 机法何故下？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' come true?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' What was your least favorite tour?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Why?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 语の授知 15．了～の心時間共变化？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 、の感？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' の、何故、化？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' How did you feel when you learned that the Filipino class Did your interest in the tours change through time?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' How and will be held online?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' why?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 語の才授の最初の1、２、 16．、来、自分語姿 の感？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' ？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 寸？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='簡单下？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Did you imagine yourself speaking Filipino in the future?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' How were your first few days of learning Filipino in an online class?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Was it difficult?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Was it easy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='の强？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' effect？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 授の中下一番、楽活動何下？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Did the virtual tours make your image stronger?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Was it 飞机法何故下？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' effective?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' What were themost enjoyable activities in the class?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Why?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 18．困難感何 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 、授前の考 ？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' L?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' What were the problems you encountered with the virtual How did you think of the Philippines before the class?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' tours?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 、、の考 19．来年以降の授、一于了一 办？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 加?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' How do you think of the Philippines now?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Would you recommend using virtual tours in future classes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 、の方·考方、授の 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='の答、何故？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 中方·考方变？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Why/Why not?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 、？' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' If your view changed, was there something in class that helped you change it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' How?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='R以外面 办?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Is there any activity you had fun except VR tour?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Roberto Bacani Figueroa Jr, Florinda Amparo Palma Gil & Hiroshi Taniguchi  26 Appendix C Observation Notes The notes of the authors based on observations during the classes and from the discus- sion and brainstorming during the weekly researchers’ meetings were used to address the challenges encountered during each tour session.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The challenges were tackled by ad- justing the design and implementation of succeeding lessons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Observation Notes TOUR 1: RIZAL TOUR 2: AURORA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Started with an orientation on how to use VR Goggles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The new grouping suits the students.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The students seem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Students had a hard time using the goggles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' happier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' They were excited and eager to do the VR Tours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Things went smoothly this time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' I guessI wasnotabletoclearlyexplainthegroupings 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=" We could finish up to after-class survey plus we were able to because at the end of the class, some students say they didn't dothelesson ofthe day." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=" knowwhattheyshoulddoorshould'vehavedoneduringthe 4." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' I heard positive feedback from the students today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' I think first VR Tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' they enjoyed the tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Some students were lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The pre-test and post-test seemed very easy for them 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Two to three students complained that they did not hear anything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=" Everything took a lot of time - couldn't do the lesson for the day,just theVR Tour." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='TheTour needmoretimeto explain- maybe next time a separate day for Orientation, Pre-VR Tour Survey, practice, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=" Around2studentsrequestforchangeofgroupingstoGroup 3(PowerPoint-basedTour)becausetheirPCcouldn'ttake the tour's memory." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=" One said she is not good with technology so she really didn't know what to do during the whole process." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' No time for after-class survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=" We couldn't do it." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  The post-test seemed very easy for them so they all got a perfect score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='I think this is because we listenedto the audio file several times and thetargetwords werewritten explicitly in thetour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Maybethelabels should betaken out fromthe nexttour TOUR 3: QUIAPO CHURCH TOUR 4: LRT 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Thepre-test and post-test seemed very easy for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Or they 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='They also enjoyed this tour because it seemed real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' are very good in guessing?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Especially for the pre-test or they They asked why there were only guys on the train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' I already know the words from other classes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Some words may explained about “women\'s car" havealreadybeenintroducedinotherclasses-needto 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=" I'm not really sure if G1 students are still using the goggles." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' considerthis in next experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' They have their videos turned off to save internet memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' I 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' A lot of students -G1 and G2 said they liked the photo - couldn\'t force them to turn it on because of school very beautiful and it was their first time “to enter"a church regulations to respect students\' feelings regarding turning on 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' One student who was supposed to be in G2 used the VR Tour and off videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' What to do about this?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' For now, just keep on linkforGl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Myfault!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Needtosendlinkexclusivelytoeach reminding them to use goggles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' G!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' students not on the group chat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Theywere shocked tohear/listentoanall-Filipinotourbut some students said it was good because they want to listen to moreFilipino ifpossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' TOUR5:PLASA TOUR 6:MUSEO 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='  Alot of students enjoyed thephoto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='They said thisand the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The last tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content='Some students who were part of Gl and G2felt Quiapa, Church are the two most beautiful photos among the sad about this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content="Actually,fromG3I didn'thearanyfeedback tours." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' all throughout the tour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=" Well, they said they really didn't 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' TheActiveLearning-VRTourusingPPTbystudentswasa want to join because they are not good with pc and that if success.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' Out of the 8 pairs, 3 pairs made their tour using the they joined,theywould havehadalot of pcproblems Filipino language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' This was not required, but they said they 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' They said they learned about the fashion of Filipino people so wanted to try using the Filipino they have learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' it was very interesting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' The tour is going smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=" We could do lessons and could I discovered that some studentsdidn't answerthe survey soI finish the post-test and after-class survey." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
+page_content=' needtogivethemafollowup onthis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9AzT4oBgHgl3EQf8f40/content/2301.01904v1.pdf'}
diff --git a/z9E5T4oBgHgl3EQfOQ4n/content/tmp_files/2301.05495v1.pdf.txt b/z9E5T4oBgHgl3EQfOQ4n/content/tmp_files/2301.05495v1.pdf.txt
new file mode 100644
index 0000000000000000000000000000000000000000..c3199d2fb22be1846396fe4ccaab94108d53128b
--- /dev/null
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@@ -0,0 +1,6463 @@
+Resampling techniques for a class of
+smooth, possibly data-adaptive
+empirical copulas
+Ivan Kojadinovic1 and Bingqing Yi1,2
+1CNRS / Universit´e de Pau et des Pays de l’Adour / E2S UPPA
+Laboratoire de math´ematiques et applications IPRA, UMR 5142
+B.P. 1155, 64013 Pau Cedex, France
+2School of Mathematics & Statistics
+The University of Melbourne
+Parkville, VIC 3010, Australia
+Abstract: We investigate the validity of two resampling techniques when
+carrying out inference on the underlying unknown copula using a recently
+proposed class of smooth, possibly data-adaptive nonparametric estimators
+that contains empirical Bernstein copulas (and thus the empirical beta cop-
+ula). Following Kiriliouk, Segers and Tsukahara (2021), the first resampling
+technique is based on drawing samples from the smooth estimator and can
+only can be used in the case of independent observations. The second tech-
+nique is a smooth extension of the so-called sequential dependent multiplier
+bootstrap and can thus be used in a time series setting and, possibly, for
+change-point analysis. The two studied resampling schemes are applied to
+confidence interval construction and the offline detection of changes in the
+cross-sectional dependence of multivariate time series, respectively. Monte
+Carlo experiments confirm the possible advantages of such smooth infer-
+ence procedures over their non-smooth counterparts. A by-product of this
+work is the study of the weak consistency and finite-sample performance of
+two classes of smooth estimators of the first-order partial derivatives of a
+copula which can have applications in mean and quantile regression.
+MSC 2010 subject classifications: Primary 62G05; secondary 62G20.
+Keywords and phrases: data-adaptive smooth empirical copulas, empir-
+ical beta copula, smooth data-adaptive estimators of the first-order partial
+derivatives of the copula, smooth bootstraps, smooth sequential dependent
+multiplier bootstraps, strong mixing.
+Contents
+1
+Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+2
+2
+Smooth, possibly data-adaptive, empirical copulas and their asymptotics
+4
+2.1
+Smooth, possibly data-adaptive, nonparametric copula estimators
+5
+2.2
+Asymptotics of related sequential processes
+. . . . . . . . . . . .
+7
+3
+Bootstrap by drawing samples from the estimators in the i.i.d. case . .
+10
+3.1
+Drawing samples from the smooth empirical copulas . . . . . . .
+11
+3.2
+Asymptotic validity results
+. . . . . . . . . . . . . . . . . . . . .
+12
+3.3
+Finite-sample comparison of two smooth bootstraps
+. . . . . . .
+13
+1
+arXiv:2301.05495v1  [math.ST]  13 Jan 2023
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+2
+4
+Smooth sequential dependent multiplier bootstraps in the time series
+case
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+16
+4.1
+Main intuition and existing work . . . . . . . . . . . . . . . . . .
+16
+4.2
+I.i.d. and dependent multiplier sequences . . . . . . . . . . . . . .
+16
+4.3
+Non-smooth sequential dependent multiplier replicates . . . . . .
+17
+4.4
+Smooth sequential dependent multiplier replicates
+. . . . . . . .
+18
+4.5
+Finite-sample comparison of three multiplier bootstraps . . . . .
+20
+4.6
+Application to change-point detection
+. . . . . . . . . . . . . . .
+25
+5
+Estimators of the first-order partial derivatives of the copula
+. . . . .
+28
+5.1
+Estimators based on finite differences of the empirical copula
+. .
+29
+5.2
+Two classes of smooth estimators . . . . . . . . . . . . . . . . . .
+30
+5.3
+Weak consistency . . . . . . . . . . . . . . . . . . . . . . . . . . .
+32
+5.4
+Finite-sample performance of selected estimators . . . . . . . . .
+34
+6
+Conclusion
+. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+38
+A Proof of Corollary 2.13 . . . . . . . . . . . . . . . . . . . . . . . . . . .
+39
+B Proofs of Proposition 3.3 and Lemma B.2 . . . . . . . . . . . . . . . .
+40
+C Proof of Theorem 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+42
+D Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+46
+E
+Proofs of Propositions 5.3, 5.5, 5.9 and 5.10 . . . . . . . . . . . . . . .
+48
+References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
+57
+1. Introduction
+Let X 1:n = (X1, . . . , Xn) be a stretch from a d-dimensional stationary time
+series (Xi)i∈Z of continuous random vectors. From a well-known theorem due
+to Sklar (1959), the multivariate distribution function (d.f.) F of each Xi can
+be expressed as
+F(x) = C{F1(x1), . . . , Fd(xd)},
+x ∈ Rd,
+(1.1)
+in terms of a unique copula C and the univariate margins F1, . . . , Fd of F.
+Representation (1.1) is at root of many applications in probability, statistics and
+related fields (see, e.g., Hofert et al., 2018, and the references therein) because
+it suggests that F can be modeled in two separate steps: the first (resp. second)
+step consists of estimating the univariate margins F1, . . . , Fd (resp. the copula
+C). This work is only concerned with the estimation of the copula.
+Statistical inference on the unknown copula C frequently involves the use of
+a nonparametric estimator of C. The best-known one is the empirical copula
+(R¨uschendorf, 1976; Deheuvels, 1979) which we shall define as the empirical
+d.f. of the multivariate ranks obtained from X 1:n scaled by 1/n. Note that the
+latter function is piecewise constant and cannot therefore be a genuine copula.
+A promising smooth nonparametric estimator of C that is a genuine copula
+when there are no ties in the components samples of X 1:n and that displays
+substantially better small-sample performance than the empirical copula is the
+empirical beta copula. This estimator was proposed by Segers, Sibuya and Tsuka-
+hara (2017) and is a particular case of the empirical Bernstein copula studied by
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+3
+Sancetta and Satchell (2004) and Janssen, Swanepoel and Veraverbeke (2012)
+when all the underlying Bernstein polynomials have degree n. Building upon
+the work of Segers, Sibuya and Tsukahara (2017), Kojadinovic and Yi (2022)
+recently studied data-adaptive generalizations of the empirical beta copula that
+can perform even better in small samples.
+Whatever nonparametric estimator of the unknown copula C in (1.1) is used
+in inference procedures, it is almost always necessary to rely on resampling tech-
+niques to compute corresponding confidence intervals or p-values. To approxi-
+mate the “sampling distribution” of the classical empirical copula, a frequently
+used approach in the literature is the so-called multiplier bootstrap (see, e.g.,
+Scaillet, 2005; R´emillard and Scaillet, 2009). When the random vectors in X 1:n
+are independent and identically distributed (i.i.d.), B¨ucher and Dette (2010)
+found the latter resampling scheme to have better finite-sample properties than
+approaches consisting of adapting the empirical (multinomial) bootstrap. The
+multiplier bootstrap was extended to the time series and sequential settings in
+B¨ucher and Kojadinovic (2016) and B¨ucher et al. (2014).
+One of the advantages of the empirical beta copula is that it is particu-
+larly easy to draw samples from it. The resulting smooth bootstrap that can
+be used to approximate the “sampling distribution” of the empirical beta cop-
+ula was recently studied both theoretically and empirically in Kiriliouk, Segers
+and Tsukahara (2021). The Monte Carlo experiments reported therein reveal
+that it is a competitive alternative to the multiplier bootstrap while being sub-
+stantially simpler to implement. One practical inconvenience however is that
+the aforementioned smooth bootstrap cannot be directly extended to the time
+series setting.
+The first aim of this work is to obtain, in the i.i.d. case, a smooth bootstrap
+`a la Kiriliouk, Segers and Tsukahara (2021) for the smooth, possibly data-
+adaptive, nonparametric estimators of the copula investigated in Kojadinovic
+and Yi (2022). The second aim is to propose smooth versions of the dependent
+multiplier bootstrap that can be used to approximate the “sampling distribu-
+tion” of the aforementioned estimators in a time series setting. Intuitively, one
+could expect that the resulting smooth inference procedures will perform better
+than corresponding non-smooth procedures in particular when the amount of
+data is low. Indeed, as already mentioned, it is when n is small that smooth cop-
+ula estimators can substantially outperform rough estimators such as the classi-
+cal empirical copula; see for instance the finite-sample experiments reported in
+Segers, Sibuya and Tsukahara (2017), Kiriliouk, Segers and Tsukahara (2021)
+or Kojadinovic and Yi (2022). Another situation where one could expect that
+the use of smooth estimators can be advantageous is when carrying out change-
+point detection. Indeed, statistics for change-point detection often involve the
+comparison of estimators computed from small subsets of observations. It is to
+be able to cover this application area that many of the theoretical investigations
+carried out in this work are of a sequential nature.
+A by-product of this work is the study of the weak consistency and finite-
+sample performance of two classes of smooth estimators of the first-order par-
+tial derivatives of the unknown copula C in (1.1) as these are needed to carry
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+4
+out the dependent multiplier bootstrap. As explained for instance in Janssen,
+Swanepoel and Veraverbeke (2016), such estimators have applications in mean
+and quantile regression as they lead to estimators of the conditional distribution
+function. From a practical perspective, our investigations lead to the proposal of
+a smooth data-adaptive estimator of the first-order partial derivatives of C that
+substantially outperforms, among others, the Bernstein estimator considered in
+Janssen, Swanepoel and Veraverbeke (2016).
+This paper is organized as follows. In the second section, we recall the def-
+inition of the broad class of smooth, possibly data adaptive, empirical copulas
+studied in Kojadinovic and Yi (2022) and the asymptotics of related sequential
+empirical processes. The third section is concerned with an extension of the
+smooth bootstrap of Kiriliouk, Segers and Tsukahara (2021) that can be used
+to approximate the “sampling distribution” of the aforementioned smooth es-
+timators in the i.i.d. case. After investigating its asymptotic validity, results of
+finite-sample experiments comparing smooth bootstraps based on the empirical
+beta copula and on its data-adaptive extension suggested in Kojadinovic and
+Yi (2022) are reported. In Section 4, to be able to cover the time series setting,
+we propose natural smooth extensions of the sequential dependent multiplier
+bootstrap. After providing asymptotic validity results, we compare the finite-
+sample performance of various versions of the multiplier bootstrap and consider
+an application to the offline detection of changes in the cross-sectional depen-
+dence of multivariate time series. The latter confirms the possible advantages
+of smooth inference procedures over their non-smooth counterparts. The fifth
+section is devoted to the study of two classes of smooth estimators of the first-
+order partial derivatives of C: their weak consistency is investigated and the
+finite-sample performance of selected estimators is studied.
+Unless stated otherwise, all convergences in the paper are as n → ∞. Also,
+in the sequel, the arrow ‘⇝’ denotes weak convergence in the sense of Defi-
+nition 1.3.3 in van der Vaart and Wellner (2000) and, given a set T, ℓ∞(T)
+(resp. C (T)) represents the space of all bounded (resp. continuous) real-valued
+functions on T equipped with the uniform metric.
+All the numerical experiments presented in the work were carried out using
+the R statistical environment (R Core Team, 2022) as well as its packages copula
+(Hofert et al., 2022) and extraDistr (Wolodzko, 2020).
+2. Smooth, possibly data-adaptive, empirical copulas and their
+asymptotics
+In this section, we start by defining the broad class of smooth, possibly data
+adaptive, empirical copulas studied in Kojadinovic and Yi (2022). We then recall
+the asymptotics of related sequential empirical processes established in the same
+reference.
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+5
+2.1. Smooth, possibly data-adaptive, nonparametric copula
+estimators
+Because the results to be stated in the next section are of a sequential nature,
+all the quantities hereafter are defined for a substretch X k:l = (Xk, . . . , Xl),
+1 ≤ k ≤ l ≤ n, of the available data X 1:n = (X1, . . . , Xn).
+For any j ∈ {1, . . . , d}, let Fk:l,j be the empirical d.f. computed from the jth
+component subsample Xkj, . . . , Xlj of X k:l. Then, Rk:l
+ij = (l−k+1)Fk:l,j(Xij) =
+�l
+t=k 1(Xtj ≤ Xij) is the (maximal) rank of Xij among Xkj, . . . , Xlj. Further-
+more, let
+Rk:l
+i
+=
+�
+Rk:l
+i1 , . . . , Rk:l
+id
+�
+and
+ˆU k:l
+i
+=
+Rk:l
+i
+l − k + 1,
+i ∈ {k, . . . , l},
+be the multivariate ranks (resp. multivariate scaled ranks) obtained from X k:l.
+Following R¨uschendorf (1976), the empirical copula Ck:l of X k:l is then defined,
+for any u = (u1, . . . , ud) ∈ [0, 1]d, by
+Ck:l(u) =
+1
+l − k + 1
+l
+�
+i=k
+d
+�
+j=1
+1
+�
+Rk:l
+ij
+l − k + 1 ≤ uj
+�
+=
+1
+l − k + 1
+l
+�
+i=k
+1( ˆU k:l
+i
+≤ u),
+(2.1)
+where inequalities between vectors are to be understood componentwise.
+As we continue, following Kojadinovic and Yi (2022), for any m ∈ N, x ∈
+(Rd)m and u ∈ [0, 1]d, νx
+u is the law of a [0, 1]d-valued mean u random vector
+W x
+u . Its components are denoted by W x
+1,u1, . . . , W x
+d,ud to indicate that the jth
+component depends on uj but not on u1, . . . , uj−1, uj+1, . . . , ud. Let p ≥ d be a
+fixed integer and let U be a p-dimensional random vector whose components are
+independent and standard uniform. The following assumption was considered
+in Kojadinovic and Yi (2022) and is likely to be non-restrictive as discussed in
+Remark 3 therein.
+Condition 2.1 (Construction of smoothing random vectors). For any m ∈ N,
+x ∈ (Rd)m and u ∈ [0, 1]d, there exists a function W x
+u : [0, 1]p → [0, 1]d such
+that W x
+u = W x
+u (U).
+To be able to define, for any n ∈ N, X 1:n and, for any m ≤ n, the random
+vectors W x
+u , x ∈ (Rd)m, u ∈ [0, 1]d, on the same probability space (Ω, A , P), we
+assume a product structure, that is, Ω = Ω0 ×Ω1 ×. . . with probability measure
+P = P0 ⊗ P1 ⊗ . . . , where Pi denotes the probability measure on Ωi, such that,
+for any ω ∈ Ω, X 1:n(ω) only depends on the first coordinate of ω, U(ω) only
+depends on the second coordinate of ω and potential “bootstrap weights” (to be
+introduced in Sections 3 and 4) only depend on one of the remaining coordinates
+of ω, implying in particular that X 1:n, U and potential bootstrap weights are
+independent. A broad class of smooth versions of Ck:l in (2.1), with possibly
+data-adaptive smoothing, is then given by
+Cν
+k:l(u) =
+�
+[0,1]d Ck:l(w)dν
+X k:l
+u
+(w),
+u ∈ [0, 1]d.
+(2.2)
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+6
+Intuitively, for a given u ∈ [0, 1]d, Cν
+k:l(u) can be thought of as a “weighted
+average” of Ck:l(w) for w “in a neighborhood of u” according to the smoothing
+distribution ν
+X k:l
+u
+(that may depend on the observations X k:l). Note that, if
+k > l, we adopt the convention that Ck:l = Cν
+k:l = 0 and that, for any u ∈ [0, 1]d,
+ν
+X k:l
+u
+is the Dirac measure at u.
+Remark 2.2. Given m ∈ N and u ∈ [0, 1]d, let µm,u be the law of the d-
+dimensional random vector (Sm,1,u1/m, . . . , Sm,d,ud/m) such that the random
+variables Sm,1,u1, . . . , Sm,d,ud are independent and, for each j ∈ {1, . . . d}, Sm,j,uj
+is Binomial(m, uj). From Section 3 of Segers, Sibuya and Tsukahara (2017), the
+empirical Bernstein copula of X k:l whose Bernstein polynomial degrees are all
+equal to m is then given by
+C
+Bern
+k:l,m(u) =
+�
+[0,1]d Ck:l(w)dµm,u(w),
+u ∈ [0, 1]d.
+(2.3)
+The latter is clearly a special case of Cν
+k:l in (2.2). If, additionally, m = l−k +1,
+that is, if the smoothing distributions satisfy ν
+X k:l
+u
+= µl−k+1,u, u ∈ [0, 1]d, Cν
+k:l
+in (2.2) or, equivalently, CBern
+k:l,m in (2.3), corresponds to the empirical beta copula
+of X k:l studied in Segers, Sibuya and Tsukahara (2017).
+For any m ∈ N, x ∈ (Rd)m, r ∈ [0, m]d and u ∈ [0, 1]d, let
+K x
+r (u) =
+�
+[0,1]d 1(r/m ≤ w)dνx
+u(w) = E {1(r/m ≤ W x
+u )} .
+(2.4)
+By linearity of the integral, we can then express Cν
+k:l in (2.2) as
+Cν
+k:l(u) =
+1
+l − k + 1
+l
+�
+i=k
+K
+X k:l
+Rk:l
+i
+(u),
+u ∈ [0, 1]d.
+(2.5)
+Since copulas have standard uniform margins, it is particularly meaningful
+to focus on estimators of the form (2.5) that have standard uniform margins.
+As verified in Section 3.1 of Kojadinovic and Yi (2022), the following two as-
+sumptions imply the latter.
+Condition 2.3 (No ties). With probability 1, there are no ties in each of the
+component samples X1j, . . . , Xnj, j ∈ {1, . . . , d}, of X 1:n.
+Condition 2.4 (Condition for uniform margins). For any m ∈ N, x ∈ (Rd)m,
+u ∈ [0, 1]d and j ∈ {1, . . . , d}, W x
+j,uj takes its values in the set {0, 1/m, . . . , (m−
+1)/m, 1}.
+Under Condition 2.4, from Section 3.2 of Kojadinovic and Yi (2022), for any
+m ∈ N, x ∈ (Rd)m, r ∈ [1, m]d and u ∈ [0, 1]d, K x
+r (u) in (2.4) can be written
+as
+K x
+r (u) = ¯
+C x
+u
+� ¯
+F x
+1,u1{(r1 − 1)/m}, . . . , ¯
+F x
+d,ud{(rd − 1)/m}
+�
+,
+(2.6)
+where
+¯
+C x
+u (resp.
+¯
+F x
+1,u1, . . . , ¯
+F x
+d,ud) is a survival copula (resp. are the marginal
+survival functions) of the random vector W x
+u . Upon additionally assuming the
+following two conditions considered in Section 3.2 of Kojadinovic and Yi (2022),
+estimators of the form (2.5) can be shown to be genuine copulas.
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+7
+Condition 2.5 (Condition on the smoothing survival margins). For any m ∈ N,
+x ∈ (Rd)m, j ∈ {1, . . . , d} and w ∈ [0, 1), the function t �→
+¯
+F x
+j,t(w) is right-
+continuous and increasing on [0, 1].
+Condition 2.6 (Condition on the smoothing survival copulas). For any m ∈ N,
+x ∈ (Rd)m and u ∈ [0, 1]d, the copulas ¯
+C x
+u in (2.6) do not depend on u, that is,
+¯
+C x
+u = ¯
+C x.
+The following result was then proven in Kojadinovic and Yi (2022); see Propo-
+sition 11 and Corollary 12 therein.
+Proposition 2.7 (Cν
+k:l is a genuine copula). Assume that Conditions 2.3,
+2.4, 2.5 and 2.6 hold. Then, the smooth empirical copula Cν
+k:l in (2.2) or in (2.5)
+can be expressed, for any u ∈ [0, 1]d, as
+Cν
+k:l(u) =
+1
+l − k + 1
+l
+�
+i=k
+¯
+C
+X k:l
+�
+¯
+F
+X k:l
+1,u1
+� Rk:l
+i1 − 1
+l − k + 1
+�
+, . . . , ¯
+F
+X k:l
+d,ud
+� Rk:l
+id − 1
+l − k + 1
+��
+,
+(2.7)
+and is a genuine copula.
+From Remark 2.2 above, we can infer that the empirical beta copula of X k:l
+studied in Segers, Sibuya and Tsukahara (2017) is of the form (2.7) with ¯
+C X k:l
+the independence copula and, for any j ∈ {1, . . . , d} and u ∈ [0, 1]
+¯
+F
+X k:l
+j,u
+the
+survival function of a scaled (by 1/(l − k + 1)) Binomial(l − k + 1, u) random
+variable. For that reason, the latter will be denoted as CBin
+k:l as we continue. As a
+possible improvement of the empirical beta copula CBin
+k:l of X k:l, Kojadinovic and
+Yi (2022) suggested to consider a smooth data-adaptive empirical copula of the
+form (2.7) with ¯
+C X k:l the empirical beta copula CBin
+k:l and, for any j ∈ {1, . . . , d}
+and u ∈ [0, 1],
+¯
+F
+X k:l
+j,u
+the survival function of a scaled (by 1/(l − k + 1)) Beta-
+Binomial(m, α, β) random variable, where m = l − k + 1, α = u(m − ρ)/(ρ − 1),
+β = (1 − u)(m − ρ)/(ρ − 1) and ρ = 4. The resulting data-adaptive estimator,
+denoted by CBetaB4
+k:l
+as we continue, was found to outperform the empirical beta
+copula CBin
+k:l in terms of integrated mean squared error in all the bivariate and
+trivariate experiments considered in Kojadinovic and Yi (2022).
+2.2. Asymptotics of related sequential processes
+We can now define the sequential empirical processes corresponding to the em-
+pirical copula in (2.1) and to its smooth generalizations in (2.2). Let Λ = {(s, t) ∈
+[0, 1]2 : s ≤ t} and let λn(s, t) = (⌊nt⌋ − ⌊ns⌋)/n, (s, t) ∈ Λ. The corresponding
+two-sided sequential empirical copula processes are given, for any (s, t) ∈ Λ and
+u ∈ [0, 1]d, by
+Cn(s, t, u) = √nλn(s, t){C⌊ns⌋+1:⌊nt⌋(u) − C(u)},
+(2.8)
+Cν
+n(s, t, u) = √nλn(s, t){Cν
+⌊ns⌋+1:⌊nt⌋(u) − C(u)},
+(2.9)
+where C⌊ns⌋+1:⌊nt⌋ and Cν
+⌊ns⌋+1:⌊nt⌋ are generically defined in (2.1) and (2.2),
+respectively. The asymptotics of Cn were established in B¨ucher and Kojadinovic
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+8
+(2016), while the asymptotics of Cν
+n (which we recall in Theorem 2.10 hereafter)
+were investigated in Kojadinovic and Yi (2022) by generalizing the arguments
+used in Segers, Sibuya and Tsukahara (2017).
+The following conditions were considered in Kojadinovic and Yi (2022).
+Condition 2.8 (Smooth partial derivatives). For any j ∈ {1, . . . , d}, the partial
+derivative ˙Cj = ∂C/∂uj exists and is continuous on the set Vj = {u ∈ [0, 1]d :
+uj ∈ (0, 1)}.
+Condition 2.9 (Variance condition). There exists a constant κ > 0 such that,
+for any n ∈ N, x ∈ (Rd)n, u ∈ [0, 1]d and j ∈ {1, . . . , d}, Var(W x
+j,uj) ≤ κuj(1 −
+uj)/n.
+The first condition was initially considered in Segers (2012) and can be con-
+sidered non-restricted as explained in the latter reference. In the rest of the
+paper, for any j ∈ {1, . . . , d}, ˙Cj is arbitrarily defined to be zero on the set
+{u ∈ [0, 1]d : uj ∈ {0, 1}}, which implies that, under Condition 2.8, ˙Cj is de-
+fined on the whole of [0, 1]d. The second condition imposes constraints on the
+spread of the smoothing distributions involved in the definition of the smooth,
+possibly data-adaptive, empirical copulas.
+Theorem 2.10 (Asymptotics of Cν
+n). Assume that Conditions 2.8 and 2.9
+hold, and that Cn ⇝ CC in ℓ∞(Λ×[0, 1]d), where the trajectories of the limiting
+process CC are continuous almost surely. Then,
+sup
+(s,t)∈Λ
+u∈[0,1]d
+|Cν
+n(s, t, u) − Cn(s, t, u)| = oP(1).
+Consequently, Cν
+n ⇝ CC in ℓ∞(Λ × [0, 1]d).
+Hence, the smooth sequential empirical copula process Cν
+n in (2.9) and the
+classical sequential empirical copula process Cn in (2.8) are asymptotically
+equivalent when the latter converges weakly to a limiting process whose tra-
+jectories are continuous almost surely. As discussed in Section 3 of B¨ucher and
+Kojadinovic (2016), for such a convergence to hold, it suffices that the corre-
+sponding “uniform multivariate sequential empirical process” converges weakly
+to a limiting process whose trajectories are continuous almost surely. Specifically,
+let U1, . . . , Un be the unobservable sample obtained from X 1:n = (X1, . . . , Xn)
+by the probability integral transformations Uij = Fj(Xij), i ∈ {1, . . . , n},
+j ∈ {1, . . . , d}, and let
+Bn(s, t, u) =
+1
+√n
+⌊nt⌋
+�
+i=⌊ns⌋+1
+{1(Ui ≤ u)−C(u)},
+(s, t, u) ∈ Λ×[0, 1]d, (2.10)
+with the convention that Bn(s, t, ·) = 0 if ⌊nt⌋ − ⌊ns⌋ = 0. The aforementioned
+sufficient condition can then be stated as follows.
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+9
+Condition 2.11 (Weak convergence of Bn(0, ·, ·)). The sequential empirical
+process Bn(0, ·, ·) converges weakly in ℓ∞([0, 1]d+1) to a tight centered Gaussian
+process ZC concentrated on
+{f ∈ C ([0, 1]d+1) : f(s, u) = 0 if one of the components of (s, u) is 0,
+and f(s, 1, . . . , 1) = 0 for all s ∈ (0, 1]}.
+Under Condition 2.11 (which holds for instance when (Xi)i∈Z is strongly mix-
+ing; see, e.g., B¨ucher (2015) as well as forthcoming Section 4.1), it immediately
+follows from the continuous mapping theorem that Bn ⇝ BC in ℓ∞(Λ × [0, 1]d),
+where
+BC(s, t, u) = ZC(t, u) − ZC(s, u),
+(s, t, u) ∈ Λ × [0, 1]d.
+(2.11)
+For any j ∈ {1, . . . , d} and any u ∈ [0, 1]d, let u(j) be the vector of [0, 1]d
+defined by u(j)
+i
+= uj if i = j and 1 otherwise. The following result is then an
+immediate consequence of Theorem 3.4 in B¨ucher and Kojadinovic (2016) and
+Proposition 3.3 of B¨ucher et al. (2014).
+Theorem 2.12 (Asymptotics of Cn). Under Conditions 2.8 and 2.11,
+sup
+(s,t,u)∈Λ×[0,1]d
+���Cn(s, t, u) − ˜Cn(s, t, u)
+��� = oP(1),
+where
+˜Cn(s, t, u) = Bn(s, t, u) −
+d
+�
+j=1
+˙Cj(u) Bn(s, t, u(j)),
+(s, t, u) ∈ Λ × [0, 1]d,
+(2.12)
+and Bn is defined in (2.10). Consequently, Cn ⇝ CC in ℓ∞(Λ × [0, 1]d), where
+CC(s, t, u) = BC(s, t, u) −
+d
+�
+j=1
+˙Cj(u) BC(s, t, u(j)),
+(s, t, u) ∈ Λ × [0, 1]d,
+(2.13)
+and BC is defined in (2.11).
+We end this section with the statement of a corollary of Theorems 2.10
+and 2.12. Having (2.4) in mind, two natural smooth extensions of the unobserv-
+able empirical process Bn in (2.10) can be defined, for any (s, t, u) ∈ Λ × [0, 1]d,
+by
+˜Bν
+n(s, t, u) =
+1
+√n
+⌊nt⌋
+�
+i=⌊ns⌋+1
+��
+[0,1]d 1(Ui ≤ w)dν
+X ⌊ns⌋+1:⌊nt⌋
+u
+(w) − C(u)
+�
+,
+(2.14)
+¯Bν
+n(s, t, u) =
+1
+√n
+⌊nt⌋
+�
+i=⌊ns⌋+1
+��
+[0,1]d 1(Ui ≤ w)dν
+X 1:n
+u
+(w) − C(u)
+�
+.
+(2.15)
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+10
+Combining Theorem 2.12 with key intermediate results used in Kojadinovic
+and Yi (2022) for proving Theorem 2.10 stated above, we obtain the follow-
+ing asymptotic representations for the smooth sequential empirical process Cν
+n
+in (2.9). The proof of this result is given in Appendix A.
+Corollary 2.13 (Asymptotic representations of Cν
+n). Under Conditions 2.8, 2.9
+and 2.11,
+sup
+(s,t,u)∈Λ×[0,1]d
+���Cν
+n(s, t, u) − ˜Cν
+n(s, t, u)
+��� = oP(1),
+sup
+(s,t,u)∈Λ×[0,1]d
+��Cν
+n(s, t, u) − ¯Cν
+n(s, t, u)
+�� = oP(1),
+where, for any (s, t, u) ∈ Λ × [0, 1]d,
+˜Cν
+n(s, t, u) = ˜Bν
+n(s, t, u) −
+d
+�
+j=1
+˙Cj(u) ˜Bν
+n(s, t, u(j)),
+¯Cν
+n(s, t, u) = ¯Bν
+n(s, t, u) −
+d
+�
+j=1
+˙Cj(u) ¯Bν
+n(s, t, u(j)).
+Remark 2.14. The previous results do not unfortunately allow us to decide
+which of the above two asymptotic representations for Cν
+n may be better. The
+knowledge of the underlying convergence rates would be needed for that. As
+we shall see in Section 4, these representations will be at the root of smooth
+proposals for bootstrapping Cν
+n in a time series context.
+3. Bootstrap by drawing samples from the estimators in the i.i.d.
+case
+The aim of this section is to study both theoretically and empirically a smooth
+bootstrap `a la Kiriliouk, Segers and Tsukahara (2021) based on drawing samples
+from the smooth estimators defined in the previous section. As hinted at in the
+introduction, such an approach can only be used in the i.i.d. case. Throughout
+this section, we thus assume that the random vectors in X 1:n are i.i.d. Notice
+that the latter implies Condition 2.3. Given that change-point analysis is es-
+sentially of interest in the time series setting, we do not consider a sequential
+setting below but instead focus only on the situation where k = 1 and l = n.
+This section is organized as follows. After describing the sampling algorithm
+on which the smooth bootstrap is based, we state conditions under which it is
+asymptotically valid and report results of finite-sample experiments comparing
+smooth bootstraps based on the empirical beta copula CBin
+1:n and on its data-
+adaptive extension CBetaB4
+1:n
+proposed in Kojadinovic and Yi (2022) and recalled
+at the end of Section 2.1.
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+11
+3.1. Drawing samples from the smooth empirical copulas
+As explained in Section 2.1, the empirical beta copula CBin
+1:n is a particular case of
+the smooth estimators Cν
+1:n defined in (2.2). From Segers, Sibuya and Tsukahara
+(2017) (see also Lemma 1 in Kojadinovic and Yi 2022), one has that
+C
+Bin
+1:n(u) = 1
+n
+n
+�
+i=1
+d
+�
+j=1
+Fn,R1:n
+ij (uj),
+u = (u1, . . . , ud) ∈ [0, 1]d,
+(3.1)
+where, for any n ∈ N and r ∈ {1, . . . , n}, Fn,r denotes the d.f. of the beta
+distribution with shape parameters α = r and β = n+1−r. It follows from (3.1)
+that CBin
+1:n is a mixture of n d-dimensional distributions having beta margins and
+whose copula is the independence copula. To generate one random variate from
+CBin
+1:n, it thus suffices to randomly select one of the n components of the mixture
+by drawing a uniform on {1, . . . , n} and then generate one random variate from
+the selected d-dimensional distribution. This is detailed in Algorithm 3.2 of
+Kiriliouk, Segers and Tsukahara (2021).
+In a related way, having (2.5) in mind, it thus suffices to assume the following
+to be able sample from Cν
+1:n.
+Condition 3.1. (Cν
+1:n is a mixture) For any n ∈ N, x ∈ (Rd)n and r ∈
+{1, . . . , n}d, K x
+r
+in (2.4) is a d.f. on [0, 1]d.
+The sampling algorithm is then conceptually the same as Algorithm 3.2 of
+Kiriliouk, Segers and Tsukahara (2021).
+Algorithm 3.2. (Sampling from Cν
+1:n under Condition 3.1)
+1. Generate I from the discrete uniform distribution on {1, . . . , n}.
+2. Generate a random variate V # from a d-dimensional distribution whose
+d.f. is K
+X 1:n
+R1:n
+I
+.
+The above algorithm can be used in practice as soon as one knows how to
+sample from the d.f.s K
+X 1:n
+R1:n
+i
+, i ∈ {1, . . . , n}.
+Interestingly enough, three of the conditions stated in Section 2.1 imply Con-
+dition 3.1 as shown in the next result proven in Appendix B.
+Proposition 3.3. Conditions 2.4, 2.5 and 2.6 imply Condition 3.1. Specif-
+ically, under Conditions 2.4, 2.5 and 2.6, for any n ∈ N, x ∈ (Rd)n and
+r ∈ {1, . . . , n}d, K x
+r
+in (2.4) is a d.f. on [0, 1]d whose d univariate margins,
+denoted by K x
+r1,1, . . . , K x
+rd,d, respectively, satisfy K x
+rj,j(u) =
+¯
+F x
+j,u{(rj − 1)/n},
+u ∈ [0, 1], j ∈ {1, . . . , d}, and whose copula is
+¯
+C x.
+Remark 3.4. The previous result leads to an alternative (and simpler) proof of
+Proposition 11 of Kojadinovic and Yi (2022). Indeed, under the assumptions
+of Proposition 3.3, Cν
+1:n in (2.5) is a convex combination of multivariate d.f.s
+on [0, 1]d and therefore a multivariate d.f. on [0, 1]d. Since Condition 2.3 holds
+in the current i.i.d. setting, from Section 3.1 in Kojadinovic and Yi (2022),
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+12
+Condition 2.4 also implies that Cν
+1:n has standard uniform margins. Hence, under
+the assumptions of Proposition 3.3, Cν
+1:n is a genuine copula.
+For any univariate d.f. H, let H−1 denote its associated quantile function
+(generalized inverse) defined by H−1(y) = inf{x ∈ R : H(x) ≥ y}, y ∈ [0, 1],
+with the convention that inf ∅ = ∞. The second step of Algorithm 3.2 can then
+be made explicit under Conditions 2.4, 2.5 and 2.6 :
+(i) Generate a random variate U # from the copula ¯
+C X 1:n independently of I.
+(ii) A random variate from the distribution whose d.f. is K
+X 1:n
+R1:n
+I
+is then
+V
+# =
+�
+K
+X 1:n,−1
+R1:n
+I1 ,1
+(U
+#
+1 ), . . . , K
+X 1:n,−1
+R1:n
+Id ,d
+(U
+#
+d )
+�
+.
+(3.2)
+We end this section by discussing how Algorithm 3.2 can be practically im-
+plemented for the smooth data-adaptive estimator CBetaB4
+1:n
+introduced in Ko-
+jadinovic and Yi (2022) as a possible improvement of the empirical beta copula
+CBin
+1:n. Recall from Section 2.1 that CBetaB4
+1:n
+is of the form (2.7) with
+¯
+C X 1:n the
+empirical beta copula CBin
+1:n and, for any j ∈ {1, . . . , d} and u ∈ [0, 1], ¯
+F
+X 1:n
+j,u
+the
+survival function of a scaled (by 1/n) Beta-Binomial(n, α, β) random variable,
+where α = u(n − ρ)/(ρ − 1), β = (1 − u)(n − ρ)/(ρ − 1) and ρ = 4. The latter
+implies that, for any i ∈ {1, . . . , n}, j ∈ {1, . . . , d} and u ∈ [0, 1],
+K
+X 1:n
+R1:n
+ij ,j(u) = ¯
+F
+X 1:n
+j,u
+{(R1:n
+ij −1)/n} = P(nW
+X 1:n
+j,u
+> R1:n
+ij −1) = ¯
+Bn,u,ρ(R1:n
+ij −1),
+(3.3)
+where ¯
+Bn,u,ρ is the survival function of the Beta-Binomial(n, α, β). As can be
+checked from Lemma 27 in Kojadinovic and Yi (2022) and Lemma B.2 in Ap-
+pendix B, the univariate d.f. K
+X 1:n
+R1:n
+ij ,j in (3.3) is continuous and strictly increas-
+ing, respectively. Hence, to compute its associated quantile function needed
+in (3.2), one can proceed numerically. In that respect, an implementation of Al-
+gorithm 3.2 for the R statistical environment for the estimators CBin
+1:n and CBetaB4
+1:n
+is available on the web page of the first author.
+3.2. Asymptotic validity results
+Building upon the work of Kiriliouk, Segers and Tsukahara (2021), we will
+now provide asymptotic validity results for a smooth bootstrap based on draw-
+ing samples from Cν
+1:n in (2.5) under Conditions 2.4, 2.5 and 2.6. Recall that,
+according to Proposition 3.3, the latter conditions imply Condition 3.1. Let
+V
+#
+1:n = (V
+#
+1 , . . . , V #
+n ) be a random sample from Cν
+1:n obtained by applying
+Algorithm 3.2 n times independently. Note that this implies that the com-
+ponent samples of V
+#
+1:n do not contain ties with probability 1. For any j ∈
+{1, . . . , d}, let G
+#
+1:n,j be the empirical d.f. computed from the jth component
+sample V
+#
+1j, . . . , V
+#
+nj of V
+#
+1:n. Then, R1:n,#
+ij
+= nG
+#
+1:n,j(V
+#
+ij ) is the rank of V
+#
+ij
+among V
+#
+1j, . . . , V
+#
+nj. The (classical) empirical copula of V
+#
+1:n is thus given by
+C
+#
+1:n(u) = 1
+n
+n
+�
+i=1
+d
+�
+j=1
+1
+�
+R1:n,#
+ij
+n
+≤ uj
+�
+,
+u ∈ [0, 1]d,
+(3.4)
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+13
+and the smooth analog of Cν
+1:n for V
+#
+1:n is
+C
+#,ν
+1:n (u) =
+�
+[0,1]d C
+#
+1:n(w)dν
+V #
+1:n
+u
+(w),
+u ∈ [0, 1]d.
+(3.5)
+To state our asymptotic validity results, we consider independent copies
+V
+#,[1]
+1:n , V
+#,[2]
+1:n , . . . of V
+#
+1:n. Let C
+#,[i]
+1:n (resp. C
+#,ν,[i]
+1:n
+) be the version of C
+#
+1:n in (3.4)
+(resp. C
+#,ν
+1:n in (3.5)) obtained from V
+#,[i]
+1:n , i ∈ N.
+The following result can be regarded as an extension of Proposition 3.3 of
+Kiriliouk, Segers and Tsukahara (2021) and is proven in Appendix C.
+Theorem 3.5. Assume that the random vectors in X 1:n are i.i.d., and that
+Conditions 2.4, 2.5, 2.6, 2.8 and 2.9 hold. Then,
+�
+Cn(0, 1, ·), √n(C
+#,[1]
+1:n − C1:n),√n(C
+#,[2]
+1:n − C1:n)
+�
+⇝
+�
+CC(0, 1, ·), C
+[1]
+C (0, 1, ·), C
+[2]
+C (0, 1, ·)
+�
+,
+�
+Cν
+n(0, 1, ·), √n(C
+#,ν,[1]
+1:n
+− Cν
+1:n),√n(C
+#,ν,[2]
+1:n
+− Cν
+1:n)
+�
+⇝
+�
+CC(0, 1, ·), C
+[1]
+C (0, 1, ·), C
+[2]
+C (0, 1, ·)
+�
+in {ℓ∞([0, 1]d)}3, where Cn and Cν
+n are defined in (2.8) and (2.9), respectively,
+and C
+[1]
+C and C
+[2]
+C are independent copies of CC defined in (2.13).
+Remark 3.6. The first joint weak convergence in Theorem 3.5 establishes the
+asymptotic validity of a smooth bootstrap for the (non-sequential) classical em-
+pirical process while the second one provides a similar results for the smooth
+empirical copula process Cν
+n(0, 1, ·). According to Lemma 3.1 in B¨ucher and
+Kojadinovic (2019), these two joint weak convergences are equivalent to simi-
+lar joint weak convergences with B ≥ 2 bootstrap replicates. In a further step,
+the latter can be transferred to the “statistic level” using the continuous map-
+ping theorem or the functional delta method, which could then be combined
+with the results in Section 4 of B¨ucher and Kojadinovic (2019) to establish
+the validity of bootstrap-based confidence intervals or tests. Note also that,
+from Lemma 3.1 in B¨ucher and Kojadinovic (2019), the unconditional asymp-
+totic validity results appearing in Theorem 3.5 are equivalent to possibly more
+classical conditional results which rely, however, on a more subtle mode of con-
+vergence. For instance, the first claim can be equivalently informally stated as
+“√n(C
+#,[1]
+1:n −C1:n) converges weakly to CC(0, 1, ·) in ℓ∞([0, 1]d) conditionally on
+the data in probability”; see, e.g., Kosorok (2008, Section 2.2.3) or Appendix C
+for a precise definition of that mode of convergence.
+3.3. Finite-sample comparison of two smooth bootstraps
+As already mentioned in the introduction, in their Monte Carlo experiments,
+Kiriliouk, Segers and Tsukahara (2021) found the smooth bootstrap based on
+the empirical beta copula CBin
+1:n to be a competitive alternative to many other re-
+sampling schemes (including the multiplier bootstrap to be studied in the forth-
+coming section). Since the data-adaptive empirical copula CBetaB4
+1:n
+was found
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+14
+Table 1
+Coverage probabilities (cov.) and average lengths (ave.) of 95%-confidence intervals for
+Kendall’s tau estimated from 1000 random samples of size n ∈ {20, 40, 80, 160} from the
+bivariate Clayton or Gumbel–Hougaard copula with a Kendall’s tau of τ ∈ {0, 0.5, 0.75, 0.9}.
+Each confidence interval was computed using 1000 smooth bootstrap samples drawn from
+either CBin
+1:n or CBetaB4
+1:n
+using Algorithm 3.2.
+Clayton
+Gumbel–Hougaard
+Bin
+BetaB4
+Bin
+BetaB4
+τ
+n
+cov.
+ave.
+cov.
+ave.
+cov.
+ave.
+cov.
+ave.
+0.00
+20
+0.973
+0.626
+0.967
+0.615
+0.969
+0.624
+0.962
+0.614
+40
+0.962
+0.428
+0.954
+0.424
+0.950
+0.430
+0.944
+0.425
+80
+0.949
+0.298
+0.943
+0.296
+0.970
+0.297
+0.961
+0.296
+160
+0.944
+0.208
+0.943
+0.207
+0.949
+0.208
+0.949
+0.208
+0.50
+20
+0.971
+0.513
+0.972
+0.493
+0.978
+0.521
+0.982
+0.498
+40
+0.958
+0.347
+0.954
+0.334
+0.959
+0.345
+0.957
+0.332
+80
+0.946
+0.239
+0.938
+0.233
+0.950
+0.237
+0.947
+0.231
+160
+0.954
+0.168
+0.957
+0.165
+0.954
+0.164
+0.958
+0.162
+0.75
+20
+0.717
+0.392
+0.899
+0.367
+0.777
+0.391
+0.927
+0.358
+40
+0.728
+0.234
+0.908
+0.221
+0.793
+0.231
+0.954
+0.211
+80
+0.798
+0.151
+0.930
+0.146
+0.844
+0.146
+0.953
+0.137
+160
+0.866
+0.101
+0.943
+0.100
+0.883
+0.098
+0.944
+0.094
+0.90
+20
+0.000
+0.315
+0.212
+0.272
+0.000
+0.317
+0.270
+0.264
+40
+0.000
+0.160
+0.475
+0.131
+0.000
+0.162
+0.593
+0.127
+80
+0.000
+0.086
+0.692
+0.074
+0.000
+0.087
+0.804
+0.069
+160
+0.000
+0.050
+0.837
+0.047
+0.000
+0.050
+0.902
+0.043
+to outperform the empirical beta copula CBin
+1:n in the experiments reported in
+Kojadinovic and Yi (2022), it seems natural to empirically investigate how the
+smooth bootstrap based on CBetaB4
+1:n
+compares to the smooth bootstrap based on
+CBin
+1:n. To do so, we reproduced some of the experiments reported in Sections 4.2
+and 4.3 of Kiriliouk, Segers and Tsukahara (2021).
+We first estimated coverage probabilities and average lengths of confidence
+intervals of level 95% for Kendall’s tau from 1000 random samples of size n ∈
+{20, 40, 80, 160} from the bivariate Clayton or Gumbel–Hougaard copula with
+a Kendall’s tau of τ ∈ {0, 0.5, 0.75, 0.9}. Each confidence interval was computed
+using 1000 smooth bootstrap samples drawn from either CBin
+1:n or CBetaB4
+1:n
+. The
+results are reported in Table 1. As one can see, under independence or moderate
+dependence (τ ∈ {0, 0.5}), the estimated coverage probabilities are overall on
+target and very similar for the two resampling schemes. The intervals obtained
+using the smooth bootstrap based on CBetaB4
+1:n
+seem nonetheless to be slightly
+shorter on average. Under strong dependence (τ = 0.75) however, the estimated
+coverage probabilities of the confidence intervals computed using the smooth
+bootstrap based on CBin
+1:n are substantially below the 0.95 target value. The
+results for τ = 0.9 actually show that the smooth bootstrap based on CBin
+1:n is
+unable to generate samples with such a very strong dependence. While its results
+are not perfect, the smooth bootstrap based on CBetaB4
+1:n
+copes much better with
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+15
+Table 2
+Coverage probabilities (cov.) and average lengths (ave.) of 95%-confidence intervals for the
+parameter of a bivariate Frank copula estimated by maximum pseudo-likelihood from 1000
+random samples of size n ∈ {20, 40, 80} from the bivariate Frank copula with a Kendall’s
+tau of τ ∈ {−0.9, −0.75, −0.5, 0, 0.5, 0.75, 0.9}. Each confidence interval was computed using
+1000 smooth bootstrap samples drawn from either CBin
+1:n or CBetaB4
+1:n
+using Algorithm 3.2.
+Bin
+BetaB4
+τ
+n
+cov.
+ave.
+cov.
+ave.
+-0.90
+20
+0.000
+0.194
+0.000
+0.133
+40
+0.000
+0.086
+0.056
+0.051
+80
+0.000
+0.039
+0.313
+0.023
+-0.75
+20
+0.731
+0.286
+0.940
+0.237
+40
+0.641
+0.153
+0.939
+0.126
+80
+0.662
+0.088
+0.947
+0.077
+-0.50
+20
+0.988
+0.548
+0.981
+0.511
+40
+0.975
+0.342
+0.957
+0.327
+80
+0.957
+0.230
+0.940
+0.224
+0.00
+20
+0.952
+1.009
+0.946
+1.002
+40
+0.937
+0.681
+0.929
+0.681
+80
+0.941
+0.467
+0.938
+0.469
+0.50
+20
+0.986
+0.542
+0.970
+0.508
+40
+0.972
+0.344
+0.949
+0.328
+80
+0.959
+0.224
+0.948
+0.219
+0.75
+20
+0.722
+0.285
+0.938
+0.235
+40
+0.634
+0.154
+0.927
+0.128
+80
+0.671
+0.088
+0.942
+0.077
+0.90
+20
+0.000
+0.193
+0.000
+0.132
+40
+0.000
+0.086
+0.046
+0.051
+80
+0.000
+0.039
+0.319
+0.023
+strong dependence. This is likely to be due to the modification of the “shape” of
+the underlying smoothing distributions using the empirical beta copula in the
+expression of CBetaB4
+1:n
+as can be deduced from (2.7).
+In a second experiment, following Kiriliouk, Segers and Tsukahara (2021,
+Section 4.3) we estimated coverage probabilities and average lengths of 95%-
+confidence intervals for the parameter of a bivariate Frank copula estimated by
+maximum pseudo-likelihood (see Genest, Ghoudi and Rivest, 1995) from 1000
+random samples of size n ∈ {20, 40, 80, 160} from the bivariate Frank copula
+with a Kendall’s tau of τ ∈ {−0.9, −0.75, −0.5, 0, 0.5, 0.75, 0.9}. Again, each
+confidence interval was computed using 1000 smooth bootstrap samples drawn
+from either CBin
+1:n or CBetaB4
+1:n
+. The results are reported in Table 2 and the main
+conclusion is qualitatively the same as for the previous experiment: the smooth
+bootstrap based on CBetaB4
+1:n
+copes much better with strong dependence than the
+smooth bootstrap based on CBin
+1:n.
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+16
+4. Smooth sequential dependent multiplier bootstraps in the time
+series case
+The smooth bootstrap investigated in the previous section can only be used in
+the case of i.i.d. observations. Fortunately, the multiplier bootstrap, one of the
+most popular approaches for bootstrapping functionals of the classical empirical
+copula, can be employed in the time series setting. In this section, after provid-
+ing some intuitions and defining multiplier sequences, we recall the non-smooth
+sequential dependent multiplier bootstrap studied in B¨ucher and Kojadinovic
+(2016). We next propose smooth extensions of the latter, provide asymptotic
+validity results and compare the finite-sample performance of three (smooth)
+multiplier bootstraps for approximating three (smooth) empirical copula pro-
+cesses. Finally, as an application, we consider a smooth version (based on the
+empirical beta copula and corresponding smooth multiplier bootstrap replicates)
+of the test for change-point detection developed in B¨ucher et al. (2014) and we
+compare its finite-sample performance to that of its non-smooth counterpart.
+4.1. Main intuition and existing work
+As mentioned in Section 2.2, Condition 2.11 holds under strong mixing. Given a
+stationary time series (Yi)i∈Z, denote by F k
+j the σ-field generated by (Yi)j≤i≤k,
+j, k ∈ Z∪{−∞, +∞}, and recall that the strong mixing coefficients correspond-
+ing to the stationary sequence (Yi)i∈Z are then defined by
+αY
+r =
+sup
+A∈F 0
+−∞,B∈F +∞
+r
+��P(A ∩ B) − P(A)P(B)
+��,
+r ∈ N, r > 0,
+and that the sequence (Yi)i∈Z is said to be strongly mixing if αY
+r → 0 as r → ∞.
+From B¨ucher (2015), Condition 2.11 holds if the strong mixing coefficients
+of the time series (Xi)i∈Z satisfy αX
+r = O(r−a) with a > 1 as r → ∞. In that
+case, Theorem 2.12 suggests that, in order to bootstrap the classical sequential
+empirical copula process Cn in (2.8) in an asymptotically valid way, it suffices to
+bootstrap the process ˜Cn in (2.12). The latter could be done by bootstrapping
+Bn in (2.10) and estimating the first-order partial derivatives ˙Cj, j ∈ {1, . . . , d},
+of C. Such an approach was initially proposed in the independent non-sequential
+setting by Scaillet (2005) and R´emillard and Scaillet (2009) who used a multi-
+plier bootstrap in the spirit of van der Vaart and Wellner (2000, Chapter 2.9)
+to resample Bn, and finite-differencing to estimate the partial derivatives ˙Cj,
+j ∈ {1, . . . , d}. This resampling scheme was extended to the time series sequen-
+tial setting in B¨ucher and Kojadinovic (2016) and B¨ucher et al. (2014).
+4.2. I.i.d. and dependent multiplier sequences
+In the case of independent observations, multiplier bootstraps are based on i.i.d.
+multiplier sequences. We say that a sequence of random variables (ξi,n)i∈Z is an
+i.i.d. multiplier sequence if:
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+17
+(M0) (ξi,n)i∈Z is i.i.d., independent of X 1:n, with distribution not changing
+with n, having mean 0, variance 1, and being such that
+� ∞
+0 {P(|ξ0,n| >
+x)}1/2dx < ∞.
+The time series extension of the multiplier bootstrap relies on the notion of
+dependent multiplier sequence. The key idea due to B¨uhlmann (1993) is to re-
+place i.i.d. multipliers by suitably serially dependent multipliers that will capture
+the serial dependence in the data. We say that a sequence of random variables
+(ξi,n)i∈Z is a dependent multiplier sequence if:
+(M1) The sequence of random variables (ξi,n)i∈Z is stationary with E(ξ0,n) = 0,
+E(ξ2
+0,n) = 1 and supn≥1 E(|ξ0,n|γ) < ∞ for all γ ≥ 1, and is independent
+of the available sample X 1:n.
+(M2) There exists a sequence ℓn → ∞ of strictly positive constants such that
+ℓn = o(n) and the sequence (ξi,n)i∈Z is ℓn-dependent, i.e., ξi,n is indepen-
+dent of ξi+h,n for all h > ℓn and i ∈ N.
+(M3) There exists a function ϕ : R → [0, 1], symmetric around 0, continuous at
+0, satisfying ϕ(0) = 1 and ϕ(x) = 0 for all |x| > 1 such that E(ξ0,nξh,n) =
+ϕ(h/ℓn) for all h ∈ Z.
+As shall become clearer for instance from (4.1) or (4.2) below, the bandwidth
+parameter ℓn defined in (M2) plays a role similar to that of the block length
+in the block bootstrap. In practice, for the non-smooth sequential dependent
+multiplier bootstrap to be presented in the forthcoming section, its value can be
+chosen in a data-driven way using the approach described in detail in B¨ucher
+and Kojadinovic (2016, Section 5); see also Section 4.6. The latter reference also
+describes in detail ways to generate dependent multiplier sequences.
+4.3. Non-smooth sequential dependent multiplier replicates
+Let (ξ
+[1]
+i,n)i∈Z, (ξ
+[2]
+i,n)i∈Z,. . . , be independent copies of the same multiplier se-
+quence. Two different multiplier bootstrap replicates of the process Bn in (2.10)
+were proposed in B¨ucher and Kojadinovic (2016) and B¨ucher et al. (2014), re-
+spectively. For any b ∈ N, (s, t) ∈ Λ and u ∈ [0, 1]d, they are defined by
+ˆB
+[b]
+n (s, t, u) =
+1
+√n
+⌊nt⌋
+�
+i=⌊ns⌋+1
+ξ
+[b]
+i,n
+�
+1( ˆU 1:n
+i
+≤ u) − C1:n(u)
+�
+(4.1)
+and
+ˇB
+[b]
+n (s, t, u) =
+1
+√n
+⌊nt⌋
+�
+i=⌊ns⌋+1
+ξ
+[b]
+i,n
+�
+1( ˆU ⌊ns⌋+1:⌊nt⌋
+i
+≤ u) − C⌊ns⌋+1:⌊nt⌋(u)
+�
+, (4.2)
+respectively, where C1:n and C⌊ns⌋+1:⌊nt⌋ are generically defined in (2.1) and
+with the convention that ˆB[b]
+n (s, t, ·) = ˇB[b]
+n (s, t, ·) = 0 if ⌊nt⌋ − ⌊ns⌋ = 0.
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+18
+In order to define multiplier bootstrap replicates of ˜Cn in (2.12), it is fur-
+ther necessary to estimate the unknown first-order partial derivatives ˙Cj, j ∈
+{1, . . . , d}, of C. In the rest of this section, ˙Cj,k:l will denote an estimator of ˙Cj
+based on a stretch X k:l = (Xk, . . . , Xl) of observations, 1 ≤ k ≤ l ≤ n, with
+the convention that ˙Cj,k:l = 0 if k > l. Then, following B¨ucher and Kojadinovic
+(2016) and B¨ucher et al. (2014), we consider two types of multiplier bootstrap
+replicates of Cn in (2.8). For any b ∈ N, (s, t) ∈ Λ and u ∈ [0, 1]d, these are
+defined by
+ˆC
+[b]
+n (s, t, u) = ˆB
+[b]
+n (s, t, u) −
+d
+�
+j=1
+˙Cj,1:n(u) ˆB
+[b]
+n (s, t, u(j))
+(4.3)
+and
+ˇC
+[b]
+n (s, t, u) = ˇB
+[b]
+n (s, t, u) −
+d
+�
+j=1
+˙Cj,⌊ns⌋+1:⌊nt⌋(u) ˇB
+[b]
+n (s, t, u(j)),
+(4.4)
+respectively, where ˆB[b]
+n (resp. ˇB[b]
+n ) is defined in (4.1) (resp. (4.2)). Clearly,
+both types of replicates coincide in a non-sequential setting as ˆC[b]
+n (0, 1, ·) =
+ˇC[b]
+n (0, 1, ·). As far as the estimators of the partial derivatives are concerned, it
+is expected that the more accurate they are, the better the approximation of
+the “sampling distribution” of Cn by the multiplier replicates will be. The latter
+aspect will be discussed in detail in Section 5, where two broad classes of smooth
+estimators will be introduced and studied both theoretically and empirically.
+4.4. Smooth sequential dependent multiplier replicates
+We now consider a similar construction but based on smooth analogs of ˆB[b]
+n
+in (4.1) and ˇB[b]
+n in (4.2). Specifically, Corollary 2.13 suggests that, to bootstrap
+Cν
+n in (2.9), a first step is to bootstrap ˜Bν
+n in (2.14) or ¯Bν
+n in (2.15). By analogy
+with (2.2) and (2.5), natural smooth analogs of ˆB[b]
+n and ˇB[b]
+n could be defined,
+for any b ∈ N, (s, t) ∈ Λ and u ∈ [0, 1]d, by
+ˆB
+[b],ν
+n
+(s, t, u) =
+�
+[0,1]d
+ˆB
+[b]
+n (s, t, w)dν
+X 1:n
+u
+(w)
+=
+1
+√n
+⌊nt⌋
+�
+i=⌊ns⌋+1
+ξ
+[b]
+i,n
+�
+K
+X 1:n
+R1:n
+i
+(u) − Cν
+1:n(u)
+�
+(4.5)
+and
+ˇB
+[b],ν
+n
+(s, t, u) =
+�
+[0,1]d
+ˇB
+[b]
+n (s, t, w)dν
+X ⌊ns⌋+1:⌊nt⌋
+u
+(w)
+=
+1
+√n
+⌊nt⌋
+�
+i=⌊ns⌋+1
+ξ
+[b]
+i,n
+�
+K
+X ⌊ns⌋+1:⌊nt⌋
+R⌊ns⌋+1:⌊nt⌋
+i
+(u) − Cν
+⌊ns⌋+1:⌊nt⌋(u)
+�
+,
+(4.6)
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+19
+respectively, where K
+X 1:n
+R1:n
+i
+and K
+X ⌊ns⌋+1:⌊nt⌋
+R⌊ns⌋+1:⌊nt⌋
+i
+are defined in (2.4). Combining
+these ingredients with estimators of the unknown partial derivatives of C, as
+smooth analogs of ˆC[b]
+n in (4.3) and ˇC[b]
+n in (4.4), we obtain
+ˆC
+[b],ν
+n
+(s, t, u) = ˆB
+[b],ν
+n
+(s, t, u) −
+d
+�
+j=1
+˙Cj,1:n(u) ˆB
+[b],ν
+n
+(s, t, u(j)),
+(4.7)
+and
+ˇC
+[b],ν
+n
+(s, t, u) = ˇB
+[b],ν
+n
+(s, t, u) −
+d
+�
+j=1
+˙Cj,⌊ns⌋+1:⌊nt⌋(u) ˇB
+[b],ν
+n
+(s, t, u(j)),
+(4.8)
+respectively, for b ∈ N, (s, t) ∈ Λ and u ∈ [0, 1]d.
+To establish the asymptotic validity of these smooth multiplier bootstrap
+replicates, it will suffice that the partial derivative estimators satisfy the follow-
+ing rather natural mild condition.
+Condition 4.1 (Bounded and weakly consistent partial derivative estimators).
+There exists a constant ζ > 0 such that, for any j ∈ {1, . . . , d} and n ∈ N,
+sup
+(s,t,u)∈Λ×[0,1]d
+��� ˙Cj,⌊ns⌋+1:⌊nt⌋(u)
+��� ≤ ζ.
+Furthermore, for any δ ∈ (0, 1), ε ∈ (0, 1/2) and j ∈ {1, . . . , d},
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+��� ˙Cj,⌊ns⌋+1:⌊nt⌋(u) − ˙Cj(u)
+��� = oP(1).
+In addition, following B¨ucher et al. (2014), we impose the following condition
+on the observations and the underlying multiplier sequences.
+Condition 4.2 (Strong mixing and multiplier conditions). One of the following
+two conditions holds:
+(i) The random vectors in X 1:n are i.i.d. and (ξ
+[1]
+i,n)i∈Z, (ξ
+[2]
+i,n)i∈Z,. . . are in-
+dependent copies of a multiplier sequence satisfying (M0).
+(ii) The stretch X 1:n is drawn from a stationary sequence (Xi)i∈Z whose
+strong mixing coefficients satisfy αX
+r = O(r−a) for some a > 3 + 3d/2 as
+r → ∞. Furthermore, (ξ
+[1]
+i,n)i∈Z, (ξ
+[2]
+i,n)i∈Z, . . . are independent copies of a
+dependent multiplier sequence satisfying (M1)–(M3) with ℓn = O(n1/2−γ)
+for some 0 < γ < 1/2.
+The following result is proven in Appendix D.
+Theorem 4.3 (Asymptotic validity of the smooth dependent multiplier boot-
+straps). Under Conditions 2.8, 2.9, 4.1 and 4.2, for any b ∈ N, there holds
+sup
+(s,t,u)∈Λ×[0,1]d
+���ˆC
+[b],ν
+n
+(s, t, u) − ˆC
+[b]
+n (s, t, u)
+��� = oP(1),
+(4.9)
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+20
+sup
+(s,t,u)∈Λ×[0,1]d
+��ˇC
+[b],ν
+n
+(s, t, u) − ˇC
+[b]
+n (s, t, u)
+�� = oP(1).
+(4.10)
+Furthermore,
+(Cν
+n, ˆC
+[1],ν
+n
+, ˆC
+[2],ν
+n
+) ⇝ (CC, C
+[1]
+C , C
+[2]
+C ),
+(Cν
+n, ˇC
+[1],ν
+n
+, ˇC
+[2],ν
+n
+) ⇝ (CC, C
+[1]
+C , C
+[2]
+C )
+in {ℓ∞(Λ × [0, 1]d)}3, where C
+[1]
+C and C
+[2]
+C are independent copies of CC defined
+in (2.13).
+4.5. Finite-sample comparison of three multiplier bootstraps
+From Theorem 2.10, we know that, under Conditions 2.8 and 2.9, the classi-
+cal sequential empirical copula process Cn in (2.8) and the smooth sequential
+empirical copula process Cν
+n in (2.9) are asymptotically equivalent. In a related
+way, Theorem 4.3 provides conditions under which corresponding multiplier
+and smooth multiplier replicates are asymptotically equivalent. Although one
+expects that Cν
+n is probably best resampled using multiplier replicates con-
+structed with the same smoothing distributions, that is, with ˆC[b],ν
+n
+in (4.7)
+or ˇC[b],ν
+n
+in (4.8), we have no asymptotic results to support this (see also Re-
+mark 2.14). Indeed, given that all versions of multiplier replicates are asymp-
+totically equivalent, it may well be that, for instance, in some cases, classical
+(non-smooth) multiplier replicates are equivalent or even preferable to smooth
+multiplier replicates when it comes to resampling Cν
+n. It is the aim of this sec-
+tion to study this empirically. For simplicity, we restrict our investigations to a
+non-sequential setting and independent observations.
+Specifically, we designed experiments to study which multiplier replicates are
+best suited to estimate certain functionals of the three (non-sequential) empirical
+copula processes defined, for any u ∈ [0, 1]d, by
+C
+Dirac
+n
+(u) = √n{C1:n(u) − C(u)} = Cn(0, 1, u),
+(4.11)
+C
+Bin
+n (u) = √n{C
+Bin
+1:n(u) − C(u)},
+(4.12)
+C
+BetaB4
+n
+(u) = √n{C
+BetaB4
+1:n
+(u) − C(u)},
+(4.13)
+where
+• Cn is the classical (non-smooth) sequential empirical copula process de-
+fined in (2.8),
+• CBin
+1:n is the empirical beta copula in (3.1) (which is obtained by considering
+smoothing distributions with scaled binomial margins and independence
+copula as explained in Section 2.1),
+• CBetaB4
+1:n
+is the version of Cν
+1:n introduced in Section 2.1 obtained by con-
+sidering smoothing distributions with scaled beta-binomial margins and
+survival copula the empirical beta copula CBin
+1:n, and found to have the
+best finite-sample performance in the numerical experiments of Kojadi-
+novic and Yi (2022).
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+21
+As already mentioned, since we are in a non-sequential setting, the two generic
+multiplier replicates defined in (4.7) and (4.8) coincide. To approximate the
+“sampling distributions” of the three empirical copula processes defined above,
+we considered as candidate bootstraps the multiplier replicates defined using
+the same smoothing distributions. They will be denoted by ˆC[b],Dirac
+n
+, ˆC[b],Bin
+n
+and
+ˆC[b],BetaB4
+n
+, b ∈ N, respectively, as we continue. To only investigate the effect
+of the choice of the smoothing distributions involved in the definition of ˆB[b],ν
+n
+in (4.5), all three multiplier replicates were computed using the true partial
+derivative ˙Cj, j ∈ {1, . . . , d}. Furthermore, since we restricted our experiments
+to independent observations, all the multiplier replicates were based on i.i.d.
+multiplier sequences defined in (M0) in Section 4.2. Following B¨ucher and Dette
+(2010), these sequences were simply taken to be random samples drawn from
+the uniform distribution on {−1, 1}.
+For the design of our experiments, we followed again B¨ucher and Dette
+(2010). First, for d = 2, we assessed how well the covariances of the empiri-
+cal processes CDirac
+n
+in (4.11), CBin
+n
+in (4.12) and CBetaB4
+n
+in (4.13) at the points
+P = {(i/3, j/3) : i, j = 1, 2} can be approximated using the three possible mul-
+tiplier bootstrap replicates. For each target empirical copula process, we began
+by precisely estimating its covariance at the points in P from 100 000 indepen-
+dent samples of size n ∈ {10, 20, 40, 80} drawn from a bivariate copula C with a
+Kendall’s tau of τ ∈ {0, 0.25, 0.5, 0.75}. For C, we considered either the Clayton
+or the Gumbel–Hougaard copula. Next, for each considered combination of C,
+n, τ, target process and multiplier process, we generated 1000 samples from C,
+and, for each sample, we computed B = 1000 multiplier bootstrap replicates.
+These B = 1000 replicates were used to obtain one estimate of the covariance
+of the target process at the points in P.
+The results when C is the Clayton copula with a Kendall’s tau of τ ∈
+{0, 0.25, 0.5, 0.75} are reported in Figure 1. The first (resp. second, third) col-
+umn of graphs reports the average of the empirical mean square errors (MSEs)
+×104 of the three candidate multiplier estimators of the covariance of CDirac
+n
+(resp. CBin
+n , CBetaB4
+n
+) at the points in P against the sample size n. Each row of
+graphs corresponds to a different value of τ. In the top-left panel for instance, the
+solid (resp. dashed, dotted) curve gives the average MSE when the covariance
+of CDirac
+n
+is estimated using ˆC[b],Dirac
+n
+(resp. ˆC[b],Bin
+n
+, ˆC[b],BetaB4
+n
+).
+As one can see, reassuringly, all the curves are globally decreasing, confirm-
+ing that, for each target process, the bootstrap approximations improve as n
+increases. A more careful inspection reveals that, in almost all settings, it is
+the multiplier bootstrap constructed with the same smoothing distributions as
+the target process that leads to the best estimation. It is actually only when
+CBetaB4
+n
+is the target process that covariance estimations based on ˆC[b],Bin
+n
+are
+sometimes better than estimations based on ˆC[b],BetaB4
+n
+. This happens mostly for
+small n and τ. Results for the Gumbel–Hougaard copula (not reported) are not
+qualitatively different.
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+22
+10
+20
+30
+40
+50
+60
+80
+0.2
+0.5
+1.0
+2.0
+5.0
+10.0
+indep. cop. / d = 2 / tau = 0
+n
+average of MSE of cov. est.
+Dirac/Dirac
+Dirac/Bin
+Dirac/BetaB4
+10
+20
+30
+40
+50
+60
+80
+0.1
+0.2
+0.5
+1.0
+2.0
+indep. cop. / d = 2 / tau = 0
+n
+average of MSE of cov. est.
+Bin/Dirac
+Bin/Bin
+Bin/BetaB4
+10
+20
+30
+40
+50
+60
+80
+0.2
+0.5
+1.0
+2.0
+indep. cop. / d = 2 / tau = 0
+n
+average of MSE of cov. est.
+BetaB4/Dirac
+BetaB4/Bin
+BetaB4/BetaB4
+10
+20
+30
+40
+50
+60
+80
+0.2
+0.5
+1.0
+2.0
+5.0
+Clayton / d = 2 / tau = 0.25
+n
+average of MSE of cov. est.
+Dirac/Dirac
+Dirac/Bin
+Dirac/BetaB4
+10
+20
+30
+40
+50
+60
+80
+0.2
+0.5
+1.0
+2.0
+Clayton / d = 2 / tau = 0.25
+n
+average of MSE of cov. est.
+Bin/Dirac
+Bin/Bin
+Bin/BetaB4
+10
+20
+30
+40
+50
+60
+80
+0.2
+0.5
+1.0
+2.0
+Clayton / d = 2 / tau = 0.25
+n
+average of MSE of cov. est.
+BetaB4/Dirac
+BetaB4/Bin
+BetaB4/BetaB4
+10
+20
+30
+40
+50
+60
+80
+0.5
+1.0
+2.0
+5.0
+Clayton / d = 2 / tau = 0.5
+n
+average of MSE of cov. est.
+Dirac/Dirac
+Dirac/Bin
+Dirac/BetaB4
+10
+20
+30
+40
+50
+60
+80
+0.2
+0.5
+1.0
+2.0
+Clayton / d = 2 / tau = 0.5
+n
+average of MSE of cov. est.
+Bin/Dirac
+Bin/Bin
+Bin/BetaB4
+10
+20
+30
+40
+50
+60
+80
+0.2
+0.5
+1.0
+2.0
+Clayton / d = 2 / tau = 0.5
+n
+average of MSE of cov. est.
+BetaB4/Dirac
+BetaB4/Bin
+BetaB4/BetaB4
+10
+20
+30
+40
+50
+60
+80
+0.2
+0.5
+1.0
+2.0
+Clayton / d = 2 / tau = 0.75
+n
+average of MSE of cov. est.
+Dirac/Dirac
+Dirac/Bin
+Dirac/BetaB4
+10
+20
+30
+40
+50
+60
+80
+0.05
+0.10
+0.20
+0.50
+1.00
+2.00
+Clayton / d = 2 / tau = 0.75
+n
+average of MSE of cov. est.
+Bin/Dirac
+Bin/Bin
+Bin/BetaB4
+10
+20
+30
+40
+50
+60
+80
+0.05
+0.10
+0.20
+0.50
+1.00
+2.00
+Clayton / d = 2 / tau = 0.75
+n
+average of MSE of cov. est.
+BetaB4/Dirac
+BetaB4/Bin
+BetaB4/BetaB4
+Fig 1. For observations generated from the bivariate Clayton copula with a Kendall’s tau of
+τ ∈ {0, 0.25, 0.5, 0.75} and for each combination of target and multiplier process, average of
+the empirical MSEs (×104) of the bootstrap estimators of the covariance of the target process
+at the points in P against the sample size n. The legend “Dirac/Bin” for instance refers to
+the situation when the target process is CDirac
+n
+and the multiplier process is ˆC
+[b],Bin
+n
+.
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+23
+In a second experiment, we assessed how well high quantiles of
+KS(fn) =
+sup
+u∈[0,1]d |fn(u)|
+and
+CvM(fn) =
+�
+[0,1]d{fn(u)}2du
+(4.14)
+for d ∈ {2, 3} and fn ∈ {CDirac
+n
+, CBin
+n , CBetaB4
+n
+} can be estimated by the three can-
+didate multiplier bootstraps. From a practical perspective, the integral in (4.14)
+was approximated by a mean using a uniform grid on (0, 1)d of size 102 when
+d = 2 and 53 when d = 3. For d ∈ {2, 3}, C the Clayton or the Gumbel-Hougaard
+copula whose bivariate margins have a Kendall’s tau of τ ∈ {0, 0.25, 0.5, 0.75}
+and n ∈ {10, 20, 40, 80}, the 90% and 95%-quantiles of CvM(fn) were first
+precisely estimated from 100 000 independent samples of size n drawn from C.
+Next, for each combination of d, C, n, τ, target process and multiplier pro-
+cess, we generated 1000 samples from C and, for each sample, we computed
+B = 1000 multiplier bootstrap replicates. These B = 1000 replicates were used
+to obtain one estimate of each of the target quantiles. Following Kojadinovic
+and Stemikovskaya (2019), all such estimations were carried out using centered
+replicates of fn. When fn = CDirac
+n
+for instance, this amounts to using, for any
+u ∈ [0, 1]d and b ∈ {1, . . . , B},
+ˆC
+[b],Dirac
+n
+(u) − 1
+B
+B
+�
+b=1
+ˆC
+[b],Dirac
+n
+(u),
+instead of ˆC[b],Dirac
+n
+(u) = ˆC[b]
+n (0, 1, u) in (4.3). The centered versions of the other
+replicates are defined analogously. The rationale behind centering is that the
+replicates, whatever their type, can be regarded as computable approximations
+of the limiting centered Gaussian process CC(0, 1, ·) in (2.13); see, for instance,
+Theorem 4.3. Note that the use of centered replicates was found to always lead to
+better finite-sample performance in the related Monte Carlo experiments carried
+out in Kojadinovic and Stemikovskaya (2019). Its use is however irrelevant in the
+previous covariance estimation experiment given the formula of the empirical
+covariance.
+The results for the 95%-quantiles of the Kolmogorov–Smirnov functionals
+when C is the trivariate Gumbel–Hougaard are reported in Figure 2. The con-
+clusions are overall similar to those obtained after the first experiment:
+• The 95%-quantile of the Kolmogorov–Smirnov functional of CDirac
+n
+is al-
+ways best estimated using the corresponding empirical quantile of the
+same functional of ˆC[b],Dirac
+n
+.
+• When the target process is CBin
+n , the best results are obtained when the
+multiplier process is ˆC[b],Bin
+n
+, except in the case of strongly dependent ob-
+servations in which case, for the sample sizes under consideration, ˆC[b],Dirac
+n
+gives better estimations.
+• When the target process is CBetaB4
+n
+, it is only when n reaches 40 or 80 that
+the best estimations are obtained using ˆC[b],BetaB4
+n
+. For smaller n, the use
+of ˆC[b],Bin
+n
+gives better results.
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+24
+10
+20
+30
+40
+50
+60
+80
+5
+10
+20
+50
+200
+500
+2000
+indep. cop. / d = 3 / tau = 0
+n
+MSE of the est. of the 0.95−quant. of KS
+Dirac/Dirac
+Dirac/Bin
+Dirac/BetaB4
+10
+20
+30
+40
+50
+60
+80
+5
+10
+20
+50
+100
+200
+500
+indep. cop. / d = 3 / tau = 0
+n
+MSE of the est. of the 0.95−quant. of KS
+Bin/Dirac
+Bin/Bin
+Bin/BetaB4
+10
+20
+30
+40
+50
+60
+80
+10
+20
+50
+100
+200
+500
+1000
+indep. cop. / d = 3 / tau = 0
+n
+MSE of the est. of the 0.95−quant. of KS
+BetaB4/Dirac
+BetaB4/Bin
+BetaB4/BetaB4
+10
+20
+30
+40
+50
+60
+80
+5
+10
+20
+50
+200
+500
+2000
+Gumbel−Hougaard / d = 3 / tau = 0.25
+n
+MSE of the est. of the 0.95−quant. of KS
+Dirac/Dirac
+Dirac/Bin
+Dirac/BetaB4
+10
+20
+30
+40
+50
+60
+80
+5
+10
+20
+50
+100
+200
+500
+Gumbel−Hougaard / d = 3 / tau = 0.25
+n
+MSE of the est. of the 0.95−quant. of KS
+Bin/Dirac
+Bin/Bin
+Bin/BetaB4
+10
+20
+30
+40
+50
+60
+80
+20
+50
+100
+200
+500
+Gumbel−Hougaard / d = 3 / tau = 0.25
+n
+MSE of the est. of the 0.95−quant. of KS
+BetaB4/Dirac
+BetaB4/Bin
+BetaB4/BetaB4
+10
+20
+30
+40
+50
+60
+80
+20
+50
+100
+500
+2000
+Gumbel−Hougaard / d = 3 / tau = 0.5
+n
+MSE of the est. of the 0.95−quant. of KS
+Dirac/Dirac
+Dirac/Bin
+Dirac/BetaB4
+10
+20
+30
+40
+50
+60
+80
+20
+50
+100
+200
+500
+1000
+Gumbel−Hougaard / d = 3 / tau = 0.5
+n
+MSE of the est. of the 0.95−quant. of KS
+Bin/Dirac
+Bin/Bin
+Bin/BetaB4
+10
+20
+30
+40
+50
+60
+80
+20
+50
+100
+200
+500
+1000
+Gumbel−Hougaard / d = 3 / tau = 0.5
+n
+MSE of the est. of the 0.95−quant. of KS
+BetaB4/Dirac
+BetaB4/Bin
+BetaB4/BetaB4
+10
+20
+30
+40
+50
+60
+80
+20
+50
+100
+200
+500
+2000
+Gumbel−Hougaard / d = 3 / tau = 0.75
+n
+MSE of the est. of the 0.95−quant. of KS
+Dirac/Dirac
+Dirac/Bin
+Dirac/BetaB4
+10
+20
+30
+40
+50
+60
+80
+20
+50
+100
+200
+500
+1000
+Gumbel−Hougaard / d = 3 / tau = 0.75
+n
+MSE of the est. of the 0.95−quant. of KS
+Bin/Dirac
+Bin/Bin
+Bin/BetaB4
+10
+20
+30
+40
+50
+60
+80
+50
+100
+200
+500
+Gumbel−Hougaard / d = 3 / tau = 0.75
+n
+MSE of the est. of the 0.95−quant. of KS
+BetaB4/Dirac
+BetaB4/Bin
+BetaB4/BetaB4
+Fig 2.
+For observations generated from the trivariate Gumbel–Hougaard copula whose bi-
+variate margins have a Kendall’s tau of τ ∈ {0, 0.25, 0.5, 0.75}, empirical MSE (×104)
+of the three candidate multiplier estimators of high quantiles of KS(fn) in (4.14) for
+fn ∈ {CDirac
+n
+, CBin
+n , CBetaB4
+n
+} against the sample size n. The legend “Dirac/Bin” for instance
+refers to the situation when the target process is CDirac
+n
+and the multiplier process is ˆC
+[b],Bin
+n
+.
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+25
+Results for the Clayton copula, 90%-quantiles, dimension d = 2 or Cram´er–von
+Mises functionals (not reported) are not qualitatively different.
+The previous experiments confirm that it seems meaningful to resample Cν
+n
+in (2.9) using multiplier replicates constructed with the same smoothing distri-
+butions, that is, with ˆC[b],ν
+n
+in (4.7) or ˇC[b],ν
+n
+in (4.8), although this choice may
+not be optimal in certain cases when n is small.
+4.6. Application to change-point detection
+A natural application area for the smooth sequential empirical copula process Cν
+n
+in (2.9) is that of change-point detection. To illustrate the possible advantages
+coming from the use of smooth empirical copulas in inference procedures, we
+first briefly explain in this section how the previous derivations can be used to
+obtain a smooth version of the test proposed in B¨ucher et al. (2014) for detecting
+changes in the cross-sectional dependence of multivariate time series. We then
+reproduce some of the experiments of B¨ucher et al. (2014) to compare the (non-
+smooth) test proposed therein with its smooth version based on the empirical
+beta copula and on corresponding smooth bootstrap replicates. Note that we
+did not consider the use of the alternative data-adaptive smoothing distributions
+considered in Kojadinovic and Yi (2022) and leading to the estimator CBetaB4
+k:l
+because they incur a substantially higher computational cost.
+The null hypothesis of such tests is that X 1:n is a stretch from a stationary
+time series (of continuous random vectors) and their aim is to be particularly
+sensitive to the alternative hypothesis
+H1 : ∃ distinct C1, C2 and k⋆ ∈ {1, . . . , n − 1} such that
+X1, . . . , Xk⋆ have copula C1 and Xk⋆+1, . . . , Xn have copula C2. (4.15)
+The ingredients of the smooth version of the test can be obtained mutatis
+mutandis from B¨ucher et al. (2014). Specifically, we consider as test statistic
+the maximally selected Cram´er–von Mises functional defined by
+Sν
+n = sup
+s∈[0,1]
+�
+[0,1]d {Dν
+n(s, u)}2 dC1:n(u),
+where
+Dν
+n(s, u) = √nλn(0, s)λn(s, 1){Cν
+1:⌊ns⌋(u)−Cν
+⌊ns⌋+1:n(u)},
+(s, u) ∈ [0, 1]d+1.
+As one can see, the latter involves comparisons of (smooth) empirical copulas
+computed from subsamples of the data. Noticing that, under the null,
+Dν
+n(s, u) = λn(s, 1) Cν
+n(0, s, u) − λn(0, s)Cν
+n(s, 1, u),
+(s, u) ∈ [0, 1]d+1,
+possible multiplier bootstrap replicates for Sν
+n can be defined either by
+ˆS
+[b],ν
+n
+= sup
+s∈[0,1]
+�
+[0,1]d{ˆD
+[b],ν
+n
+(s, u)}2dC1:n(u),
+b ∈ N,
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+26
+or by
+ˇS
+[b],ν
+n
+= sup
+s∈[0,1]
+�
+[0,1]d{ˇD
+[b],ν
+n
+(s, u)}2dC1:n(u),
+b ∈ N,
+(4.16)
+where, for any (s, u) ∈ [0, 1]d+1,
+ˆD
+[b],ν
+n
+(s, u) = λn(s, 1) ˆC
+[b],ν
+n
+(0, s, u) − λn(0, s) ˆC
+[b],ν
+n
+(s, 1, u),
+ˇD
+[b],ν
+n
+(s, u) = λn(s, 1) ˇC
+[b],ν
+n
+(0, s, u) − λn(0, s) ˇC
+[b],ν
+n
+(s, 1, u),
+with ˆC[b],ν
+n
+and ˇC[b],ν
+n
+defined in (4.7) and (4.8), respectively. Note that, in the
+expressions of the multiplier replicates of Cν
+n, as estimators of the first-order
+partial derivatives of the copula, we use the “truncated” finite-difference based
+estimators defined in (5.8) of the forthcoming section with bandwidths h = h′ =
+min{(l − k + 1)−1/2, 1/2}. As we will see from Proposition 5.5, the latter can
+satisfy Condition 4.1. Finally, as in B¨ucher et al. (2014), approximate p-values
+for Sν
+n can be computed via either
+1
+B
+B
+�
+b=1
+1
+�
+ˆS
+[b],ν
+n
+≥ Sν
+n
+�
+or
+1
+B
+B
+�
+b=1
+1
+�
+ˇS
+[b],ν
+n
+≥ Sν
+n
+�
+,
+for some large integer B. Theoretical results confirming that the above way of
+proceeding is asymptotically valid under the null can be obtained by starting
+from Theorem 4.3, proceeding as in B¨ucher et al. (2014) and finally using results
+stated in Section 4 of B¨ucher and Kojadinovic (2019).
+If, for any m ∈ N, the underlying smoothing distributions νx
+u, x ∈ (Rd)m,
+u ∈ [0, 1]d, are Dirac measures at u, the previous ingredients are non-smooth
+and the resulting test coincides exactly with the test studied in B¨ucher et al.
+(2014). The test statistic will naturally be denoted by SDirac
+n
+in that case. As
+alternative smoothing distributions, we considered those leading to the empirical
+beta copula and specified in Remark 2.2 as well as at the end of Section 2.1.
+The resulting statistic will then naturally be denoted by SBin
+n .
+To compare the test based on SBin
+n
+to the test based on SDirac
+n
+, we consid-
+ered experiments similar to those reported in Section 5 of B¨ucher et al. (2014).
+Both tests were carried out at the 5% significance level using replicates of the
+form (4.16) as these seemed to lead to better results. The dependent multiplier
+sequences necessary to carry out the tests were generated as explained in the last
+paragraph of Appendix C of B¨ucher et al. (2014). The value of the bandwidth
+parameter ℓn appearing in (M2) and (M3) in Section 4.2 was chosen using the
+procedure described in B¨ucher and Kojadinovic (2016, Section 5) (although this
+way of proceeding may not be “optimal” for the smooth multiplier bootstrap
+replicates).
+As a first experiment, we estimated the percentages of rejection of the null
+hypothesis of stationarity for data generated under the null. As data generating
+model, we used a bivariate AR(1) model. Specifically, let Ui, i ∈ {−100, . . . , n},
+be a bivariate i.i.d. sample from a copula C. Then, set ϵi = (Φ−1(Ui1), Φ−1(Ui2)),
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+27
+Table 3
+Percentages of rejection of the null hypothesis of stationarity computed from 1000 samples
+of size n ∈ {25, 50, 100, 200} generated as explained in Section 4.6, where C is the bivariate
+Frank copula with a Kendall’s tau of τ ∈ {0, 0.33, 0.66}.
+n = 25
+n = 50
+n = 100
+n = 200
+β
+τ
+SDirac
+n
+SBin
+n
+SDirac
+n
+SBin
+n
+SDirac
+n
+SBin
+n
+SDirac
+n
+SBin
+n
+0
+0.00
+17.5
+13.3
+7.7
+8.0
+5.5
+5.8
+3.8
+4.4
+0.33
+18.7
+13.4
+7.6
+7.3
+4.9
+6.3
+4.2
+4.0
+0.66
+21.1
+11.7
+5.6
+4.9
+3.0
+3.1
+3.2
+3.8
+0.3
+0.00
+18.8
+16.1
+6.2
+7.4
+4.3
+4.7
+6.4
+6.0
+0.33
+21.4
+16.4
+7.8
+8.7
+5.2
+5.9
+5.4
+5.4
+0.66
+25.3
+16.8
+5.4
+5.9
+2.1
+3.0
+1.2
+1.4
+0.5
+0.00
+26.1
+22.8
+11.4
+11.7
+6.1
+6.6
+6.2
+7.2
+0.33
+22.9
+23.0
+10.3
+11.2
+5.5
+7.2
+2.4
+3.6
+0.66
+27.5
+20.1
+10.5
+11.0
+2.2
+3.6
+1.6
+1.6
+where Φ is the d.f. of the standard normal distribution, and X−100 = ϵ−100.
+Finally, for any j ∈ {1, 2} and i ∈ {−99, . . . , n}, compute recursively
+Xij = βXi−1,j + ϵij,
+where the first 100 observations are used as a burn-out sample.
+We considered n ∈ {25, 50, 100, 200}, C to be bivariate Frank copula with
+a Kendall’s tau of τ ∈ {0, 0.33, 0.66} and β ∈ {0, 0.3, 0.5}. The corresponding
+rejection percentages are reported in Table 3. As one can see, both tests appear
+to hold their level reasonably well when n ∈ {100, 200}. The tests should however
+clearly not be used when n = 25 but might be employed when n = 50 in the
+case of weakly serially dependent data.
+As a second experiment, we estimated rejection percentages of the null hy-
+pothesis of stationarity for data generated under H1 in (4.15). To do so, we
+considered a similar data generating model as in the first experiment except
+that the Ui’s for i ∈ {−100, . . . , k⋆} are i.i.d. from a copula C1 while the Ui’s
+for i ∈ {k⋆ + 1, . . . , n} are i.i.d. from a copula C2 ̸= C1. Following B¨ucher
+et al. (2014), we set k⋆ = ⌊nt⌋ with t ∈ {0.1, 0.25, 0.5} and considered n ∈
+{50, 100, 200}, C1 the bivariate Frank copula with a Kendall’s tau 0.2 and C2
+the bivariate Frank copula with a Kendall’s tau in {0.4, 0.6}. The results are
+reported in Table 4. As one can see, the test based on SBin
+n
+appears overall to
+be more powerful than the one based on SDirac
+n
+. The largest differences in power
+tend to occur for τ = 0.6 and t ∈ {0.1, 0.25} which corresponds to the situation
+when the test statistic should be the largest because of a difference between an
+empirical copula computed from a small number of observations (approximately
+⌊nt⌋) and an empirical copula computed from the remaining observations. While
+one cannot conclude that smooth change-point detection tests such as the one
+based on SBin
+n
+will be more powerful than the non-smooth test based on SDirac
+n
+in all situations, the obtained results confirm in part the intuition that smooth
+tests might be more sensitive to changes at the beginning or at the end of the
+data sequence.
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+28
+Table 4
+Percentages of rejection of the null hypothesis of stationarity computed from 1000 samples
+of size n ∈ {50, 100, 200} generated under H1 as explained in Section 4.6, where k⋆ = ⌊nt⌋,
+C1 and C2 are both bivariate Frank copulas such that C1 has a Kendall’s tau of 0.2 and C2
+a Kendall’s tau of τ ∈ {0.4, 0.6}.
+β = 0
+β = 0.3
+τ
+n
+t
+SDirac
+n
+SBin
+n
+SDirac
+n
+SBin
+n
+0.4
+50
+0.10
+8.8
+8.7
+8.0
+8.1
+0.25
+13.5
+16.1
+14.0
+15.5
+0.50
+14.7
+15.3
+17.5
+18.4
+100
+0.10
+4.0
+4.9
+5.5
+7.6
+0.25
+16.9
+19.3
+14.8
+17.9
+0.50
+26.6
+28.8
+22.5
+25.3
+200
+0.10
+6.6
+7.4
+5.6
+6.6
+0.25
+29.4
+31.8
+22.0
+24.2
+0.50
+51.4
+53.8
+42.0
+43.8
+0.6
+50
+0.10
+10.2
+13.0
+9.1
+11.6
+0.25
+33.0
+39.8
+31.6
+39.1
+0.50
+53.0
+56.8
+47.0
+51.1
+100
+0.10
+12.1
+16.6
+8.6
+12.5
+0.25
+62.6
+70.9
+51.9
+60.3
+0.50
+83.1
+84.9
+75.0
+78.6
+200
+0.10
+30.4
+37.8
+21.0
+28.8
+0.25
+95.2
+97.0
+87.8
+91.0
+0.50
+99.4
+99.4
+97.0
+97.2
+5. Estimators of the first-order partial derivatives of the copula
+The multiplier bootstrap replicates defined in the previous section all depend
+on the choice of estimators of the first-order partial derivatives of C. For asymp-
+totic reasons, the latter were required to satisfy Condition 4.1. Obviously, the
+more accurate such estimators, the better we can expect the multiplier boot-
+straps to behave, whether they involve smoothing or not. After recalling existing
+definitions of such estimators based on finite differences of the classical empir-
+ical copula, we define two related classes of smooth estimators. Then, upon an
+appropriate choice of the underlying bandwidth parameters, we establish their
+weak consistency in a sequential setting which implies that many of the con-
+sidered estimators satisfy Condition 4.1. In the last subsection, we report the
+results of bivariate and trivariate Monte Carlo experiments comparing selected
+estimators in terms of integrated mean squared error.
+Note that, as already mentioned in the introduction, the results of this sec-
+tion can be of independent interest since, as discussed for instance in Janssen,
+Swanepoel and Veraverbeke (2016), estimators of the first-order partial deriva-
+tives of a copula have applications in mean and quantile regression as they lead
+to estimators of the conditional distribution function. In particular, as we shall
+see, several estimators considered in our Monte Carlo experiments display a bet-
+ter finite-sample performance than the Bernstein estimator studied in Janssen,
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+29
+Swanepoel and Veraverbeke (2016).
+5.1. Estimators based on finite differences of the empirical copula
+As already mentioned in Section 4.3, in their seminal work on the multiplier
+bootstrap for the classical empirical copula process, R´emillard and Scaillet
+(2009) considered estimators of the first-order partial derivatives ˙Cj, j ∈ {1, . . . , d},
+of C based on finite-differences of the empirical copula. In a sequential context,
+given a stretch X k:l = (Xk, . . . , Xl), 1 ≤ k ≤ l ≤ n, of observations and two
+bandwidth parameters h and h′ in [0, 1/2] such that h + h′ > 0, a slightly more
+general definition of the aforementioned estimators is
+˙C
+∇
+j,k:l,h,h′(u) = Ck:l{(u + hej) ∧ 1} − Ck:l{(u − h′ej) ∨ 0}
+h + h′
+,
+u ∈ [0, 1]d,
+(5.1)
+where ej is the jth vector of the canonical basis of Rd, 0 = (0, . . . , 0), 1 =
+(1, . . . , 1) ∈ Rd, ∧ (resp. ∨) denotes the minimum (resp. maximum) component-
+wise operator and Ck:l is the classical empirical copula of X k:l defined in (2.1).
+The symbol ∇ indicates that the estimators are based on finite-differences of
+Ck:l with “right” (resp. “left”) bandwidth h (resp. h′).
+In order to reduce the bias of the previous estimator for evaluation points
+u ∈ [0, 1] with uj ∈ [0, h′) ∪ (1 − h, 1], Kojadinovic, Segers and Yan (2011)
+considered the following minor variation of (5.1):
+˙C
+∆
+j,k:l,h,h′(u) = Ck:l{(u + hej) ∧ 1} − Ck:l{(u − h′ej) ∨ 0}
+(uj + h) ∧ 1 − (uj − h′) ∨ 0
+,
+u ∈ [0, 1]d.
+(5.2)
+Note the use of the symbol ∆ still referring to finite-differences but upside-down
+compared to ∇ to distinguish (5.2) from (5.1).
+As is well known, in general, ˙Cj exists almost everywhere on [0, 1]d and, for
+those u ∈ [0, 1]d for which it exists, 0 ≤
+˙Cj(u) ≤ 1 (see e.g., Nelsen, 2006,
+Theorem 2.2.7). A natural modification of the estimators ˙C∇
+j,k:l,h,h′ in (5.1) and
+˙C∆
+j,k:l,h,h′ in (5.2) thus consists of ensuring that they take their values in [0, 1]
+by truncating them:
+˙C
+∇
+j,k:l,h,h′ = ( ˙C
+∇
+j,k:l,h,h′ ∨ 0) ∧ 1,
+(5.3)
+˙C
+∆
+j,k:l,h,h′ = ( ˙C
+∆
+j,k:l,h,h′ ∨ 0) ∧ 1.
+(5.4)
+Notice that taking the maximum with 0 in the previous expressions is actually
+not necessary as the estimators in (5.1) and (5.2) cannot be negative since the
+empirical copula Ck:l is a multivariate d.f. We nonetheless keep (5.3) and (5.4) as
+they are to be consistent with certain forthcoming definitions. More generally, in
+the rest of this section, underlining will be used to denote estimators constrained
+to take their values in [0, 1].
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+30
+5.2. Two classes of smooth estimators
+To obtain smooth estimators of the first-order partial derivatives of C, the pro-
+posals in (5.1) and (5.2) can be extended in two natural ways. The first approach
+consists of considering finite-differences of smooth estimators of C. Given a
+stretch X k:l, 1 ≤ k ≤ l ≤ n, of observations and two bandwidth parameters h
+and h′ in [0, 1/2] such that h + h′ > 0, this leads to the estimators
+˙C
+ν,∇
+j,k:l,h,h′(u) = Cν
+k:l{(u + hej) ∧ 1} − Cν
+k:l{(u − h′ej) ∨ 0}
+h + h′
+,
+u ∈ [0, 1]d,
+(5.5)
+˙C
+ν,∆
+j,k:l,h,h′(u) = Cν
+k:l{(u + hej) ∧ 1} − Cν
+k:l{(u − h′ej) ∨ 0}
+(uj + h) ∧ 1 − (uj − h′) ∨ 0
+,
+u ∈ [0, 1]d,
+(5.6)
+where Cν
+k:l is the smooth empirical copula of X k:l defined in (2.2). Notice the
+order of the symbols ν and ∇ (resp. ∆) indicating that the empirical copula is
+first smoothed before finite-differencing is applied. Clearly, (5.1) (resp. (5.2)) is
+a particular case of (5.5) (resp. (5.6)) when the smoothing distributions ν
+X k:l
+u
+,
+u ∈ [0, 1]d, in (2.2) are chosen to be Dirac measures at u ∈ [0, 1]d.
+Remark 5.1. Since C is a multivariate d.f. with standard uniform margins, we
+have that, for any j ∈ {1, . . . , d} and u ∈ Vj (where Vj is defined in Con-
+dition 2.8),
+˙Cj(u(j)) = limh→0{C(u(j) + hej) − C(u(j))}/h = 1 (where the
+notation u(j) is defined above Theorem 2.12). Interestingly enough, the estima-
+tor ˙C
+ν,∆
+j,k:l,h,h′ in (5.6) can satisfy this boundary constraint, that is, we can have
+˙C
+ν,∆
+j,k:l,h,h′(u(j)) = 1. This will indeed happen if Cν
+k:l is a genuine copula, which
+according to Proposition 2.7, can occur under Condition 2.3 for specific choices
+of the smoothing distributions in (2.2) such as those leading to the empirical
+copulas CBin
+k:l or CBetaB4
+k:l
+defined in Section 2.1.
+By analogy with (5.3) and (5.4), it is straightforward to define truncated
+versions of the estimators in (5.5) and (5.6) by
+˙C
+ν,∇
+j,k:l,h,h′ = ( ˙C
+ν,∇
+j,k:l,h,h′ ∨ 0) ∧ 1,
+(5.7)
+˙C
+ν,∆
+j,k:l,h,h′ = ( ˙C
+ν,∆
+j,k:l,h,h′ ∨ 0) ∧ 1.
+(5.8)
+Remark 5.2. As discussed in Remark 5.1, the smoothing distributions ν
+X k:l
+u
+,
+u ∈ [0, 1]d, can be chosen such that Cν
+k:l is a genuine copula under Condi-
+tion 2.3. In that case, using the fact that Cν
+k:l is a multivariate d.f. with standard
+uniform margins, we immediately obtain (see, e.g., Durante and Sempi, 2015,
+Lemma 1.2.14) that, for any u ∈ [0, 1]d and h, h′ in [0, 1/2] such that h+h′ > 0,
+Cν
+k:l{(u + hej) ∧ 1} − Cν
+k:l{(u − h′ej) ∨ 0} ≤ (uj + h) ∧ 1 − (uj − h′) ∨ 0,
+which implies that 0 ≤ ˙C
+ν,∆
+j,k:l,h,h′ ≤ 1 and thus that truncation of ˙C
+ν,∆
+j,k:l,h,h′ is
+not necessary in that case since ˙C
+ν,∆
+j,k:l,h,h′ in (5.8) is equal to ˙C
+ν,∆
+j,k:l,h,h′.
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+31
+By analogy with (2.2), a second natural approach to obtain smooth partial
+derivative estimators consists of directly smoothing (5.1) and (5.2) and leads to
+the estimators
+˙C
+∇,ν
+j,k:l,h,h′(u) =
+�
+[0,1]d
+˙C
+∇
+j,k:l,h,h′(w)dν
+X k:l
+u
+(w),
+u ∈ [0, 1]d,
+(5.9)
+˙C
+∆,ν
+j,k:l,h,h′(u) =
+�
+[0,1]d
+˙C
+∆
+j,k:l,h,h′(w)dν
+X k:l
+u
+(w),
+u ∈ [0, 1]d.
+(5.10)
+This time the order of the symbols ν and ∇ (resp. ∆) is reversed indicating
+that it is the finite-differences-based estimator ˙C∇
+j,k:l,h,h′ in (5.1) (resp. ˙C∆
+j,k:l,h,h′
+in (5.2)) that is smoothed. Versions of these estimators that necessarily take
+their values in [0, 1] can be obtained by constructing them from the truncated
+estimators (5.3) and (5.4) instead, leading respectively to
+˙C
+∇,ν
+j,k:l,h,h′(u) =
+�
+[0,1]d
+˙C
+∇
+j,k:l,h,h′(w)dν
+X k:l
+u
+(w),
+u ∈ [0, 1]d,
+(5.11)
+˙C
+∆,ν
+j,k:l,h,h′(u) =
+�
+[0,1]d
+˙C
+∆
+j,k:l,h,h′(w)dν
+X k:l
+u
+(w),
+u ∈ [0, 1]d.
+(5.12)
+Note that a third approach to obtain a smooth estimator of the jth partial
+derivative ˙Cj would consist of attempting to directly differentiate Cν
+k:l in (2.2)
+with respect to its jth argument (provided of course that Cν
+k:l is differentiable).
+The resulting estimator
+˙Cν
+j,k:l = ∂Cν
+k:l
+∂uj
+may exist only on the set Vj defined in Condition 2.8. This is the path followed
+by Janssen, Swanepoel and Veraverbeke (2016), who, for some integer m ≥ 2,
+started from the empirical Bernstein copula CBern
+k:l,m in (2.3) (which, as discussed
+in Remark 2.2, is a particular case of Cν
+k:l in (2.2)). Let ˙CBern
+j,k:l,m = ∂CBern
+k:l,m/∂uj be
+the resulting estimator. Interestingly enough, from Lemma E.1 in Appendix E,
+we have that
+˙C
+Bern
+j,k:l,m(u) =
+�
+[0,1]d
+˙C
+∇
+j,k:l, 1
+m ,0(w)d˜µj,m,u(w),
+u ∈ Vj,
+(5.13)
+where
+˙C∇
+j,k:l, 1
+m ,0 is given by (5.1) with h = 1/m and h′ = 0 and, for any
+u ∈ [0, 1]d, ˜µj,m,u is the law of the random vector ( ˜Sm,1,u1/m, . . . , ˜Sm,d,ud/m)
+whose components are independent such that, for i ∈ {1, . . . , d} \ {j}, ˜Sm,i,ui is
+Binomial(m, ui) while ˜Sm,j,uj is Binomial(m − 1, uj). In other words, differenti-
+ating directly the empirical Bernstein copula CBern
+k:l,m in (2.3) with respect to its
+jth argument leads to a special case of the estimator in (5.9). Notice that, since
+the measures ˜µj,m,u are well-defined for any u ∈ [0, 1]d, the integral in (5.13) is
+actually well-defined for any u ∈ [0, 1]d. Hence, as we continue, we take (5.13)
+with u ∈ [0, 1]d as the definition of ˙CBern
+j,k:l,m.
+The following result, proven in Appendix E, shows that ˙CBern
+j,k:l,m can be easily
+computed.
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+32
+Proposition 5.3. Given a stretch X k:l, 1 ≤ k ≤ l ≤ n, of observations, we
+have that, for any j ∈ {1, . . . , d}, u ∈ [0, 1]d and integer m ≥ 2,
+˙C
+Bern
+j,k:l,m(u) =
+m
+l − k + 1
+l
+�
+i=k
+bm−1,uj
+�
+⌈mRk:l
+ij /(l − k + 1)⌉ − 1
+�
+×
+d
+�
+t=1
+t̸=j
+¯Bm,ut
+�
+⌈mRk:l
+it /(l − k + 1)⌉ − 1
+�
+,
+(5.14)
+where ⌈·⌉ denotes the ceiling function and, for any p ∈ N and u ∈ [0, 1], ¯Bp,u
+(resp. bp,u) is the survival (resp. probability mass) function of the Binomial(p, u).
+5.3. Weak consistency
+In order to study the weak consistency of the estimators of the partial derivatives
+of C defined in the previous subsection, it is necessary to link the bandwidth
+parameters in their expressions to the data (or, at least, to the amount of data)
+from which these estimators are computed. As we continue, for any n ∈ N and
+any potential d-dimensional data set x ∈ (Rd)n, h(x) and h′(x) will denote the
+values of the left and right bandwidths for the data set x. With this in mind,
+in the rest of this subsection, for the sake of a more compact notation, we shall
+write
+˙C
+ν,∇
+j,k:l (resp.
+˙C
+ν,∆
+j,k:l,
+˙C
+∇,ν
+j,k:l, . . . ) for
+˙C
+ν,∇
+j,k:l,h,h′ (resp.
+˙C
+ν,∆
+j,k:l,h,h′,
+˙C
+∇,ν
+j,k:l,h,h′,
+. . . ) with the understanding that h = h(X k:l) and h′ = h′(X k:l) are random
+variables. We impose in addition the following condition on the bandwidths.
+Condition 5.4 (Bandwidth condition). There exists positive sequences bn ↓ 0
+and b′
+n ↓ 0 and constants L2 ≥ L1 > 0 such that, for all n ∈ N, bn +b′
+n ≥ n−1/2,
+and, for any x ∈ (Rd)n, L1bn ≤ h(x) ≤ (L2bn) ∧ 1/2 and L1b′
+n ≤ h′(x) ≤
+(L2b′
+n) ∧ 1/2.
+As we shall see in Section 5.4, one meaningful possibility among many others
+is to consider that, for any n ∈ N and x ∈ Rd, the left and right bandwidths for
+the data set x are defined by h(x) = h′(x) = [M2{1−|τ(x)|}a+M1]n−1/2∧1/2,
+where M1, M2 > 0 are constants, τ(x) ∈ [−1, 1] is the value of the sample
+version of a suitable multivariate extension of Kendall’s tau for the data set x
+and a ∈ (0, ∞) is a fixed power. Roughly speaking, the bandwidths will be larger
+(resp. smaller) in the case of weakly (resp. strongly) cross-sectionally dependent
+data. It is easy to verify that Condition 5.4 holds for the previous definitions.
+The following result, proven in Appendix E, establishes the weak consistency
+of the smooth estimators of the first class in a sequential setting.
+Proposition 5.5 (Weak consistency in a sequential setting for the first class
+of smooth estimators). Under Conditions 2.8, 2.9, 2.11 and 5.4, for any j ∈
+{1, . . . , d}, δ ∈ (0, 1) and ε ∈ (0, 1/2),
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+��� ˙Cν,∆
+j,⌊ns⌋+1:⌊nt⌋(u) − ˙Cj(u)
+��� = oP(1),
+(5.15)
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+33
+where ˙Cν,∆
+j,k:l is defined in (5.6), and similarly for ˙C
+ν,∇
+j,k:l in (5.5), ˙C
+ν,∇
+j,k:l in (5.7)
+and ˙C
+ν,∆
+j,k:l in (5.8).
+Remark 5.6. An inspection of the proof of the previous result reveals that the
+second supremum in (5.15) can be replaced by a supremum over u ∈ [0, 1]d if
+˙Cj happens to be continuous on [0, 1]d instead of only satisfying Condition 2.8;
+see also Kojadinovic, Segers and Yan (2011).
+As a consequence of the previous proposition, we have that, under the con-
+ditions of Proposition 5.5, the estimators ˙C
+ν,∇
+j,k:l and ˙C
+ν,∆
+j,k:l satisfy Condition 4.1
+since they are bounded in absolute value (by one) by construction.
+The next result is an immediate corollary of Proposition 5.5 since the esti-
+mator in (5.1) (resp. (5.2)) is a particular case of the one in (5.5) (resp. (5.6))
+when the smoothing distributions ν
+X k:l
+u
+, u ∈ [0, 1]d, in (2.2) are chosen to be
+Dirac measures at u ∈ [0, 1]d (the latter clearly satisfy Condition 2.9).
+Corollary 5.7 (Weak consistency in a sequential setting for the non-smooth
+finite-differences-based estimators). Under Conditions 2.8, 2.11 and 5.4, for
+any j ∈ {1, . . . , d}, δ ∈ (0, 1) and ε ∈ (0, 1/2),
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+��� ˙C
+∆
+j,⌊ns⌋+1:⌊nt⌋(u) − ˙Cj(u)
+��� = oP(1),
+where ˙C∆
+j,k:l is defined in (5.2), and similarly for ˙C∇
+j,k:l in (5.1), ˙C
+∇
+j,k:l in (5.3)
+and ˙C
+∆
+j,k:l in (5.4).
+We now move to the second class of smooth estimators of the partial deriva-
+tives. As we shall see below, to establish their weak consistency, it suffices,
+among others, that the underlying smoothing distributions satisfy the following
+weaker version of Condition 2.9.
+Condition 5.8 (Weak variance condition). There exists a positive sequence
+an ↓ 0 such that, for any n ∈ N, x ∈ (Rd)n, u ∈ [0, 1]d and j ∈ {1, . . . , d},
+Var(W x
+j,uj) ≤ an.
+The following result is proven in Appendix E.
+Proposition 5.9 (Weak consistency in a sequential setting for the second class
+of smooth estimators). Under Conditions 2.8, 2.11, 5.4 and 5.8, for any j ∈
+{1, . . . , d}, δ ∈ (0, 1) and ε ∈ (0, 1/2),
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+��� ˙C
+∆,ν
+j,⌊ns⌋+1:⌊nt⌋(u) − ˙Cj(u)
+��� = oP(1),
+(5.16)
+where ˙C
+∆,ν
+j,k:l is defined in (5.12), and similarly for ˙C
+∇,ν
+j,k:l in (5.11).
+One may wonder why the estimators ˙C
+∇,ν
+j,k:l in (5.9) and ˙C
+∆,ν
+j,k:l in (5.10) are
+not included in the previous proposition. Actually, upon additionally imposing
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+34
+that the left and right bandwidths of ˙C∇
+j,k:l in (5.1) and ˙C∆
+j,k:l in (5.2) (which are
+to be smoothed) are equal and in the absence of ties (see Condition 2.3), weak
+consistency can also be proven for the estimators ˙C
+∇,ν
+j,k:l and ˙C
+∆,ν
+j,k:l using the same
+technique of proof. An inspection of the proof and some additional thinking
+reveals that this follows from the fact that these estimators are bounded on
+[0, 1]d in this case. When one of the bandwidths is zero, this is not necessarily
+the case anymore. This is also why the previous proposition cannot be directly
+used to establish the weak consistency of the Bernstein estimator in (5.13).
+For this estimator, one additionally needs to rely on the fact that the finite
+difference-based estimator that is smoothed is bounded on the support of the
+smoothing distributions. This is used in the proof in Appendix E of the next
+proposition.
+Proposition 5.10 (Weak consistency of the Bernstein estimator in a sequential
+setting). Assume that Conditions 2.3, 2.8 and 2.11 hold and, for any i ∈ N,
+let mi = ⌊Liθ⌋ ∨ 2 for some constants L > 0 and θ ∈ (0, 1/2]. Then, for any
+j ∈ {1, . . . , d}, δ ∈ (0, 1) and ε ∈ (0, 1/2),
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+��� ˙C
+Bern
+j,⌊ns⌋+1:⌊nt⌋,m⌊nt⌋−⌊ns⌋ − ˙Cj(u)
+��� = oP(1),
+(5.17)
+where ˙CBern
+j,k:l,m is defined in (5.13). In addition, for any n ∈ N,
+sup
+(s,t,u)∈Λ×[0,1]d
+��� ˙C
+Bern
+j,⌊ns⌋+1:⌊nt⌋,m⌊nt⌋−⌊ns⌋(u)
+��� ≤ 1 + L ∨ 2.
+(5.18)
+Note that our technique of proof allows us to establish uniform weak con-
+sistency of the estimator even for θ = 1/2, whereas the approach used in the
+proof of Proposition 3.1 of Bouezmarni, El Ghouch and Taamouti (2013) (for
+Bernstein copula density estimators and which could be adapted to first-order
+partial derivative estimators) leads to the result only for θ < 1/2. Another con-
+sequence of the previous result is that the Bernstein partial derivative estimator
+as parametrized in Proposition 5.10 can satisfy Condition 4.1.
+5.4. Finite-sample performance of selected estimators
+The aim of this subsection is to compare the finite-sample performance of some
+of the estimators introduced previously. Specifically, for each n ∈ {10, 20, . . . , 100},
+each data generating copula C and each partial derivative estimator ˙Cj,1:n under
+investigation, we estimated its integrated mean squared error
+IMSE( ˙Cj,1:n) =
+�
+[0,1]d E
+��
+˙Cj,1:n(u) − ˙Cj(u)
+�2�
+du.
+To do so, we applied the trick described in detail in Appendix B of Segers, Sibuya
+and Tsukahara (2017) allowing to compute IMSE( ˙Cj,1:n) as a single expectation
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+35
+and proceeded by Monte Carlo simulation using 20 000 independent random
+samples of size n from C.
+In a first experiment, we compared estimators of the form ˙C
+ν,∇
+j,1:n,h,h′ in
+(5.7) to estimators of the form ˙C
+ν,∆
+j,1:n,h,h′ in (5.8) for deterministic bandwidths
+h = h′ = n−1/2 ∧ 1/2. Specifically, we considered estimators based, respec-
+tively, on finite-differences of the classical empirical copula C1:n in (2.1), on
+finite-differences of the empirical beta copula CBin
+1:n defined in (3.1), and on
+finite-differences of its data-adaptive extension CBetaB4
+1:n
+defined at the end of Sec-
+tion 2.1. As data-generating copula C, we considered the bivariate or trivariate
+Clayton or Gumbel–Hougaard copula with bivariate margins with a Kendall’s
+tau of τ ∈ {0, 0.25, 0.5, 0.75} as well as the bivariate Frank copula with a
+Kendall’s tau of τ ∈ {0, −0.25, −0.5, −0.75}. Note that, since all data-generating
+copulas are exchangeable, it suffices to focus on only one partial derivative es-
+timator, say the first one. As expected, the integrated mean squared error of
+estimators of the form ˙C
+ν,∆
+j,1:n,h,h′ was always found to be (substantially) below
+that of the corresponding estimator ˙C
+ν,∇
+j,1:n,h,h′, confirming that the adjusted
+numerator in (5.6) compared to the one in (5.5) helps indeed to improve the
+finite-sample performance of finite-difference-based estimators.
+In a second experiment, we compared the aforementioned three estimators
+of the form ˙C
+ν,∆
+j,1:n,h,h′ in (5.8). They will be denoted as ˙C
+∆
+j,1:n,h,h′ = ˙C
+Dirac,∆
+j,1:n,h,h′,
+˙C
+Bin,∆
+j,1:n,h,h′ and ˙C
+BetaB4,∆
+j,1:n,h,h′ as we continue. As could have been expected from the
+experiments of Kojadinovic and Yi (2022) comparing the underlying copula esti-
+mators, it is the estimator ˙C
+BetaB4,∆
+j,1:n,h,h′ that always displayed the lowest integrated
+mean squared error, followed by ˙C
+Bin,∆
+j,1:n,h,h′ and ˙C
+∆
+j,1:n,h,h′.
+We next investigated the influence of the bandwidths on the integrated mean
+squared error of ˙C
+BetaB4,∆
+j,1:n,h,h′. Deterministic bandwidths of the form h = h′ =
+(Ln−1/2) ∧ 1/2 were considered with L ∈ {0.5, 1, 2, 4}. The corresponding in-
+tegrated mean squared errors are represented in the first column of graphs of
+Figure 3 (resp. Figure 4) when the data-generating copula is the bivariate Frank
+copula with negative dependence (resp. the trivariate Gumbel–Hougaard cop-
+ula). The legend “BetaB 0.5” refers to the estimator with L = 0.5 and so on. As
+one can see, the weaker the cross-sectional dependence, the larger the (constant
+L in the expression of the) bandwidths should be.
+An inspection of (5.9) and (5.10) and some thinking reveals that estimators
+from the second class can be difficult to compute in practice. For that reason, in
+our experiments, we solely focused on the Bernstein estimator ˙CBern
+j,1:n,m in (5.13)
+which can be computed using (5.14). Mimicking the previous experiment, we
+considered a deterministic choice for the parameter m of the form m = ⌊Ln1/2⌋∨
+2 with L ∈ {0.5, 1, 2, 4}. The corresponding integrated mean squared errors are
+represented in the second column of graphs of Figure 3 (resp. Figure 4) when the
+data-generating copula is the bivariate Frank copula with negative dependence
+(resp. the trivariate Gumbel–Hougaard copula). The legend “Bern 0.5” refers
+to the estimator with L = 0.5 and so on. As one can see, this time, the stronger
+the cross-sectional dependence, the larger the (constant L in the expression of
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+36
+20
+40
+60
+80
+100
+0.000
+0.002
+0.004
+0.006
+0.008
+indep. cop. / d = 2 / tau = 0
+n
+integrated mean squared error
+BetaB 0.5
+BetaB 1
+BetaB 2
+BetaB 4
+20
+40
+60
+80
+100
+0.000
+0.002
+0.004
+0.006
+0.008
+Frank / d = 2 / tau = −0.25
+n
+integrated mean squared error
+BetaB 0.5
+BetaB 1
+BetaB 2
+BetaB 4
+20
+40
+60
+80
+100
+0.000
+0.005
+0.010
+0.015
+0.020
+Frank / d = 2 / tau = −0.5
+n
+integrated mean squared error
+BetaB 0.5
+BetaB 1
+BetaB 2
+BetaB 4
+20
+40
+60
+80
+100
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+Frank / d = 2 / tau = −0.75
+n
+integrated mean squared error
+BetaB 0.5
+BetaB 1
+BetaB 2
+BetaB 4
+20
+40
+60
+80
+100
+0.000
+0.010
+0.020
+0.030
+indep. cop. / d = 2 / tau = 0
+n
+integrated mean squared error
+Bern 0.5
+Bern 1
+Bern 2
+Bern 4
+20
+40
+60
+80
+100
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+Frank / d = 2 / tau = −0.25
+n
+integrated mean squared error
+Bern 0.5
+Bern 1
+Bern 2
+Bern 4
+20
+40
+60
+80
+100
+0.00
+0.01
+0.02
+0.03
+0.04
+0.05
+Frank / d = 2 / tau = −0.5
+n
+integrated mean squared error
+Bern 0.5
+Bern 1
+Bern 2
+Bern 4
+20
+40
+60
+80
+100
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+Frank / d = 2 / tau = −0.75
+n
+integrated mean squared error
+Bern 0.5
+Bern 1
+Bern 2
+Bern 4
+20
+40
+60
+80
+100
+0.000
+0.005
+0.010
+0.015
+indep. cop. / d = 2 / tau = 0
+n
+integrated mean squared error
+Dirac
+Bin
+Adap BetaB4
+Adap Bern
+20
+40
+60
+80
+100
+0.000
+0.005
+0.010
+0.015
+Frank / d = 2 / tau = −0.25
+n
+integrated mean squared error
+Dirac
+Bin
+Adap BetaB4
+Adap Bern
+20
+40
+60
+80
+100
+0.000
+0.005
+0.010
+0.015
+Frank / d = 2 / tau = −0.5
+n
+integrated mean squared error
+Dirac
+Bin
+Adap BetaB4
+Adap Bern
+20
+40
+60
+80
+100
+0.000
+0.010
+0.020
+0.030
+Frank / d = 2 / tau = −0.75
+n
+integrated mean squared error
+Dirac
+Bin
+Adap BetaB4
+Adap Bern
+Fig 3.
+Estimated integrated mean squared errors against n
+∈
+{10, 20, . . . , 100} of
+four estimators of
+˙C1 when C is the bivariate Frank copula with a Kendall’s tau in
+{0, −0.25, −0.5, −0.75}.
+the) parameter m should be. This is of course not surprising given the previous
+experiment and since 1/m plays the role of a bandwidth.
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+37
+20
+40
+60
+80
+100
+0.000
+0.002
+0.004
+0.006
+indep. cop. / d = 3 / tau = 0
+n
+integrated mean squared error
+BetaB 0.5
+BetaB 1
+BetaB 2
+BetaB 4
+20
+40
+60
+80
+100
+0.000
+0.002
+0.004
+0.006
+0.008
+Gumbel−Hougaard / d = 3 / tau = 0.25
+n
+integrated mean squared error
+BetaB 0.5
+BetaB 1
+BetaB 2
+BetaB 4
+20
+40
+60
+80
+100
+0.000
+0.005
+0.010
+0.015
+0.020
+Gumbel−Hougaard / d = 3 / tau = 0.5
+n
+integrated mean squared error
+BetaB 0.5
+BetaB 1
+BetaB 2
+BetaB 4
+20
+40
+60
+80
+100
+0.00
+0.01
+0.02
+0.03
+0.04
+Gumbel−Hougaard / d = 3 / tau = 0.75
+n
+integrated mean squared error
+BetaB 0.5
+BetaB 1
+BetaB 2
+BetaB 4
+20
+40
+60
+80
+100
+0.000
+0.005
+0.010
+0.015
+0.020
+indep. cop. / d = 3 / tau = 0
+n
+integrated mean squared error
+Bern 0.5
+Bern 1
+Bern 2
+Bern 4
+20
+40
+60
+80
+100
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+Gumbel−Hougaard / d = 3 / tau = 0.25
+n
+integrated mean squared error
+Bern 0.5
+Bern 1
+Bern 2
+Bern 4
+20
+40
+60
+80
+100
+0.00
+0.01
+0.02
+0.03
+0.04
+Gumbel−Hougaard / d = 3 / tau = 0.5
+n
+integrated mean squared error
+Bern 0.5
+Bern 1
+Bern 2
+Bern 4
+20
+40
+60
+80
+100
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+Gumbel−Hougaard / d = 3 / tau = 0.75
+n
+integrated mean squared error
+Bern 0.5
+Bern 1
+Bern 2
+Bern 4
+20
+40
+60
+80
+100
+0.000
+0.005
+0.010
+0.015
+indep. cop. / d = 3 / tau = 0
+n
+integrated mean squared error
+Dirac
+Bin
+Adap BetaB4
+Adap Bern
+20
+40
+60
+80
+100
+0.000
+0.005
+0.010
+0.015
+Gumbel−Hougaard / d = 3 / tau = 0.25
+n
+integrated mean squared error
+Dirac
+Bin
+Adap BetaB4
+Adap Bern
+20
+40
+60
+80
+100
+0.000
+0.005
+0.010
+0.015
+0.020
+Gumbel−Hougaard / d = 3 / tau = 0.5
+n
+integrated mean squared error
+Dirac
+Bin
+Adap BetaB4
+Adap Bern
+20
+40
+60
+80
+100
+0.00
+0.01
+0.02
+0.03
+Gumbel−Hougaard / d = 3 / tau = 0.75
+n
+integrated mean squared error
+Dirac
+Bin
+Adap BetaB4
+Adap Bern
+Fig 4.
+Estimated integrated mean squared errors against n ∈ {10, 20, . . . , 100} of four es-
+timators of
+˙C1 when C is the trivariate Gumbel–Hougaard copula whose bivariate margins
+have a Kendall’s tau in {0, 0.25, 0.5, 0.75}.
+The two previous experiments suggested to focus on data-adaptive band-
+widths for the estimators ˙C
+BetaB4,∆
+j,1:n,h,h′ and ˙CBern
+j,1:n,m. Specifically, for d ∈ {2, 3}
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+38
+and a data set x ∈ (Rd)n, we considered the settings h(x) = h′(x) = ([4{1 −
+|τ(x)|}6+1/2]n−1/2)∧1/2 for ˙C
+BetaB4,∆
+j,1:n,h,h′ and m(x) = ⌊{4|τ(x)|3/2+1/2}n1/2⌋∨2
+for ˙CBern
+j,1:n,m, where τ(x) ∈ [−1, 1] is the average of the values of the sample ver-
+sion of Kendall’s tau computed from the bivariate margins of the data set x.
+The integrated mean squared errors of the resulting data-adaptive estimators
+are represented in the third column of graphs of Figure 3 (resp. Figure 4)
+when the data-generating copula is the bivariate Frank copula with negative
+dependence (resp. the trivariate Gumbel–Hougaard copula). The legends “Adap
+BetaB4” and “Adap Bern” refer to the above-mentioned versions of the estima-
+tors ˙C
+BetaB4,∆
+j,1:n,h,h′ and ˙CBern
+j,1:n,m, respectively, while “Dirac” and “Bin” refer to the
+benchmark estimators ˙C
+∆
+j,1:n,h,h′ = ˙C
+Dirac,∆
+j,1:n,h,h′ and ˙C
+Bin,∆
+j,1:n,h,h′ with deterministic
+bandwidths h = h′ = n−1/2 ∧1/2 based on the empirical copula and the empiri-
+cal beta copula, respectively. Overall, it is the data-adaptive estimator ˙C
+BetaB4,∆
+j,1:n,h,h′
+that displays the best finite-sample behavior. The data-adaptive Bernstein esti-
+mator ˙CBern
+j,1:n,m appears to be competitive only when the data-generating copula
+is close to the independence copula.
+6. Conclusion
+Smooth nonparametric copula estimators, such as the empirical beta copula
+proposed by Segers, Sibuya and Tsukahara (2017) or its data-adaptive extension
+studied in Kojadinovic and Yi (2022), can be substantially better estimators
+than the classical empirical copula in small samples. To use such estimators in
+inference procedures, one typically needs to rely on resampling techniques.
+As investigated in Section 3, in the case of i.i.d. observations, a smooth boot-
+strap `a la Kiriliouk, Segers and Tsukahara (2021) can be asymptotically valid
+for a large class of smooth estimators that can be expressed as mixtures of d.f.s.
+When based on the empirical beta copula, Kiriliouk, Segers and Tsukahara
+(2021) found such a smooth bootstrap to be a competitive alternative to the
+multiplier bootstrap while being substantially simpler to implement. An empiri-
+cal finding of this work is that the smooth bootstrap based on the data-adaptive
+extension of the empirical beta copula proposed in Kojadinovic and Yi (2022)
+seems to lead to even better-behaved inference procedures than the former as it
+copes better with stronger dependence.
+Unfortunately, such smooth bootstraps cannot be used anymore in the time
+series setting. A second contribution of this work was to study both theoreti-
+cally and empirically smooth extensions of the sequential dependent multiplier
+bootstrap of B¨ucher and Kojadinovic (2016). As illustrated at the end of the
+fourth section, the latter can for instance be used to derive smooth change-point
+detection tests which are likely to be more sensitive to early or late changes than
+their non-smooth counterparts since, as already mentioned, smooth estimators
+are likely to be more accurate than the empirical copula when computed from
+small subsets of observations.
+In connection with the multiplier bootstrap, a third contribution of this work
+was the study of the weak consistency and finite-sample performance of two
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+39
+classes of smooth estimators of the first-order partial derivatives of the copula.
+The obtained results may be of independent interest since such estimators have
+applications in mean and quantile regression as they lead to estimators of the
+conditional distribution function. From an empirical perspective, our investiga-
+tions led to the proposal of a smooth data-adaptive estimator of the first-order
+partial derivatives of the copula that substantially outperforms, among others,
+the Bernstein estimator studied in Janssen, Swanepoel and Veraverbeke (2016).
+Appendix A: Proof of Corollary 2.13
+The proof of Corollary 2.13 is based on the following two lemmas.
+Lemma A.1. Let Xn be a process in ℓ∞(Λ × [0, 1]d) such that for all u ∈
+[0, 1]d and s ∈ [0, 1], Xn(s, s, u) = 0. Furthermore, assume that Xn ⇝ X in
+ℓ∞(Λ × [0, 1]d) where X has continuous trajectories almost surely. Then, under
+Condition 5.8 (which is implied by Condition 2.9),
+sup
+(s,t,u)∈Λ×[0,1]d
+�����
+�
+[0,1]d Xn(s, t, w)dν
+X ⌊ns⌋+1:⌊nt⌋
+u
+(w) − Xn(s, t, u)
+����� = oP(1), (A.1)
+sup
+(s,t,u)∈Λ×[0,1]d
+�����
+�
+[0,1]d Xn(s, t, w)dν
+X 1:n
+u
+(w) − Xn(s, t, u)
+����� = oP(1). (A.2)
+Proof. The first claim was proven in the proof of Lemma 32 of Kojadinovic and
+Yi (2022). The proof of (A.2) is very similar.
+■
+Lemma A.2. Assume that Conditions 2.8 and 2.9 hold. Then, almost surely,
+sup
+(s,t,u)∈Λ×[0,1]d
+√nλn(s, t)
+�����
+�
+[0,1]d C(w)dν
+X ⌊ns⌋+1:⌊nt⌋
+u
+(w) − C(u)
+����� = o(1), (A.3)
+sup
+(s,t,u)∈Λ×[0,1]d
+√nλn(s, t)
+�����
+�
+[0,1]d C(w)dν
+X 1:n
+u
+(w) − C(u)
+����� = o(1).
+(A.4)
+Proof. The first claim was proven in the proof of Lemma 33 of Kojadinovic and
+Yi (2022). The proof of (A.4) is an immediate consequence of the fact that the
+left-hand side of (A.4) is almost surely smaller than
+sup
+u∈[0,1]d
+√n
+�����
+�
+[0,1]d C(w)dν
+X 1:n
+u
+(w) − C(u)
+����� ,
+which is smaller than the left-hand side of (A.3) with probability 1.
+■
+Proof of Corollary 2.13. Combining Theorem 2.10 and Theorem 2.12, by the
+triangle inequality, we immediately obtain that
+sup
+(s,t,u)∈Λ×[0,1]d
+���Cν
+n(s, t, u) − ˜Cn(s, t, u)
+��� = oP(1),
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+40
+where ˜Cn is defined in (2.12). It thus remains to show that
+sup
+(s,t,u)∈Λ×[0,1]d
+���˜Cn(s, t, u) − ˜Cν
+n(s, t, u)
+��� = oP(1),
+sup
+(s,t,u)∈Λ×[0,1]d
+���˜Cn(s, t, u) − ¯Cν
+n(s, t, u)
+��� = oP(1).
+We only prove the first claim, the proof of the second one being similar. For any
+(s, t, u) ∈ Λ × [0, 1]d, let
+˘Bν
+n(s, t, u) =
+�
+[0,1]d Bn(s, t, w)dν
+X ⌊ns⌋+1:⌊nt⌋
+u
+(w).
+Under Condition 2.11, Bn ⇝ BC in ℓ∞(Λ × [0, 1]d), where BC has continuous
+trajectories almost surely. We then obtain from (A.1) in Lemma A.1 that
+sup
+(s,t,u)∈Λ×[0,1]d
+���˘Bν
+n(s, t, u) − Bn(s, t, u)
+��� = oP(1),
+and, furthermore, since Conditions 2.8 and 2.9 hold, from (A.3) in Lemma A.2,
+that
+sup
+(s,t,u)∈Λ×[0,1]d
+���˘Bν
+n(s, t, u) − ˜Bν
+n(s, t, u)
+��� = o(1),
+with probability one, where ˜Bν
+n is defined in (2.14), which implies that
+sup
+(s,t,u)∈Λ×[0,1]d
+���˜Bν
+n(s, t, u) − Bn(s, t, u)
+��� = oP(1).
+(A.5)
+Moreover, from the triangle inequality, we have
+sup
+(s,t)∈Λ
+u∈[0,1]d
+���˜Cν
+n(s, t, u) − ˜Cn(s, t, u)
+��� ≤
+sup
+(s,t)∈Λ
+u∈[0,1]d
+���˜Bν
+n(s, t, u) − Bn(s, t, u)
+���
++
+d
+�
+j=1
+sup
+u∈[0,1]d
+��� ˙Cj(u)
+���
+sup
+(s,t)∈Λ
+u∈[0,1]d
+���˜Bν
+n(s, t, u(j)) − Bn(s, t, u(j))
+��� .
+The terms on the right-hand side of the previous display converge to zero in
+probability as a consequence of (A.5) and the fact that 0 ≤ ˙Cj ≤ 1.
+■
+Appendix B: Proofs of Proposition 3.3 and Lemma B.2
+Proof of Proposition 3.3. Fix n ∈ N, x ∈ (Rd)n and r ∈ {1, . . . , n}d and let
+us check that K x
+r , which can be expressed as in (2.6) under Condition 2.4, is a
+multivariate d.f.
+By Condition 2.5, for any j ∈ {1, . . . , d}, the function K x
+rj,j defined by
+K x
+rj,j(u) = ¯
+F x
+j,u{(rj−1)/n}, u ∈ [0, 1], is a univariate d.f. on [0, 1]. Indeed, K x
+rj,j
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+41
+is right-continuous and increasing on [0, 1] and, by properties of the smoothing
+distributions,
+K x
+rj,j(0) = ¯
+F x
+j,0{(rj − 1)/n} = P{W x
+j,0 > (rj − 1)/n} = P{0 > (rj − 1)/n} = 0,
+K x
+rj,j(1) = ¯
+F x
+j,1{(rj − 1)/n} = P{W x
+j,1 > (rj − 1)/n} = P{1 > (rj − 1)/n} = 1.
+Using additionally Condition 2.6, the expression of K x
+r
+in (2.6) can then by
+further simplified to
+K x
+r (u) = ¯
+C x{K x
+r1,1(u1), . . . , K x
+rd,d(ud)},
+u ∈ [0, 1]d.
+From Sklar’s Theorem (Sklar, 1959), K x
+r is thus a d.f. on [0, 1]d with univariate
+margins K x
+r1,1, . . . , K x
+rd,d and copula ¯
+C x.
+■
+The proof of Lemma B.2 below is based on the following lemma.
+Lemma B.1. For any n ∈ N and t ∈ [0, 1], let ¯Bn,t be the survival function of
+a Binomial(n, t). Then, for any n ∈ N and w ∈ [0, n), the function t �→ ¯Bn,t(w)
+is strictly increasing on [0, 1].
+Proof. Fix n ∈ N and w ∈ [0, n). Since t �→ ¯Bn,t(w) is continuous on [0, 1], it
+suffices to prove that, for any t ∈ (0, 1),
+∂
+∂t
+� ¯Bn,t(w)
+�
+> 0. We have
+∂
+∂t
+� ¯Bn,t(w)
+�
+= ∂
+∂t
+�
+�
+�
+n
+�
+k=⌊w⌋+1
+�
+n
+k
+�
+tk(1 − t)n−k
+�
+�
+� = ∂
+∂t
+�
+�
+�
+n
+�
+k=⌊w⌋+1
+�
+n
+k
+�
+tk(1 − t)n−k
+�
+�
+�
+=
+n
+�
+k=⌊w⌋+1
+n!
+k!(n − k)!{ktk−1(1 − t)n−k − (n − k)tk(1 − t)n−k−1}
+=
+n
+�
+k=⌊w⌋+1
+n!
+k!(n − k)!ktk−1(1 − t)n−k −
+n−1
+�
+k=⌊w⌋+1
+n!
+k!(n − k)!(n − k)tk(1 − t)n−k−1
+=
+n
+�
+k=⌊w⌋+1
+n!
+(k − 1)!(n − k)!tk−1(1 − t)n−k −
+n−1
+�
+k=⌊w⌋+1
+n!
+k!(n − k − 1)!tk(1 − t)n−k−1
+=
+n
+�
+k=⌊w⌋+1
+n!
+(k − 1)!(n − k)!tk−1(1 − t)n−k −
+n
+�
+k=⌊w⌋+2
+n!
+(k − 1)!(n − k)!tk−1(1 − t)n−k
+=
+n!
+⌊w⌋!(n − ⌊w⌋ − 1)!t⌊w⌋(1 − t)n−⌊w⌋−1 > 0.
+■
+Lemma B.2. For any n ∈ N, t ∈ [0, 1] and ρ ∈ (1, n), let ¯
+Bn,t,ρ be the survival
+function of a Beta-Binomial(n, α, β), where α = t(n − ρ)/(ρ − 1) and β =
+(1 − t)(n − ρ)/(ρ − 1). Then, for any n ∈ N, ρ ∈ (1, n) and w ∈ [0, n), the
+function t �→ ¯
+Bn,t,ρ(w) is strictly increasing on [0, 1].
+Proof. First, notice that, for any t ∈ [0, 1], ρ ∈ (1, n) and w ∈ [0, n), by definition
+of the beta-binomial distribution,
+¯
+Bn,t,ρ(w) = EΘ{ ¯Bn,Θ(w)},
+(B.1)
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+42
+where ¯Bn,t is the survival function of a Binomial(n, t) and Θ is Beta(α, β) with
+α = t(n−ρ)/(ρ−1) and β = (1−t)(n−ρ)/(ρ−1). From Lemma 30 in Kojadinovic
+and Yi (2022), we have that, for any n ∈ N, ρ ∈ (1, n) and w ∈ [0, n), the function
+t �→ ¯
+Bn,t,ρ(w) is increasing on [0, 1]. It thus suffices to show strict increasingness.
+Let us prove this by contradiction. Suppose that there exists 0 ≤ t1 < t2 ≤ 1
+such that
+¯
+Bn,t1,ρ(w) = ¯
+Bn,t2,ρ(w) for some n ∈ N, ρ ∈ (1, n) and w ∈ [0, n).
+Then, from (B.1), we have that
+EΘ1{ ¯Bn,Θ1(w)} = EΘ2{ ¯Bn,Θ2(w)},
+(B.2)
+where Θ1 (resp. Θ2) is Beta(α1, β1) (resp. Beta(α2, β2)) with α1 = t1(n−ρ)/(ρ−
+1) and β1 = (1 − t1)(n − ρ)/(ρ − 1) (resp. α2 = t2(n − ρ)/(ρ − 1) and β2 =
+(1 − t2)(n − ρ)/(ρ − 1)). From the proof of Lemma 29 in Kojadinovic and Yi
+(2022), we have that Θ1 ≤st Θ2, where ≤st denotes the usual stochastic order.
+Using additionally (B.2) and the fact that the function t �→ ¯Bn,t(w) is strictly
+increasing on [0, 1] from Lemma B.1, we have, according to Theorem 1.A.8 of
+Shaked and Shanthikumar (2007), that Θ1 and Θ2 have the same distribution.
+This contradicts the fact that t1 < t2.
+■
+Appendix C: Proof of Theorem 3.5
+The proof of Theorem 3.5 is based on two lemmas which we show first.
+Let Φ be the map from ℓ∞([0, 1]d) to ℓ∞([0, 1]d) defined for any d.f H on
+[0, 1]d whose univariate margins H1, . . . , Hd do not assign mass at zero by
+Φ(H)(u) = H{H−1
+1 (u1), . . . , H−1
+d (ud)},
+u ∈ [0, 1]d.
+(C.1)
+Lemma C.1. Assume that the random vectors in X 1:n are i.i.d. and that
+Condition 5.8 holds. Then, almost surely,
+sup
+u∈[0,1]d |Cν
+1:n(u) − C(u)| = o(1),
+(C.2)
+where Cν
+1:n is defined in (2.2).
+Proof. The supremum on the left-hand side of (C.2) is smaller than In + Jn,
+where
+In =
+sup
+u∈[0,1]d
+�����
+�
+[0,1]d C1:n(w)dν
+X 1:n
+u
+(w) −
+�
+[0,1]d C(w)dν
+X 1:n
+u
+(w)
+����� ,
+Jn =
+sup
+u∈[0,1]d
+�����
+�
+[0,1]d C(w)dν
+X 1:n
+u
+(w) − C(u)
+����� .
+Term In: From the triangle inequality, In is smaller than
+sup
+u∈[0,1]d|C1:n(u) − C(u)| ≤ I′
+n + I′′
+n + I′′′
+n ,
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+43
+where
+I′
+n =
+sup
+u∈[0,1]d |C1:n(u) − Φ(G1:n)(u)|,
+I′′
+n =
+sup
+u∈[0,1]d |Φ(G1:n)(u) − G1:n(u)|,
+I′′′
+n =
+sup
+u∈[0,1]d |G1:n(u) − C(u)|,
+where the map Φ is defined in (C.1) and G1:n is empirical d.f. of the unobserv-
+able random sample U1, . . . , Un obtained from X 1:n by the probability integral
+transformations Uij = Fj(Xij), i ∈ {1, . . . , n}, j ∈ {1, . . . , d}. Using the well-
+known facts (see, e.g., Segers, 2012) that Φ(G1:n) = Φ(F1:n), where F1:n is the
+empirical d.f. of X 1:n, and
+sup
+u∈[0,1]d |C1:n(u) − Φ(F1:n)(u)| ≤ d
+n,
+we obtain that I′
+n = o(1). Furthermore, from the Glivenko-Cantelli lemma (see,
+e.g., van der Vaart, 1998, Theorem 19.1), I′′′
+n
+= o(1) with probability one.
+Finally, using a well-known property of multivariate d.f.s (see, e.g., Durante
+and Sempi, 2015, Lemma 1.2.14), the well-known fact, for any j ∈ {1, . . . , d},
+supu∈[0,1] |G−1
+1:n,j(u)−u| = supu∈[0,1] |G1:n,j(u)−u| (see, e.g., Shorack and Well-
+ner, 1986, Chapter 3) and, again, the Glivenko-Cantelli lemma, we obtain that,
+almost surely,
+I′′
+n ≤
+d
+�
+j=1
+sup
+u∈[0,1]
+|G−1
+1:n,j(u) − u| =
+d
+�
+j=1
+sup
+u∈[0,1]
+|G1:n,j(u) − u| = o(1).
+Term Jn: We proceed as in the proof of Lemma 3.2 of Segers, Sibuya and
+Tsukahara (2017). Fix η > 0 and let us show that, with probability one, Jn
+can be made smaller than η provided n is large enough. Let | · |∞ denote the
+maximum norm on Rd. For any ε > 0, we have that
+Jn =
+sup
+u∈[0,1]d
+�����
+�
+[0,1]d{C(w) − C(u)}dν
+X 1:n
+u
+(w)
+�����
+≤
+sup
+u∈[0,1]d
+�����
+�
+{w∈[0,1]d:|u−w|∞≤ε}
+{C(w) − C(u)}dν
+X 1:n
+u
+(w)
+�����
++
+sup
+u∈[0,1]d
+�����
+�
+{w∈[0,1]d:|u−w|∞>ε}
+{C(w) − C(u)}dν
+X 1:n
+u
+(w)
+����� ≤ J′
+n + J′′
+n,
+where
+J′
+n =
+sup
+(u,w)∈[0,1]2d
+|u−w|∞≤ε
+|C(w) − C(u)| ,
+J′′
+n =
+sup
+u∈[0,1]d ν
+X 1:n
+u
+({w ∈ [0, 1]d : |u − w|∞ > ε}).
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+44
+Let ε = η/(2d). Then, from the Lipschitz continuity of C, J′
+n ≤ η/2. As far as J′′
+n
+is concerned, conditionally on X1, X2, . . . , for almost any sequence X1, X2, . . . ,
+using Chebyshev’s inequality and Condition 5.8, we have that
+J′′
+n =
+sup
+u∈[0,1]d P {|W
+X 1:n
+u
+− u)|∞ > ε | X 1:n}
+=
+sup
+u∈[0,1]d P
+�
+�
+d�
+j=1
+����W
+X 1:n
+j,uj
+− uj
+��� > ε
+�
+| X 1:n
+�
+�
+≤
+sup
+u∈[0,1]d
+d
+�
+j=1
+P
+����W
+X 1:n
+j,uj
+− uj
+��� > ε | X 1:n
+�
+≤
+sup
+u∈[0,1]d
+d
+�
+j=1
+Var(W
+X 1:n
+j,uj
+| X 1:n)
+ε2
+≤ dan
+ε2 .
+which implies that, for n sufficiently large, J′′
+n ≤ η/2. The latter holds condi-
+tionally on X1, X2, . . . for almost any sequence X1, X2, . . . , which completes
+the proof.
+■
+Next, we recall the mode of convergence classically used to state asymptotic
+validity results of resampling techniques when dealing with empirical processes;
+see, e.g., van der Vaart and Wellner (2000, Chapter 2.9) or Kosorok (2008,
+Section 2.2.3). Let
+BL1 = {h : ℓ∞([0, 1]d) → [−1, 1] such that,
+for all x, y ∈ ℓ∞([0, 1]d), |h(x) − h(y)| ≤
+sup
+u∈[0,1]d |x(u) − y(u)|}.
+Let Xn = Xn(X 1:n, Wn) be a sequence of bootstrapped empirical processes in
+ℓ∞([0, 1]d) depending on an additional source of randomness Wn (often called
+the “bootstrap weights”). For the smooth bootstraps under consideration, Wn
+is independent of the data X 1:n and corresponds to n independent copies of the
+independent random variables I and U # necessary to carry out Algorithm 3.2
+(see also (3.2)) n times independently. The notation Xn
+P⇝
+W X then means that
+• suph∈BL1 |EW {h(Xn)} − E{h(X)} → 0 in outer probability,
+• EW {h(Xn)∗} − EW {h(Xn)∗}
+P→ 0 for all h ∈ BL1,
+where EW denotes an expectation with respect to the bootstrap weights Wn only
+and h(Xn)∗ and h(Xn)∗ denote the minimal measurable majorant and maximal
+measurable minorant with respect to (X 1:n, Wn).
+The next lemma is very closely related to Proposition 3.3 of Kiriliouk, Segers
+and Tsukahara (2021).
+Lemma C.2. Assume that the random vectors in X 1:n are i.i.d., and that
+Conditions 2.4, 2.5, 2.6, 2.8 and 2.9 hold. Then,
+√n(C
+#
+1:n − Cν
+1:n)
+P⇝
+W CC(0, 1, ·),
+(C.3)
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+45
+where CC is defined in (2.13).
+Proof. Let G
+#
+1:n be the empirical d.f. of V
+#
+1:n. Using Lemma C.1 and proceeding
+as in Step 1 of the proof of Proposition 3.3 of Kiriliouk, Segers and Tsukahara
+(2021), one obtains that
+√n(G
+#
+1:n − Cν
+1:n)
+P⇝
+W BC(0, 1, ·),
+(C.4)
+where BC is defined in (2.11). Then, proceeding as in Step 2 of the proof of
+Proposition 3.3 of Kiriliouk, Segers and Tsukahara (2021), that is, combin-
+ing (C.4) with the Hadamard differentiability of the map Φ in (C.1) established
+in Theorem 2.4 of B¨ucher and Volgushev (2013), the functional delta method
+for the bootstrap “in probability” (van der Vaart and Wellner, 2000, Theo-
+rem 3.9.11) and the fact Φ(Cν
+1:n) = Cν
+1:n (since Cν
+1:n has standard uniform
+margins under Condition 2.4 in the considered i.i.d. setting), one obtains
+√n(Φ(G
+#
+1:n) − Cν
+1:n)
+P⇝
+W CC(0, 1, ·).
+(C.5)
+The desired result finally follows from (C.5) and the well-known fact that
+sup
+u∈[0,1]d |C
+#
+1:n(u) − Φ(G
+#
+1:n)(u)| ≤ d
+n
+since the components samples of V
+#
+1:n contain no ties with probability one.
+■
+Proof of Theorem 3.5. Combining Lemma C.2 with Lemma 3.1 of B¨ucher
+and Kojadinovic (2019), we obtain that (C.3) is equivalent to
+�
+Cn(0, 1, ·), √n(C
+#,[1]
+1:n − Cν
+1:n), √n(C
+#,[2]
+1:n − Cν
+1:n)
+�
+⇝
+�
+CC(0, 1, ·), C
+[1]
+C (0, 1, ·), C
+[2]
+C (0, 1, ·)
+�
+(C.6)
+in {ℓ∞([0, 1]d)}3, where Cn is defined in (2.8). From Theorem 2.10, we have that
+sup
+u∈[0,1]d
+√n|Cν
+1:n(u) − C1:n(u)| =
+sup
+u∈[0,1]d |Cν
+n(0, 1, u) − Cn(0, 1, u)| = oP(1),
+(C.7)
+where Cν
+n is defined in (2.9). The first joint weak convergence in Theorem 3.5
+then follows from (C.6) and (C.7).
+Fix j ∈ {1, 2}. Since (C.7) holds, to establish the second joint weak conver-
+gence from the first, it suffices to show that
+sup
+u∈[0,1]d
+��√n{C
+#,ν,[j]
+1:n
+(u) − Cν
+1:n(u)} − √n{C
+#,[j]
+1:n (u) − C1:n(u)}
+�� = oP(1).
+(C.8)
+The supremum on the left hand-side of (C.8) is smaller than In + Jn, where
+In =
+sup
+u∈[0,1]d
+�����
+�
+[0,1]d
+√n{C
+#,[j]
+1:n (w) − C1:n(w)}dν
+V #,[j]
+1:n
+u
+(w) − √n{C
+#,[j]
+1:n (u) − C1:n(u)}
+����� ,
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+46
+Jn =√n
+sup
+u∈[0,1]d
+�����
+�
+[0,1]d C1:n(w)dν
+V #,[j]
+1:n
+u
+(w) −
+�
+[0,1]d C1:n(w)dν
+X 1:n
+u
+(w)
+����� .
+Since, according to the first claim of Theorem 3.5, √n(C
+#,[j]
+1:n −C1:n) ⇝ CC(0, 1, ·)
+in ℓ∞([0, 1]d) and CC(0, 1, ·) has continuous trajectories almost surely, it can be
+verified by proceeding as in the proof of Lemma A.1 that In = oP(1). For the
+term Jn, we have that Jn ≤ Kn + Ln, where
+Kn =√n
+sup
+u∈[0,1]d
+�����
+�
+[0,1]d{C1:n(w) − C(w)}dν
+V #,[j]
+1:n
+u
+(w)
+−
+�
+[0,1]d{C1:n(w) − C(w)}dν
+X 1:n
+u
+(w)
+����� ,
+Ln =√n
+sup
+u∈[0,1]d
+�����
+�
+[0,1]d C(w)dν
+V #,[j]
+1:n
+u
+(w) −
+�
+[0,1]d C(w)dν
+X 1:n
+u
+(w)
+����� .
+The term Kn is smaller than K′
+n + K′′
+n, where
+K′
+n =
+sup
+u∈[0,1]d
+�����
+�
+[0,1]d Cn(0, 1, w)dν
+V #,[j]
+1:n
+u
+(w) − Cn(0, 1, u)
+����� ,
+K′′
+n =
+sup
+u∈[0,1]d
+�����Cn(0, 1, u) −
+�
+[0,1]d Cn(0, 1, w)dν
+X 1:n
+u
+(w)
+����� .
+From (A.2) in Lemma A.1, K′′
+n = oP(1) and, proceeding again as in the proof
+of the latter lemma, it can be verified that K′
+n = oP(1). The term Ln is smaller
+than L′
+n + L′′
+n, where
+L′
+n = √n
+sup
+u∈[0,1]d
+�����
+�
+[0,1]d C(w)dν
+V #,[j]
+1:n
+u
+(w) − C(u)
+����� ,
+L′′
+n = √n
+sup
+u∈[0,1]d
+�����C(u) −
+�
+[0,1]d C(w)dν
+X 1:n
+u
+(w)
+����� .
+The term L′′
+n converges almost surely to zero as a consequence of (A.4) in
+Lemma A.2. The proof of the latter result can be adapted to verify that the term
+L′
+n also converges almost surely to zero. Hence, (C.8) holds, which completes
+the proof.
+■
+Appendix D: Proof of Theorem 4.3
+The proof of Theorem 4.3 is based on the following lemma which we prove first.
+Lemma D.1. Under Conditions 4.2 and 5.8 (the latter is implied by Condi-
+tion 2.9), for any b ∈ N,
+sup
+(s,t,u)∈Λ×[0,1]d
+���ˆB
+[b],ν
+n
+(s, t, u) − ˆB
+[b]
+n (s, t, u)
+��� = oP(1),
+(D.1)
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+47
+sup
+(s,t,u)∈Λ×[0,1]d
+��ˇB
+[b],ν
+n
+(s, t, u) − ˇB
+[b]
+n (s, t, u)
+�� = oP(1).
+(D.2)
+Proof. Fix b ∈ N. We first prove (D.1). Starting from (4.5), we have that
+sup
+(s,t,u)∈Λ×[0,1]d
+���ˆB
+[b],ν
+n
+(s, t, u) − ˆB
+[b]
+n (s, t, u)
+���
+=
+sup
+(s,t,u)∈Λ×[0,1]d
+�����
+�
+[0,1]d
+ˆB
+[b]
+n (s, t, w)dν
+X 1:n
+u
+(w) − ˆB
+[b]
+n (s, t, u)
+����� = oP(1),
+where the last equality follows from (A.2) in Lemma A.1 since ˆB[b]
+n ⇝ BC in
+ℓ∞(Λ × [0, 1]d), where BC is defined in (2.11) and has continuous trajectories
+almost surely under Condition 4.2. Under Condition 4.2 (ii), the latter is a
+consequence of Lemmas D.1 and D.2 in B¨ucher and Kojadinovic (2016) as well
+as Theorem 2.1 in B¨ucher and Kojadinovic (2016). Under Condition 4.2 (i), one
+can rely on Theorem 1 of Holmes, Kojadinovic and Quessy (2013) instead.
+The proof of (D.2) is similar. Starting from (4.6), we have that
+sup
+(s,t,u)∈Λ×[0,1]d
+��ˇB
+[b],ν
+n
+(s, t, u) − ˇB
+[b]
+n (s, t, u)
+��
+=
+sup
+(s,t,u)∈Λ×[0,1]d
+�����
+�
+[0,1]d
+ˇB
+[b]
+n (s, t, w)dν
+X ⌊ns⌋+1:⌊nt⌋
+u
+(w) − ˇB
+[b]
+n (s, t, u)
+����� = oP(1),
+where the last equality follows from (A.1) in Lemma A.1 since ˇB[b]
+n ⇝ BC in
+ℓ∞(Λ × [0, 1]d). Under Condition 4.2 (ii), the latter is a consequence of (B.3)
+in the proof of Proposition 4.3 in B¨ucher et al. (2014) and Theorem 2.1 in
+B¨ucher and Kojadinovic (2016). Under Condition 4.2 (i), one can rely again on
+Theorem 1 of Holmes, Kojadinovic and Quessy (2013) instead.
+■
+Proof of Theorem 4.3. Fix b ∈ N. We only prove (4.10), the proof of (4.9)
+being simpler. Starting from (4.4) and (4.8), we have that
+sup
+(s,t)∈Λ
+u∈[0,1]d
+��ˇC
+[b]
+n (s, t, u) − ˇC
+[b],ν
+n
+(s, t, u)
+�� ≤
+sup
+(s,t)∈Λ
+u∈[0,1]d
+��ˇB
+[b]
+n (s, t, u) − ˇB
+[b],ν
+n
+(s, t, u)
+��
++
+d
+�
+j=1
+sup
+(s,t)∈Λ
+u∈[0,1]d
+��� ˙Cj,⌊ns⌋+1:⌊nt⌋(u)
+���
+sup
+(s,t)∈Λ
+u∈[0,1]d
+���ˇB
+[b]
+n (s, t, u(j)) − ˇB
+[b],ν
+n
+(s, t, u(j))
+��� .
+The terms on the right-hand side of the previous display converge to zero
+in probability as a consequence of (D.2) in Lemma D.1 and the fact that
+sup(s,t,u)∈Λ×[0,1]d
+��� ˙Cj,⌊ns⌋+1:⌊nt⌋(u)
+��� ≤ ζ from Condition 4.1.
+The last two claims of the theorem are an immediate consequence of (4.9),
+(4.10) and straightforward extensions of Propositions 4.2 and 4.3 in B¨ucher et al.
+(2014) for non-smooth multiplier replicates based on arbitrary partial derivative
+estimators satisfying Condition 4.1.
+■
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+48
+Appendix E: Proofs of Propositions 5.3, 5.5, 5.9 and 5.10
+Lemma E.1. Let f be any function from [0, 1]d to [0, 1], let m ∈ N, m ≥ 2
+and recall that, for any u ∈ [0, 1]d, µm,u is the law of the random vector
+(Sm,1,u1/m, . . . , Sm,d,ud/m), where Sm,1,u1, . . . Sm,d,ud are independent, and for
+each k ∈ {1, . . . , d}, Sm,k,uk is Binomial(m, uk). Moreover, recall that, for any
+j ∈ {1, . . . , d}, ˜µj,m,u is the law of the random vector ( ˜Sm,1,u1/m, . . . , ˜Sm,d,ud/m)
+whose components are independent and, for i ∈ {1, . . . , d} \ {j}, ˜Sm,i,ui is
+Binomial(m, ui), whereas ˜Sm,j,uj is Binomial(m − 1, uj). Then, for any j ∈
+{1, . . . , d} and u ∈ [0, 1]d such that uj ∈ (0, 1),
+∂uj
+��
+[0,1]d f(w)dµm,u(w)
+�
+= m
+�
+[0,1]d {f(w + ej/m) − f(w)} d˜µj,m,u(w).
+Proof. Fix m ≥ 2 and, without loss of generality, fix j = 1. Also, for any
+u ∈ [0, 1], let bm,u(s) =
+�m
+s
+�
+us(1−u)m−s, s ∈ {0, . . . , m}. Then, for all u ∈ [0, 1]d
+such that u1 ∈ (0, 1),
+∂u1
+��
+[0,1]d f(w)dµm,u(w)
+�
+= ∂u1
+�
+�
+�
+m
+�
+s1=0
+· · ·
+m
+�
+sd=0
+f
+�s1
+m , . . . , sd
+m
+�
+d
+�
+j=1
+bm,uj(sj)
+�
+�
+�
+=
+m
+�
+s1=0
+· · ·
+m
+�
+sd=0
+f
+�s1
+m , . . . , sd
+m
+�
+∂u1bm,u1(s1)
+d
+�
+j=2
+bm,uj(sj)
+=
+m
+�
+s1=0
+· · ·
+m
+�
+sd=0
+f
+�s1
+m , . . . , sd
+m
+�
+×
+�
+m
+s1
+�
+�
+s1us1−1
+1
+(1 − u1)m−s1 − (m − s1)us1
+1 (1 − u1)m−s1−1�
+d
+�
+j=2
+bm,uj(sj)
+= m
+m
+�
+s1=1
+· · ·
+m
+�
+sd=0
+f
+�s1
+m , . . . , sd
+m
+�
+(m − 1)!
+(s1 − 1)!(m − s1)!us1−1
+1
+(1 − u1)m−s1
+d
+�
+j=2
+bm,uj(sj)
+− m
+m−1
+�
+s1=0
+· · ·
+m
+�
+sd=0
+f
+�s1
+m , . . . , sd
+m
+�
+(m − 1)!
+s1!(m − s1 − 1)!us1
+1 (1 − u1)m−s1−1
+d
+�
+j=2
+bm,uj(sj)
+= m
+m−1
+�
+s1=0
+· · ·
+m
+�
+sd=0
+f
+�s1 + 1
+m
+, . . . , sd
+m
+�
+(m − 1)!
+s1!(m − s1 − 1)!us1
+1 (1 − u1)m−s1−1
+d
+�
+j=2
+bm,uj(sj)
+− m
+m−1
+�
+s1=0
+· · ·
+m
+�
+sd=0
+f
+�s1
+m , . . . , sd
+m
+�
+(m − 1)!
+s1!(m − s1 − 1)!us1
+1 (1 − u1)m−s1−1
+d
+�
+j=2
+bm,uj(sj)
+= m
+m−1
+�
+s1=0
+· · ·
+m
+�
+sd=0
+�
+f
+�s1 + 1
+m
+, . . . , sd
+m
+�
+− f
+�s1
+m , . . . , sd
+m
+��
+bm−1,u1(s1)
+d
+�
+j=2
+bm,uj(sj)
+= m
+�
+[0,1]d {f(w + e1/m) − f(w)} d˜µ1,m,u(w).
+■
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+49
+Proof of Proposition 5.3. Fix j ∈ {1, . . . , d}, u ∈ [0, 1]d and m ≥ 2, and
+recall the definition of the measure ˜µj,m,u given in Lemma E.1. From (5.13), we
+have that
+˙C
+Bern
+j,k:l,m(u) = m
+�
+[0,1]d Ck:l(w+ej/m)d˜µj,m,u(w)−m
+�
+[0,1]d Ck:l(w)d˜µj,m,u(w).
+(E.1)
+Let ˜S = ( ˜Sm,1,u1, . . . , ˜Sm,d,ud) so that ˜S/m is a random vector with law ˜µj,m,u.
+Then, combined with the definition of Ck:l in (2.1), the first integral on the
+right-hand side of (E.1) can be rewritten as
+�
+[0,1]d
+1
+l − k + 1
+l
+�
+i=k
+1
+�
+Rk:l
+i /(l − k + 1) ≤ w + ej/m
+�
+d˜µj,m,u(w)
+=
+1
+l − k + 1
+l
+�
+i=k
+�
+[0,1]d 1
+�
+Rk:l
+i /(l − k + 1) − ej/m ≤ w
+�
+d˜µj,m,u(w)
+=
+1
+l − k + 1
+l
+�
+i=k
+P
+�
+˜S ≥ mRk:l
+i /(l − k + 1) − ej | X k:l
+�
+=
+1
+l − k + 1
+l
+�
+i=k
+P
+�
+˜Sm,j,uj ≥ mRk:l
+ij /(l − k + 1) − 1 | X k:l
+�
+×
+d
+�
+t=1
+t̸=j
+P
+�
+˜Sm,t,ut ≥ mRk:l
+it /(l − k + 1) | X k:l
+�
+=
+1
+l − k + 1
+l
+�
+i=k
+P
+�
+˜Sm,j,uj > ⌈mRk:l
+ij /(l − k + 1) − 1⌉ − 1 | X k:l
+�
+×
+d
+�
+t=1
+t̸=j
+P
+�
+˜Sm,t,ut > ⌈mRk:l
+it /(l − k + 1)⌉ − 1 | X k:l
+�
+=
+1
+l − k + 1
+l
+�
+i=k
+¯Bm−1,uj
+�
+⌈mRk:l
+ij /(l − k + 1)⌉ − 2
+�
+×
+d
+�
+t=1
+t̸=j
+¯Bm,ut
+�
+⌈mRk:l
+it /(l − k + 1)⌉ − 1
+�
+,
+where we have used the fact that, for any t ∈ {1, . . . , d} and x ∈ R, P( ˜Sm,t,ut ≥
+x) = P( ˜Sm,t,ut > ⌈x⌉ − 1) and ⌈x − 1⌉ = ⌈x⌉ − 1. Similarly, for the second
+integral on the right-hand side of (E.1), we have
+�
+[0,1]dCk:l(w)d˜µj,m,u(w)
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+50
+=
+1
+l − k + 1
+�
+[0,1]d
+l
+�
+i=k
+1
+�
+Rk:l
+i /(l − k + 1) ≤ w
+�
+d˜µj,m,u(w)
+=
+1
+l − k + 1
+l
+�
+i=k
+P
+�
+˜S ≥ mRk:l
+i /(l − k + 1) | X k:l
+�
+=
+1
+l − k + 1
+l
+�
+i=k
+d
+�
+t=1
+P
+�
+˜Sm,t,ut ≥ mRk:l
+it /(l − k + 1) | X k:l
+�
+=
+1
+l − k + 1
+l
+�
+i=k
+¯Bm−1,uj
+�
+⌈mRk:l
+ij /(l − k + 1)⌉ − 1
+�
+×
+d
+�
+t=1
+t̸=j
+¯Bm,ut
+�
+⌈mRk:l
+it /(l − k + 1)⌉ − 1
+�
+.
+The desired result finally follows by noticing that
+¯Bm−1,uj
+�
+⌈mRk:l
+ij /(l − k + 1)⌉ − 2
+�
+− ¯Bm−1,uj
+�
+⌈mRk:l
+ij /(l − k + 1)⌉ − 1
+�
+= Bm−1,uj
+�
+⌈mRk:l
+ij /(l − k + 1)⌉ − 1
+�
+− Bm−1,uj
+�
+⌈mRk:l
+ij /(l − k + 1)⌉ − 2
+�
+= bm−1,uj
+�
+⌈mRk:l
+ij /(l − k + 1)⌉ − 1
+�
+.
+■
+Lemma E.2. Under Condition 5.4, for any j ∈ {1, . . . , d}, δ ∈ (0, 1) and
+ε ∈ (0, 1/2), with probability 1,
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+��� ˙C
+ν,∇
+j,⌊ns⌋+1:⌊nt⌋(u) − ˙C
+ν,∆
+j,⌊ns⌋+1:⌊nt⌋(u)
+��� = o(1).
+Proof. Fix j ∈ {1, . . . , d}, δ ∈ (0, 1) as well as ε ∈ (0, 1/2) and assume that
+n is large enough so that, for any (s, t) ∈ Λ such that t − s > δ, L2b⌊nt⌋−⌊ns⌋
+and L2b′
+⌊nt⌋−⌊ns⌋ are smaller than ε. Then, using the fact that Cν
+k:l in (2.2) is
+between 0 and 1, we obtain that, with probability 1,
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+��� ˙C
+ν,∇
+j,⌊ns⌋+1:⌊nt⌋(u) − ˙C
+ν,∆
+j,⌊ns⌋+1:⌊nt⌋(u)
+���
+≤
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+����
+1
+h + h′ −
+1
+(uj + h) ∧ 1 − (uj − h′) ∨ 0
+���� = 0.
+■
+Proof of Proposition 5.5. Fix j ∈ {1, . . . , d} and let us first prove (5.15)
+by proceeding along the lines of the proof of (B.4) in B¨ucher et al. (2014).
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+51
+From (2.9), notice that, for any (s, t, u) ∈ Λ × [0, 1]d such that ⌊ns⌋ < ⌊nt⌋,
+Cν
+⌊ns⌋+1:⌊nt⌋(u) = C(u) +
+1
+√nλn(s, t)Cν
+n(s, t, u).
+Fix δ ∈ (0, 1) and notice that, by Condition 5.4,
+dn =
+sup
+(s,t)∈Λ
+t−s≥δ
+(b⌊nt⌋−⌊ns⌋ + b′
+⌊nt⌋−⌊ns⌋) ≤
+sup
+k≥⌊nδ⌋−1
+(bk + b′
+k) → 0.
+(E.2)
+Next, fix ε ∈ (0, 1/2) and assume that n is large enough so that, for any t−s > δ,
+L2b⌊nt⌋−⌊ns⌋ and L2b′
+⌊nt⌋−⌊ns⌋ are smaller than ε/2. Then, for any t − s > δ and
+u ∈ [0, 1]d such that uj ∈ [ε, 1 − ε],
+˙C
+ν,∆
+j,⌊ns⌋+1:⌊nt⌋(u) =
+1
+h + h′ {C(u + hej) − C(u − h′ej)}
++
+1
+(h + h′)√nλn(s, t) {Cν
+n(s, t, u + hej) − Cν
+n(s, t, u − h′ej)} .
+(E.3)
+Since, by Condition 2.8, ˙Cj exists (and is continuous) on the set {u ∈ [0, 1]d :
+uj ∈ [ε/2, 1 − ε/2]}, from the mean value theorem, for any t − s > δ and
+u ∈ [0, 1]d such that uj ∈ [ε, 1 − ε],
+1
+h + h′ {C(u + hej) − C(u − h′ej)} = ˙Cj(u∗
+n,s,t),
+where u∗
+n,s,t is between u − h′ej and u + hej almost surely. Hence, with prob-
+ability 1,
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+����
+1
+h + h′ {C(u + hej) − C(u − h′ej)} − ˙Cj(u)
+����
+=
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+��� ˙Cj(u∗
+n,s,t) − ˙Cj(u)
+��� ≤
+sup
+(u,v)∈[0,1]2d
+uj,vj∈[ε/2,1−ε/2]
+|u−v|∞≤L2dn
+��� ˙Cj(u) − ˙Cj(v)
+��� → 0,
+(E.4)
+where dn is defined in (E.2). Furthermore, since, by Condition 2.9 and as a result
+of Theorem 2.10, Cν
+n is asymptotically uniformly equicontinuous in probability,
+we have that
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+|Cν
+n(s, t, u + hej) − Cν
+n(s, t, u − h′ej)|
+≤
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+(u,v)∈[0,1]2d
+uj,vj∈[ε/2,1−ε/2]
+|u−v|∞≤L2dn
+|Cν
+n(s, t, u) − Cν
+n(s, t, v)| = oP(1).
+(E.5)
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+52
+The fact that (5.15) holds is then an immediate consequence of (E.3), (E.4),
+(E.5) and the fact that, from Condition 5.4,
+sup
+(s,t)∈Λ
+t−s≥δ
+1
+(h + h′) √nλn(s, t) ≤
+sup
+(s,t)∈Λ
+t−s≥δ
+1
+L1(b⌊nt⌋−⌊ns⌋ + b′
+⌊nt⌋−⌊ns⌋)√nλn(s, t)
+≤
+sup
+(s,t)∈Λ
+t−s≥δ
+1
+L1(⌊nt⌋ − ⌊ns⌋)−1/2√nλn(s, t)
+=
+sup
+(s,t)∈Λ
+t−s≥δ
+1
+L1
+�
+λn(s, t)
+≤
+1
+L1
+�
+δ − 1/n
+.
+The claim for ˙C
+ν,∇
+j,k:l (resp. for ˙C
+ν,∆
+j,k:l and ˙C
+ν,∇
+j,k:l) follows from Lemma E.2 (resp.
+the continuous mapping theorem).
+■
+Proof of Proposition 5.9. Fix j ∈ {1, . . . , d}, δ ∈ (0, 1) and ε ∈ (0, 1/2). We
+first prove (5.16). From (5.12) and the triangle inequality, we have that
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+��� ˙C
+∆,ν
+j,⌊ns⌋+1:⌊nt⌋(u) − ˙Cj(u)
+��� ≤ Ij,n,δ,ε + Jj,n,δ,ε,
+where
+Ij,n,δ,ε =
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+�����
+�
+[0,1]d
+�
+˙C
+∆
+j,⌊ns⌋+1:⌊nt⌋(w) − ˙Cj(w)
+�
+dν
+X ⌊ns⌋+1:⌊nt⌋
+u
+(w)
+����� ,
+Jj,n,δ,ε =
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+�����
+�
+[0,1]d
+˙Cj(w)dν
+X ⌊ns⌋+1:⌊nt⌋
+u
+(w) − ˙Cj(u)
+����� ,
+where ˙C
+∆
+j,k:l is defined in (5.4). We shall now show that both Ij,n,δ,ε = oP(1)
+and Jj,n,δ,ε = oP(1).
+Term Ij,n,δ,ε: From the triangle inequality and the fact that 0 ≤ ˙C
+∆
+j,k:l ≤ 1
+and 0 ≤ ˙Cj ≤ 1, we have that Ij,n,δ,ε is smaller than
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+������
+�
+{w∈[0,1]d:
+wj∈[ε/2,1−ε/2]}
+�
+˙C
+∆
+j,⌊ns⌋+1:⌊nt⌋(w) − ˙Cj(w)
+�
+dν
+X ⌊ns⌋+1:⌊nt⌋
+u
+(w)
+������
++
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+������
+�
+{w∈[0,1]d:
+wj<ε/2}
+�
+˙C
+∆
+j,⌊ns⌋+1:⌊nt⌋(w) − ˙Cj(w)
+�
+dν
+X ⌊ns⌋+1:⌊nt⌋
+u
+(w)
+������
++
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+������
+�
+{w∈[0,1]d:
+wj>1−ε/2}
+�
+˙C
+∆
+j,⌊ns⌋+1:⌊nt⌋(w) − ˙Cj(w)
+�
+dν
+X ⌊ns⌋+1:⌊nt⌋
+u
+(w)
+������
+≤ I′
+j,n,δ,ε + I′′
+j,n,δ,ε + I′′′
+j,n,δ,ε,
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+53
+where
+I′
+j,n,δ,ε = sup
+(s,t)∈Λ
+t−s≥δ
+sup
+w∈[0,1]d
+wj∈[ε/2,1−ε/2]
+��� ˙C
+∆
+j,⌊ns⌋+1:⌊nt⌋(w) − ˙Cj(w)
+��� ,
+I′′
+j,n,δ,ε = sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+ν
+X ⌊ns⌋+1:⌊nt⌋
+u
+�
+w ∈ [0, 1]d : wj < ε/2
+�
+,
+I′′′
+j,n,δ,ε = sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+ν
+X ⌊ns⌋+1:⌊nt⌋
+u
+�
+w ∈ [0, 1]d : wj > 1 − ε/2
+�
+.
+We have that I′
+j,n,δ,ε = oP(1) as a consequence of Corollary 5.7. We shall
+now show that both I′′
+j,n,δ,ε and I′′′
+j,n,δ,ε converge almost surely to zero. To do
+so, it suffices to show that I′′
+j,n,δ,ε and I′′′
+j,n,δ,ε converge to zero conditionally
+on X1, X2, . . . for almost any sequence X1, X2, . . . . Concerning I′′
+j,n,δ,ε, using
+Chebyshev’s inequality and Condition 5.8, for almost any sequence X1, X2, . . . ,
+conditionally on X1, X2, . . . , we obtain that
+I′′
+j,n,δ,ε =
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+uj∈[ε,1−ε]
+P
+�
+W
+X ⌊ns⌋+1:⌊nt⌋
+j,uj
+< ε/2 | X ⌊ns⌋+1:⌊nt⌋
+�
+=
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+uj∈[ε,1−ε]
+P
+�
+W
+X ⌊ns⌋+1:⌊nt⌋
+j,uj
+− uj < ε/2 − uj | X ⌊ns⌋+1:⌊nt⌋
+�
+≤
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+uj∈[ε,1−ε]
+P
+�
+−
+���W
+X ⌊ns⌋+1:⌊nt⌋
+j,uj
+− uj
+��� ≤ −uj + ε/2 | X ⌊ns⌋+1:⌊nt⌋
+�
+≤
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+uj∈[ε,1−ε]
+Var
+�
+W
+X ⌊ns⌋+1:⌊nt⌋
+j,uj
+| X ⌊ns⌋+1:⌊nt⌋
+�
+(uj − ε/2)2
+≤
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+uj∈[ε,1−ε]
+a⌊nt⌋−⌊ns⌋
+(uj − ε/2)2 ≤
+sup
+(s,t)∈Λ
+t−s≥δ
+a⌊nt⌋−⌊ns⌋
+sup
+uj∈[ε,1−ε]
+1
+(uj − ε/2)2
+≤ 4
+ε2
+sup
+k≥⌊nδ⌋−1
+ak → 0.
+Similarly, concerning I′′′
+j,n,δ,ε, for almost any sequence X1, X2, . . . , conditionally
+on X1, X2, . . . , we obtain that
+I′′′
+j,n,δ,ε =
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+uj∈[ε,1−ε]
+P
+�
+W
+X ⌊ns⌋+1:⌊nt⌋
+j,uj
+> 1 − ε/2 | X ⌊ns⌋+1:⌊nt⌋
+�
+=
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+uj∈[ε,1−ε]
+P
+�
+W
+X ⌊ns⌋+1:⌊nt⌋
+j,uj
+− uj > 1 − ε/2 − uj | X ⌊ns⌋+1:⌊nt⌋
+�
+≤
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+uj∈[ε,1−ε]
+P
+����W
+X ⌊ns⌋+1:⌊nt⌋
+j,uj
+− uj
+��� ≥ 1 − ε/2 − uj | X ⌊ns⌋+1:⌊nt⌋
+�
+≤
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+uj∈[ε,1−ε]
+Var
+�
+W
+X ⌊ns⌋+1:⌊nt⌋
+j,uj
+| X ⌊ns⌋+1:⌊nt⌋
+�
+(1 − ε/2 − uj)2
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+54
+≤
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+uj∈[ε,1−ε]
+a⌊nt⌋−⌊ns⌋
+(1 − ε/2 − uj)2
+≤
+sup
+(s,t)∈Λ
+t−s≥δ
+a⌊nt⌋−⌊ns⌋
+sup
+uj∈[ε,1−ε]
+1
+(1 − ε/2 − uj)2 ≤ 4
+ε2
+sup
+k≥⌊nδ⌋−1
+ak → 0.
+Term Jj,n,δ,ε: Let η > 0 and let us show that Jj,n,δ,ε ≤ η for n sufficiently
+large. For any ρ ∈ (0, 1), from the triangle inequality and the fact that 0 ≤ ˙Cj ≤
+1, we have that Jj,n,δ,ε is smaller than
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+�����
+�
+{w∈[0,1]d:|w−u|∞≤ρ}
+{ ˙Cj(w) − ˙Cj(u)}dν
+X ⌊ns⌋+1:⌊nt⌋
+u
+(w)
+�����
++ sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+�����
+�
+{w∈[0,1]d:|w−u|∞>ρ}
+{ ˙Cj(w) − ˙Cj(u)}dν
+X ⌊ns⌋+1:⌊nt⌋
+u
+(w)
+�����
+≤ J′
+j,ε,ρ + J′′
+j,n,δ,ρ,
+where
+J′
+j,ε,ρ =
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+sup
+w∈[0,1]d
+|w−u|∞≤ρ
+��� ˙Cj(w) − ˙Cj(u)
+��� ,
+J′′
+j,n,δ,ρ =
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+�
+[0,1]d 1{|w − u|∞ > ρ}dν
+X ⌊ns⌋+1:⌊nt⌋
+u
+(w).
+From Condition 2.8, ˙Cj is uniformly continuous on the set {u ∈ [0, 1]d : uj ∈
+[ε/2, 1 − ε/2]}. We then choose ρ = ρ(ε, η) > 0 sufficiently small such that
+J′
+j,ε,ρ =
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+sup
+w∈[0,1]d
+|w−u|∞≤ρ
+��� ˙Cj(w) − ˙Cj(u)
+��� ≤ η
+2.
+(E.6)
+As far as J′′
+j,n,δ,ρ is concerned, using Chebyshev’s inequality and Condition 5.8,
+for almost any sequence X1, X2, . . . , conditionally on X1, X2, . . . , we obtain
+that
+J′′
+j,n,δ,ρ =
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d ν
+X ⌊ns⌋+1:⌊nt⌋
+u
+({w ∈ [0, 1]d : |u − w|∞ > ρ})
+=
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d P
+�
+�
+d�
+j=1
+����W
+X ⌊ns⌋+1:⌊nt⌋
+j,uj
+− uj
+��� > ρ
+�
+| X ⌊ns⌋+1:⌊nt⌋
+�
+�
+≤
+d
+�
+j=1
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d P
+����W
+X ⌊ns⌋+1:⌊nt⌋
+j,uj
+− uj
+��� > ρ | X ⌊ns⌋+1:⌊nt⌋
+�
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+55
+≤
+d
+�
+j=1
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+Var
+�
+W
+X ⌊ns⌋+1:⌊nt⌋
+j,uj
+| X ⌊ns⌋+1:⌊nt⌋
+�
+ρ2
+≤ d
+ρ2
+sup
+(s,t)∈Λ
+t−s≥δ
+a⌊nt⌋−⌊ns⌋ ≤ d
+ρ2
+sup
+k≥⌊nδ⌋−1
+ak → 0,
+which implies that, for n sufficiently large, with probability 1, J′′
+j,n,δ,ρ ≤ η/2.
+Using additionally (E.6), we obtain that Jj,n,δ,ε converges almost surely to zero,
+which concludes the proof of (5.16). The proof of the analogous result for ˙C
+∇,ν
+j,k:l
+in (5.11) is almost identical.
+■
+Proof of Proposition 5.10. Fix j ∈ {1, . . . , d}, δ ∈ (0, 1) and ε ∈ (0, 1/2).
+From (5.13), we have that, for any (s, t) ∈ Λ and u ∈ [0, 1]d,
+˙C
+Bern
+j,⌊ns⌋+1:⌊nt⌋,m⌊nt⌋−⌊ns⌋(u)
+=
+�
+[0,1]d
+˙C
+∇
+j,⌊ns⌋+1:⌊nt⌋,1/m⌊nt⌋−⌊ns⌋,0(w)d˜µj,m⌊nt⌋−⌊ns⌋,u(w),
+where ˙C∇
+j,k:l,1/m,0 is defined in (5.1) and, for any m ≥ 2, ˜µj,m,u is the law of the
+random vector ( ˜Sm,1,u1/m, . . . , ˜Sm,d,ud/m) whose components are independent
+such that, for i ∈ {1, . . . , d} \ {j}, ˜Sm,i,ui is Binomial(m, ui) while ˜Sm,j,uj is
+Binomial(m − 1, uj). It follows that, for any (s, t) ∈ Λ and u ∈ [0, 1]d,
+˙C
+Bern
+j,⌊ns⌋+1:⌊nt⌋,m⌊nt⌋−⌊ns⌋(u)
+=
+�
+Wj,n,s,t
+˙C
+∇
+j,⌊ns⌋+1:⌊nt⌋,1/m⌊nt⌋−⌊ns⌋,0(w)d˜µj,m⌊nt⌋−⌊ns⌋,u(w),
+(E.7)
+where Wj,n,s,t = {w ∈ [0, 1]d : wj ≤ 1 − 1/m⌊nt⌋−⌊ns⌋}. For the sake of a more
+compact notation, from now on, we shall write ms,t for m⌊nt⌋−⌊ns⌋, (s, t) ∈
+Λ. From the triangle inequality, the left-hand side of (5.17) is smaller than
+Ij,n,δ,ε + Jj,n,δ,ε, where
+Ij,n,δ,ε =
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+�����
+�
+Wj,n,s,t
+�
+˙C
+∇
+j,⌊ns⌋+1:⌊nt⌋,1/ms,t,0(w)
+−
+˙Cj(w)
+�
+d˜µj,ms,t,u(w)
+��� ,
+Jj,n,δ,ε =
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+�����
+�
+[0,1]d
+˙Cj(w)d˜µj,ms,t,u(w) − ˙Cj(u)
+����� .
+For any n ∈ N, x ∈ (Rd)n and u ∈ [0, 1]d, let νx
+u = ˜µj,⌊Lnθ⌋∨2,u. With this
+notation, Condition 5.8 holds for the considered smoothing distributions and
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+56
+it can be verified that Jj,n,δ,ε = oP(1) by proceeding exactly as in the proof of
+Proposition 5.9 for the analogous term. It thus remain to show that Ij,n,δ,ε =
+oP(1).
+From the triangle inequality, we have that Ij,n,δ,ε is smaller than
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+������
+�
+{w∈Wj,n,s,t:
+wj∈[ε/2,1−ε/2]}
+�
+˙C
+∇
+j,⌊ns⌋+1:⌊nt⌋,1/ms,t,0(w) − ˙Cj(w)
+�
+d˜µj,ms,t,u(w)
+������
++
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+������
+�
+{w∈Wj,n,s,t:
+wj<ε/2}
+�
+˙C
+∇
+j,⌊ns⌋+1:⌊nt⌋,1/ms,t,0(w) − ˙Cj(w)
+�
+d˜µj,ms,t,u(w)
+������
++
+sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+������
+�
+{w∈Wj,n,s,t:
+wj>1−ε/2}
+�
+˙C
+∇
+j,⌊ns⌋+1:⌊nt⌋,1/ms,t,0(w) − ˙Cj(w)
+�
+d˜µj,ms,t,u(w)
+������
+≤ I′
+j,n,δ,ε + MnI′′
+j,n,δ,ε + MnI′′′
+j,n,δ,ε,
+where
+I′
+j,n,δ,ε = sup
+(s,t)∈Λ
+t−s≥δ
+sup
+w∈[0,1]d
+wj∈[ε/2,1−ε/2]
+��� ˙C
+∇
+j,⌊ns⌋+1:⌊nt⌋,1/ms,t,0(w) − ˙Cj(w)
+��� ,
+I′′
+j,n,δ,ε = sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+˜µj,ms,t,u
+�
+w ∈ [0, 1]d : wj < ε/2
+�
+,
+I′′′
+j,n,δ,ε = sup
+(s,t)∈Λ
+t−s≥δ
+sup
+u∈[0,1]d
+uj∈[ε,1−ε]
+˜µj,ms,t,u
+�
+w ∈ [0, 1]d : wj > 1 − ε/2
+�
+,
+Mn = sup
+(s,t)∈Λ
+sup
+w∈Wj,n,s,t
+��� ˙C
+∇
+j,⌊ns⌋+1:⌊nt⌋,1/ms,t,0(w)
+��� .
+Since the conditions of the proposition imply that Condition 5.4 holds with
+h(x) = 1/(⌊Lnθ⌋ ∨ 2) and h′(x) = 0 for all n ∈ N and x ∈ (Rd)n, we have that
+I′
+j,n,δ,ε = oP(1) as a consequence of Corollary 5.7. Also, given that Condition 5.8
+holds for the considered smoothing distributions, it can be verified that I′′
+j,n,δ,ε
+and I′′′
+j,n,δ,ε converge almost surely to zero by proceeding exactly as in the proof
+of Proposition 5.9 for the analogous terms. To complete the proof of (5.17), it
+suffices to show that, there exists a constant ζ > 0 such that, for any n ∈ N,
+Mn < ζ almost surely.
+Fix n ∈ N. From the adopted conventions, we have that ˙C∇
+j,⌊ns⌋+1:⌊nt⌋,1/ms,t,0 =
+0 for all (s, t) ∈ Λ such that ⌊ns⌋ = ⌊nt⌋. Fix (s, t) ∈ Λ such that ⌊ns⌋ < ⌊nt⌋
+and let p = ⌊nt⌋ − ⌊ns⌋. The empirical copula C⌊ns⌋+1:⌊nt⌋, generically defined
+in (2.1), is a multivariate d.f. whose d univariate margins, under Condition 2.3,
+are all equal to G⌊ns⌋+1:⌊nt⌋, where G⌊ns⌋+1:⌊nt⌋(u) = ⌊pu⌋/p, u ∈ [0, 1]. As a
+consequence of a well-known property of multivariate d.f.s (see, e.g., Durante
+
+/Resampling techniques for smooth, possibly data-adaptive empirical copulas
+57
+and Sempi, 2015, Lemma 1.2.14), we have that
+��C⌊ns⌋+1:⌊nt⌋(u) − C⌊ns⌋+1:⌊nt⌋(v)
+�� ≤
+d
+�
+j=1
+��G⌊ns⌋+1:⌊nt⌋(uj) − G⌊ns⌋+1:⌊nt⌋(vj)
+��
+for all u, v ∈ [0, 1]d. We then obtain that, for any u ∈ Wj,n,s,t,
+��C⌊ns⌋+1:⌊nt⌋(u + ej/ms,t) − C⌊ns⌋+1:⌊nt⌋(u)
+��
+≤
+��G⌊ns⌋+1:⌊nt⌋(uj + 1/ms,t) − G⌊ns⌋+1:⌊nt⌋(uj)
+�� ,
+which implies that
+��� ˙Cj,⌊ns⌋+1:⌊nt⌋(u)
+��� ≤
+��G⌊ns⌋+1:⌊nt⌋(uj + 1/ms,t) − G⌊ns⌋+1:⌊nt⌋(uj)
+��
+1/ms,t
+= ms,t
+�⌊p(uj + 1/ms,t)⌋
+p
+− ⌊puj⌋
+p
+�
+≤ ms,t
+�p(uj + 1/ms,t)
+p
+− puj − 1
+p
+�
+≤ ms,t
+� 1
+ms,t
++ 1
+p
+�
+≤ 1 + ms,t
+p
+= 1 + ⌊Lpθ⌋ ∨ 2
+p
+≤ 1 + Lpθ−1 ∨ (2/p) ≤ 1 + L ∨ 2,
+which completes the proof of (5.17). The fact that (5.18) holds is finally an
+immediate consequence of the previous centered display and (E.7).
+■
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+
diff --git a/z9E5T4oBgHgl3EQfOQ4n/content/tmp_files/load_file.txt b/z9E5T4oBgHgl3EQfOQ4n/content/tmp_files/load_file.txt
new file mode 100644
index 0000000000000000000000000000000000000000..fe75e9783ac75079a22aa22dfe7dd1244745a686
--- /dev/null
+++ b/z9E5T4oBgHgl3EQfOQ4n/content/tmp_files/load_file.txt
@@ -0,0 +1,3746 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf,len=3745
+page_content='Resampling techniques for a class of  smooth, possibly data-adaptive  empirical copulas  Ivan Kojadinovic1 and Bingqing Yi1,2  1CNRS / Universit´e de Pau et des Pays de l’Adour / E2S UPPA  Laboratoire de math´ematiques et applications IPRA, UMR 5142  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' 1155, 64013 Pau Cedex, France  2School of Mathematics & Statistics  The University of Melbourne  Parkville, VIC 3010, Australia  Abstract: We investigate the validity of two resampling techniques when  carrying out inference on the underlying unknown copula using a recently  proposed class of smooth, possibly data-adaptive nonparametric estimators  that contains empirical Bernstein copulas (and thus the empirical beta cop-  ula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Following Kiriliouk, Segers and Tsukahara (2021), the first resampling  technique is based on drawing samples from the smooth estimator and can  only can be used in the case of independent observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The second tech-  nique is a smooth extension of the so-called sequential dependent multiplier  bootstrap and can thus be used in a time series setting and, possibly, for  change-point analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The two studied resampling schemes are applied to  confidence interval construction and the offline detection of changes in the  cross-sectional dependence of multivariate time series, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Monte  Carlo experiments confirm the possible advantages of such smooth infer-  ence procedures over their non-smooth counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' A by-product of this  work is the study of the weak consistency and finite-sample performance of  two classes of smooth estimators of the first-order partial derivatives of a  copula which can have applications in mean and quantile regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  MSC 2010 subject classifications: Primary 62G05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' secondary 62G20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Keywords and phrases: data-adaptive smooth empirical copulas, empir-  ical beta copula, smooth data-adaptive estimators of the first-order partial  derivatives of the copula, smooth bootstraps, smooth sequential dependent  multiplier bootstraps, strong mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Contents  1  Introduction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  2  2  Smooth, possibly data-adaptive, empirical copulas and their asymptotics  4  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1  Smooth, possibly data-adaptive, nonparametric copula estimators  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  Asymptotics of related sequential processes  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  7  3  Bootstrap by drawing samples from the estimators in the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' case .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  10  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1  Drawing samples from the smooth empirical copulas .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='  11  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  Asymptotic validity results  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='  12  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3  Finite-sample comparison of two smooth bootstraps  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='  13  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  16  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1  Main intuition and existing work .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  16  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' and dependent multiplier sequences .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  16  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3  Non-smooth sequential dependent multiplier replicates .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  17  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4  Smooth sequential dependent multiplier replicates  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  18  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  Finite-sample comparison of three multiplier bootstraps .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  20  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6  Application to change-point detection  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  25  5  Estimators of the first-order partial derivatives of the copula  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  28  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1  Estimators based on finite differences of the empirical copula  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  29  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  Two classes of smooth estimators .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  30  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3  Weak consistency .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  32  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4  Finite-sample performance of selected estimators .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='  34  6  Conclusion  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='  38  A Proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='  39  B Proofs of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3 and Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='  40  C Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='  42  D Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='  46  E  Proofs of Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='  48  References .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='  57  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Introduction  Let X 1:n = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content=' , Xn) be a stretch from a d-dimensional stationary time  series (Xi)i∈Z of continuous random vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' From a well-known theorem due  to Sklar (1959), the multivariate distribution function (d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=') F of each Xi can  be expressed as  F(x) = C{F1(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , Fd(xd)},  x ∈ Rd,  (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1)  in terms of a unique copula C and the univariate margins F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , Fd of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Representation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) is at root of many applications in probability, statistics and  related fields (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=', Hofert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=', 2018, and the references therein) because  it suggests that F can be modeled in two separate steps: the first (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' second)  step consists of estimating the univariate margins F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , Fd (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' the copula  C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' This work is only concerned with the estimation of the copula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Statistical inference on the unknown copula C frequently involves the use of  a nonparametric estimator of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The best-known one is the empirical copula  (R¨uschendorf, 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Deheuvels, 1979) which we shall define as the empirical  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of the multivariate ranks obtained from X 1:n scaled by 1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Note that the  latter function is piecewise constant and cannot therefore be a genuine copula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  A promising smooth nonparametric estimator of C that is a genuine copula  when there are no ties in the components samples of X 1:n and that displays  substantially better small-sample performance than the empirical copula is the  empirical beta copula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' This estimator was proposed by Segers, Sibuya and Tsuka-  hara (2017) and is a particular case of the empirical Bernstein copula studied by  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  3  Sancetta and Satchell (2004) and Janssen, Swanepoel and Veraverbeke (2012)  when all the underlying Bernstein polynomials have degree n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Building upon  the work of Segers, Sibuya and Tsukahara (2017), Kojadinovic and Yi (2022)  recently studied data-adaptive generalizations of the empirical beta copula that  can perform even better in small samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Whatever nonparametric estimator of the unknown copula C in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) is used  in inference procedures, it is almost always necessary to rely on resampling tech-  niques to compute corresponding confidence intervals or p-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' To approxi-  mate the “sampling distribution” of the classical empirical copula, a frequently  used approach in the literature is the so-called multiplier bootstrap (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=',  Scaillet, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' R´emillard and Scaillet, 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' When the random vectors in X 1:n  are independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' ), B¨ucher and Dette (2010)  found the latter resampling scheme to have better finite-sample properties than  approaches consisting of adapting the empirical (multinomial) bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The  multiplier bootstrap was extended to the time series and sequential settings in  B¨ucher and Kojadinovic (2016) and B¨ucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  One of the advantages of the empirical beta copula is that it is particu-  larly easy to draw samples from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The resulting smooth bootstrap that can  be used to approximate the “sampling distribution” of the empirical beta cop-  ula was recently studied both theoretically and empirically in Kiriliouk, Segers  and Tsukahara (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The Monte Carlo experiments reported therein reveal  that it is a competitive alternative to the multiplier bootstrap while being sub-  stantially simpler to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' One practical inconvenience however is that  the aforementioned smooth bootstrap cannot be directly extended to the time  series setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The first aim of this work is to obtain, in the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' case, a smooth bootstrap  `a la Kiriliouk, Segers and Tsukahara (2021) for the smooth, possibly data-  adaptive, nonparametric estimators of the copula investigated in Kojadinovic  and Yi (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The second aim is to propose smooth versions of the dependent  multiplier bootstrap that can be used to approximate the “sampling distribu-  tion” of the aforementioned estimators in a time series setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Intuitively, one  could expect that the resulting smooth inference procedures will perform better  than corresponding non-smooth procedures in particular when the amount of  data is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Indeed, as already mentioned, it is when n is small that smooth cop-  ula estimators can substantially outperform rough estimators such as the classi-  cal empirical copula;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' see for instance the finite-sample experiments reported in  Segers, Sibuya and Tsukahara (2017), Kiriliouk, Segers and Tsukahara (2021)  or Kojadinovic and Yi (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Another situation where one could expect that  the use of smooth estimators can be advantageous is when carrying out change-  point detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Indeed, statistics for change-point detection often involve the  comparison of estimators computed from small subsets of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' It is to  be able to cover this application area that many of the theoretical investigations  carried out in this work are of a sequential nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  A by-product of this work is the study of the weak consistency and finite-  sample performance of two classes of smooth estimators of the first-order par-  tial derivatives of the unknown copula C in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) as these are needed to carry  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  4  out the dependent multiplier bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As explained for instance in Janssen,  Swanepoel and Veraverbeke (2016), such estimators have applications in mean  and quantile regression as they lead to estimators of the conditional distribution  function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' From a practical perspective, our investigations lead to the proposal of  a smooth data-adaptive estimator of the first-order partial derivatives of C that  substantially outperforms, among others, the Bernstein estimator considered in  Janssen, Swanepoel and Veraverbeke (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' In the second section, we recall the def-  inition of the broad class of smooth, possibly data adaptive, empirical copulas  studied in Kojadinovic and Yi (2022) and the asymptotics of related sequential  empirical processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The third section is concerned with an extension of the  smooth bootstrap of Kiriliouk, Segers and Tsukahara (2021) that can be used  to approximate the “sampling distribution” of the aforementioned smooth es-  timators in the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' After investigating its asymptotic validity, results of  finite-sample experiments comparing smooth bootstraps based on the empirical  beta copula and on its data-adaptive extension suggested in Kojadinovic and  Yi (2022) are reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' In Section 4, to be able to cover the time series setting,  we propose natural smooth extensions of the sequential dependent multiplier  bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' After providing asymptotic validity results, we compare the finite-  sample performance of various versions of the multiplier bootstrap and consider  an application to the offline detection of changes in the cross-sectional depen-  dence of multivariate time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The latter confirms the possible advantages  of smooth inference procedures over their non-smooth counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The fifth  section is devoted to the study of two classes of smooth estimators of the first-  order partial derivatives of C: their weak consistency is investigated and the  finite-sample performance of selected estimators is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Unless stated otherwise, all convergences in the paper are as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Also,  in the sequel, the arrow ‘⇝’ denotes weak convergence in the sense of Defi-  nition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3 in van der Vaart and Wellner (2000) and, given a set T, ℓ∞(T)  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' C (T)) represents the space of all bounded (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' continuous) real-valued  functions on T equipped with the uniform metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  All the numerical experiments presented in the work were carried out using  the R statistical environment (R Core Team, 2022) as well as its packages copula  (Hofert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=', 2022) and extraDistr (Wolodzko, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Smooth, possibly data-adaptive, empirical copulas and their  asymptotics  In this section, we start by defining the broad class of smooth, possibly data  adaptive, empirical copulas studied in Kojadinovic and Yi (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' We then recall  the asymptotics of related sequential empirical processes established in the same  reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Smooth, possibly data-adaptive, nonparametric copula  estimators  Because the results to be stated in the next section are of a sequential nature,  all the quantities hereafter are defined for a substretch X k:l = (Xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , Xl),  1 ≤ k ≤ l ≤ n, of the available data X 1:n = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  For any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, let Fk:l,j be the empirical d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' computed from the jth  component subsample Xkj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , Xlj of X k:l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then, Rk:l  ij = (l−k+1)Fk:l,j(Xij) =  �l  t=k 1(Xtj ≤ Xij) is the (maximal) rank of Xij among Xkj, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , Xlj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Further-  more, let  Rk:l  i  =  �  Rk:l  i1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , Rk:l  id  �  and  ˆU k:l  i  =  Rk:l  i  l − k + 1,  i ∈ {k, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , l},  be the multivariate ranks (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' multivariate scaled ranks) obtained from X k:l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Following R¨uschendorf (1976), the empirical copula Ck:l of X k:l is then defined,  for any u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , ud) ∈ [0, 1]d, by  Ck:l(u) =  1  l − k + 1  l  �  i=k  d  �  j=1  1  �  Rk:l  ij  l − k + 1 ≤ uj  �  =  1  l − k + 1  l  �  i=k  1( ˆU k:l  i  ≤ u),  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1)  where inequalities between vectors are to be understood componentwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  As we continue, following Kojadinovic and Yi (2022), for any m ∈ N, x ∈  (Rd)m and u ∈ [0, 1]d, νx  u is the law of a [0, 1]d-valued mean u random vector  W x  u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Its components are denoted by W x  1,u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , W x  d,ud to indicate that the jth  component depends on uj but not on u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , uj−1, uj+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , ud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Let p ≥ d be a  fixed integer and let U be a p-dimensional random vector whose components are  independent and standard uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The following assumption was considered  in Kojadinovic and Yi (2022) and is likely to be non-restrictive as discussed in  Remark 3 therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 (Construction of smoothing random vectors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For any m ∈ N,  x ∈ (Rd)m and u ∈ [0, 1]d, there exists a function W x  u : [0, 1]p → [0, 1]d such  that W x  u = W x  u (U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  To be able to define, for any n ∈ N, X 1:n and, for any m ≤ n, the random  vectors W x  u , x ∈ (Rd)m, u ∈ [0, 1]d, on the same probability space (Ω, A , P), we  assume a product structure, that is, Ω = Ω0 ×Ω1 ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' with probability measure  P = P0 ⊗ P1 ⊗ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , where Pi denotes the probability measure on Ωi, such that,  for any ω ∈ Ω, X 1:n(ω) only depends on the first coordinate of ω, U(ω) only  depends on the second coordinate of ω and potential “bootstrap weights” (to be  introduced in Sections 3 and 4) only depend on one of the remaining coordinates  of ω, implying in particular that X 1:n, U and potential bootstrap weights are  independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' A broad class of smooth versions of Ck:l in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1), with possibly  data-adaptive smoothing, is then given by  Cν  k:l(u) =  �  [0,1]d Ck:l(w)dν  X k:l  u  (w),  u ∈ [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2)  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  6  Intuitively, for a given u ∈ [0, 1]d, Cν  k:l(u) can be thought of as a “weighted  average” of Ck:l(w) for w “in a neighborhood of u” according to the smoothing  distribution ν  X k:l  u  (that may depend on the observations X k:l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Note that, if  k > l, we adopt the convention that Ck:l = Cν  k:l = 0 and that, for any u ∈ [0, 1]d,  ν  X k:l  u  is the Dirac measure at u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Given m ∈ N and u ∈ [0, 1]d, let µm,u be the law of the d-  dimensional random vector (Sm,1,u1/m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , Sm,d,ud/m) such that the random  variables Sm,1,u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , Sm,d,ud are independent and, for each j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' d}, Sm,j,uj  is Binomial(m, uj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' From Section 3 of Segers, Sibuya and Tsukahara (2017), the  empirical Bernstein copula of X k:l whose Bernstein polynomial degrees are all  equal to m is then given by  C  Bern  k:l,m(u) =  �  [0,1]d Ck:l(w)dµm,u(w),  u ∈ [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3)  The latter is clearly a special case of Cν  k:l in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' If, additionally, m = l−k +1,  that is, if the smoothing distributions satisfy ν  X k:l  u  = µl−k+1,u, u ∈ [0, 1]d, Cν  k:l  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) or, equivalently, CBern  k:l,m in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3), corresponds to the empirical beta copula  of X k:l studied in Segers, Sibuya and Tsukahara (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  For any m ∈ N, x ∈ (Rd)m, r ∈ [0, m]d and u ∈ [0, 1]d, let  K x  r (u) =  �  [0,1]d 1(r/m ≤ w)dνx  u(w) = E {1(r/m ≤ W x  u )} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4)  By linearity of the integral, we can then express Cν  k:l in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) as  Cν  k:l(u) =  1  l − k + 1  l  �  i=k  K  X k:l  Rk:l  i  (u),  u ∈ [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5)  Since copulas have standard uniform margins, it is particularly meaningful  to focus on estimators of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5) that have standard uniform margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  As verified in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 of Kojadinovic and Yi (2022), the following two as-  sumptions imply the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3 (No ties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' With probability 1, there are no ties in each of the  component samples X1j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , Xnj, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, of X 1:n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4 (Condition for uniform margins).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For any m ∈ N, x ∈ (Rd)m,  u ∈ [0, 1]d and j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, W x  j,uj takes its values in the set {0, 1/m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , (m−  1)/m, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Under Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4, from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 of Kojadinovic and Yi (2022), for any  m ∈ N, x ∈ (Rd)m, r ∈ [1, m]d and u ∈ [0, 1]d, K x  r (u) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4) can be written  as  K x  r (u) = ¯  C x  u  � ¯  F x  1,u1{(r1 − 1)/m}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , ¯  F x  d,ud{(rd − 1)/m}  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6)  where  ¯  C x  u (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ¯  F x  1,u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , ¯  F x  d,ud) is a survival copula (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' are the marginal  survival functions) of the random vector W x  u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Upon additionally assuming the  following two conditions considered in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 of Kojadinovic and Yi (2022),  estimators of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5) can be shown to be genuine copulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  7  Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5 (Condition on the smoothing survival margins).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For any m ∈ N,  x ∈ (Rd)m, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d} and w ∈ [0, 1), the function t �→  ¯  F x  j,t(w) is right-  continuous and increasing on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6 (Condition on the smoothing survival copulas).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For any m ∈ N,  x ∈ (Rd)m and u ∈ [0, 1]d, the copulas ¯  C x  u in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6) do not depend on u, that is,  ¯  C x  u = ¯  C x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The following result was then proven in Kojadinovic and Yi (2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' see Propo-  sition 11 and Corollary 12 therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7 (Cν  k:l is a genuine copula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Assume that Conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3,  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then, the smooth empirical copula Cν  k:l in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) or in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5)  can be expressed, for any u ∈ [0, 1]d, as  Cν  k:l(u) =  1  l − k + 1  l  �  i=k  ¯  C  X k:l  �  ¯  F  X k:l  1,u1  � Rk:l  i1 − 1  l − k + 1  �  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , ¯  F  X k:l  d,ud  � Rk:l  id − 1  l − k + 1  ��  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7)  and is a genuine copula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  From Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 above, we can infer that the empirical beta copula of X k:l  studied in Segers, Sibuya and Tsukahara (2017) is of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7) with ¯  C X k:l  the independence copula and, for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d} and u ∈ [0, 1]  ¯  F  X k:l  j,u  the  survival function of a scaled (by 1/(l − k + 1)) Binomial(l − k + 1, u) random  variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For that reason, the latter will be denoted as CBin  k:l as we continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As a  possible improvement of the empirical beta copula CBin  k:l of X k:l, Kojadinovic and  Yi (2022) suggested to consider a smooth data-adaptive empirical copula of the  form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7) with ¯  C X k:l the empirical beta copula CBin  k:l and, for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}  and u ∈ [0, 1],  ¯  F  X k:l  j,u  the survival function of a scaled (by 1/(l − k + 1)) Beta-  Binomial(m, α, β) random variable, where m = l − k + 1, α = u(m − ρ)/(ρ − 1),  β = (1 − u)(m − ρ)/(ρ − 1) and ρ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The resulting data-adaptive estimator,  denoted by CBetaB4  k:l  as we continue, was found to outperform the empirical beta  copula CBin  k:l in terms of integrated mean squared error in all the bivariate and  trivariate experiments considered in Kojadinovic and Yi (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Asymptotics of related sequential processes  We can now define the sequential empirical processes corresponding to the em-  pirical copula in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) and to its smooth generalizations in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Let Λ = {(s, t) ∈  [0, 1]2 : s ≤ t} and let λn(s, t) = (⌊nt⌋ − ⌊ns⌋)/n, (s, t) ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The corresponding  two-sided sequential empirical copula processes are given, for any (s, t) ∈ Λ and  u ∈ [0, 1]d, by  Cn(s, t, u) = √nλn(s, t){C⌊ns⌋+1:⌊nt⌋(u) − C(u)},  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8)  Cν  n(s, t, u) = √nλn(s, t){Cν  ⌊ns⌋+1:⌊nt⌋(u) − C(u)},  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9)  where C⌊ns⌋+1:⌊nt⌋ and Cν  ⌊ns⌋+1:⌊nt⌋ are generically defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2),  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The asymptotics of Cn were established in B¨ucher and Kojadinovic  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  8  (2016), while the asymptotics of Cν  n (which we recall in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10 hereafter)  were investigated in Kojadinovic and Yi (2022) by generalizing the arguments  used in Segers, Sibuya and Tsukahara (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The following conditions were considered in Kojadinovic and Yi (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8 (Smooth partial derivatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, the partial  derivative ˙Cj = ∂C/∂uj exists and is continuous on the set Vj = {u ∈ [0, 1]d :  uj ∈ (0, 1)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9 (Variance condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' There exists a constant κ > 0 such that,  for any n ∈ N, x ∈ (Rd)n, u ∈ [0, 1]d and j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, Var(W x  j,uj) ≤ κuj(1 −  uj)/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The first condition was initially considered in Segers (2012) and can be con-  sidered non-restricted as explained in the latter reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' In the rest of the  paper, for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, ˙Cj is arbitrarily defined to be zero on the set  {u ∈ [0, 1]d : uj ∈ {0, 1}}, which implies that, under Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8, ˙Cj is de-  fined on the whole of [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The second condition imposes constraints on the  spread of the smoothing distributions involved in the definition of the smooth,  possibly data-adaptive, empirical copulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10 (Asymptotics of Cν  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Assume that Conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9  hold, and that Cn ⇝ CC in ℓ∞(Λ×[0, 1]d), where the trajectories of the limiting  process CC are continuous almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then,  sup  (s,t)∈Λ  u∈[0,1]d  |Cν  n(s, t, u) − Cn(s, t, u)| = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Consequently, Cν  n ⇝ CC in ℓ∞(Λ × [0, 1]d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Hence, the smooth sequential empirical copula process Cν  n in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9) and the  classical sequential empirical copula process Cn in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8) are asymptotically  equivalent when the latter converges weakly to a limiting process whose tra-  jectories are continuous almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As discussed in Section 3 of B¨ucher and  Kojadinovic (2016), for such a convergence to hold, it suffices that the corre-  sponding “uniform multivariate sequential empirical process” converges weakly  to a limiting process whose trajectories are continuous almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Specifically,  let U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , Un be the unobservable sample obtained from X 1:n = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , Xn)  by the probability integral transformations Uij = Fj(Xij), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , n},  j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, and let  Bn(s, t, u) =  1  √n  ⌊nt⌋  �  i=⌊ns⌋+1  {1(Ui ≤ u)−C(u)},  (s, t, u) ∈ Λ×[0, 1]d, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10)  with the convention that Bn(s, t, ·) = 0 if ⌊nt⌋ − ⌊ns⌋ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The aforementioned  sufficient condition can then be stated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  9  Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11 (Weak convergence of Bn(0, ·, ·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The sequential empirical  process Bn(0, ·, ·) converges weakly in ℓ∞([0, 1]d+1) to a tight centered Gaussian  process ZC concentrated on  {f ∈ C ([0, 1]d+1) : f(s, u) = 0 if one of the components of (s, u) is 0,  and f(s, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , 1) = 0 for all s ∈ (0, 1]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Under Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11 (which holds for instance when (Xi)i∈Z is strongly mix-  ing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=', B¨ucher (2015) as well as forthcoming Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1), it immediately  follows from the continuous mapping theorem that Bn ⇝ BC in ℓ∞(Λ × [0, 1]d),  where  BC(s, t, u) = ZC(t, u) − ZC(s, u),  (s, t, u) ∈ Λ × [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11)  For any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d} and any u ∈ [0, 1]d, let u(j) be the vector of [0, 1]d  defined by u(j)  i  = uj if i = j and 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The following result is then an  immediate consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4 in B¨ucher and Kojadinovic (2016) and  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3 of B¨ucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='12 (Asymptotics of Cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Under Conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11,  sup  (s,t,u)∈Λ×[0,1]d  ���Cn(s, t, u) − ˜Cn(s, t, u)  ��� = oP(1),  where  ˜Cn(s, t, u) = Bn(s, t, u) −  d  �  j=1  ˙Cj(u) Bn(s, t, u(j)),  (s, t, u) ∈ Λ × [0, 1]d,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='12)  and Bn is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Consequently, Cn ⇝ CC in ℓ∞(Λ × [0, 1]d), where  CC(s, t, u) = BC(s, t, u) −  d  �  j=1  ˙Cj(u) BC(s, t, u(j)),  (s, t, u) ∈ Λ × [0, 1]d,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13)  and BC is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  We end this section with the statement of a corollary of Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10  and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Having (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4) in mind, two natural smooth extensions of the unobserv-  able empirical process Bn in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10) can be defined, for any (s, t, u) ∈ Λ × [0, 1]d,  by  ˜Bν  n(s, t, u) =  1  √n  ⌊nt⌋  �  i=⌊ns⌋+1  ��  [0,1]d 1(Ui ≤ w)dν  X ⌊ns⌋+1:⌊nt⌋  u  (w) − C(u)  �  ,  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='14)  ¯Bν  n(s, t, u) =  1  √n  ⌊nt⌋  �  i=⌊ns⌋+1  ��  [0,1]d 1(Ui ≤ w)dν  X 1:n  u  (w) − C(u)  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='15)  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  10  Combining Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='12 with key intermediate results used in Kojadinovic  and Yi (2022) for proving Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10 stated above, we obtain the follow-  ing asymptotic representations for the smooth sequential empirical process Cν  n  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The proof of this result is given in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13 (Asymptotic representations of Cν  n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Under Conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9  and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11,  sup  (s,t,u)∈Λ×[0,1]d  ���Cν  n(s, t, u) − ˜Cν  n(s, t, u)  ��� = oP(1),  sup  (s,t,u)∈Λ×[0,1]d  ��Cν  n(s, t, u) − ¯Cν  n(s, t, u)  �� = oP(1),  where, for any (s, t, u) ∈ Λ × [0, 1]d,  ˜Cν  n(s, t, u) = ˜Bν  n(s, t, u) −  d  �  j=1  ˙Cj(u) ˜Bν  n(s, t, u(j)),  ¯Cν  n(s, t, u) = ¯Bν  n(s, t, u) −  d  �  j=1  ˙Cj(u) ¯Bν  n(s, t, u(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The previous results do not unfortunately allow us to decide  which of the above two asymptotic representations for Cν  n may be better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The  knowledge of the underlying convergence rates would be needed for that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As  we shall see in Section 4, these representations will be at the root of smooth  proposals for bootstrapping Cν  n in a time series context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Bootstrap by drawing samples from the estimators in the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  case  The aim of this section is to study both theoretically and empirically a smooth  bootstrap `a la Kiriliouk, Segers and Tsukahara (2021) based on drawing samples  from the smooth estimators defined in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As hinted at in the  introduction, such an approach can only be used in the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Throughout  this section, we thus assume that the random vectors in X 1:n are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Notice  that the latter implies Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Given that change-point analysis is es-  sentially of interest in the time series setting, we do not consider a sequential  setting below but instead focus only on the situation where k = 1 and l = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  This section is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' After describing the sampling algorithm  on which the smooth bootstrap is based, we state conditions under which it is  asymptotically valid and report results of finite-sample experiments comparing  smooth bootstraps based on the empirical beta copula CBin  1:n and on its data-  adaptive extension CBetaB4  1:n  proposed in Kojadinovic and Yi (2022) and recalled  at the end of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  11  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Drawing samples from the smooth empirical copulas  As explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1, the empirical beta copula CBin  1:n is a particular case of  the smooth estimators Cν  1:n defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' From Segers, Sibuya and Tsukahara  (2017) (see also Lemma 1 in Kojadinovic and Yi 2022), one has that  C  Bin  1:n(u) = 1  n  n  �  i=1  d  �  j=1  Fn,R1:n  ij (uj),  u = (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , ud) ∈ [0, 1]d,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1)  where, for any n ∈ N and r ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , n}, Fn,r denotes the d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of the beta  distribution with shape parameters α = r and β = n+1−r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1)  that CBin  1:n is a mixture of n d-dimensional distributions having beta margins and  whose copula is the independence copula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' To generate one random variate from  CBin  1:n, it thus suffices to randomly select one of the n components of the mixture  by drawing a uniform on {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , n} and then generate one random variate from  the selected d-dimensional distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' This is detailed in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 of  Kiriliouk, Segers and Tsukahara (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  In a related way, having (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5) in mind, it thus suffices to assume the following  to be able sample from Cν  1:n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (Cν  1:n is a mixture) For any n ∈ N, x ∈ (Rd)n and r ∈  {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , n}d, K x  r  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4) is a d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' on [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The sampling algorithm is then conceptually the same as Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 of  Kiriliouk, Segers and Tsukahara (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (Sampling from Cν  1:n under Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1)  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Generate I from the discrete uniform distribution on {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Generate a random variate V # from a d-dimensional distribution whose  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' is K  X 1:n  R1:n  I  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The above algorithm can be used in practice as soon as one knows how to  sample from the d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='s K  X 1:n  R1:n  i  , i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Interestingly enough, three of the conditions stated in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 imply Con-  dition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 as shown in the next result proven in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6 imply Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Specif-  ically, under Conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6, for any n ∈ N, x ∈ (Rd)n and  r ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , n}d, K x  r  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4) is a d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' on [0, 1]d whose d univariate margins,  denoted by K x  r1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , K x  rd,d, respectively, satisfy K x  rj,j(u) =  ¯  F x  j,u{(rj − 1)/n},  u ∈ [0, 1], j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, and whose copula is  ¯  C x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The previous result leads to an alternative (and simpler) proof of  Proposition 11 of Kojadinovic and Yi (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Indeed, under the assumptions  of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3, Cν  1:n in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5) is a convex combination of multivariate d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='s  on [0, 1]d and therefore a multivariate d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' on [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Since Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3 holds  in the current i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' setting, from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 in Kojadinovic and Yi (2022),  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  12  Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4 also implies that Cν  1:n has standard uniform margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Hence, under  the assumptions of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3, Cν  1:n is a genuine copula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  For any univariate d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' H, let H−1 denote its associated quantile function  (generalized inverse) defined by H−1(y) = inf{x ∈ R : H(x) ≥ y}, y ∈ [0, 1],  with the convention that inf ∅ = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The second step of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 can then  be made explicit under Conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6 :  (i) Generate a random variate U # from the copula ¯  C X 1:n independently of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (ii) A random variate from the distribution whose d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' is K  X 1:n  R1:n  I  is then  V  # =  �  K  X 1:n,−1  R1:n  I1 ,1  (U  #  1 ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , K  X 1:n,−1  R1:n  Id ,d  (U  #  d )  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2)  We end this section by discussing how Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 can be practically im-  plemented for the smooth data-adaptive estimator CBetaB4  1:n  introduced in Ko-  jadinovic and Yi (2022) as a possible improvement of the empirical beta copula  CBin  1:n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Recall from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 that CBetaB4  1:n  is of the form (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7) with  ¯  C X 1:n the  empirical beta copula CBin  1:n and, for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d} and u ∈ [0, 1], ¯  F  X 1:n  j,u  the  survival function of a scaled (by 1/n) Beta-Binomial(n, α, β) random variable,  where α = u(n − ρ)/(ρ − 1), β = (1 − u)(n − ρ)/(ρ − 1) and ρ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The latter  implies that, for any i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , n}, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d} and u ∈ [0, 1],  K  X 1:n  R1:n  ij ,j(u) = ¯  F  X 1:n  j,u  {(R1:n  ij −1)/n} = P(nW  X 1:n  j,u  > R1:n  ij −1) = ¯  Bn,u,ρ(R1:n  ij −1),  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3)  where ¯  Bn,u,ρ is the survival function of the Beta-Binomial(n, α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As can be  checked from Lemma 27 in Kojadinovic and Yi (2022) and Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 in Ap-  pendix B, the univariate d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' K  X 1:n  R1:n  ij ,j in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3) is continuous and strictly increas-  ing, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Hence, to compute its associated quantile function needed  in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2), one can proceed numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' In that respect, an implementation of Al-  gorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 for the R statistical environment for the estimators CBin  1:n and CBetaB4  1:n  is available on the web page of the first author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Asymptotic validity results  Building upon the work of Kiriliouk, Segers and Tsukahara (2021), we will  now provide asymptotic validity results for a smooth bootstrap based on draw-  ing samples from Cν  1:n in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5) under Conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Recall that,  according to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3, the latter conditions imply Condition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Let  V  #  1:n = (V  #  1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , V #  n ) be a random sample from Cν  1:n obtained by applying  Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 n times independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Note that this implies that the com-  ponent samples of V  #  1:n do not contain ties with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For any j ∈  {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, let G  #  1:n,j be the empirical d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' computed from the jth component  sample V  #  1j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , V  #  nj of V  #  1:n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then, R1:n,#  ij  = nG  #  1:n,j(V  #  ij ) is the rank of V  #  ij  among V  #  1j, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , V  #  nj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The (classical) empirical copula of V  #  1:n is thus given by  C  #  1:n(u) = 1  n  n  �  i=1  d  �  j=1  1  �  R1:n,#  ij  n  ≤ uj  �  ,  u ∈ [0, 1]d,  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4)  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  13  and the smooth analog of Cν  1:n for V  #  1:n is  C  #,ν  1:n (u) =  �  [0,1]d C  #  1:n(w)dν  V #  1:n  u  (w),  u ∈ [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5)  To state our asymptotic validity results, we consider independent copies  V  #,[1]  1:n , V  #,[2]  1:n , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of V  #  1:n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Let C  #,[i]  1:n (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' C  #,ν,[i]  1:n  ) be the version of C  #  1:n in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4)  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' C  #,ν  1:n in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5)) obtained from V  #,[i]  1:n , i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The following result can be regarded as an extension of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3 of  Kiriliouk, Segers and Tsukahara (2021) and is proven in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Assume that the random vectors in X 1:n are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=', and that  Conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then,  �  Cn(0, 1, ·), √n(C  #,[1]  1:n − C1:n),√n(C  #,[2]  1:n − C1:n)  �  ⇝  �  CC(0, 1, ·), C  [1]  C (0, 1, ·), C  [2]  C (0, 1, ·)  �  ,  �  Cν  n(0, 1, ·), √n(C  #,ν,[1]  1:n  − Cν  1:n),√n(C  #,ν,[2]  1:n  − Cν  1:n)  �  ⇝  �  CC(0, 1, ·), C  [1]  C (0, 1, ·), C  [2]  C (0, 1, ·)  �  in {ℓ∞([0, 1]d)}3, where Cn and Cν  n are defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9), respectively,  and C  [1]  C and C  [2]  C are independent copies of CC defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The first joint weak convergence in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5 establishes the  asymptotic validity of a smooth bootstrap for the (non-sequential) classical em-  pirical process while the second one provides a similar results for the smooth  empirical copula process Cν  n(0, 1, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' According to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 in B¨ucher and  Kojadinovic (2019), these two joint weak convergences are equivalent to simi-  lar joint weak convergences with B ≥ 2 bootstrap replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' In a further step,  the latter can be transferred to the “statistic level” using the continuous map-  ping theorem or the functional delta method, which could then be combined  with the results in Section 4 of B¨ucher and Kojadinovic (2019) to establish  the validity of bootstrap-based confidence intervals or tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Note also that,  from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 in B¨ucher and Kojadinovic (2019), the unconditional asymp-  totic validity results appearing in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5 are equivalent to possibly more  classical conditional results which rely, however, on a more subtle mode of con-  vergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For instance, the first claim can be equivalently informally stated as  “√n(C  #,[1]  1:n −C1:n) converges weakly to CC(0, 1, ·) in ℓ∞([0, 1]d) conditionally on  the data in probability”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=', Kosorok (2008, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3) or Appendix C  for a precise definition of that mode of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Finite-sample comparison of two smooth bootstraps  As already mentioned in the introduction, in their Monte Carlo experiments,  Kiriliouk, Segers and Tsukahara (2021) found the smooth bootstrap based on  the empirical beta copula CBin  1:n to be a competitive alternative to many other re-  sampling schemes (including the multiplier bootstrap to be studied in the forth-  coming section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Since the data-adaptive empirical copula CBetaB4  1:n  was found  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  14  Table 1  Coverage probabilities (cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=') and average lengths (ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=') of 95%-confidence intervals for  Kendall’s tau estimated from 1000 random samples of size n ∈ {20, 40, 80, 160} from the  bivariate Clayton or Gumbel–Hougaard copula with a Kendall’s tau of τ ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Each confidence interval was computed using 1000 smooth bootstrap samples drawn from  either CBin  1:n or CBetaB4  1:n  using Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Clayton  Gumbel–Hougaard  Bin  BetaB4  Bin  BetaB4  τ  n  cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='043  to outperform the empirical beta copula CBin  1:n in the experiments reported in  Kojadinovic and Yi (2022), it seems natural to empirically investigate how the  smooth bootstrap based on CBetaB4  1:n  compares to the smooth bootstrap based on  CBin  1:n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' To do so, we reproduced some of the experiments reported in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3 of Kiriliouk, Segers and Tsukahara (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  We first estimated coverage probabilities and average lengths of confidence  intervals of level 95% for Kendall’s tau from 1000 random samples of size n ∈  {20, 40, 80, 160} from the bivariate Clayton or Gumbel–Hougaard copula with  a Kendall’s tau of τ ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Each confidence interval was computed  using 1000 smooth bootstrap samples drawn from either CBin  1:n or CBetaB4  1:n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The  results are reported in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As one can see, under independence or moderate  dependence (τ ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5}), the estimated coverage probabilities are overall on  target and very similar for the two resampling schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The intervals obtained  using the smooth bootstrap based on CBetaB4  1:n  seem nonetheless to be slightly  shorter on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Under strong dependence (τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75) however, the estimated  coverage probabilities of the confidence intervals computed using the smooth  bootstrap based on CBin  1:n are substantially below the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='95 target value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The  results for τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9 actually show that the smooth bootstrap based on CBin  1:n is  unable to generate samples with such a very strong dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' While its results  are not perfect, the smooth bootstrap based on CBetaB4  1:n  copes much better with  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  15  Table 2  Coverage probabilities (cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=') and average lengths (ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=') of 95%-confidence intervals for the  parameter of a bivariate Frank copula estimated by maximum pseudo-likelihood from 1000  random samples of size n ∈ {20, 40, 80} from the bivariate Frank copula with a Kendall’s  tau of τ ∈ {−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content=' Each confidence interval was computed using  1000 smooth bootstrap samples drawn from either CBin  1:n or CBetaB4  1:n  using Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='319  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='023  strong dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' This is likely to be due to the modification of the “shape” of  the underlying smoothing distributions using the empirical beta copula in the  expression of CBetaB4  1:n  as can be deduced from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  In a second experiment, following Kiriliouk, Segers and Tsukahara (2021,  Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3) we estimated coverage probabilities and average lengths of 95%-  confidence intervals for the parameter of a bivariate Frank copula estimated by  maximum pseudo-likelihood (see Genest, Ghoudi and Rivest, 1995) from 1000  random samples of size n ∈ {20, 40, 80, 160} from the bivariate Frank copula  with a Kendall’s tau of τ ∈ {−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Again, each  confidence interval was computed using 1000 smooth bootstrap samples drawn  from either CBin  1:n or CBetaB4  1:n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The results are reported in Table 2 and the main  conclusion is qualitatively the same as for the previous experiment: the smooth  bootstrap based on CBetaB4  1:n  copes much better with strong dependence than the  smooth bootstrap based on CBin  1:n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  16  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Smooth sequential dependent multiplier bootstraps in the time  series case  The smooth bootstrap investigated in the previous section can only be used in  the case of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Fortunately, the multiplier bootstrap, one of the  most popular approaches for bootstrapping functionals of the classical empirical  copula, can be employed in the time series setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' In this section, after provid-  ing some intuitions and defining multiplier sequences, we recall the non-smooth  sequential dependent multiplier bootstrap studied in B¨ucher and Kojadinovic  (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' We next propose smooth extensions of the latter, provide asymptotic  validity results and compare the finite-sample performance of three (smooth)  multiplier bootstraps for approximating three (smooth) empirical copula pro-  cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Finally, as an application, we consider a smooth version (based on the  empirical beta copula and corresponding smooth multiplier bootstrap replicates)  of the test for change-point detection developed in B¨ucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (2014) and we  compare its finite-sample performance to that of its non-smooth counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Main intuition and existing work  As mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2, Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11 holds under strong mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Given a  stationary time series (Yi)i∈Z, denote by F k  j the σ-field generated by (Yi)j≤i≤k,  j, k ∈ Z∪{−∞, +∞}, and recall that the strong mixing coefficients correspond-  ing to the stationary sequence (Yi)i∈Z are then defined by  αY  r =  sup  A∈F 0  −∞,B∈F +∞  r  ��P(A ∩ B) − P(A)P(B)  ��,  r ∈ N, r > 0,  and that the sequence (Yi)i∈Z is said to be strongly mixing if αY  r → 0 as r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  From B¨ucher (2015), Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11 holds if the strong mixing coefficients  of the time series (Xi)i∈Z satisfy αX  r = O(r−a) with a > 1 as r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' In that  case, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='12 suggests that, in order to bootstrap the classical sequential  empirical copula process Cn in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8) in an asymptotically valid way, it suffices to  bootstrap the process ˜Cn in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The latter could be done by bootstrapping  Bn in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10) and estimating the first-order partial derivatives ˙Cj, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d},  of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Such an approach was initially proposed in the independent non-sequential  setting by Scaillet (2005) and R´emillard and Scaillet (2009) who used a multi-  plier bootstrap in the spirit of van der Vaart and Wellner (2000, Chapter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9)  to resample Bn, and finite-differencing to estimate the partial derivatives ˙Cj,  j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' This resampling scheme was extended to the time series sequen-  tial setting in B¨ucher and Kojadinovic (2016) and B¨ucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' and dependent multiplier sequences  In the case of independent observations, multiplier bootstraps are based on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  multiplier sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' We say that a sequence of random variables (ξi,n)i∈Z is an  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' multiplier sequence if:  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  17  (M0) (ξi,n)i∈Z is i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=', independent of X 1:n, with distribution not changing  with n, having mean 0, variance 1, and being such that  � ∞  0 {P(|ξ0,n| >  x)}1/2dx < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The time series extension of the multiplier bootstrap relies on the notion of  dependent multiplier sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The key idea due to B¨uhlmann (1993) is to re-  place i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' multipliers by suitably serially dependent multipliers that will capture  the serial dependence in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' We say that a sequence of random variables  (ξi,n)i∈Z is a dependent multiplier sequence if:  (M1) The sequence of random variables (ξi,n)i∈Z is stationary with E(ξ0,n) = 0,  E(ξ2  0,n) = 1 and supn≥1 E(|ξ0,n|γ) < ∞ for all γ ≥ 1, and is independent  of the available sample X 1:n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (M2) There exists a sequence ℓn → ∞ of strictly positive constants such that  ℓn = o(n) and the sequence (ξi,n)i∈Z is ℓn-dependent, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=', ξi,n is indepen-  dent of ξi+h,n for all h > ℓn and i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (M3) There exists a function ϕ : R → [0, 1], symmetric around 0, continuous at  0, satisfying ϕ(0) = 1 and ϕ(x) = 0 for all |x| > 1 such that E(ξ0,nξh,n) =  ϕ(h/ℓn) for all h ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  As shall become clearer for instance from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) or (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) below, the bandwidth  parameter ℓn defined in (M2) plays a role similar to that of the block length  in the block bootstrap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' In practice, for the non-smooth sequential dependent  multiplier bootstrap to be presented in the forthcoming section, its value can be  chosen in a data-driven way using the approach described in detail in B¨ucher  and Kojadinovic (2016, Section 5);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' see also Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The latter reference also  describes in detail ways to generate dependent multiplier sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Non-smooth sequential dependent multiplier replicates  Let (ξ  [1]  i,n)i∈Z, (ξ  [2]  i,n)i∈Z,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , be independent copies of the same multiplier se-  quence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Two different multiplier bootstrap replicates of the process Bn in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10)  were proposed in B¨ucher and Kojadinovic (2016) and B¨ucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (2014), re-  spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For any b ∈ N, (s, t) ∈ Λ and u ∈ [0, 1]d, they are defined by  ˆB  [b]  n (s, t, u) =  1  √n  ⌊nt⌋  �  i=⌊ns⌋+1  ξ  [b]  i,n  �  1( ˆU 1:n  i  ≤ u) − C1:n(u)  �  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1)  and  ˇB  [b]  n (s, t, u) =  1  √n  ⌊nt⌋  �  i=⌊ns⌋+1  ξ  [b]  i,n  �  1( ˆU ⌊ns⌋+1:⌊nt⌋  i  ≤ u) − C⌊ns⌋+1:⌊nt⌋(u)  �  , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2)  respectively, where C1:n and C⌊ns⌋+1:⌊nt⌋ are generically defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) and  with the convention that ˆB[b]  n (s, t, ·) = ˇB[b]  n (s, t, ·) = 0 if ⌊nt⌋ − ⌊ns⌋ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  18  In order to define multiplier bootstrap replicates of ˜Cn in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='12), it is fur-  ther necessary to estimate the unknown first-order partial derivatives ˙Cj, j ∈  {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' In the rest of this section, ˙Cj,k:l will denote an estimator of ˙Cj  based on a stretch X k:l = (Xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , Xl) of observations, 1 ≤ k ≤ l ≤ n, with  the convention that ˙Cj,k:l = 0 if k > l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then, following B¨ucher and Kojadinovic  (2016) and B¨ucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (2014), we consider two types of multiplier bootstrap  replicates of Cn in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For any b ∈ N, (s, t) ∈ Λ and u ∈ [0, 1]d, these are  defined by  ˆC  [b]  n (s, t, u) = ˆB  [b]  n (s, t, u) −  d  �  j=1  ˙Cj,1:n(u) ˆB  [b]  n (s, t, u(j))  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3)  and  ˇC  [b]  n (s, t, u) = ˇB  [b]  n (s, t, u) −  d  �  j=1  ˙Cj,⌊ns⌋+1:⌊nt⌋(u) ˇB  [b]  n (s, t, u(j)),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4)  respectively, where ˆB[b]  n (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' ˇB[b]  n ) is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Clearly,  both types of replicates coincide in a non-sequential setting as ˆC[b]  n (0, 1, ·) =  ˇC[b]  n (0, 1, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As far as the estimators of the partial derivatives are concerned, it  is expected that the more accurate they are, the better the approximation of  the “sampling distribution” of Cn by the multiplier replicates will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The latter  aspect will be discussed in detail in Section 5, where two broad classes of smooth  estimators will be introduced and studied both theoretically and empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Smooth sequential dependent multiplier replicates  We now consider a similar construction but based on smooth analogs of ˆB[b]  n  in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) and ˇB[b]  n in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Specifically, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13 suggests that, to bootstrap  Cν  n in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9), a first step is to bootstrap ˜Bν  n in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='14) or ¯Bν  n in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' By analogy  with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5), natural smooth analogs of ˆB[b]  n and ˇB[b]  n could be defined,  for any b ∈ N, (s, t) ∈ Λ and u ∈ [0, 1]d, by  ˆB  [b],ν  n  (s, t, u) =  �  [0,1]d  ˆB  [b]  n (s, t, w)dν  X 1:n  u  (w)  =  1  √n  ⌊nt⌋  �  i=⌊ns⌋+1  ξ  [b]  i,n  �  K  X 1:n  R1:n  i  (u) − Cν  1:n(u)  �  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5)  and  ˇB  [b],ν  n  (s, t, u) =  �  [0,1]d  ˇB  [b]  n (s, t, w)dν  X ⌊ns⌋+1:⌊nt⌋  u  (w)  =  1  √n  ⌊nt⌋  �  i=⌊ns⌋+1  ξ  [b]  i,n  �  K  X ⌊ns⌋+1:⌊nt⌋  R⌊ns⌋+1:⌊nt⌋  i  (u) − Cν  ⌊ns⌋+1:⌊nt⌋(u)  �  ,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6)  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  19  respectively, where K  X 1:n  R1:n  i  and K  X ⌊ns⌋+1:⌊nt⌋  R⌊ns⌋+1:⌊nt⌋  i  are defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Combining  these ingredients with estimators of the unknown partial derivatives of C, as  smooth analogs of ˆC[b]  n in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3) and ˇC[b]  n in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4), we obtain  ˆC  [b],ν  n  (s, t, u) = ˆB  [b],ν  n  (s, t, u) −  d  �  j=1  ˙Cj,1:n(u) ˆB  [b],ν  n  (s, t, u(j)),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7)  and  ˇC  [b],ν  n  (s, t, u) = ˇB  [b],ν  n  (s, t, u) −  d  �  j=1  ˙Cj,⌊ns⌋+1:⌊nt⌋(u) ˇB  [b],ν  n  (s, t, u(j)),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8)  respectively, for b ∈ N, (s, t) ∈ Λ and u ∈ [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  To establish the asymptotic validity of these smooth multiplier bootstrap  replicates, it will suffice that the partial derivative estimators satisfy the follow-  ing rather natural mild condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Condition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 (Bounded and weakly consistent partial derivative estimators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  There exists a constant ζ > 0 such that, for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d} and n ∈ N,  sup  (s,t,u)∈Λ×[0,1]d  ��� ˙Cj,⌊ns⌋+1:⌊nt⌋(u)  ��� ≤ ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Furthermore, for any δ ∈ (0, 1), ε ∈ (0, 1/2) and j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d},  sup  (s,t)∈Λ  t−s≥δ  sup  u∈[0,1]d  uj∈[ε,1−ε]  ��� ˙Cj,⌊ns⌋+1:⌊nt⌋(u) − ˙Cj(u)  ��� = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  In addition, following B¨ucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (2014), we impose the following condition  on the observations and the underlying multiplier sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Condition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 (Strong mixing and multiplier conditions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' One of the following  two conditions holds:  (i) The random vectors in X 1:n are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' and (ξ  [1]  i,n)i∈Z, (ξ  [2]  i,n)i∈Z,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' are in-  dependent copies of a multiplier sequence satisfying (M0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (ii) The stretch X 1:n is drawn from a stationary sequence (Xi)i∈Z whose  strong mixing coefficients satisfy αX  r = O(r−a) for some a > 3 + 3d/2 as  r → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Furthermore, (ξ  [1]  i,n)i∈Z, (ξ  [2]  i,n)i∈Z, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' are independent copies of a  dependent multiplier sequence satisfying (M1)–(M3) with ℓn = O(n1/2−γ)  for some 0 < γ < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The following result is proven in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3 (Asymptotic validity of the smooth dependent multiplier boot-  straps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Under Conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2, for any b ∈ N, there holds  sup  (s,t,u)∈Λ×[0,1]d  ���ˆC  [b],ν  n  (s, t, u) − ˆC  [b]  n (s, t, u)  ��� = oP(1),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9)  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  20  sup  (s,t,u)∈Λ×[0,1]d  ��ˇC  [b],ν  n  (s, t, u) − ˇC  [b]  n (s, t, u)  �� = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10)  Furthermore,  (Cν  n, ˆC  [1],ν  n  , ˆC  [2],ν  n  ) ⇝ (CC, C  [1]  C , C  [2]  C ),  (Cν  n, ˇC  [1],ν  n  , ˇC  [2],ν  n  ) ⇝ (CC, C  [1]  C , C  [2]  C )  in {ℓ∞(Λ × [0, 1]d)}3, where C  [1]  C and C  [2]  C are independent copies of CC defined  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Finite-sample comparison of three multiplier bootstraps  From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10, we know that, under Conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9, the classi-  cal sequential empirical copula process Cn in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8) and the smooth sequential  empirical copula process Cν  n in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9) are asymptotically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' In a related  way, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3 provides conditions under which corresponding multiplier  and smooth multiplier replicates are asymptotically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Although one  expects that Cν  n is probably best resampled using multiplier replicates con-  structed with the same smoothing distributions, that is, with ˆC[b],ν  n  in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7)  or ˇC[b],ν  n  in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8), we have no asymptotic results to support this (see also Re-  mark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Indeed, given that all versions of multiplier replicates are asymp-  totically equivalent, it may well be that, for instance, in some cases, classical  (non-smooth) multiplier replicates are equivalent or even preferable to smooth  multiplier replicates when it comes to resampling Cν  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' It is the aim of this sec-  tion to study this empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For simplicity, we restrict our investigations to a  non-sequential setting and independent observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Specifically, we designed experiments to study which multiplier replicates are  best suited to estimate certain functionals of the three (non-sequential) empirical  copula processes defined, for any u ∈ [0, 1]d, by  C  Dirac  n  (u) = √n{C1:n(u) − C(u)} = Cn(0, 1, u),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11)  C  Bin  n (u) = √n{C  Bin  1:n(u) − C(u)},  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='12)  C  BetaB4  n  (u) = √n{C  BetaB4  1:n  (u) − C(u)},  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13)  where  Cn is the classical (non-smooth) sequential empirical copula process de-  fined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8),  CBin  1:n is the empirical beta copula in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) (which is obtained by considering  smoothing distributions with scaled binomial margins and independence  copula as explained in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1),  CBetaB4  1:n  is the version of Cν  1:n introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 obtained by con-  sidering smoothing distributions with scaled beta-binomial margins and  survival copula the empirical beta copula CBin  1:n, and found to have the  best finite-sample performance in the numerical experiments of Kojadi-  novic and Yi (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  21  As already mentioned, since we are in a non-sequential setting, the two generic  multiplier replicates defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8) coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' To approximate the  “sampling distributions” of the three empirical copula processes defined above,  we considered as candidate bootstraps the multiplier replicates defined using  the same smoothing distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' They will be denoted by ˆC[b],Dirac  n  , ˆC[b],Bin  n  and  ˆC[b],BetaB4  n  , b ∈ N, respectively, as we continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' To only investigate the effect  of the choice of the smoothing distributions involved in the definition of ˆB[b],ν  n  in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5), all three multiplier replicates were computed using the true partial  derivative ˙Cj, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Furthermore, since we restricted our experiments  to independent observations, all the multiplier replicates were based on i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  multiplier sequences defined in (M0) in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Following B¨ucher and Dette  (2010), these sequences were simply taken to be random samples drawn from  the uniform distribution on {−1, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  For the design of our experiments, we followed again B¨ucher and Dette  (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' First, for d = 2, we assessed how well the covariances of the empiri-  cal processes CDirac  n  in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11), CBin  n  in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='12) and CBetaB4  n  in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13) at the points  P = {(i/3, j/3) : i, j = 1, 2} can be approximated using the three possible mul-  tiplier bootstrap replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For each target empirical copula process, we began  by precisely estimating its covariance at the points in P from 100 000 indepen-  dent samples of size n ∈ {10, 20, 40, 80} drawn from a bivariate copula C with a  Kendall’s tau of τ ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For C, we considered either the Clayton  or the Gumbel–Hougaard copula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Next, for each considered combination of C,  n, τ, target process and multiplier process, we generated 1000 samples from C,  and, for each sample, we computed B = 1000 multiplier bootstrap replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  These B = 1000 replicates were used to obtain one estimate of the covariance  of the target process at the points in P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The results when C is the Clayton copula with a Kendall’s tau of τ ∈  {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75} are reported in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The first (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' second, third) col-  umn of graphs reports the average of the empirical mean square errors (MSEs)  ×104 of the three candidate multiplier estimators of the covariance of CDirac  n  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' CBin  n , CBetaB4  n  ) at the points in P against the sample size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Each row of  graphs corresponds to a different value of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' In the top-left panel for instance, the  solid (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' dashed, dotted) curve gives the average MSE when the covariance  of CDirac  n  is estimated using ˆC[b],Dirac  n  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' ˆC[b],Bin  n  , ˆC[b],BetaB4  n  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  As one can see, reassuringly, all the curves are globally decreasing, confirm-  ing that, for each target process, the bootstrap approximations improve as n  increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' A more careful inspection reveals that, in almost all settings, it is  the multiplier bootstrap constructed with the same smoothing distributions as  the target process that leads to the best estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' It is actually only when  CBetaB4  n  is the target process that covariance estimations based on ˆC[b],Bin  n  are  sometimes better than estimations based on ˆC[b],BetaB4  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' This happens mostly for  small n and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Results for the Gumbel–Hougaard copula (not reported) are not  qualitatively different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  22  10  20  30  40  50  60  80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' cop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' / d = 2 / tau = 0  n  average of MSE of cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Dirac/Dirac  Dirac/Bin  Dirac/BetaB4  10  20  30  40  50  60  80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='0  indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' cop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' / d = 2 / tau = 0  n  average of MSE of cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Bin/Dirac  Bin/Bin  Bin/BetaB4  10  20  30  40  50  60  80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' cop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' / d = 2 / tau = 0  n  average of MSE of cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  BetaB4/Dirac  BetaB4/Bin  BetaB4/BetaB4  10  20  30  40  50  60  80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  Clayton / d = 2 / tau = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25  n  average of MSE of cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Dirac/Dirac  Dirac/Bin  Dirac/BetaB4  10  20  30  40  50  60  80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  Clayton / d = 2 / tau = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25  n  average of MSE of cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Bin/Dirac  Bin/Bin  Bin/BetaB4  10  20  30  40  50  60  80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  Clayton / d = 2 / tau = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25  n  average of MSE of cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  BetaB4/Dirac  BetaB4/Bin  BetaB4/BetaB4  10  20  30  40  50  60  80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  Clayton / d = 2 / tau = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  n  average of MSE of cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Dirac/Dirac  Dirac/Bin  Dirac/BetaB4  10  20  30  40  50  60  80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  Clayton / d = 2 / tau = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  n  average of MSE of cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Bin/Dirac  Bin/Bin  Bin/BetaB4  10  20  30  40  50  60  80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  Clayton / d = 2 / tau = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  n  average of MSE of cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  BetaB4/Dirac  BetaB4/Bin  BetaB4/BetaB4  10  20  30  40  50  60  80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  Clayton / d = 2 / tau = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75  n  average of MSE of cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Dirac/Dirac  Dirac/Bin  Dirac/BetaB4  10  20  30  40  50  60  80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='00  Clayton / d = 2 / tau = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75  n  average of MSE of cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Bin/Dirac  Bin/Bin  Bin/BetaB4  10  20  30  40  50  60  80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='50  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='00  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='00  Clayton / d = 2 / tau = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75  n  average of MSE of cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  BetaB4/Dirac  BetaB4/Bin  BetaB4/BetaB4  Fig 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For observations generated from the bivariate Clayton copula with a Kendall’s tau of  τ ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75} and for each combination of target and multiplier process, average of  the empirical MSEs (×104) of the bootstrap estimators of the covariance of the target process  at the points in P against the sample size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The legend “Dirac/Bin” for instance refers to  the situation when the target process is CDirac  n  and the multiplier process is ˆC  [b],Bin  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  23  In a second experiment, we assessed how well high quantiles of  KS(fn) =  sup  u∈[0,1]d |fn(u)|  and  CvM(fn) =  �  [0,1]d{fn(u)}2du  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='14)  for d ∈ {2, 3} and fn ∈ {CDirac  n  , CBin  n , CBetaB4  n  } can be estimated by the three can-  didate multiplier bootstraps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' From a practical perspective, the integral in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='14)  was approximated by a mean using a uniform grid on (0, 1)d of size 102 when  d = 2 and 53 when d = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For d ∈ {2, 3}, C the Clayton or the Gumbel-Hougaard  copula whose bivariate margins have a Kendall’s tau of τ ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75}  and n ∈ {10, 20, 40, 80}, the 90% and 95%-quantiles of CvM(fn) were first  precisely estimated from 100 000 independent samples of size n drawn from C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Next, for each combination of d, C, n, τ, target process and multiplier pro-  cess, we generated 1000 samples from C and, for each sample, we computed  B = 1000 multiplier bootstrap replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' These B = 1000 replicates were used  to obtain one estimate of each of the target quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Following Kojadinovic  and Stemikovskaya (2019), all such estimations were carried out using centered  replicates of fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' When fn = CDirac  n  for instance, this amounts to using, for any  u ∈ [0, 1]d and b ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , B},  ˆC  [b],Dirac  n  (u) − 1  B  B  �  b=1  ˆC  [b],Dirac  n  (u),  instead of ˆC[b],Dirac  n  (u) = ˆC[b]  n (0, 1, u) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The centered versions of the other  replicates are defined analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The rationale behind centering is that the  replicates, whatever their type, can be regarded as computable approximations  of the limiting centered Gaussian process CC(0, 1, ·) in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' see, for instance,  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Note that the use of centered replicates was found to always lead to  better finite-sample performance in the related Monte Carlo experiments carried  out in Kojadinovic and Stemikovskaya (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Its use is however irrelevant in the  previous covariance estimation experiment given the formula of the empirical  covariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The results for the 95%-quantiles of the Kolmogorov–Smirnov functionals  when C is the trivariate Gumbel–Hougaard are reported in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The con-  clusions are overall similar to those obtained after the first experiment:  The 95%-quantile of the Kolmogorov–Smirnov functional of CDirac  n  is al-  ways best estimated using the corresponding empirical quantile of the  same functional of ˆC[b],Dirac  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  When the target process is CBin  n , the best results are obtained when the  multiplier process is ˆC[b],Bin  n  , except in the case of strongly dependent ob-  servations in which case, for the sample sizes under consideration, ˆC[b],Dirac  n  gives better estimations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  When the target process is CBetaB4  n  , it is only when n reaches 40 or 80 that  the best estimations are obtained using ˆC[b],BetaB4  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For smaller n, the use  of ˆC[b],Bin  n  gives better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  24  10  20  30  40  50  60  80  5  10  20  50  200  500  2000  indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' cop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' / d = 3 / tau = 0  n  MSE of the est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='95−quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of KS  Dirac/Dirac  Dirac/Bin  Dirac/BetaB4  10  20  30  40  50  60  80  5  10  20  50  100  200  500  indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' cop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' / d = 3 / tau = 0  n  MSE of the est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='95−quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of KS  Bin/Dirac  Bin/Bin  Bin/BetaB4  10  20  30  40  50  60  80  10  20  50  100  200  500  1000  indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' cop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' / d = 3 / tau = 0  n  MSE of the est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='95−quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of KS  BetaB4/Dirac  BetaB4/Bin  BetaB4/BetaB4  10  20  30  40  50  60  80  5  10  20  50  200  500  2000  Gumbel−Hougaard / d = 3 / tau = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25  n  MSE of the est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='95−quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of KS  Dirac/Dirac  Dirac/Bin  Dirac/BetaB4  10  20  30  40  50  60  80  5  10  20  50  100  200  500  Gumbel−Hougaard / d = 3 / tau = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25  n  MSE of the est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='95−quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of KS  Bin/Dirac  Bin/Bin  Bin/BetaB4  10  20  30  40  50  60  80  20  50  100  200  500  Gumbel−Hougaard / d = 3 / tau = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25  n  MSE of the est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='95−quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of KS  BetaB4/Dirac  BetaB4/Bin  BetaB4/BetaB4  10  20  30  40  50  60  80  20  50  100  500  2000  Gumbel−Hougaard / d = 3 / tau = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  n  MSE of the est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='95−quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of KS  Dirac/Dirac  Dirac/Bin  Dirac/BetaB4  10  20  30  40  50  60  80  20  50  100  200  500  1000  Gumbel−Hougaard / d = 3 / tau = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  n  MSE of the est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='95−quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of KS  Bin/Dirac  Bin/Bin  Bin/BetaB4  10  20  30  40  50  60  80  20  50  100  200  500  1000  Gumbel−Hougaard / d = 3 / tau = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  n  MSE of the est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='95−quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of KS  BetaB4/Dirac  BetaB4/Bin  BetaB4/BetaB4  10  20  30  40  50  60  80  20  50  100  200  500  2000  Gumbel−Hougaard / d = 3 / tau = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75  n  MSE of the est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='95−quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of KS  Dirac/Dirac  Dirac/Bin  Dirac/BetaB4  10  20  30  40  50  60  80  20  50  100  200  500  1000  Gumbel−Hougaard / d = 3 / tau = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75  n  MSE of the est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='95−quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of KS  Bin/Dirac  Bin/Bin  Bin/BetaB4  10  20  30  40  50  60  80  50  100  200  500  Gumbel−Hougaard / d = 3 / tau = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75  n  MSE of the est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='95−quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of KS  BetaB4/Dirac  BetaB4/Bin  BetaB4/BetaB4  Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  For observations generated from the trivariate Gumbel–Hougaard copula whose bi-  variate margins have a Kendall’s tau of τ ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75}, empirical MSE (×104)  of the three candidate multiplier estimators of high quantiles of KS(fn) in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='14) for  fn ∈ {CDirac  n  , CBin  n , CBetaB4  n  } against the sample size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The legend “Dirac/Bin” for instance  refers to the situation when the target process is CDirac  n  and the multiplier process is ˆC  [b],Bin  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  25  Results for the Clayton copula, 90%-quantiles, dimension d = 2 or Cram´er–von  Mises functionals (not reported) are not qualitatively different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The previous experiments confirm that it seems meaningful to resample Cν  n  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9) using multiplier replicates constructed with the same smoothing distri-  butions, that is, with ˆC[b],ν  n  in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7) or ˇC[b],ν  n  in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8), although this choice may  not be optimal in certain cases when n is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Application to change-point detection  A natural application area for the smooth sequential empirical copula process Cν  n  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9) is that of change-point detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' To illustrate the possible advantages  coming from the use of smooth empirical copulas in inference procedures, we  first briefly explain in this section how the previous derivations can be used to  obtain a smooth version of the test proposed in B¨ucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (2014) for detecting  changes in the cross-sectional dependence of multivariate time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' We then  reproduce some of the experiments of B¨ucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (2014) to compare the (non-  smooth) test proposed therein with its smooth version based on the empirical  beta copula and on corresponding smooth bootstrap replicates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Note that we  did not consider the use of the alternative data-adaptive smoothing distributions  considered in Kojadinovic and Yi (2022) and leading to the estimator CBetaB4  k:l  because they incur a substantially higher computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The null hypothesis of such tests is that X 1:n is a stretch from a stationary  time series (of continuous random vectors) and their aim is to be particularly  sensitive to the alternative hypothesis  H1 : ∃ distinct C1, C2 and k⋆ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , n − 1} such that  X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , Xk⋆ have copula C1 and Xk⋆+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , Xn have copula C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='15)  The ingredients of the smooth version of the test can be obtained mutatis  mutandis from B¨ucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Specifically, we consider as test statistic  the maximally selected Cram´er–von Mises functional defined by  Sν  n = sup  s∈[0,1]  �  [0,1]d {Dν  n(s, u)}2 dC1:n(u),  where  Dν  n(s, u) = √nλn(0, s)λn(s, 1){Cν  1:⌊ns⌋(u)−Cν  ⌊ns⌋+1:n(u)},  (s, u) ∈ [0, 1]d+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  As one can see, the latter involves comparisons of (smooth) empirical copulas  computed from subsamples of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Noticing that, under the null,  Dν  n(s, u) = λn(s, 1) Cν  n(0, s, u) − λn(0, s)Cν  n(s, 1, u),  (s, u) ∈ [0, 1]d+1,  possible multiplier bootstrap replicates for Sν  n can be defined either by  ˆS  [b],ν  n  = sup  s∈[0,1]  �  [0,1]d{ˆD  [b],ν  n  (s, u)}2dC1:n(u),  b ∈ N,  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  26  or by  ˇS  [b],ν  n  = sup  s∈[0,1]  �  [0,1]d{ˇD  [b],ν  n  (s, u)}2dC1:n(u),  b ∈ N,  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='16)  where, for any (s, u) ∈ [0, 1]d+1,  ˆD  [b],ν  n  (s, u) = λn(s, 1) ˆC  [b],ν  n  (0, s, u) − λn(0, s) ˆC  [b],ν  n  (s, 1, u),  ˇD  [b],ν  n  (s, u) = λn(s, 1) ˇC  [b],ν  n  (0, s, u) − λn(0, s) ˇC  [b],ν  n  (s, 1, u),  with ˆC[b],ν  n  and ˇC[b],ν  n  defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Note that, in the  expressions of the multiplier replicates of Cν  n, as estimators of the first-order  partial derivatives of the copula, we use the “truncated” finite-difference based  estimators defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8) of the forthcoming section with bandwidths h = h′ =  min{(l − k + 1)−1/2, 1/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As we will see from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, the latter can  satisfy Condition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Finally, as in B¨ucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (2014), approximate p-values  for Sν  n can be computed via either  1  B  B  �  b=1  1  �  ˆS  [b],ν  n  ≥ Sν  n  �  or  1  B  B  �  b=1  1  �  ˇS  [b],ν  n  ≥ Sν  n  �  ,  for some large integer B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Theoretical results confirming that the above way of  proceeding is asymptotically valid under the null can be obtained by starting  from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3, proceeding as in B¨ucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (2014) and finally using results  stated in Section 4 of B¨ucher and Kojadinovic (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  If, for any m ∈ N, the underlying smoothing distributions νx  u, x ∈ (Rd)m,  u ∈ [0, 1]d, are Dirac measures at u, the previous ingredients are non-smooth  and the resulting test coincides exactly with the test studied in B¨ucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The test statistic will naturally be denoted by SDirac  n  in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As  alternative smoothing distributions, we considered those leading to the empirical  beta copula and specified in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 as well as at the end of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The resulting statistic will then naturally be denoted by SBin  n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  To compare the test based on SBin  n  to the test based on SDirac  n  , we consid-  ered experiments similar to those reported in Section 5 of B¨ucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Both tests were carried out at the 5% significance level using replicates of the  form (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='16) as these seemed to lead to better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The dependent multiplier  sequences necessary to carry out the tests were generated as explained in the last  paragraph of Appendix C of B¨ucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The value of the bandwidth  parameter ℓn appearing in (M2) and (M3) in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 was chosen using the  procedure described in B¨ucher and Kojadinovic (2016, Section 5) (although this  way of proceeding may not be “optimal” for the smooth multiplier bootstrap  replicates).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  As a first experiment, we estimated the percentages of rejection of the null  hypothesis of stationarity for data generated under the null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As data generating  model, we used a bivariate AR(1) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Specifically, let Ui, i ∈ {−100, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , n},  be a bivariate i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' sample from a copula C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then, set ϵi = (Φ−1(Ui1), Φ−1(Ui2)),  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  27  Table 3  Percentages of rejection of the null hypothesis of stationarity computed from 1000 samples  of size n ∈ {25, 50, 100, 200} generated as explained in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6, where C is the bivariate  Frank copula with a Kendall’s tau of τ ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='33, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='66}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  n = 25  n = 50  n = 100  n = 200  β  τ  SDirac  n  SBin  n  SDirac  n  SBin  n  SDirac  n  SBin  n  SDirac  n  SBin  n  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='00  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='33  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='6  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='66  21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='8  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='2  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='4  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='66  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='00  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='2  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='33  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9  23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='66  27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6  where Φ is the d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of the standard normal distribution, and X−100 = ϵ−100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Finally, for any j ∈ {1, 2} and i ∈ {−99, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , n}, compute recursively  Xij = βXi−1,j + ϵij,  where the first 100 observations are used as a burn-out sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  We considered n ∈ {25, 50, 100, 200}, C to be bivariate Frank copula with  a Kendall’s tau of τ ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='33, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='66} and β ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The corresponding  rejection percentages are reported in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As one can see, both tests appear  to hold their level reasonably well when n ∈ {100, 200}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The tests should however  clearly not be used when n = 25 but might be employed when n = 50 in the  case of weakly serially dependent data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  As a second experiment, we estimated rejection percentages of the null hy-  pothesis of stationarity for data generated under H1 in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' To do so, we  considered a similar data generating model as in the first experiment except  that the Ui’s for i ∈ {−100, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , k⋆} are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' from a copula C1 while the Ui’s  for i ∈ {k⋆ + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , n} are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' from a copula C2 ̸= C1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Following B¨ucher  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (2014), we set k⋆ = ⌊nt⌋ with t ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5} and considered n ∈  {50, 100, 200}, C1 the bivariate Frank copula with a Kendall’s tau 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 and C2  the bivariate Frank copula with a Kendall’s tau in {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The results are  reported in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As one can see, the test based on SBin  n  appears overall to  be more powerful than the one based on SDirac  n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The largest differences in power  tend to occur for τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6 and t ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25} which corresponds to the situation  when the test statistic should be the largest because of a difference between an  empirical copula computed from a small number of observations (approximately  ⌊nt⌋) and an empirical copula computed from the remaining observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' While  one cannot conclude that smooth change-point detection tests such as the one  based on SBin  n  will be more powerful than the non-smooth test based on SDirac  n  in all situations, the obtained results confirm in part the intuition that smooth  tests might be more sensitive to changes at the beginning or at the end of the  data sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  28  Table 4  Percentages of rejection of the null hypothesis of stationarity computed from 1000 samples  of size n ∈ {50, 100, 200} generated under H1 as explained in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6, where k⋆ = ⌊nt⌋,  C1 and C2 are both bivariate Frank copulas such that C1 has a Kendall’s tau of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 and C2  a Kendall’s tau of τ ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  β = 0  β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3  τ  n  t  SDirac  n  SBin  n  SDirac  n  SBin  n  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4  50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='50  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25  16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9  19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3  14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8  17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='50  26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6  28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8  22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content=' Estimators of the first-order partial derivatives of the copula  The multiplier bootstrap replicates defined in the previous section all depend  on the choice of estimators of the first-order partial derivatives of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For asymp-  totic reasons, the latter were required to satisfy Condition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Obviously, the  more accurate such estimators, the better we can expect the multiplier boot-  straps to behave, whether they involve smoothing or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' After recalling existing  definitions of such estimators based on finite differences of the classical empir-  ical copula, we define two related classes of smooth estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then, upon an  appropriate choice of the underlying bandwidth parameters, we establish their  weak consistency in a sequential setting which implies that many of the con-  sidered estimators satisfy Condition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' In the last subsection, we report the  results of bivariate and trivariate Monte Carlo experiments comparing selected  estimators in terms of integrated mean squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Note that, as already mentioned in the introduction, the results of this sec-  tion can be of independent interest since, as discussed for instance in Janssen,  Swanepoel and Veraverbeke (2016), estimators of the first-order partial deriva-  tives of a copula have applications in mean and quantile regression as they lead  to estimators of the conditional distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' In particular, as we shall  see, several estimators considered in our Monte Carlo experiments display a bet-  ter finite-sample performance than the Bernstein estimator studied in Janssen,  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  29  Swanepoel and Veraverbeke (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Estimators based on finite differences of the empirical copula  As already mentioned in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3, in their seminal work on the multiplier  bootstrap for the classical empirical copula process, R´emillard and Scaillet  (2009) considered estimators of the first-order partial derivatives ˙Cj, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d},  of C based on finite-differences of the empirical copula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' In a sequential context,  given a stretch X k:l = (Xk, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , Xl), 1 ≤ k ≤ l ≤ n, of observations and two  bandwidth parameters h and h′ in [0, 1/2] such that h + h′ > 0, a slightly more  general definition of the aforementioned estimators is  ˙C  ∇  j,k:l,h,h′(u) = Ck:l{(u + hej) ∧ 1} − Ck:l{(u − h′ej) ∨ 0}  h + h′  ,  u ∈ [0, 1]d,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1)  where ej is the jth vector of the canonical basis of Rd, 0 = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , 0), 1 =  (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , 1) ∈ Rd, ∧ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' ∨) denotes the minimum (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' maximum) component-  wise operator and Ck:l is the classical empirical copula of X k:l defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The symbol ∇ indicates that the estimators are based on finite-differences of  Ck:l with “right” (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' “left”) bandwidth h (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' h′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  In order to reduce the bias of the previous estimator for evaluation points  u ∈ [0, 1] with uj ∈ [0, h′) ∪ (1 − h, 1], Kojadinovic, Segers and Yan (2011)  considered the following minor variation of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1):  ˙C  ∆  j,k:l,h,h′(u) = Ck:l{(u + hej) ∧ 1} − Ck:l{(u − h′ej) ∨ 0}  (uj + h) ∧ 1 − (uj − h′) ∨ 0  ,  u ∈ [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2)  Note the use of the symbol ∆ still referring to finite-differences but upside-down  compared to ∇ to distinguish (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  As is well known, in general, ˙Cj exists almost everywhere on [0, 1]d and, for  those u ∈ [0, 1]d for which it exists, 0 ≤  ˙Cj(u) ≤ 1 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=', Nelsen, 2006,  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' A natural modification of the estimators ˙C∇  j,k:l,h,h′ in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) and  ˙C∆  j,k:l,h,h′ in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) thus consists of ensuring that they take their values in [0, 1]  by truncating them:  ˙C  ∇  j,k:l,h,h′ = ( ˙C  ∇  j,k:l,h,h′ ∨ 0) ∧ 1,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3)  ˙C  ∆  j,k:l,h,h′ = ( ˙C  ∆  j,k:l,h,h′ ∨ 0) ∧ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4)  Notice that taking the maximum with 0 in the previous expressions is actually  not necessary as the estimators in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) cannot be negative since the  empirical copula Ck:l is a multivariate d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' We nonetheless keep (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4) as  they are to be consistent with certain forthcoming definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' More generally, in  the rest of this section, underlining will be used to denote estimators constrained  to take their values in [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  30  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Two classes of smooth estimators  To obtain smooth estimators of the first-order partial derivatives of C, the pro-  posals in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) can be extended in two natural ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The first approach  consists of considering finite-differences of smooth estimators of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Given a  stretch X k:l, 1 ≤ k ≤ l ≤ n, of observations and two bandwidth parameters h  and h′ in [0, 1/2] such that h + h′ > 0, this leads to the estimators  ˙C  ν,∇  j,k:l,h,h′(u) = Cν  k:l{(u + hej) ∧ 1} − Cν  k:l{(u − h′ej) ∨ 0}  h + h′  ,  u ∈ [0, 1]d,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5)  ˙C  ν,∆  j,k:l,h,h′(u) = Cν  k:l{(u + hej) ∧ 1} − Cν  k:l{(u − h′ej) ∨ 0}  (uj + h) ∧ 1 − (uj − h′) ∨ 0  ,  u ∈ [0, 1]d,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6)  where Cν  k:l is the smooth empirical copula of X k:l defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Notice the  order of the symbols ν and ∇ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' ∆) indicating that the empirical copula is  first smoothed before finite-differencing is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Clearly, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2)) is  a particular case of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6)) when the smoothing distributions ν  X k:l  u  ,  u ∈ [0, 1]d, in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) are chosen to be Dirac measures at u ∈ [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Since C is a multivariate d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' with standard uniform margins, we  have that, for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d} and u ∈ Vj (where Vj is defined in Con-  dition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8),  ˙Cj(u(j)) = limh→0{C(u(j) + hej) − C(u(j))}/h = 1 (where the  notation u(j) is defined above Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Interestingly enough, the estima-  tor ˙C  ν,∆  j,k:l,h,h′ in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6) can satisfy this boundary constraint, that is, we can have  ˙C  ν,∆  j,k:l,h,h′(u(j)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' This will indeed happen if Cν  k:l is a genuine copula, which  according to Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7, can occur under Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3 for specific choices  of the smoothing distributions in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) such as those leading to the empirical  copulas CBin  k:l or CBetaB4  k:l  defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  By analogy with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4), it is straightforward to define truncated  versions of the estimators in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6) by  ˙C  ν,∇  j,k:l,h,h′ = ( ˙C  ν,∇  j,k:l,h,h′ ∨ 0) ∧ 1,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7)  ˙C  ν,∆  j,k:l,h,h′ = ( ˙C  ν,∆  j,k:l,h,h′ ∨ 0) ∧ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8)  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As discussed in Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1, the smoothing distributions ν  X k:l  u  ,  u ∈ [0, 1]d, can be chosen such that Cν  k:l is a genuine copula under Condi-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' In that case, using the fact that Cν  k:l is a multivariate d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' with standard  uniform margins, we immediately obtain (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=', Durante and Sempi, 2015,  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='14) that, for any u ∈ [0, 1]d and h, h′ in [0, 1/2] such that h+h′ > 0,  Cν  k:l{(u + hej) ∧ 1} − Cν  k:l{(u − h′ej) ∨ 0} ≤ (uj + h) ∧ 1 − (uj − h′) ∨ 0,  which implies that 0 ≤ ˙C  ν,∆  j,k:l,h,h′ ≤ 1 and thus that truncation of ˙C  ν,∆  j,k:l,h,h′ is  not necessary in that case since ˙C  ν,∆  j,k:l,h,h′ in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8) is equal to ˙C  ν,∆  j,k:l,h,h′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  31  By analogy with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2), a second natural approach to obtain smooth partial  derivative estimators consists of directly smoothing (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) and leads to  the estimators  ˙C  ∇,ν  j,k:l,h,h′(u) =  �  [0,1]d  ˙C  ∇  j,k:l,h,h′(w)dν  X k:l  u  (w),  u ∈ [0, 1]d,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9)  ˙C  ∆,ν  j,k:l,h,h′(u) =  �  [0,1]d  ˙C  ∆  j,k:l,h,h′(w)dν  X k:l  u  (w),  u ∈ [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10)  This time the order of the symbols ν and ∇ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' ∆) is reversed indicating  that it is the finite-differences-based estimator ˙C∇  j,k:l,h,h′ in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' ˙C∆  j,k:l,h,h′  in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2)) that is smoothed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Versions of these estimators that necessarily take  their values in [0, 1] can be obtained by constructing them from the truncated  estimators (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4) instead, leading respectively to  ˙C  ∇,ν  j,k:l,h,h′(u) =  �  [0,1]d  ˙C  ∇  j,k:l,h,h′(w)dν  X k:l  u  (w),  u ∈ [0, 1]d,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11)  ˙C  ∆,ν  j,k:l,h,h′(u) =  �  [0,1]d  ˙C  ∆  j,k:l,h,h′(w)dν  X k:l  u  (w),  u ∈ [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='12)  Note that a third approach to obtain a smooth estimator of the jth partial  derivative ˙Cj would consist of attempting to directly differentiate Cν  k:l in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2)  with respect to its jth argument (provided of course that Cν  k:l is differentiable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The resulting estimator  ˙Cν  j,k:l = ∂Cν  k:l  ∂uj  may exist only on the set Vj defined in Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' This is the path followed  by Janssen, Swanepoel and Veraverbeke (2016), who, for some integer m ≥ 2,  started from the empirical Bernstein copula CBern  k:l,m in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3) (which, as discussed  in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2, is a particular case of Cν  k:l in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Let ˙CBern  j,k:l,m = ∂CBern  k:l,m/∂uj be  the resulting estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Interestingly enough, from Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 in Appendix E,  we have that  ˙C  Bern  j,k:l,m(u) =  �  [0,1]d  ˙C  ∇  j,k:l, 1  m ,0(w)d˜µj,m,u(w),  u ∈ Vj,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13)  where  ˙C∇  j,k:l, 1  m ,0 is given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) with h = 1/m and h′ = 0 and, for any  u ∈ [0, 1]d, ˜µj,m,u is the law of the random vector ( ˜Sm,1,u1/m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , ˜Sm,d,ud/m)  whose components are independent such that, for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d} \\ {j}, ˜Sm,i,ui is  Binomial(m, ui) while ˜Sm,j,uj is Binomial(m − 1, uj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' In other words, differenti-  ating directly the empirical Bernstein copula CBern  k:l,m in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3) with respect to its  jth argument leads to a special case of the estimator in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Notice that, since  the measures ˜µj,m,u are well-defined for any u ∈ [0, 1]d, the integral in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13) is  actually well-defined for any u ∈ [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Hence, as we continue, we take (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13)  with u ∈ [0, 1]d as the definition of ˙CBern  j,k:l,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The following result, proven in Appendix E, shows that ˙CBern  j,k:l,m can be easily  computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  32  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Given a stretch X k:l, 1 ≤ k ≤ l ≤ n, of observations, we  have that, for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, u ∈ [0, 1]d and integer m ≥ 2,  ˙C  Bern  j,k:l,m(u) =  m  l − k + 1  l  �  i=k  bm−1,uj  �  ⌈mRk:l  ij /(l − k + 1)⌉ − 1  �  ×  d  �  t=1  t̸=j  ¯Bm,ut  �  ⌈mRk:l  it /(l − k + 1)⌉ − 1  �  ,  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='14)  where ⌈·⌉ denotes the ceiling function and, for any p ∈ N and u ∈ [0, 1], ¯Bp,u  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' bp,u) is the survival (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' probability mass) function of the Binomial(p, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Weak consistency  In order to study the weak consistency of the estimators of the partial derivatives  of C defined in the previous subsection, it is necessary to link the bandwidth  parameters in their expressions to the data (or, at least, to the amount of data)  from which these estimators are computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As we continue, for any n ∈ N and  any potential d-dimensional data set x ∈ (Rd)n, h(x) and h′(x) will denote the  values of the left and right bandwidths for the data set x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' With this in mind,  in the rest of this subsection, for the sake of a more compact notation, we shall  write  ˙C  ν,∇  j,k:l (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ˙C  ν,∆  j,k:l,  ˙C  ∇,ν  j,k:l, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' ) for  ˙C  ν,∇  j,k:l,h,h′ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ˙C  ν,∆  j,k:l,h,h′,  ˙C  ∇,ν  j,k:l,h,h′,  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' ) with the understanding that h = h(X k:l) and h′ = h′(X k:l) are random  variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' We impose in addition the following condition on the bandwidths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4 (Bandwidth condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' There exists positive sequences bn ↓ 0  and b′  n ↓ 0 and constants L2 ≥ L1 > 0 such that, for all n ∈ N, bn +b′  n ≥ n−1/2,  and, for any x ∈ (Rd)n, L1bn ≤ h(x) ≤ (L2bn) ∧ 1/2 and L1b′  n ≤ h′(x) ≤  (L2b′  n) ∧ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  As we shall see in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4, one meaningful possibility among many others  is to consider that, for any n ∈ N and x ∈ Rd, the left and right bandwidths for  the data set x are defined by h(x) = h′(x) = [M2{1−|τ(x)|}a+M1]n−1/2∧1/2,  where M1, M2 > 0 are constants, τ(x) ∈ [−1, 1] is the value of the sample  version of a suitable multivariate extension of Kendall’s tau for the data set x  and a ∈ (0, ∞) is a fixed power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Roughly speaking, the bandwidths will be larger  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' smaller) in the case of weakly (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' strongly) cross-sectionally dependent  data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' It is easy to verify that Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4 holds for the previous definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The following result, proven in Appendix E, establishes the weak consistency  of the smooth estimators of the first class in a sequential setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5 (Weak consistency in a sequential setting for the first class  of smooth estimators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Under Conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4, for any j ∈  {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, δ ∈ (0, 1) and ε ∈ (0, 1/2),  sup  (s,t)∈Λ  t−s≥δ  sup  u∈[0,1]d  uj∈[ε,1−ε]  ��� ˙Cν,∆  j,⌊ns⌋+1:⌊nt⌋(u) − ˙Cj(u)  ��� = oP(1),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='15)  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  33  where ˙Cν,∆  j,k:l is defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6), and similarly for ˙C  ν,∇  j,k:l in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5), ˙C  ν,∇  j,k:l in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7)  and ˙C  ν,∆  j,k:l in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' An inspection of the proof of the previous result reveals that the  second supremum in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='15) can be replaced by a supremum over u ∈ [0, 1]d if  ˙Cj happens to be continuous on [0, 1]d instead of only satisfying Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  see also Kojadinovic, Segers and Yan (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  As a consequence of the previous proposition, we have that, under the con-  ditions of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, the estimators ˙C  ν,∇  j,k:l and ˙C  ν,∆  j,k:l satisfy Condition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1  since they are bounded in absolute value (by one) by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The next result is an immediate corollary of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5 since the esti-  mator in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2)) is a particular case of the one in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6))  when the smoothing distributions ν  X k:l  u  , u ∈ [0, 1]d, in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) are chosen to be  Dirac measures at u ∈ [0, 1]d (the latter clearly satisfy Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7 (Weak consistency in a sequential setting for the non-smooth  finite-differences-based estimators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Under Conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4, for  any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, δ ∈ (0, 1) and ε ∈ (0, 1/2),  sup  (s,t)∈Λ  t−s≥δ  sup  u∈[0,1]d  uj∈[ε,1−ε]  ��� ˙C  ∆  j,⌊ns⌋+1:⌊nt⌋(u) − ˙Cj(u)  ��� = oP(1),  where ˙C∆  j,k:l is defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2), and similarly for ˙C∇  j,k:l in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1), ˙C  ∇  j,k:l in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3)  and ˙C  ∆  j,k:l in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  We now move to the second class of smooth estimators of the partial deriva-  tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As we shall see below, to establish their weak consistency, it suffices,  among others, that the underlying smoothing distributions satisfy the following  weaker version of Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8 (Weak variance condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' There exists a positive sequence  an ↓ 0 such that, for any n ∈ N, x ∈ (Rd)n, u ∈ [0, 1]d and j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d},  Var(W x  j,uj) ≤ an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The following result is proven in Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9 (Weak consistency in a sequential setting for the second class  of smooth estimators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Under Conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8, for any j ∈  {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, δ ∈ (0, 1) and ε ∈ (0, 1/2),  sup  (s,t)∈Λ  t−s≥δ  sup  u∈[0,1]d  uj∈[ε,1−ε]  ��� ˙C  ∆,ν  j,⌊ns⌋+1:⌊nt⌋(u) − ˙Cj(u)  ��� = oP(1),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='16)  where ˙C  ∆,ν  j,k:l is defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='12), and similarly for ˙C  ∇,ν  j,k:l in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  One may wonder why the estimators ˙C  ∇,ν  j,k:l in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9) and ˙C  ∆,ν  j,k:l in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10) are  not included in the previous proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Actually, upon additionally imposing  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  34  that the left and right bandwidths of ˙C∇  j,k:l in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) and ˙C∆  j,k:l in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) (which are  to be smoothed) are equal and in the absence of ties (see Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3), weak  consistency can also be proven for the estimators ˙C  ∇,ν  j,k:l and ˙C  ∆,ν  j,k:l using the same  technique of proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' An inspection of the proof and some additional thinking  reveals that this follows from the fact that these estimators are bounded on  [0, 1]d in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' When one of the bandwidths is zero, this is not necessarily  the case anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' This is also why the previous proposition cannot be directly  used to establish the weak consistency of the Bernstein estimator in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  For this estimator, one additionally needs to rely on the fact that the finite  difference-based estimator that is smoothed is bounded on the support of the  smoothing distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' This is used in the proof in Appendix E of the next  proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10 (Weak consistency of the Bernstein estimator in a sequential  setting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Assume that Conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11 hold and, for any i ∈ N,  let mi = ⌊Liθ⌋ ∨ 2 for some constants L > 0 and θ ∈ (0, 1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then, for any  j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, δ ∈ (0, 1) and ε ∈ (0, 1/2),  sup  (s,t)∈Λ  t−s≥δ  sup  u∈[0,1]d  uj∈[ε,1−ε]  ��� ˙C  Bern  j,⌊ns⌋+1:⌊nt⌋,m⌊nt⌋−⌊ns⌋ − ˙Cj(u)  ��� = oP(1),  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='17)  where ˙CBern  j,k:l,m is defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' In addition, for any n ∈ N,  sup  (s,t,u)∈Λ×[0,1]d  ��� ˙C  Bern  j,⌊ns⌋+1:⌊nt⌋,m⌊nt⌋−⌊ns⌋(u)  ��� ≤ 1 + L ∨ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='18)  Note that our technique of proof allows us to establish uniform weak con-  sistency of the estimator even for θ = 1/2, whereas the approach used in the  proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 of Bouezmarni, El Ghouch and Taamouti (2013) (for  Bernstein copula density estimators and which could be adapted to first-order  partial derivative estimators) leads to the result only for θ < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Another con-  sequence of the previous result is that the Bernstein partial derivative estimator  as parametrized in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10 can satisfy Condition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Finite-sample performance of selected estimators  The aim of this subsection is to compare the finite-sample performance of some  of the estimators introduced previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Specifically, for each n ∈ {10, 20, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , 100},  each data generating copula C and each partial derivative estimator ˙Cj,1:n under  investigation, we estimated its integrated mean squared error  IMSE( ˙Cj,1:n) =  �  [0,1]d E  ��  ˙Cj,1:n(u) − ˙Cj(u)  �2�  du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  To do so, we applied the trick described in detail in Appendix B of Segers, Sibuya  and Tsukahara (2017) allowing to compute IMSE( ˙Cj,1:n) as a single expectation  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  35  and proceeded by Monte Carlo simulation using 20 000 independent random  samples of size n from C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  In a first experiment, we compared estimators of the form ˙C  ν,∇  j,1:n,h,h′ in  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7) to estimators of the form ˙C  ν,∆  j,1:n,h,h′ in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8) for deterministic bandwidths  h = h′ = n−1/2 ∧ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Specifically, we considered estimators based, respec-  tively, on finite-differences of the classical empirical copula C1:n in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1), on  finite-differences of the empirical beta copula CBin  1:n defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1), and on  finite-differences of its data-adaptive extension CBetaB4  1:n  defined at the end of Sec-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As data-generating copula C, we considered the bivariate or trivariate  Clayton or Gumbel–Hougaard copula with bivariate margins with a Kendall’s  tau of τ ∈ {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75} as well as the bivariate Frank copula with a  Kendall’s tau of τ ∈ {0, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Note that, since all data-generating  copulas are exchangeable, it suffices to focus on only one partial derivative es-  timator, say the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As expected, the integrated mean squared error of  estimators of the form ˙C  ν,∆  j,1:n,h,h′ was always found to be (substantially) below  that of the corresponding estimator ˙C  ν,∇  j,1:n,h,h′, confirming that the adjusted  numerator in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6) compared to the one in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5) helps indeed to improve the  finite-sample performance of finite-difference-based estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  In a second experiment, we compared the aforementioned three estimators  of the form ˙C  ν,∆  j,1:n,h,h′ in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' They will be denoted as ˙C  ∆  j,1:n,h,h′ = ˙C  Dirac,∆  j,1:n,h,h′,  ˙C  Bin,∆  j,1:n,h,h′ and ˙C  BetaB4,∆  j,1:n,h,h′ as we continue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As could have been expected from the  experiments of Kojadinovic and Yi (2022) comparing the underlying copula esti-  mators, it is the estimator ˙C  BetaB4,∆  j,1:n,h,h′ that always displayed the lowest integrated  mean squared error, followed by ˙C  Bin,∆  j,1:n,h,h′ and ˙C  ∆  j,1:n,h,h′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  We next investigated the influence of the bandwidths on the integrated mean  squared error of ˙C  BetaB4,∆  j,1:n,h,h′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Deterministic bandwidths of the form h = h′ =  (Ln−1/2) ∧ 1/2 were considered with L ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, 1, 2, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The corresponding in-  tegrated mean squared errors are represented in the first column of graphs of  Figure 3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Figure 4) when the data-generating copula is the bivariate Frank  copula with negative dependence (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' the trivariate Gumbel–Hougaard cop-  ula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The legend “BetaB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5” refers to the estimator with L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As  one can see, the weaker the cross-sectional dependence, the larger the (constant  L in the expression of the) bandwidths should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  An inspection of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10) and some thinking reveals that estimators  from the second class can be difficult to compute in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For that reason, in  our experiments, we solely focused on the Bernstein estimator ˙CBern  j,1:n,m in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13)  which can be computed using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Mimicking the previous experiment, we  considered a deterministic choice for the parameter m of the form m = ⌊Ln1/2⌋∨  2 with L ∈ {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, 1, 2, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The corresponding integrated mean squared errors are  represented in the second column of graphs of Figure 3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Figure 4) when the  data-generating copula is the bivariate Frank copula with negative dependence  (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' the trivariate Gumbel–Hougaard copula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The legend “Bern 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5” refers  to the estimator with L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5 and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As one can see, this time, the stronger  the cross-sectional dependence, the larger the (constant L in the expression of  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  36  20  40  60  80  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='008  indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' cop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' / d = 2 / tau = 0  n  integrated mean squared error  BetaB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  BetaB 1  BetaB 2  BetaB 4  20  40  60  80  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='002  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='004  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='006  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='008  Frank / d = 2 / tau = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25  n  integrated mean squared error  BetaB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  BetaB 1  BetaB 2  BetaB 4  20  40  60  80  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='020  Frank / d = 2 / tau = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  n  integrated mean squared error  BetaB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  BetaB 1  BetaB 2  BetaB 4  20  40  60  80  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='05  Frank / d = 2 / tau = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75  n  integrated mean squared error  BetaB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  BetaB 1  BetaB 2  BetaB 4  20  40  60  80  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='030  indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' cop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' / d = 2 / tau = 0  n  integrated mean squared error  Bern 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  Bern 1  Bern 2  Bern 4  20  40  60  80  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='025  Frank / d = 2 / tau = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25  n  integrated mean squared error  Bern 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  Bern 1  Bern 2  Bern 4  20  40  60  80  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='05  Frank / d = 2 / tau = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  n  integrated mean squared error  Bern 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  Bern 1  Bern 2  Bern 4  20  40  60  80  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10  Frank / d = 2 / tau = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75  n  integrated mean squared error  Bern 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  Bern 1  Bern 2  Bern 4  20  40  60  80  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='015  indep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' cop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' / d = 2 / tau = 0  n  integrated mean squared error  Dirac  Bin  Adap BetaB4  Adap Bern  20  40  60  80  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='015  Frank / d = 2 / tau = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25  n  integrated mean squared error  Dirac  Bin  Adap BetaB4  Adap Bern  20  40  60  80  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='005  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='015  Frank / d = 2 / tau = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  n  integrated mean squared error  Dirac  Bin  Adap BetaB4  Adap Bern  20  40  60  80  100  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='000  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='010  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='020  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='030  Frank / d = 2 / tau = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75  n  integrated mean squared error  Dirac  Bin  Adap BetaB4  Adap Bern  Fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Estimated integrated mean squared errors against n  ∈  {10, 20, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , 100} of  four estimators of  ˙C1 when C is the bivariate Frank copula with a Kendall’s tau in  {0, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='  the) parameter m should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content=' / d = 3 / tau = 0  n  integrated mean squared error  BetaB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content='03  Gumbel−Hougaard / d = 3 / tau = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75  n  integrated mean squared error  Dirac  Bin  Adap BetaB4  Adap Bern  Fig 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Estimated integrated mean squared errors against n ∈ {10, 20, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , 100} of four es-  timators of  ˙C1 when C is the trivariate Gumbel–Hougaard copula whose bivariate margins  have a Kendall’s tau in {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='75}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The two previous experiments suggested to focus on data-adaptive band-  widths for the estimators ˙C  BetaB4,∆  j,1:n,h,h′ and ˙CBern  j,1:n,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Specifically, for d ∈ {2, 3}  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  38  and a data set x ∈ (Rd)n, we considered the settings h(x) = h′(x) = ([4{1 −  |τ(x)|}6+1/2]n−1/2)∧1/2 for ˙C  BetaB4,∆  j,1:n,h,h′ and m(x) = ⌊{4|τ(x)|3/2+1/2}n1/2⌋∨2  for ˙CBern  j,1:n,m, where τ(x) ∈ [−1, 1] is the average of the values of the sample ver-  sion of Kendall’s tau computed from the bivariate margins of the data set x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The integrated mean squared errors of the resulting data-adaptive estimators  are represented in the third column of graphs of Figure 3 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Figure 4)  when the data-generating copula is the bivariate Frank copula with negative  dependence (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' the trivariate Gumbel–Hougaard copula).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The legends “Adap  BetaB4” and “Adap Bern” refer to the above-mentioned versions of the estima-  tors ˙C  BetaB4,∆  j,1:n,h,h′ and ˙CBern  j,1:n,m, respectively, while “Dirac” and “Bin” refer to the  benchmark estimators ˙C  ∆  j,1:n,h,h′ = ˙C  Dirac,∆  j,1:n,h,h′ and ˙C  Bin,∆  j,1:n,h,h′ with deterministic  bandwidths h = h′ = n−1/2 ∧1/2 based on the empirical copula and the empiri-  cal beta copula, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Overall, it is the data-adaptive estimator ˙C  BetaB4,∆  j,1:n,h,h′  that displays the best finite-sample behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The data-adaptive Bernstein esti-  mator ˙CBern  j,1:n,m appears to be competitive only when the data-generating copula  is close to the independence copula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Conclusion  Smooth nonparametric copula estimators, such as the empirical beta copula  proposed by Segers, Sibuya and Tsukahara (2017) or its data-adaptive extension  studied in Kojadinovic and Yi (2022), can be substantially better estimators  than the classical empirical copula in small samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' To use such estimators in  inference procedures, one typically needs to rely on resampling techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  As investigated in Section 3, in the case of i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' observations, a smooth boot-  strap `a la Kiriliouk, Segers and Tsukahara (2021) can be asymptotically valid  for a large class of smooth estimators that can be expressed as mixtures of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  When based on the empirical beta copula, Kiriliouk, Segers and Tsukahara  (2021) found such a smooth bootstrap to be a competitive alternative to the  multiplier bootstrap while being substantially simpler to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' An empiri-  cal finding of this work is that the smooth bootstrap based on the data-adaptive  extension of the empirical beta copula proposed in Kojadinovic and Yi (2022)  seems to lead to even better-behaved inference procedures than the former as it  copes better with stronger dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Unfortunately, such smooth bootstraps cannot be used anymore in the time  series setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' A second contribution of this work was to study both theoreti-  cally and empirically smooth extensions of the sequential dependent multiplier  bootstrap of B¨ucher and Kojadinovic (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As illustrated at the end of the  fourth section, the latter can for instance be used to derive smooth change-point  detection tests which are likely to be more sensitive to early or late changes than  their non-smooth counterparts since, as already mentioned, smooth estimators  are likely to be more accurate than the empirical copula when computed from  small subsets of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  In connection with the multiplier bootstrap, a third contribution of this work  was the study of the weak consistency and finite-sample performance of two  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  39  classes of smooth estimators of the first-order partial derivatives of the copula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The obtained results may be of independent interest since such estimators have  applications in mean and quantile regression as they lead to estimators of the  conditional distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' From an empirical perspective, our investiga-  tions led to the proposal of a smooth data-adaptive estimator of the first-order  partial derivatives of the copula that substantially outperforms, among others,  the Bernstein estimator studied in Janssen, Swanepoel and Veraverbeke (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Appendix A: Proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13  The proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13 is based on the following two lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Let Xn be a process in ℓ∞(Λ × [0, 1]d) such that for all u ∈  [0, 1]d and s ∈ [0, 1], Xn(s, s, u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Furthermore, assume that Xn ⇝ X in  ℓ∞(Λ × [0, 1]d) where X has continuous trajectories almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then, under  Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8 (which is implied by Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9),  sup  (s,t,u)∈Λ×[0,1]d  �����  �  [0,1]d Xn(s, t, w)dν  X ⌊ns⌋+1:⌊nt⌋  u  (w) − Xn(s, t, u)  ����� = oP(1), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1)  sup  (s,t,u)∈Λ×[0,1]d  �����  �  [0,1]d Xn(s, t, w)dν  X 1:n  u  (w) − Xn(s, t, u)  ����� = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The first claim was proven in the proof of Lemma 32 of Kojadinovic and  Yi (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The proof of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) is very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ■  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Assume that Conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then, almost surely,  sup  (s,t,u)∈Λ×[0,1]d  √nλn(s, t)  �����  �  [0,1]d C(w)dν  X ⌊ns⌋+1:⌊nt⌋  u  (w) − C(u)  ����� = o(1), (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3)  sup  (s,t,u)∈Λ×[0,1]d  √nλn(s, t)  �����  �  [0,1]d C(w)dν  X 1:n  u  (w) − C(u)  ����� = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The first claim was proven in the proof of Lemma 33 of Kojadinovic and  Yi (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The proof of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4) is an immediate consequence of the fact that the  left-hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4) is almost surely smaller than  sup  u∈[0,1]d  √n  �����  �  [0,1]d C(w)dν  X 1:n  u  (w) − C(u)  ����� ,  which is smaller than the left-hand side of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3) with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ■  Proof of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Combining Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='12, by the  triangle inequality, we immediately obtain that  sup  (s,t,u)∈Λ×[0,1]d  ���Cν  n(s, t, u) − ˜Cn(s, t, u)  ��� = oP(1),  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  40  where ˜Cn is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' It thus remains to show that  sup  (s,t,u)∈Λ×[0,1]d  ���˜Cn(s, t, u) − ˜Cν  n(s, t, u)  ��� = oP(1),  sup  (s,t,u)∈Λ×[0,1]d  ���˜Cn(s, t, u) − ¯Cν  n(s, t, u)  ��� = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  We only prove the first claim, the proof of the second one being similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For any  (s, t, u) ∈ Λ × [0, 1]d, let  ˘Bν  n(s, t, u) =  �  [0,1]d Bn(s, t, w)dν  X ⌊ns⌋+1:⌊nt⌋  u  (w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Under Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11, Bn ⇝ BC in ℓ∞(Λ × [0, 1]d), where BC has continuous  trajectories almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' We then obtain from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 that  sup  (s,t,u)∈Λ×[0,1]d  ���˘Bν  n(s, t, u) − Bn(s, t, u)  ��� = oP(1),  and, furthermore, since Conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9 hold, from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3) in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2,  that  sup  (s,t,u)∈Λ×[0,1]d  ���˘Bν  n(s, t, u) − ˜Bν  n(s, t, u)  ��� = o(1),  with probability one, where ˜Bν  n is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='14), which implies that  sup  (s,t,u)∈Λ×[0,1]d  ���˜Bν  n(s, t, u) − Bn(s, t, u)  ��� = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5)  Moreover, from the triangle inequality, we have  sup  (s,t)∈Λ  u∈[0,1]d  ���˜Cν  n(s, t, u) − ˜Cn(s, t, u)  ��� ≤  sup  (s,t)∈Λ  u∈[0,1]d  ���˜Bν  n(s, t, u) − Bn(s, t, u)  ���  +  d  �  j=1  sup  u∈[0,1]d  ��� ˙Cj(u)  ���  sup  (s,t)∈Λ  u∈[0,1]d  ���˜Bν  n(s, t, u(j)) − Bn(s, t, u(j))  ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The terms on the right-hand side of the previous display converge to zero in  probability as a consequence of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5) and the fact that 0 ≤ ˙Cj ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ■  Appendix B: Proofs of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3 and Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Fix n ∈ N, x ∈ (Rd)n and r ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , n}d and let  us check that K x  r , which can be expressed as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6) under Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4, is a  multivariate d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  By Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, the function K x  rj,j defined by  K x  rj,j(u) = ¯  F x  j,u{(rj−1)/n}, u ∈ [0, 1], is a univariate d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Indeed, K x  rj,j  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  41  is right-continuous and increasing on [0, 1] and, by properties of the smoothing  distributions,  K x  rj,j(0) = ¯  F x  j,0{(rj − 1)/n} = P{W x  j,0 > (rj − 1)/n} = P{0 > (rj − 1)/n} = 0,  K x  rj,j(1) = ¯  F x  j,1{(rj − 1)/n} = P{W x  j,1 > (rj − 1)/n} = P{1 > (rj − 1)/n} = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Using additionally Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6, the expression of K x  r  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6) can then by  further simplified to  K x  r (u) = ¯  C x{K x  r1,1(u1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , K x  rd,d(ud)},  u ∈ [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  From Sklar’s Theorem (Sklar, 1959), K x  r is thus a d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' on [0, 1]d with univariate  margins K x  r1,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , K x  rd,d and copula ¯  C x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ■  The proof of Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 below is based on the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For any n ∈ N and t ∈ [0, 1], let ¯Bn,t be the survival function of  a Binomial(n, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then, for any n ∈ N and w ∈ [0, n), the function t �→ ¯Bn,t(w)  is strictly increasing on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Fix n ∈ N and w ∈ [0, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Since t �→ ¯Bn,t(w) is continuous on [0, 1], it  suffices to prove that, for any t ∈ (0, 1),  ∂  ∂t  � ¯Bn,t(w)  �  > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' We have  ∂  ∂t  � ¯Bn,t(w)  �  = ∂  ∂t  �  �  �  n  �  k=⌊w⌋+1  �  n  k  �  tk(1 − t)n−k  �  �  � = ∂  ∂t  �  �  �  n  �  k=⌊w⌋+1  �  n  k  �  tk(1 − t)n−k  �  �  �  =  n  �  k=⌊w⌋+1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' {ktk−1(1 − t)n−k − (n − k)tk(1 − t)n−k−1}  =  n  �  k=⌊w⌋+1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ktk−1(1 − t)n−k −  n−1  �  k=⌊w⌋+1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (n − k)tk(1 − t)n−k−1  =  n  �  k=⌊w⌋+1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='tk−1(1 − t)n−k −  n−1  �  k=⌊w⌋+1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (n − k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='tk(1 − t)n−k−1  =  n  �  k=⌊w⌋+1  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='tk−1(1 − t)n−k −  n  �  k=⌊w⌋+2  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (n − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='tk−1(1 − t)n−k  =  n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ⌊w⌋!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (n − ⌊w⌋ − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t⌊w⌋(1 − t)n−⌊w⌋−1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ■  Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For any n ∈ N, t ∈ [0, 1] and ρ ∈ (1, n), let ¯  Bn,t,ρ be the survival  function of a Beta-Binomial(n, α, β), where α = t(n − ρ)/(ρ − 1) and β =  (1 − t)(n − ρ)/(ρ − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then, for any n ∈ N, ρ ∈ (1, n) and w ∈ [0, n), the  function t �→ ¯  Bn,t,ρ(w) is strictly increasing on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' First, notice that, for any t ∈ [0, 1], ρ ∈ (1, n) and w ∈ [0, n), by definition  of the beta-binomial distribution,  ¯  Bn,t,ρ(w) = EΘ{ ¯Bn,Θ(w)},  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1)  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  42  where ¯Bn,t is the survival function of a Binomial(n, t) and Θ is Beta(α, β) with  α = t(n−ρ)/(ρ−1) and β = (1−t)(n−ρ)/(ρ−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' From Lemma 30 in Kojadinovic  and Yi (2022), we have that, for any n ∈ N, ρ ∈ (1, n) and w ∈ [0, n), the function  t �→ ¯  Bn,t,ρ(w) is increasing on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' It thus suffices to show strict increasingness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Let us prove this by contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Suppose that there exists 0 ≤ t1 < t2 ≤ 1  such that  ¯  Bn,t1,ρ(w) = ¯  Bn,t2,ρ(w) for some n ∈ N, ρ ∈ (1, n) and w ∈ [0, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Then, from (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1), we have that  EΘ1{ ¯Bn,Θ1(w)} = EΘ2{ ¯Bn,Θ2(w)},  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2)  where Θ1 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Θ2) is Beta(α1, β1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Beta(α2, β2)) with α1 = t1(n−ρ)/(ρ−  1) and β1 = (1 − t1)(n − ρ)/(ρ − 1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' α2 = t2(n − ρ)/(ρ − 1) and β2 =  (1 − t2)(n − ρ)/(ρ − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' From the proof of Lemma 29 in Kojadinovic and Yi  (2022), we have that Θ1 ≤st Θ2, where ≤st denotes the usual stochastic order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Using additionally (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) and the fact that the function t �→ ¯Bn,t(w) is strictly  increasing on [0, 1] from Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1, we have, according to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8 of  Shaked and Shanthikumar (2007), that Θ1 and Θ2 have the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  This contradicts the fact that t1 < t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ■  Appendix C: Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  The proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5 is based on two lemmas which we show first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Let Φ be the map from ℓ∞([0, 1]d) to ℓ∞([0, 1]d) defined for any d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f H on  [0, 1]d whose univariate margins H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , Hd do not assign mass at zero by  Φ(H)(u) = H{H−1  1 (u1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , H−1  d (ud)},  u ∈ [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1)  Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Assume that the random vectors in X 1:n are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' and that  Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then, almost surely,  sup  u∈[0,1]d |Cν  1:n(u) − C(u)| = o(1),  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2)  where Cν  1:n is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The supremum on the left-hand side of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) is smaller than In + Jn,  where  In =  sup  u∈[0,1]d  �����  �  [0,1]d C1:n(w)dν  X 1:n  u  (w) −  �  [0,1]d C(w)dν  X 1:n  u  (w)  ����� ,  Jn =  sup  u∈[0,1]d  �����  �  [0,1]d C(w)dν  X 1:n  u  (w) − C(u)  ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Term In: From the triangle inequality, In is smaller than  sup  u∈[0,1]d|C1:n(u) − C(u)| ≤ I′  n + I′′  n + I′′′  n ,  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  43  where  I′  n =  sup  u∈[0,1]d |C1:n(u) − Φ(G1:n)(u)|,  I′′  n =  sup  u∈[0,1]d |Φ(G1:n)(u) − G1:n(u)|,  I′′′  n =  sup  u∈[0,1]d |G1:n(u) − C(u)|,  where the map Φ is defined in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) and G1:n is empirical d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of the unobserv-  able random sample U1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , Un obtained from X 1:n by the probability integral  transformations Uij = Fj(Xij), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , n}, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Using the well-  known facts (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=', Segers, 2012) that Φ(G1:n) = Φ(F1:n), where F1:n is the  empirical d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of X 1:n, and  sup  u∈[0,1]d |C1:n(u) − Φ(F1:n)(u)| ≤ d  n,  we obtain that I′  n = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Furthermore, from the Glivenko-Cantelli lemma (see,  e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=', van der Vaart, 1998, Theorem 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1), I′′′  n  = o(1) with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Finally, using a well-known property of multivariate d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='s (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=', Durante  and Sempi, 2015, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='14), the well-known fact, for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d},  supu∈[0,1] |G−1  1:n,j(u)−u| = supu∈[0,1] |G1:n,j(u)−u| (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=', Shorack and Well-  ner, 1986, Chapter 3) and, again, the Glivenko-Cantelli lemma, we obtain that,  almost surely,  I′′  n ≤  d  �  j=1  sup  u∈[0,1]  |G−1  1:n,j(u) − u| =  d  �  j=1  sup  u∈[0,1]  |G1:n,j(u) − u| = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Term Jn: We proceed as in the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 of Segers, Sibuya and  Tsukahara (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Fix η > 0 and let us show that, with probability one, Jn  can be made smaller than η provided n is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Let | · |∞ denote the  maximum norm on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For any ε > 0, we have that  Jn =  sup  u∈[0,1]d  �����  �  [0,1]d{C(w) − C(u)}dν  X 1:n  u  (w)  �����  ≤  sup  u∈[0,1]d  �����  �  {w∈[0,1]d:|u−w|∞≤ε}  {C(w) − C(u)}dν  X 1:n  u  (w)  �����  +  sup  u∈[0,1]d  �����  �  {w∈[0,1]d:|u−w|∞>ε}  {C(w) − C(u)}dν  X 1:n  u  (w)  ����� ≤ J′  n + J′′  n,  where  J′  n =  sup  (u,w)∈[0,1]2d  |u−w|∞≤ε  |C(w) − C(u)| ,  J′′  n =  sup  u∈[0,1]d ν  X 1:n  u  ({w ∈ [0, 1]d : |u − w|∞ > ε}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  44  Let ε = η/(2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then, from the Lipschitz continuity of C, J′  n ≤ η/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As far as J′′  n  is concerned, conditionally on X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , for almost any sequence X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' ,  using Chebyshev’s inequality and Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8, we have that  J′′  n =  sup  u∈[0,1]d P {|W  X 1:n  u  − u)|∞ > ε | X 1:n}  =  sup  u∈[0,1]d P  �  �  d�  j=1  ����W  X 1:n  j,uj  − uj  ��� > ε  �  | X 1:n  �  �  ≤  sup  u∈[0,1]d  d  �  j=1  P  ����W  X 1:n  j,uj  − uj  ��� > ε | X 1:n  �  ≤  sup  u∈[0,1]d  d  �  j=1  Var(W  X 1:n  j,uj  | X 1:n)  ε2  ≤ dan  ε2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  which implies that, for n sufficiently large, J′′  n ≤ η/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The latter holds condi-  tionally on X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' for almost any sequence X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , which completes  the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ■  Next, we recall the mode of convergence classically used to state asymptotic  validity results of resampling techniques when dealing with empirical processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=', van der Vaart and Wellner (2000, Chapter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9) or Kosorok (2008,  Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Let  BL1 = {h : ℓ∞([0, 1]d) → [−1, 1] such that,  for all x, y ∈ ℓ∞([0, 1]d), |h(x) − h(y)| ≤  sup  u∈[0,1]d |x(u) − y(u)|}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Let Xn = Xn(X 1:n, Wn) be a sequence of bootstrapped empirical processes in  ℓ∞([0, 1]d) depending on an additional source of randomness Wn (often called  the “bootstrap weights”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For the smooth bootstraps under consideration, Wn  is independent of the data X 1:n and corresponds to n independent copies of the  independent random variables I and U # necessary to carry out Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2  (see also (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2)) n times independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The notation Xn  P⇝  W X then means that  suph∈BL1 |EW {h(Xn)} − E{h(X)} → 0 in outer probability,  EW {h(Xn)∗} − EW {h(Xn)∗}  P→ 0 for all h ∈ BL1,  where EW denotes an expectation with respect to the bootstrap weights Wn only  and h(Xn)∗ and h(Xn)∗ denote the minimal measurable majorant and maximal  measurable minorant with respect to (X 1:n, Wn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The next lemma is very closely related to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3 of Kiriliouk, Segers  and Tsukahara (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Assume that the random vectors in X 1:n are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=', and that  Conditions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then,  √n(C  #  1:n − Cν  1:n)  P⇝  W CC(0, 1, ·),  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3)  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  45  where CC is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Let G  #  1:n be the empirical d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' of V  #  1:n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Using Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 and proceeding  as in Step 1 of the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3 of Kiriliouk, Segers and Tsukahara  (2021), one obtains that  √n(G  #  1:n − Cν  1:n)  P⇝  W BC(0, 1, ·),  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4)  where BC is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then, proceeding as in Step 2 of the proof of  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3 of Kiriliouk, Segers and Tsukahara (2021), that is, combin-  ing (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4) with the Hadamard differentiability of the map Φ in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) established  in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4 of B¨ucher and Volgushev (2013), the functional delta method  for the bootstrap “in probability” (van der Vaart and Wellner, 2000, Theo-  rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11) and the fact Φ(Cν  1:n) = Cν  1:n (since Cν  1:n has standard uniform  margins under Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4 in the considered i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' setting), one obtains  √n(Φ(G  #  1:n) − Cν  1:n)  P⇝  W CC(0, 1, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5)  The desired result finally follows from (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5) and the well-known fact that  sup  u∈[0,1]d |C  #  1:n(u) − Φ(G  #  1:n)(u)| ≤ d  n  since the components samples of V  #  1:n contain no ties with probability one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ■  Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Combining Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 with Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 of B¨ucher  and Kojadinovic (2019), we obtain that (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3) is equivalent to  �  Cn(0, 1, ·), √n(C  #,[1]  1:n − Cν  1:n), √n(C  #,[2]  1:n − Cν  1:n)  �  ⇝  �  CC(0, 1, ·), C  [1]  C (0, 1, ·), C  [2]  C (0, 1, ·)  �  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6)  in {ℓ∞([0, 1]d)}3, where Cn is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10, we have that  sup  u∈[0,1]d  √n|Cν  1:n(u) − C1:n(u)| =  sup  u∈[0,1]d |Cν  n(0, 1, u) − Cn(0, 1, u)| = oP(1),  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7)  where Cν  n is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The first joint weak convergence in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5  then follows from (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6) and (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Fix j ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Since (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7) holds, to establish the second joint weak conver-  gence from the first, it suffices to show that  sup  u∈[0,1]d  ��√n{C  #,ν,[j]  1:n  (u) − Cν  1:n(u)} − √n{C  #,[j]  1:n (u) − C1:n(u)}  �� = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8)  The supremum on the left hand-side of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8) is smaller than In + Jn, where  In =  sup  u∈[0,1]d  �����  �  [0,1]d  √n{C  #,[j]  1:n (w) − C1:n(w)}dν  V #,[j]  1:n  u  (w) − √n{C  #,[j]  1:n (u) − C1:n(u)}  ����� ,  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  46  Jn =√n  sup  u∈[0,1]d  �����  �  [0,1]d C1:n(w)dν  V #,[j]  1:n  u  (w) −  �  [0,1]d C1:n(w)dν  X 1:n  u  (w)  ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Since, according to the first claim of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, √n(C  #,[j]  1:n −C1:n) ⇝ CC(0, 1, ·)  in ℓ∞([0, 1]d) and CC(0, 1, ·) has continuous trajectories almost surely, it can be  verified by proceeding as in the proof of Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 that In = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For the  term Jn, we have that Jn ≤ Kn + Ln, where  Kn =√n  sup  u∈[0,1]d  �����  �  [0,1]d{C1:n(w) − C(w)}dν  V #,[j]  1:n  u  (w)  −  �  [0,1]d{C1:n(w) − C(w)}dν  X 1:n  u  (w)  ����� ,  Ln =√n  sup  u∈[0,1]d  �����  �  [0,1]d C(w)dν  V #,[j]  1:n  u  (w) −  �  [0,1]d C(w)dν  X 1:n  u  (w)  ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The term Kn is smaller than K′  n + K′′  n, where  K′  n =  sup  u∈[0,1]d  �����  �  [0,1]d Cn(0, 1, w)dν  V #,[j]  1:n  u  (w) − Cn(0, 1, u)  ����� ,  K′′  n =  sup  u∈[0,1]d  �����Cn(0, 1, u) −  �  [0,1]d Cn(0, 1, w)dν  X 1:n  u  (w)  ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  From (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1, K′′  n = oP(1) and, proceeding again as in the proof  of the latter lemma, it can be verified that K′  n = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The term Ln is smaller  than L′  n + L′′  n, where  L′  n = √n  sup  u∈[0,1]d  �����  �  [0,1]d C(w)dν  V #,[j]  1:n  u  (w) − C(u)  ����� ,  L′′  n = √n  sup  u∈[0,1]d  �����C(u) −  �  [0,1]d C(w)dν  X 1:n  u  (w)  ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The term L′′  n converges almost surely to zero as a consequence of (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4) in  Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The proof of the latter result can be adapted to verify that the term  L′  n also converges almost surely to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Hence, (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8) holds, which completes  the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ■  Appendix D: Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3  The proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3 is based on the following lemma which we prove first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Under Conditions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8 (the latter is implied by Condi-  tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9), for any b ∈ N,  sup  (s,t,u)∈Λ×[0,1]d  ���ˆB  [b],ν  n  (s, t, u) − ˆB  [b]  n (s, t, u)  ��� = oP(1),  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1)  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  47  sup  (s,t,u)∈Λ×[0,1]d  ��ˇB  [b],ν  n  (s, t, u) − ˇB  [b]  n (s, t, u)  �� = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Fix b ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' We first prove (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Starting from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5), we have that  sup  (s,t,u)∈Λ×[0,1]d  ���ˆB  [b],ν  n  (s, t, u) − ˆB  [b]  n (s, t, u)  ���  =  sup  (s,t,u)∈Λ×[0,1]d  �����  �  [0,1]d  ˆB  [b]  n (s, t, w)dν  X 1:n  u  (w) − ˆB  [b]  n (s, t, u)  ����� = oP(1),  where the last equality follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 since ˆB[b]  n ⇝ BC in  ℓ∞(Λ × [0, 1]d), where BC is defined in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11) and has continuous trajectories  almost surely under Condition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Under Condition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 (ii), the latter is a  consequence of Lemmas D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 in B¨ucher and Kojadinovic (2016) as well  as Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 in B¨ucher and Kojadinovic (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Under Condition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 (i), one  can rely on Theorem 1 of Holmes, Kojadinovic and Quessy (2013) instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The proof of (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Starting from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6), we have that  sup  (s,t,u)∈Λ×[0,1]d  ��ˇB  [b],ν  n  (s, t, u) − ˇB  [b]  n (s, t, u)  ��  =  sup  (s,t,u)∈Λ×[0,1]d  �����  �  [0,1]d  ˇB  [b]  n (s, t, w)dν  X ⌊ns⌋+1:⌊nt⌋  u  (w) − ˇB  [b]  n (s, t, u)  ����� = oP(1),  where the last equality follows from (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) in Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 since ˇB[b]  n ⇝ BC in  ℓ∞(Λ × [0, 1]d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Under Condition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 (ii), the latter is a consequence of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3)  in the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3 in B¨ucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (2014) and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 in  B¨ucher and Kojadinovic (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Under Condition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 (i), one can rely again on  Theorem 1 of Holmes, Kojadinovic and Quessy (2013) instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ■  Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Fix b ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' We only prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10), the proof of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9)  being simpler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Starting from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8), we have that  sup  (s,t)∈Λ  u∈[0,1]d  ��ˇC  [b]  n (s, t, u) − ˇC  [b],ν  n  (s, t, u)  �� ≤  sup  (s,t)∈Λ  u∈[0,1]d  ��ˇB  [b]  n (s, t, u) − ˇB  [b],ν  n  (s, t, u)  ��  +  d  �  j=1  sup  (s,t)∈Λ  u∈[0,1]d  ��� ˙Cj,⌊ns⌋+1:⌊nt⌋(u)  ���  sup  (s,t)∈Λ  u∈[0,1]d  ���ˇB  [b]  n (s, t, u(j)) − ˇB  [b],ν  n  (s, t, u(j))  ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The terms on the right-hand side of the previous display converge to zero  in probability as a consequence of (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) in Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1 and the fact that  sup(s,t,u)∈Λ×[0,1]d  ��� ˙Cj,⌊ns⌋+1:⌊nt⌋(u)  ��� ≤ ζ from Condition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The last two claims of the theorem are an immediate consequence of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9),  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10) and straightforward extensions of Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3 in B¨ucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (2014) for non-smooth multiplier replicates based on arbitrary partial derivative  estimators satisfying Condition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ■  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  48  Appendix E: Proofs of Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10  Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Let f be any function from [0, 1]d to [0, 1], let m ∈ N, m ≥ 2  and recall that, for any u ∈ [0, 1]d, µm,u is the law of the random vector  (Sm,1,u1/m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , Sm,d,ud/m), where Sm,1,u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Sm,d,ud are independent, and for  each k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, Sm,k,uk is Binomial(m, uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Moreover, recall that, for any  j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, ˜µj,m,u is the law of the random vector ( ˜Sm,1,u1/m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , ˜Sm,d,ud/m)  whose components are independent and, for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d} \\ {j}, ˜Sm,i,ui is  Binomial(m, ui), whereas ˜Sm,j,uj is Binomial(m − 1, uj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then, for any j ∈  {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d} and u ∈ [0, 1]d such that uj ∈ (0, 1),  ∂uj  ��  [0,1]d f(w)dµm,u(w)  �  = m  �  [0,1]d {f(w + ej/m) − f(w)} d˜µj,m,u(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Fix m ≥ 2 and, without loss of generality, fix j = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Also, for any  u ∈ [0, 1], let bm,u(s) =  �m  s  �  us(1−u)m−s, s ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then, for all u ∈ [0, 1]d  such that u1 ∈ (0, 1),  ∂u1  ��  [0,1]d f(w)dµm,u(w)  �  = ∂u1  �  �  �  m  �  s1=0  · ·  m  �  sd=0  f  �s1  m , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , sd  m  �  d  �  j=1  bm,uj(sj)  �  �  �  =  m  �  s1=0  · ·  m  �  sd=0  f  �s1  m , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , sd  m  �  ∂u1bm,u1(s1)  d  �  j=2  bm,uj(sj)  =  m  �  s1=0  · ·  m  �  sd=0  f  �s1  m , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , sd  m  �  ×  �  m  s1  �  �  s1us1−1  1  (1 − u1)m−s1 − (m − s1)us1  1 (1 − u1)m−s1−1�  d  �  j=2  bm,uj(sj)  = m  m  �  s1=1  · ·  m  �  sd=0  f  �s1  m , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , sd  m  �  (m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (s1 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (m − s1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='us1−1  1  (1 − u1)m−s1  d  �  j=2  bm,uj(sj)  − m  m−1  �  s1=0  · ·  m  �  sd=0  f  �s1  m , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , sd  m  �  (m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  s1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (m − s1 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='us1  1 (1 − u1)m−s1−1  d  �  j=2  bm,uj(sj)  = m  m−1  �  s1=0  · ·  m  �  sd=0  f  �s1 + 1  m  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , sd  m  �  (m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  s1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (m − s1 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='us1  1 (1 − u1)m−s1−1  d  �  j=2  bm,uj(sj)  − m  m−1  �  s1=0  · ·  m  �  sd=0  f  �s1  m , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , sd  m  �  (m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  s1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (m − s1 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='us1  1 (1 − u1)m−s1−1  d  �  j=2  bm,uj(sj)  = m  m−1  �  s1=0  · ·  m  �  sd=0  �  f  �s1 + 1  m  , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , sd  m  �  − f  �s1  m , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , sd  m  ��  bm−1,u1(s1)  d  �  j=2  bm,uj(sj)  = m  �  [0,1]d {f(w + e1/m) − f(w)} d˜µ1,m,u(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ■  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  49  Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Fix j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, u ∈ [0, 1]d and m ≥ 2, and  recall the definition of the measure ˜µj,m,u given in Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13), we  have that  ˙C  Bern  j,k:l,m(u) = m  �  [0,1]d Ck:l(w+ej/m)d˜µj,m,u(w)−m  �  [0,1]d Ck:l(w)d˜µj,m,u(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1)  Let ˜S = ( ˜Sm,1,u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , ˜Sm,d,ud) so that ˜S/m is a random vector with law ˜µj,m,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Then, combined with the definition of Ck:l in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1), the first integral on the  right-hand side of (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) can be rewritten as  �  [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d  1  l − k + 1  l  �  i=k  1  �  Rk:l  i /(l − k + 1) ≤ w + ej/m  �  d˜µj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='u(w)  =  1  l − k + 1  l  �  i=k  �  [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d 1  �  Rk:l  i /(l − k + 1) − ej/m ≤ w  �  d˜µj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='u(w)  =  1  l − k + 1  l  �  i=k  P  �  ˜S ≥ mRk:l  i /(l − k + 1) − ej | X k:l  �  =  1  l − k + 1  l  �  i=k  P  �  ˜Sm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='uj ≥ mRk:l  ij /(l − k + 1) − 1 | X k:l  �  ×  d  �  t=1  t̸=j  P  �  ˜Sm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ut ≥ mRk:l  it /(l − k + 1) | X k:l  �  =  1  l − k + 1  l  �  i=k  P  �  ˜Sm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='uj > ⌈mRk:l  ij /(l − k + 1) − 1⌉ − 1 | X k:l  �  ×  d  �  t=1  t̸=j  P  �  ˜Sm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ut > ⌈mRk:l  it /(l − k + 1)⌉ − 1 | X k:l  �  =  1  l − k + 1  l  �  i=k  ¯Bm−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='uj  �  ⌈mRk:l  ij /(l − k + 1)⌉ − 2  �  ×  d  �  t=1  t̸=j  ¯Bm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ut  �  ⌈mRk:l  it /(l − k + 1)⌉ − 1  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  where we have used the fact that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' for any t ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d} and x ∈ R, P( ˜Sm,t,ut ≥  x) = P( ˜Sm,t,ut > ⌈x⌉ − 1) and ⌈x − 1⌉ = ⌈x⌉ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Similarly, for the second  integral on the right-hand side of (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' we have  �  [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]dCk:l(w)d˜µj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='u(w)  /Resampling techniques for smooth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' possibly data-adaptive empirical copulas  50  =  1  l − k + 1  �  [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d  l  �  i=k  1  �  Rk:l  i /(l − k + 1) ≤ w  �  d˜µj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='u(w)  =  1  l − k + 1  l  �  i=k  P  �  ˜S ≥ mRk:l  i /(l − k + 1) | X k:l  �  =  1  l − k + 1  l  �  i=k  d  �  t=1  P  �  ˜Sm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ut ≥ mRk:l  it /(l − k + 1) | X k:l  �  =  1  l − k + 1  l  �  i=k  ¯Bm−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='uj  �  ⌈mRk:l  ij /(l − k + 1)⌉ − 1  �  ×  d  �  t=1  t̸=j  ¯Bm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ut  �  ⌈mRk:l  it /(l − k + 1)⌉ − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The desired result finally follows by noticing that  ¯Bm−1,uj  �  ⌈mRk:l  ij /(l − k + 1)⌉ − 2  �  − ¯Bm−1,uj  �  ⌈mRk:l  ij /(l − k + 1)⌉ − 1  �  = Bm−1,uj  �  ⌈mRk:l  ij /(l − k + 1)⌉ − 1  �  − Bm−1,uj  �  ⌈mRk:l  ij /(l − k + 1)⌉ − 2  �  = bm−1,uj  �  ⌈mRk:l  ij /(l − k + 1)⌉ − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ■  Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Under Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4, for any j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, δ ∈ (0, 1) and  ε ∈ (0, 1/2), with probability 1,  sup  (s,t)∈Λ  t−s≥δ  sup  u∈[0,1]d  uj∈[ε,1−ε]  ��� ˙C  ν,∇  j,⌊ns⌋+1:⌊nt⌋(u) − ˙C  ν,∆  j,⌊ns⌋+1:⌊nt⌋(u)  ��� = o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Fix j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, δ ∈ (0, 1) as well as ε ∈ (0, 1/2) and assume that  n is large enough so that, for any (s, t) ∈ Λ such that t − s > δ, L2b⌊nt⌋−⌊ns⌋  and L2b′  ⌊nt⌋−⌊ns⌋ are smaller than ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then, using the fact that Cν  k:l in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2) is  between 0 and 1, we obtain that, with probability 1,  sup  (s,t)∈Λ  t−s≥δ  sup  u∈[0,1]d  uj∈[ε,1−ε]  ��� ˙C  ν,∇  j,⌊ns⌋+1:⌊nt⌋(u) − ˙C  ν,∆  j,⌊ns⌋+1:⌊nt⌋(u)  ���  ≤  sup  (s,t)∈Λ  t−s≥δ  sup  u∈[0,1]d  uj∈[ε,1−ε]  ����  1  h + h′ −  1  (uj + h) ∧ 1 − (uj − h′) ∨ 0  ���� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ■  Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Fix j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d} and let us first prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='15)  by proceeding along the lines of the proof of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4) in B¨ucher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  51  From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9), notice that, for any (s, t, u) ∈ Λ × [0, 1]d such that ⌊ns⌋ < ⌊nt⌋,  Cν  ⌊ns⌋+1:⌊nt⌋(u) = C(u) +  1  √nλn(s, t)Cν  n(s, t, u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Fix δ ∈ (0, 1) and notice that, by Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4,  dn =  sup  (s,t)∈Λ  t−s≥δ  (b⌊nt⌋−⌊ns⌋ + b′  ⌊nt⌋−⌊ns⌋) ≤  sup  k≥⌊nδ⌋−1  (bk + b′  k) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2)  Next, fix ε ∈ (0, 1/2) and assume that n is large enough so that, for any t−s > δ,  L2b⌊nt⌋−⌊ns⌋ and L2b′  ⌊nt⌋−⌊ns⌋ are smaller than ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Then, for any t − s > δ and  u ∈ [0, 1]d such that uj ∈ [ε, 1 − ε],  ˙C  ν,∆  j,⌊ns⌋+1:⌊nt⌋(u) =  1  h + h′ {C(u + hej) − C(u − h′ej)}  +  1  (h + h′)√nλn(s, t) {Cν  n(s, t, u + hej) − Cν  n(s, t, u − h′ej)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3)  Since, by Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8, ˙Cj exists (and is continuous) on the set {u ∈ [0, 1]d :  uj ∈ [ε/2, 1 − ε/2]}, from the mean value theorem, for any t − s > δ and  u ∈ [0, 1]d such that uj ∈ [ε, 1 − ε],  1  h + h′ {C(u + hej) − C(u − h′ej)} = ˙Cj(u∗  n,s,t),  where u∗  n,s,t is between u − h′ej and u + hej almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Hence, with prob-  ability 1,  sup  (s,t)∈Λ  t−s≥δ  sup  u∈[0,1]d  uj∈[ε,1−ε]  ����  1  h + h′ {C(u + hej) − C(u − h′ej)} − ˙Cj(u)  ����  =  sup  (s,t)∈Λ  t−s≥δ  sup  u∈[0,1]d  uj∈[ε,1−ε]  ��� ˙Cj(u∗  n,s,t) − ˙Cj(u)  ��� ≤  sup  (u,v)∈[0,1]2d  uj,vj∈[ε/2,1−ε/2]  |u−v|∞≤L2dn  ��� ˙Cj(u) − ˙Cj(v)  ��� → 0,  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4)  where dn is defined in (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Furthermore, since, by Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9 and as a result  of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10, Cν  n is asymptotically uniformly equicontinuous in probability,  we have that  sup  (s,t)∈Λ  t−s≥δ  sup  u∈[0,1]d  uj∈[ε,1−ε]  |Cν  n(s, t, u + hej) − Cν  n(s, t, u − h′ej)|  ≤  sup  (s,t)∈Λ  t−s≥δ  sup  (u,v)∈[0,1]2d  uj,vj∈[ε/2,1−ε/2]  |u−v|∞≤L2dn  |Cν  n(s, t, u) − Cν  n(s, t, v)| = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5)  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  52  The fact that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='15) holds is then an immediate consequence of (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3), (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4),  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='5) and the fact that, from Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4,  sup  (s,t)∈Λ  t−s≥δ  1  (h + h′) √nλn(s, t) ≤  sup  (s,t)∈Λ  t−s≥δ  1  L1(b⌊nt⌋−⌊ns⌋ + b′  ⌊nt⌋−⌊ns⌋)√nλn(s, t)  ≤  sup  (s,t)∈Λ  t−s≥δ  1  L1(⌊nt⌋ − ⌊ns⌋)−1/2√nλn(s, t)  =  sup  (s,t)∈Λ  t−s≥δ  1  L1  �  λn(s, t)  ≤  1  L1  �  δ − 1/n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  The claim for ˙C  ν,∇  j,k:l (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' for ˙C  ν,∆  j,k:l and ˙C  ν,∇  j,k:l) follows from Lemma E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  the continuous mapping theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ■  Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Fix j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, δ ∈ (0, 1) and ε ∈ (0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' We  first prove (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='12) and the triangle inequality, we have that  sup  (s,t)∈Λ  t−s≥δ  sup  u∈[0,1]d  uj∈[ε,1−ε]  ��� ˙C  ∆,ν  j,⌊ns⌋+1:⌊nt⌋(u) − ˙Cj(u)  ��� ≤ Ij,n,δ,ε + Jj,n,δ,ε,  where  Ij,n,δ,ε =  sup  (s,t)∈Λ  t−s≥δ  sup  u∈[0,1]d  uj∈[ε,1−ε]  �����  �  [0,1]d  �  ˙C  ∆  j,⌊ns⌋+1:⌊nt⌋(w) − ˙Cj(w)  �  dν  X ⌊ns⌋+1:⌊nt⌋  u  (w)  ����� ,  Jj,n,δ,ε =  sup  (s,t)∈Λ  t−s≥δ  sup  u∈[0,1]d  uj∈[ε,1−ε]  �����  �  [0,1]d  ˙Cj(w)dν  X ⌊ns⌋+1:⌊nt⌋  u  (w) − ˙Cj(u)  ����� ,  where ˙C  ∆  j,k:l is defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' We shall now show that both Ij,n,δ,ε = oP(1)  and Jj,n,δ,ε = oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Term Ij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ε: From the triangle inequality and the fact that 0 ≤ ˙C  ∆  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='k:l ≤ 1  and 0 ≤ ˙Cj ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' we have that Ij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ε is smaller than  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  u∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  ������  �  {w∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d:  wj∈[ε/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε/2]}  �  ˙C  ∆  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='⌊ns⌋+1:⌊nt⌋(w) − ˙Cj(w)  �  dν  X ⌊ns⌋+1:⌊nt⌋  u  (w)  ������  +  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  u∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  ������  �  {w∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d:  wj<ε/2}  �  ˙C  ∆  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='⌊ns⌋+1:⌊nt⌋(w) − ˙Cj(w)  �  dν  X ⌊ns⌋+1:⌊nt⌋  u  (w)  ������  +  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  u∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  ������  �  {w∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d:  wj>1−ε/2}  �  ˙C  ∆  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='⌊ns⌋+1:⌊nt⌋(w) − ˙Cj(w)  �  dν  X ⌊ns⌋+1:⌊nt⌋  u  (w)  ������  ≤ I′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ε + I′′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ε + I′′′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  /Resampling techniques for smooth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' possibly data-adaptive empirical copulas  53  where  I′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ε = sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  w∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d  wj∈[ε/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε/2]  ��� ˙C  ∆  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='⌊ns⌋+1:⌊nt⌋(w) − ˙Cj(w)  ��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  I′′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ε = sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  u∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  ν  X ⌊ns⌋+1:⌊nt⌋  u  �  w ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' 1]d : wj < ε/2  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  I′′′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ε = sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  u∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  ν  X ⌊ns⌋+1:⌊nt⌋  u  �  w ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' 1]d : wj > 1 − ε/2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  We have that I′  j,n,δ,ε = oP(1) as a consequence of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' We shall  now show that both I′′  j,n,δ,ε and I′′′  j,n,δ,ε converge almost surely to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' To do  so, it suffices to show that I′′  j,n,δ,ε and I′′′  j,n,δ,ε converge to zero conditionally  on X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' for almost any sequence X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Concerning I′′  j,n,δ,ε, using  Chebyshev’s inequality and Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8, for almost any sequence X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' ,  conditionally on X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' we obtain that  I′′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ε =  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  P  �  W  X ⌊ns⌋+1:⌊nt⌋  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='uj  < ε/2 | X ⌊ns⌋+1:⌊nt⌋  �  =  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  P  �  W  X ⌊ns⌋+1:⌊nt⌋  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='uj  − uj < ε/2 − uj | X ⌊ns⌋+1:⌊nt⌋  �  ≤  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  P  �  −  ���W  X ⌊ns⌋+1:⌊nt⌋  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='uj  − uj  ��� ≤ −uj + ε/2 | X ⌊ns⌋+1:⌊nt⌋  �  ≤  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  Var  �  W  X ⌊ns⌋+1:⌊nt⌋  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='uj  | X ⌊ns⌋+1:⌊nt⌋  �  (uj − ε/2)2  ≤  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  a⌊nt⌋−⌊ns⌋  (uj − ε/2)2 ≤  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  a⌊nt⌋−⌊ns⌋  sup  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  1  (uj − ε/2)2  ≤ 4  ε2  sup  k≥⌊nδ⌋−1  ak → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Similarly, concerning I′′′  j,n,δ,ε, for almost any sequence X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , conditionally  on X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' we obtain that  I′′′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ε =  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  P  �  W  X ⌊ns⌋+1:⌊nt⌋  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='uj  > 1 − ε/2 | X ⌊ns⌋+1:⌊nt⌋  �  =  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  P  �  W  X ⌊ns⌋+1:⌊nt⌋  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='uj  − uj > 1 − ε/2 − uj | X ⌊ns⌋+1:⌊nt⌋  �  ≤  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  P  ����W  X ⌊ns⌋+1:⌊nt⌋  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='uj  − uj  ��� ≥ 1 − ε/2 − uj | X ⌊ns⌋+1:⌊nt⌋  �  ≤  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  Var  �  W  X ⌊ns⌋+1:⌊nt⌋  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='uj  | X ⌊ns⌋+1:⌊nt⌋  �  (1 − ε/2 − uj)2  /Resampling techniques for smooth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' possibly data-adaptive empirical copulas  54  ≤  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  a⌊nt⌋−⌊ns⌋  (1 − ε/2 − uj)2  ≤  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  a⌊nt⌋−⌊ns⌋  sup  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  1  (1 − ε/2 − uj)2 ≤ 4  ε2  sup  k≥⌊nδ⌋−1  ak → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Term Jj,n,δ,ε: Let η > 0 and let us show that Jj,n,δ,ε ≤ η for n sufficiently  large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For any ρ ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' from the triangle inequality and the fact that 0 ≤ ˙Cj ≤  1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' we have that Jj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ε is smaller than  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  u∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  �����  �  {w∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d:|w−u|∞≤ρ}  { ˙Cj(w) − ˙Cj(u)}dν  X ⌊ns⌋+1:⌊nt⌋  u  (w)  �����  + sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  u∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  �����  �  {w∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d:|w−u|∞>ρ}  { ˙Cj(w) − ˙Cj(u)}dν  X ⌊ns⌋+1:⌊nt⌋  u  (w)  �����  ≤ J′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ρ + J′′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  where  J′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ρ =  sup  u∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  sup  w∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d  |w−u|∞≤ρ  ��� ˙Cj(w) − ˙Cj(u)  ��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  J′′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ρ =  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  u∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d  �  [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d 1{|w − u|∞ > ρ}dν  X ⌊ns⌋+1:⌊nt⌋  u  (w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  From Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8, ˙Cj is uniformly continuous on the set {u ∈ [0, 1]d : uj ∈  [ε/2, 1 − ε/2]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' We then choose ρ = ρ(ε, η) > 0 sufficiently small such that  J′  j,ε,ρ =  sup  u∈[0,1]d  uj∈[ε,1−ε]  sup  w∈[0,1]d  |w−u|∞≤ρ  ��� ˙Cj(w) − ˙Cj(u)  ��� ≤ η  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6)  As far as J′′  j,n,δ,ρ is concerned, using Chebyshev’s inequality and Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8,  for almost any sequence X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , conditionally on X1, X2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' we obtain  that  J′′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ρ =  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  u∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d ν  X ⌊ns⌋+1:⌊nt⌋  u  ({w ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' 1]d : |u − w|∞ > ρ})  =  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  u∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d P  �  �  d�  j=1  ����W  X ⌊ns⌋+1:⌊nt⌋  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='uj  − uj  ��� > ρ  �  | X ⌊ns⌋+1:⌊nt⌋  �  �  ≤  d  �  j=1  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  u∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d P  ����W  X ⌊ns⌋+1:⌊nt⌋  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='uj  − uj  ��� > ρ | X ⌊ns⌋+1:⌊nt⌋  �  /Resampling techniques for smooth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' possibly data-adaptive empirical copulas  55  ≤  d  �  j=1  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  u∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d  Var  �  W  X ⌊ns⌋+1:⌊nt⌋  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='uj  | X ⌊ns⌋+1:⌊nt⌋  �  ρ2  ≤ d  ρ2  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  a⌊nt⌋−⌊ns⌋ ≤ d  ρ2  sup  k≥⌊nδ⌋−1  ak → 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  which implies that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' for n sufficiently large,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' with probability 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' J′′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ρ ≤ η/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Using additionally (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='6), we obtain that Jj,n,δ,ε converges almost surely to zero,  which concludes the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The proof of the analogous result for ˙C  ∇,ν  j,k:l  in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='11) is almost identical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  ■  Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Fix j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d}, δ ∈ (0, 1) and ε ∈ (0, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  From (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='13), we have that, for any (s, t) ∈ Λ and u ∈ [0, 1]d,  ˙C  Bern  j,⌊ns⌋+1:⌊nt⌋,m⌊nt⌋−⌊ns⌋(u)  =  �  [0,1]d  ˙C  ∇  j,⌊ns⌋+1:⌊nt⌋,1/m⌊nt⌋−⌊ns⌋,0(w)d˜µj,m⌊nt⌋−⌊ns⌋,u(w),  where ˙C∇  j,k:l,1/m,0 is defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1) and, for any m ≥ 2, ˜µj,m,u is the law of the  random vector ( ˜Sm,1,u1/m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , ˜Sm,d,ud/m) whose components are independent  such that, for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' , d} \\ {j}, ˜Sm,i,ui is Binomial(m, ui) while ˜Sm,j,uj is  Binomial(m − 1, uj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' It follows that, for any (s, t) ∈ Λ and u ∈ [0, 1]d,  ˙C  Bern  j,⌊ns⌋+1:⌊nt⌋,m⌊nt⌋−⌊ns⌋(u)  =  �  Wj,n,s,t  ˙C  ∇  j,⌊ns⌋+1:⌊nt⌋,1/m⌊nt⌋−⌊ns⌋,0(w)d˜µj,m⌊nt⌋−⌊ns⌋,u(w),  (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7)  where Wj,n,s,t = {w ∈ [0, 1]d : wj ≤ 1 − 1/m⌊nt⌋−⌊ns⌋}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' For the sake of a more  compact notation, from now on, we shall write ms,t for m⌊nt⌋−⌊ns⌋, (s, t) ∈  Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' From the triangle inequality, the left-hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='17) is smaller than  Ij,n,δ,ε + Jj,n,δ,ε, where  Ij,n,δ,ε =  sup  (s,t)∈Λ  t−s≥δ  sup  u∈[0,1]d  uj∈[ε,1−ε]  �����  �  Wj,n,s,t  �  ˙C  ∇  j,⌊ns⌋+1:⌊nt⌋,1/ms,t,0(w)  −  ˙Cj(w)  �  d˜µj,ms,t,u(w)  ��� ,  Jj,n,δ,ε =  sup  (s,t)∈Λ  t−s≥δ  sup  u∈[0,1]d  uj∈[ε,1−ε]  �����  �  [0,1]d  ˙Cj(w)d˜µj,ms,t,u(w) − ˙Cj(u)  ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  For any n ∈ N, x ∈ (Rd)n and u ∈ [0, 1]d, let νx  u = ˜µj,⌊Lnθ⌋∨2,u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' With this  notation, Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8 holds for the considered smoothing distributions and  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  56  it can be verified that Jj,n,δ,ε = oP(1) by proceeding exactly as in the proof of  Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9 for the analogous term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' It thus remain to show that Ij,n,δ,ε =  oP(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  From the triangle inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' we have that Ij,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ε is smaller than  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  u∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  ������  �  {w∈Wj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t:  wj∈[ε/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε/2]}  �  ˙C  ∇  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='⌊ns⌋+1:⌊nt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1/ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0(w) − ˙Cj(w)  �  d˜µj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='u(w)  ������  +  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  u∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  ������  �  {w∈Wj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t:  wj<ε/2}  �  ˙C  ∇  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='⌊ns⌋+1:⌊nt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1/ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0(w) − ˙Cj(w)  �  d˜µj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='u(w)  ������  +  sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  u∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  ������  �  {w∈Wj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t:  wj>1−ε/2}  �  ˙C  ∇  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='⌊ns⌋+1:⌊nt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1/ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0(w) − ˙Cj(w)  �  d˜µj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='u(w)  ������  ≤ I′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ε + MnI′′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ε + MnI′′′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  where  I′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ε = sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  w∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d  wj∈[ε/2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε/2]  ��� ˙C  ∇  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='⌊ns⌋+1:⌊nt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1/ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0(w) − ˙Cj(w)  ��� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  I′′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ε = sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  u∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  ˜µj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='u  �  w ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' 1]d : wj < ε/2  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  I′′′  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ε = sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  t−s≥δ  sup  u∈[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1]d  uj∈[ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1−ε]  ˜µj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='u  �  w ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' 1]d : wj > 1 − ε/2  �  ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Mn = sup  (s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t)∈Λ  sup  w∈Wj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t  ��� ˙C  ∇  j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='⌊ns⌋+1:⌊nt⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1/ms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='0(w)  ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Since the conditions of the proposition imply that Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='4 holds with  h(x) = 1/(⌊Lnθ⌋ ∨ 2) and h′(x) = 0 for all n ∈ N and x ∈ (Rd)n, we have that  I′  j,n,δ,ε = oP(1) as a consequence of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Also, given that Condition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='8  holds for the considered smoothing distributions, it can be verified that I′′  j,n,δ,ε  and I′′′  j,n,δ,ε converge almost surely to zero by proceeding exactly as in the proof  of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='9 for the analogous terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' To complete the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='17), it  suffices to show that, there exists a constant ζ > 0 such that, for any n ∈ N,  Mn < ζ almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='  Fix n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' From the adopted conventions, we have that ˙C∇  j,⌊ns⌋+1:⌊nt⌋,1/ms,t,0 =  0 for all (s, t) ∈ Λ such that ⌊ns⌋ = ⌊nt⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' Fix (s, t) ∈ Λ such that ⌊ns⌋ < ⌊nt⌋  and let p = ⌊nt⌋ − ⌊ns⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The empirical copula C⌊ns⌋+1:⌊nt⌋, generically defined  in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='1), is a multivariate d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' whose d univariate margins, under Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='3,  are all equal to G⌊ns⌋+1:⌊nt⌋, where G⌊ns⌋+1:⌊nt⌋(u) = ⌊pu⌋/p, u ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' As a  consequence of a well-known property of multivariate d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='s (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=', Durante  /Resampling techniques for smooth, possibly data-adaptive empirical copulas  57  and Sempi, 2015, Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='14), we have that  ��C⌊ns⌋+1:⌊nt⌋(u) − C⌊ns⌋+1:⌊nt⌋(v)  �� ≤  d  �  j=1  ��G⌊ns⌋+1:⌊nt⌋(uj) − G⌊ns⌋+1:⌊nt⌋(vj)  ��  for all u, v ∈ [0, 1]d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' We then obtain that, for any u ∈ Wj,n,s,t,  ��C⌊ns⌋+1:⌊nt⌋(u + ej/ms,t) − C⌊ns⌋+1:⌊nt⌋(u)  ��  ≤  ��G⌊ns⌋+1:⌊nt⌋(uj + 1/ms,t) − G⌊ns⌋+1:⌊nt⌋(uj)  �� ,  which implies that  ��� ˙Cj,⌊ns⌋+1:⌊nt⌋(u)  ��� ≤  ��G⌊ns⌋+1:⌊nt⌋(uj + 1/ms,t) − G⌊ns⌋+1:⌊nt⌋(uj)  ��  1/ms,t  = ms,t  �⌊p(uj + 1/ms,t)⌋  p  − ⌊puj⌋  p  �  ≤ ms,t  �p(uj + 1/ms,t)  p  − puj − 1  p  �  ≤ ms,t  � 1  ms,t  + 1  p  �  ≤ 1 + ms,t  p  = 1 + ⌊Lpθ⌋ ∨ 2  p  ≤ 1 + Lpθ−1 ∨ (2/p) ≤ 1 + L ∨ 2,  which completes the proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content=' The fact that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='18) holds is finally an  immediate consequence of the previous centered display and (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
+page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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+page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/z9E5T4oBgHgl3EQfOQ4n/content/2301.05495v1.pdf'}
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diff --git a/zNE5T4oBgHgl3EQfNw6V/content/tmp_files/2301.05492v1.pdf.txt b/zNE5T4oBgHgl3EQfNw6V/content/tmp_files/2301.05492v1.pdf.txt
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+Disentangled Representation for Diversified
+Recommendations
+Xiaoying Zhang
+zhangxiaoying.xy@bytedance.com
+AI Lab, Bytedance Inc.
+Hongning Wang
+hw5x@virginia.edu
+Department of Computer Science
+University of Virginia, USA
+Hang Li
+lihang.lh@bytedance.com
+AI Lab, Bytedance Inc.
+Abstract
+Accuracy and diversity have long been considered to be two conflicting goals for recom-
+mendations. We point out, however, that as the diversity is typically measured by certain
+pre-selected item attributes, e.g., category as the most popularly employed one, improved
+diversity can be achieved without sacrificing recommendation accuracy, as long as the di-
+versification respects the user’s preference about the pre-selected attributes. This calls for a
+fine-grained understanding of a user’s preferences over items, where one needs to recognize
+the user’s choice is driven by the quality of the item itself, or the pre-selected attributes of
+the item.
+In this work, we focus on diversity defined on item categories. We propose a general di-
+versification framework agnostic to the choice of recommendation algorithms. Our solution
+disentangles the learnt user representation in the recommendation module into category-
+independent and category-dependent components to differentiate a user’s preference over
+items from two orthogonal perspectives. Experimental results on three benchmark datasets
+and online A/B test demonstrate the effectiveness of our solution in improving both rec-
+ommendation accuracy and diversity. In-depth analysis suggests that the improvement is
+due to our improved modeling of users’ categorical preferences and refined ranking within
+item categories.
+Keywords:
+Recommender system, recommendation diversity, disentangled user repre-
+sentation
+1. Introduction
+Recommender systems learn users’ interests from historical observations (e.g., their clicks,
+bookmarked or purchased items, etc.)
+so as to identify the items that best suit users’
+preferences. The success of recommender system in enhancing user experience and boosting
+platform utility has been witnessed in a number of scenarios including e-commerce (Zhou
+et al., 2018; He and Chua, 2017), online news recommendation (Wu et al., 2019a) and
+streaming services (Covington et al., 2016).
+Recommendation accuracy, which measures whether a recommendation model can rec-
+ommend items that users will like, serves as the dominant target or even the only target
+in most previous work (Zhou et al., 2018; Wang et al., 2021; He and Chua, 2017; Guo
+et al., 2017; Covington et al., 2016). Various complicated models (Zhou et al., 2018; Guo
+et al., 2017; Covington et al., 2016) have been proposed for higher accuracy. While recom-
+mendation accuracy has been shown to be closely related to user satisfaction, it is never
+1
+arXiv:2301.05492v1  [cs.IR]  13 Jan 2023
+
+Figure 1: Illustration of recommendation accuracy and diversity optimization in different
+recommendation models.
+the only rule of thumb. Recent work found the recommendation diversity, which measures
+the dissimilarity among recommended items regarding certain pre-selected item attributes
+(e.g., item category) also plays an important role in the overall user experience (Wilhelm
+et al., 2018; Kapoor et al., 2015; Zhou et al., 2010). For example, even if a user is a fan of
+basketball, he/she can still get bored with recommendations only about basketball videos
+or news, which increases the risk of user attrition.
+Following previous work (Steck, 2018; Wang et al., 2021; Zheng et al., 2021a), we fo-
+cus on diversity defined on item categories in this paper and aim to address the so-called
+accuracy-diversity dilemma (Zheng et al., 2021a). On one hand, recommendation models
+with accuracy as their primary target often lose diversity to some extent, due to overly
+emphasizing items in the dominant categories in a user’s interaction history (Steck, 2018;
+Wang et al., 2021). Figure 1(a) illustrates this issue with an example in movie recommenda-
+tion, where 70% of the movies watched by a user are action movies, which leads 90% of the
+system’s recommendations to fall in the action movie category. Worse still, because of the
+feedback loop (Chaney et al., 2018), the emphasis on the dominant categories in the system’s
+recommendations will be further intensified when the user follows the recommendations,
+causing further decreased recommendation diversity and issues like filter bubbles (Nguyen
+et al., 2014) and echo chambers (Ge et al., 2020). On the other hand, simply diversifying
+recommendations over all item categories without considering the user’s categorical prefer-
+ence hurts the accuracy of generated recommendations (Wilhelm et al., 2018; Ziegler et al.,
+2005; Qin and Zhu, 2013; Zheng et al., 2021a). As shown in Figure 1(b), although the
+recommendation list is diverse by covering all four categories, negative feedback is more
+2
+
+Action movie
+Romancemovie
+Children's movie
+Documentary
+User browsing history
+Diversifying across
+Diversifying within the
+0.7
+distribution
+all item categories
+user's preferred categories
+0.2
+0.1
+Feedback loop
+Accuracy-targeted
+recommendation models
+KIDS
+X
+0.9
+distribution
+X
+0.1
+User feedback
+(a)
+(b)
+(c)likely on the categories where the user interacted less often or negative feedback already
+prevailed, e.g., children’s movies and documentary movies respectively in this example.
+Clearly one should not recklessly increase diversity. For categories the user is less likely
+to be interested in, the risk of making a bad recommendation overweights the benefit of
+increased diversity.
+Thus, this paper focuses on conducting diversification only among
+item categories that the user prefers, suggesting the possibility to improve recommendation
+diversity without sacrificing recommendation accuracy. Figure 1(c) gives an example rec-
+ommendation list following such a strategy, where the recommended items mainly fall in
+action and romance movies, the two preferred categories inferred from the user’s interaction
+history. This strategy requires the recommendation model to clearly distinguish whether the
+user’s positive/negative feedback is due to the item’s category or other category-independent
+features of the item (e.g., the item’s own quality), which was ignored by previous recom-
+mendation models.
+In this paper, we propose a general and model-agnostic framework to disentangle a
+user’s category-dependent and category-independent preferences for an accurate and diver-
+sified recommender system (DCRS). Specifically, DCRS takes a user’s preference over an
+item as a product of: (1) the user’s preference over the item’s category; and (2) the user’s
+preference over category-independent features of the item, e.g., the item’s quality. Such
+disentanglement suggests a hierarchical decision making process by the user: If a user has
+a strong preference over a particular category of items, he/she may still enjoy items of this
+category, even though their qualities are not perfect. However, if the probability that a
+user likes a category is low, only items of high quality in this category could have a chance
+to be considered. The disentanglement ensures items of the same quality, but in different
+categories that a user prefers similarly, have equal probabilities to be recommended. It
+naturally avoids overly recommending items from the dominant categories in the user’s in-
+teraction history. The main challenge therefore lies in how to disentangle a user’s preference
+regarding the aforementioned two orthogonal perspectives, given his/her preference over the
+item categories is not observable. This makes naive solutions like using different supervision
+signals to separately train users’ representations (Zheng et al., 2021b), or separating items’
+feature vectors into category dependent and independent segments, ineffective.
+DCRS is agnostic to the choice of recommendation module, which is supposed to learn
+informative representations of users and items. In particular, DCRS adopts a discriminator
+to disentangle the learnt representation into category-independent and category-dependent
+segments respectively. The recommendation module and discriminator are learnt simulta-
+neously to ensure the effectiveness of disentangled representation learning for accurate and
+diverse recommendations. To evaluate the proposed DCRS solution, we conduct both of-
+fline experiments on three benchmark datasets and online A/B test on Toutiao app, one of
+the largest news recommendation platforms in China. Experiment results demonstrate that
+DCRS can successfully recommend diverse items that users prefer, and thus improve both
+recommendation accuracy and diversity. In-depth analysis and case studies suggest strong
+evidence showing: (1) the disentangled category-independent representation from DCRS
+can distinguish the user’s preference within category more accurately; and (2) DCRS can
+capture a user’s diverse preferences in historical interactions more thoroughly. All codes
+and data can be found in https://github.com/Xiaoyinggit/DCRS.git.
+Overall, our contribution of this work is as follows:
+3
+
+• We demonstrate that accuracy and diversity are not conflicting goals for recommen-
+dation, as long as the diversification respects the user’s categorical preference.
+• To capture a user’s latent preferences on item categories more accurately, our pro-
+posed DCRS disentangles the user’s preference into category-dependent and category-
+independent components.
+• Experiments on three benchmark datasets and online A/B test demonstrate the effec-
+tiveness of DCRS in improving both recommendation accuracy and diversity. In-depth
+analysis further demonstrates the improvement comes from more accurate modeling
+of the user’s preference both over and within categories.
+2. Framework
+In this section, we describe how the proposed DCRS solution disentangles a user’s category
+dependent and independent preferences to simultaneously improve recommendation accu-
+racy and diversity. For the ease of illustration, we first briefly describe a general architecture
+which covers almost all popularly used recommendation models. We then depict how to
+smoothly integrate DCRS into such a general architecture to diversify its recommendations.
+2.1 Preliminary: A General Recommendation Architecture
+In a recommendation task, we are given a user behavior dataset X that contains interactions
+between N users and M items. The interaction between user u and item i is represented
+as a tuple (u, i, yu,i) ∈ X. Here yu,i ∈ {0, 1} denotes user u’s feedback to item i, where
+yu,i = 1 denotes positive feedback (e,g., a click or a positive rating), and yu,i = 0 denotes
+negative feedback. Generally speaking, a recommendation model will first learn a user-item
+representation to capture the user’s preference over the item:
+hu,i = f(u, i, θ) ∈ Rd,
+(1)
+where θ denotes a set of trainable parameters in the recommendation model.
+Various
+architectures (Zhou et al., 2018; Wang et al., 2021; He and Chua, 2017; Guo et al., 2017)
+have been proposed to implement f(u, i, θ), ranging from the simple matrix factorization
+algorithm (Mnih and Salakhutdinov, 2007) that directly takes the element-wise product of
+user and item embeddings to form the representation, to complex architectures such as the
+bi-interaction layer in NFM (He and Chua, 2017). Let ˆpu,i denote the probability that user
+u gives positive feedback to item i. The goal of the recommendation model is to use the
+learnt user-item representation to estimate ˆpu,i, either by directly summing up elements in
+hu,i as in matrix factorization, or through a learnable projection layer as follows:
+ˆpu,i = P (Yu,i = 1|u, i) = σ
+�
+W ⊤hu,i
+�
+,
+(2)
+where Yu,i is a random variable representing the feedback from user u on item i; W ∈ Rd×1
+is the learnable weight vector of the projection layer, and σ(·) is the sigmoid function. The
+parameters of the recommendation model are then optimized by minimizing the following
+4
+
+loss:
+L(X, θ, W ) =
+1
+|X|
+�
+(u,i,yu,i)∈X
+Lrec(yu,i, ˆpu,i),
+(3)
+where Lrec(·, ·) represents the chosen loss function. Various loss functions have been explored
+in literature, inlcuding cross entropy loss, Mean Squared Error (MSE) and BPR loss (Rendle
+et al., 2012). In this work, we will use the cross entropy loss by default.
+2.2 Disentangle Category Dependent and Independent Representations
+We consider a user’s feedback on an item as a mixture reflecting his/her preference over
+the item’s category and category-independent properties, e.g., the item’s intrinsic quality.
+As shown in Figure 2, the first action movie that receives positive feedback can very likely
+be caused by the user’s strong preference over the category of action movies, while his/her
+positive feedback on the second romance movie is more likely to be caused by its high quality
+that makes up the low probability that the user likes romance movies. In order to diversify
+the recommendations with respect to a user’s preferred categories, the recommendation
+model needs to clearly distinguish the effect of item category and other category-independent
+properties on a user’s decision making. To make our method description general enough
+to cover situations where an item can associate with multiple categories, we take item i’s
+category as the set that contains all categories that the item relates to, and denote it as ti.
+For example, assume there are three categories {c1, c2, c3} in a dataset. If item i is related
+to the first category, then ti = {c1}. And if item i is associated with the first two categories,
+then ti = {c1, c2}.
+We propose to disentangle a user’s preference over an item into two parts :
+• Category-dependent preference: it captures the user’s preference over the item’s
+category;
+• Category-independent preference: it depicts how category-independent features
+affect the user’s preference about the item.
+Such a disentanglement can be explained through a probabilistic view about the generation
+of user u’s feedback on item i. Let Y C
+u,i denote the binary random variable indicating user
+u’s feedback on item i’s category. We have the following,
+P(Yu,i = 1|u, i) = P(Yu,i = 1, Y C
+u,i = 1|u, i)
+(4a)
+= P(Yu,i = 1|u, i, Y C
+u,i = 1)P(Y C
+u,i = 1|u, i)
+(4b)
+= P(Yu,i = 1|u, i, Y C
+u,i = 1)P(Y C
+u,i = 1|u, ti)
+(4c)
+In particular, Eq.(4a) is due to the assumption that user u gives positive feedback to item i
+only if user u likes item i’s category, i.e., P(Yu,i = 1, Y C
+u,i = 1|u, i) = 1 and P(Yu,i = 1, Y C
+u,i =
+0|u, i) = 0. Eq.(4b) follows the chain rule. And Eq.(4c) is because Y C
+u,i only depends on the
+item’s category, instead of specific items.
+The first term in Eq.(4c) depicts how likely user u will give positive feedback to item
+i when he/she is interested in item i’s category; and the second term models how likely
+user u is interested in item i’s category. Given that user u likes the category of item i, the
+5
+
+Figure 2: Hierarchical decision making process of DCRS framework. Each feedback is de-
+termined by: (1) the user’s preference over the item’s category; and (2) the user’s
+preference over category-independent features of the item.
+probability in the first term only depends on the category-independent features of item i,
+such as item i’s quality, price, etc. Thus, under the disentangled user-item representations,
+we can compute the first term by the probability P(Y ⊥C
+u,i
+= 1|u, i), which depicts user u’s
+preference over item i driven by the category-independent features. Thus, Eq.(4c) can be
+rewritten as:
+P(Yu,i = 1|u, i) = P(Y ⊥C
+u,i
+= 1|u, i)P(Y C
+u,i = 1|u, ti).
+(5)
+Eq.(5) depicts a hierarchical decision making process illustrated in Figure 2. If user u likes
+item i’s category with a higher probability P(Y C
+u,i = 1|u, ti), he/she may still enjoy item
+i even though item i’s quality is not perfect, indicated by a lower P(Y ⊥C
+u,i
+= 1|u, i). For
+example, the positive feedback of the first action movie in Figure 2 is generated under such
+a scenario. Meanwhile, if there is only a small probability that user u would be interested in
+item i’s category (i.e., low P(Y C
+u,i = 1|u, ti)), item i must be of high quality to get positive
+feedback, i.e., high P(Y ⊥C
+u,i
+= 1|u, i). The positive feedback on the second romance movie
+in Figure 2 is a good example of this case.
+Eq.(5) also suggests why disentanglement makes recommendations diversified within a
+user’s preferred categories. Assume there are two categories c1 and c2 on which the user
+has similar preference. Instead of recommending more items from the dominant category
+(either c1 or c2), via the disentanglement in Eq.(5), items of the same quality within c1 and
+c2 will have an equal chance to be recommended, thus diversifying the recommendations.
+6
+
+Action movie
+Romancemovie
+Probability that the user
+recommendation list
+The user's preference
+likeeachcategory
+within category
+Probability
+0.9
+Probability
+0.3
+Items
+Itemcategory
+Probability
+ItemsUnfortunately, both terms in Eq.(5) cannot be learnt via direct supervision signals, since
+neither user u’s feedback on item i’s category nor feedback driven by category-independent
+features of item i can be observed. Classical solutions would appeal to Expectation Maxi-
+mization type algorithms (Dempster et al., 1977) to estimate the two terms in an iterative
+manner. However, given modern recommendation algorithms are usually realized via com-
+plex deep neural networks, posterior inference becomes cumbersome and also leads to slow
+convergence. Instead, DCRS implements Eq.(5) by simultaneously learning two disentan-
+gled representations for estimating the two terms separately. Specifically, DCRS learns two
+disentangled representations by:
+��
+h⊥C
+u,i
+�⊤ ,
+�
+hC
+u,i
+�⊤�⊤
+= f(u, i, θ) ∈ R2d,
+(6)
+where h⊥C
+u,i ∈ Rd aims to capture user u’s preference over category-independent features to
+estimate P(Y ⊥C
+u,i
+= 1|u, i), and hC
+u,i ∈ Rd depicts user u’s preference over item i’s category
+ti, aiming to estimate P(Y C
+u,i = 1|u, ti).
+Simply splitting item i’s feature vector into two parts, even with separate networks,
+cannot ensure complete disentanglement. Instead, in addition to requiring the learnt the
+representations to best capture the user’s preference, we employ an adversarial discriminator
+that enforces the learnt h⊥C
+u,i and hC
+u,i to be category-independent and category-dependent
+respectively.
+Discriminator Module. The discriminator D(·) acts as a category classifier, which takes
+one segment of disentangled representation, such as hC
+u,i or h⊥C
+u,i , as input, and aims to
+predict the category of item i (i.e., ti). However, it is hard for the discriminator to directly
+predict ti, since ti can take 2K-1 values, where K is the number of unique categories available
+in the dataset. For ease of learning, we represent ti by a vector over K unique categories,
+denoted as ˜ti.
+Again, assume there are three categories {c1, c2, c3}, if t1 = {c1}, then
+˜ti = [1, 0, 0]⊤. And if t1 = {c1, c2}, then ˜ti = [0.5, 0.5, 0]⊤. Specifically, when relevance
+between item i and each associated category can be measured (Pu et al., 2020), a more
+accurate ˜ti can be achieved by making the j-th element of ˜ti proportional to the relevance
+between item i and the j-th category. Otherwise, ˜ti can be simply assumed to be evenly
+distributed among related categories, which is also the default setting in our experiments.
+The discriminator then takes hC
+u,i or h⊥C
+u,i as input to predict ˜ti. In our experiments, the
+discriminator D(·) is implemented via a fully connected layer, and it should enforce the
+following:
+• Given hC
+u,i is closely related to item i’s category, the discriminator should predict ˜ti
+accurately based on hC
+u,i, i.e., the following loss should be minimized:
+min LC
+D(u, i) = LCE
+�
+D(hC
+u,i),˜ti
+�
+,
+where LCE is the cross entropy loss.
+• Given h⊥C
+u,i is independent from item category, h⊥C
+u,i should fool the discriminator by
+maximizing the classification loss:
+max L⊥C
+D (u, i) = LCE
+�
+D(h⊥C
+u,i ),˜ti
+�
+.
+7
+
+Figure 3: The architecture of DCRS, which disentangles the user u’s preference on item
+i into category-dependent segment hC
+u,i and category-independent segment h⊥C
+u,i
+for diverse and accurate recommendations.
+We leverage a Gradient Reverse Layer (GRL) (Ganin and Lempitsky, 2015) to implement
+above requirements due to its simplicity. More specifically, we insert a Gradient Reverse
+Layer between h⊥C
+u,i and the discriminator, as shown in Figure 3. During back propagation,
+the gradients for minimizing the discriminator loss ∂L⊥C
+D (u,i)
+∂h⊥C
+u,i
+flow backward through the dis-
+criminator. After the GRL, the gradients will be reversed, i.e., becoming − ∂L⊥C
+D (u,i)
+∂h⊥C
+u,i
+. Thus,
+we perform gradient descent on parameters of the discriminator for accurately predicting
+item i’s category, while performing gradient ascent on h⊥C
+u,i , so that h⊥C
+u,i cannot predict
+item i’s category.
+Learning category-independent representation. h⊥C
+u,i should be optimized under two
+objectives: (1) it can accurately estimate the first term P(Y ⊥C
+u,i
+= 1|u, i) in Eq.(5) by:
+ˆp⊥C
+u,i = P
+�
+Y ⊥C
+u,i
+= 1|u, i
+�
+= σ
+�
+W1⊤h⊥C
+u,i
+�
+;
+(7)
+and (2) it needs to be independent from item categories. Thus we minimize the following
+loss for its learning:
+Lrec
+�
+ˆp⊥C
+u,i , yu,i
+�
+− λL⊥C
+D (u, i)
+(8)
+where the two terms optimize two distinct objectives respectively, and λ is a hyper-parameter
+that controls the strength of category-independent constraint on h⊥C
+u,i .
+Learning category-dependent representation. While user u’s preference on item i’s
+category is unobservable, P(Y C
+u,i = 1|u, ti) can be estimated by fixing the learnt category-
+independent representation h⊥C
+u,i and estimating the overall probability that user u gives
+8
+
+aLrec
+aL(u, i)
+Oh.
+u,i
+User/item
+Encoder
+L(u,i)
+G
+feature
+R
+LC (u, i)
+2
+L
+aLrec
+a (u, i)
+L(u, i)
+Discriminator
+Oht?
+Loss
+sye
+u,ipositive feedback to item i:
+ˆpu,i = P(Yu,i = 1|u, i) = σ
+�
+W2⊤
+�stop gradient(h⊥C
+u,i )
+hC
+u,i
+��
+, W2 ∈ R2d×1
+(9)
+where stop gradient
+�
+h⊥C
+u,i
+�
+implies that h⊥C
+u,i will not be updated by this prediction. In
+other words, given the learnt user u’s preference over category-independent features of item
+i, only user u’s preference over item i’s category is optimized to accurately predict the
+overall feedback of user u to item i, by minimizing the loss:
+Lrec (ˆpu,i, yu,i) + λLC
+D(u, i),
+(10)
+where the second loss forces hC
+u,i to predict item i’s category accurately with λ representing
+the strength of the constraint.
+Overall, combining Eq.(8) and Eq.(10), given a user behavior dataset X, DCRS learns
+a disentangled recommendation model as in Eq.(5) by minimizing the following loss:
+L(X, θ, W1, W2) = 1
+|X|
+�
+(u,i,yu,i)∈X
+Lrec (ˆpu,i, yu,i)
++ Lrec
+�
+ˆp⊥C
+u,i , yu,i
+�
+− λL⊥C
+D (u, i) + λLC
+D(u, i).
+Inference. At the inference stage, we leverage ˆpu,i in Eq.(9) as the predicted preference of
+user u over item i to rank items. We adopt ˆpu,i in Eq.(9) since it considers both the category
+dependent and independent preference of the user, while ˆp⊥C
+u,i in Eq.(7) only captures the
+user’s preference over category-independent features.
+3. Offline Experiments
+In this section, we conduct experiments on several public offline datasets to demonstrate
+the effectiveness of DCRS. We mainly investigate from two perspectives:
+• How does the proposed DCRS perform in terms of recommendation accuracy and
+diversity?
+• Can the disentangled category-independent representation accurately distinguish a
+user’s preference within item categories?
+A case study is also conducted to illustrate the effectiveness of the proposed DCRS more
+explicitly.
+3.1 Experimental Settings
+Dataset. We use three widely-used datasets under different recommendation scenarios for
+evaluation.
+• ML-1M1: This dataset contains 1 million ratings from 6040 users on 3883 movies
+from the online movie recommendation service MovieLens. It also contains rich user
+1. https://grouplens.org/datasets/movielens/1m/
+9
+
+Table 1: Statistics of Three Datasets
+Dataset
+#Users
+#Items
+#Interactions
+#Group
+ML-1M
+6040
+3883
+1000209
+18
+ML-10M
+69878
+10680
+10000047
+19
+Amazon-Books
+22929
+33130
+1178117
+141
+features (e.g., age, gender, etc.) and movie features (e.g., titles). We encode user and
+movie features following previous work (Zhou et al., 2018; Wang et al., 2021). We
+take yu,i = 1, if user u gives item i a rating greater than 3, otherwise yu,i = 0.
+• ML-10M2: This dataset is also from MovieLens. It contains 10 million ratings from
+69878 users on 10680 movies. Similarly, we take yu,i = 1, if user u gives item i a rating
+greater than 3, otherwise yu,i = 0.
+• Amazon-Books3: This dataset contains reviews and metadata of books from Ama-
+zon. To ensure data quality, we only keep categories that link to more than 20 books
+with 141 categories, and adopt the 20-core settings (Wang et al., 2021), i.e., discarding
+users and books with less than 20 interactions. To make the number of positive and
+negative samples balanced, we take yu,i = 1, if user u gives item i a rating greater
+than 4, otherwise yu,i = 0.
+The statistics of the three datasets are summarized in Table 1.
+On each dataset, we also randomly sampled items that the user did not interact with
+as negative instances. We then sorted the user-item interactions by timestamps, and split
+them into training, validation, and testing datasets with the ratio of 80%, 10%, and 10%.
+Baselines. The proposed DCRS is a general and model-agnostic framework to disentangle
+category dependent and independent representations for accurate and diverse recommenda-
+tions. In this paper, we instantiated it with Neural Factorization Machine (NFM) (He and
+Chua, 2017), one representative recommendation model that has been widely used. NFM
+was also taken as the backbone model in several closely related work for diversified recom-
+mendations (Grgic-Hlaca et al., 2016; Wang et al., 2021). We compared DCRS with the
+following algorithms that have different focuses on recommendation diversity and accuracy.
+• NFM (He and Chua, 2017): The state-of-the-art recommendation model serving
+as the backbone model of DCRS.
+• Unawareness (Grgic-Hlaca et al., 2016): It also takes NFM as the backbone
+model and tries to improve diversity by directly removing categorical features of items
+from model input.
+• IPS (Saito et al., 2020): It is a state-of-the-art technique of improving diversity by
+boosting item categories that a user interacted with less often, while suppressing the
+dominant categories in the user’s interaction history. Specifically, it takes the category
+distribution in a user’s historical interactions as propensity scores to reweigh items of
+2. https://grouplens.org/datasets/movielens/10m/
+3. https://jmcauley.ucsd.edu/data/amazon/
+10
+
+this category during training. Propensity clipping (Saito et al., 2020) is also employed
+to reduce the variance with clipping threshold searched in {0.001, 0.005, 0.01, 0.05,
+0.1}.
+• MMR (Carbonell and Goldstein, 1998): One of the state-of-art post-processing
+methods for diversified recommendations. It re-ranks the recommended items gener-
+ated by NFM by a greedy strategy to reduce redundancy.
+• DPP (Chen et al., 2018): An effective post-processing method for diversified rec-
+ommendations. It selects a diverse set of items from the recommended items generated
+by NFM by balancing the relevance of items and their similarities.
+• PD GAN (Wu et al., 2019c): A recent work that leverages the generative adver-
+sarial networks (GAN) framework to generate diverse and relevant recommendations.
+Its discriminator aims to distinguish the generated diverse set of items by its generator
+from the ground-truth sets randomly sampled from the observed data of the user.
+• DGCN (Zheng et al., 2021a): A recent work that leverages rebalanced neighbor
+discovering, category-boosted negative sampling and adversarial learning on top of
+Graph Convolutional Networks (GCN) for diversified recommendations.
+• DecRS (Wang et al., 2021): A recent work for alleviating the bias that previous
+recommendation models over-recommend items of the dominant categories in a user’s
+interaction history from a causal view. It aims at improving both recommendation
+accuracy and diversity.
+• DCRS CI: A variant of DCRS that leverages ˆp⊥C
+u,i in Eq.(7) for item ranking without
+considering the user’s preference over categories. Its comparison with DCRS CI can
+reveal the importance of modeling users’ categorical preference.
+Implementation Details. Following previous work (Wang et al., 2021; He and Chua,
+2017), we set the embedding size of user/item features to 64 (i.e., d = 64), and used
+AdaGrad (Duchi et al., 2011) for optimization. We used grid search to select the hyper-
+parameters based on the model’s performance on validation dataset: the learning rate was
+searched in {0.005, 0.01, 0.05}; the normalization coefficient was searched in {0, 0.1, 0.2};
+the dropout ratio was searched in {0.2, 0.3, ..., 0.5}; λ for controlling strength of category
+independent and dependent constraints was searched in {0.01, 0.05, 0.1, 0.5, 1}. For base-
+line algorithms, when evaluating on the dataset the algorithms were also evaluated in their
+original papers, we adopted the recommended hyperparameters from the original paper;
+otherwise we performed a similar grid search as above with the search range following the
+original paper.
+3.2 Performance on Recommendation Accuracy & Diversity
+We first evaluate all algorithms in terms of recommendation accuracy and diversity.
+Evaluation Metrics. We evaluate the accuracy of a recommendation model from two
+perspectives: (1) Whether the model can rank positively interacted items of a user before
+those negatively interacted ones accurately in the testing dataset; (2) Whether the model
+11
+
+Dataset
+Method
+AUC
+UAUC
+RelaImpr
+R@10
+NDCG@10
+CE@10
+CC@10
+R@20
+NDCG@20
+CE@20
+CC@20
+ML 1M
+NFM
+0.8461
+0.8224
+0.00%
+0.0522
+0.0572
+1.8056
+0.4741
+0.0908
+0.0681
+1.9764
+0.6185
+UnAwareness
+0.8414−
+0.8134−
+-2.79%
+0.0512−
+0.0568−
+1.8919+
+0.4998+
+0.0880−
+0.0669−
+2.0513+
+0.6419+
+IPS
+0.8446−
+0.8210−
+-0.43%
+0.0513−
+0.0572
+1.7929−
+0.4713−
+0.0890−
+0.0681
+1.9759−
+0.6225+
+MMR
+−
+0.8194−
+-0.93%
+0.0501−
+0.0545−
+2.1279+
+0.5886+
+0.0902−
+0.0670−
+2.2224+
+0.7244+
+DPP
+−
+0.6021−
+-68.3%
+0.0454−
+0.0518−
+2.4119+
+0.7315+
+0.0770−
+0.0601−
+2.5974+
+0.9586+
+PD GAN
+−
+−
+−
+0.0326−
+0.0347−
+2.5495+
+0.8347+
+0.0503−
+0.0386−
+2.6650+
+0.9393+
+DGCN
+0.7949−
+0.7759−
+-14.4%
+0.0365−
+0.0402−
+1.9133+
+0.5088+
+0.0640−
+0.0482−
+2.0748+
+0.6466+
+DecRS
+0.8462+
+0.8202−
+-0.07%
+0.0537+
+0.0588+
+1.8560+
+0.4876+
+0.0919+
+0.0694+
+2.0378+
+0.6365+
+DCRS CI
+0.8332−
+0.8096−
+-3.90%
+0.0530+
+0.0581+
+1.7606−
+0.4468−
+0.0936+
+0.0699+
+1.9108−
+0.5766−
+DCRS
+0.8483+
+0.8237+
+0.40%
+0.0551+
+0.0602+
+1.8877+
+0.4909+
+0.0960+
+0.0722+
+2.0620+
+0.6368+
+ML 10M
+NFM
+0.8346
+0.8193
+0.00%
+0.0474
+0.0448
+1.9351
+0.5127
+0.0797
+0.0547
+2.0877
+0.6504
+UnAwareness
+0.8274−
+0.8078−
+-3.60%
+0.0394−
+0.0363−
+2.0308+
+0.5410+
+0.0659−
+0.0446−
+2.2036+
+0.6891+
+IPS
+0.8378+
+0.8218+
+0.78%
+0.0469−
+0.0441−
+1.9280−
+0.5070−
+0.0783−
+0.0538−
+2.0913+
+0.6491−
+MMR
+−
+0.8084−
+-3.41%
+0.0436−
+0.0418−
+2.2941+
+0.6629+
+0.0762−
+0.0521−
+2.3451+
+0.7639+
+DPP
+−
+0.6459−
+-54.3%
+0.0390−
+0.0392−
+2.5014+
+0.7740+
+0.0629−
+0.0459−
+2.6248+
+0.9376+
+PD GAN
+−
+−
+−
+0.0108−
+0.0119−
+2.3134+
+0.7164+
+0.0176−
+0.0136−
+2.4446+
+0.8606+
+DGCN
+0.8069−
+0.8081−
+-3.51%
+0.0425−
+0.0380−
+2.0459+
+0.5530+
+0.0740−
+0.0482−
+2.1925+
+0.6934+
+DecRS
+0.8417+
+0.8261+
+2.12%
+0.0477+
+0.0445−
+1.9401+
+0.5048−
+0.0814+
+0.0551+
+2.1181+
+0.6480−
+DCRS CI
+0.8357+
+0.8197+
+0.12%
+0.0478+
+0.0448
+1.9810+
+0.5269+
+0.0813+
+0.0554+
+2.1322+
+0.6635+
+DCRS
+0.8447+
+0.8301+
+3.38%
+0.0499+
+0.0465+
+2.0050+
+0.5327+
+0.0838+
+0.0572+
+2.1655+
+0.6733+
+Amazon-Books
+NFM
+0.6667
+0.6289
+0.00%
+0.0076
+0.0052
+1.6722
+0.0495
+0.0118
+0.0066
+1.9551
+0.0740
+UnAwareness
+0.6267−
+0.5687−
+-46.7%
+0.0064−
+0.0043−
+1.6660−
+0.0524+
+0.0097−
+0.0054−
+1.8762−
+0.0721−
+IPS
+0.6650−
+0.6269−
+-1.55%
+0.0078+
+0.0053+
+1.5969−
+0.0453−
+0.0115−
+0.0066
+1.9148−
+0.0704−
+MMR
+−
+0.6096−
+-15.0%
+0.0067−
+0.0045−
+2.2899+
+0.0864+
+0.0109−
+0.0060−
+2.5119+
+0.1278+
+DPP
+−
+0.5300−
+-76.7%
+0.0054−
+0.0040−
+2.5184+
+0.1005+
+0.0081−
+0.0049−
+2.8741+
+0.1645+
+PD GAN
+−
+−
+−
+0.0004−
+0.0003−
+2.3179+
+0.0920+
+0.0016−
+0.0007−
+2.7648+
+0.1545+
+DGCN
+0.6747+
+0.6404+
+8.92%
+0.0071−
+0.0044−
+2.0003+
+0.0698+
+0.0122+
+0.0061−
+2.2842+
+0.1074+
+DecRS
+0.6964+
+0.6558+
+20.8%
+0.0074−
+0.0051−
+1.8207+
+0.0601+
+0.0111−
+0.0063−
+2.0973+
+0.0918+
+DCRS CI
+0.6893+
+0.6546+
+19.9%
+0.0057−
+0.0036−
+2.0172+
+0.0704+
+0.0095−
+0.0049−
+2.2799+
+0.1068+
+DCRS
+0.6974+
+0.6573+
+22.0%
+0.0079+
+0.0052
+1.8639+
+0.0622+
+0.0123+
+0.0067+
+2.1415+
+0.0953+
+Table 2: Experimental results regarding to recommendation accuracy and diversity. Im-
+proved (or dropped) performance over the base NFM model under the same setting
+is marked as + (or −).
+ML-1M
+ML-10M
+Amazon-Books
+Category
+Method
+AUC
+UAUC
+R@20
+NDGG@20
+AUC
+UAUC
+R@20
+NDGG@20
+AUC
+UAUC
+R@20
+NDGG@20
+1st ranked cat
+NFM
+0.8547
+0.8180
+0.3034
+0.1678
+0.8498
+0.8229
+0.2814
+0.1453
+0.6474
+0.5976
+0.0679
+0.0343
+DecRS
+0.8563
+0.8135
+0.3079
+0.1718
+0.8545
+0.8273
+0.3031
+0.1572
+0.6724
+0.6113
+0.0698
+0.0331
+DCRS CI
+0.8606
+0.8241
+0.3230
+0.1783
+0.8608
+0.8340
+0.3210
+0.1659
+0.6730
+0.6178
+0.0730
+0.0345
+2nd ranked cat
+NFM
+0.8403
+0.8009
+0.3820
+0.1962
+0.8372
+0.8078
+0.3434
+0.1655
+0.6637
+0.5489
+0.0536
+0.0302
+DecRS
+0.8407
+0.8014
+0.3817
+0.1982
+0.8407
+0.8088
+0.3558
+0.1718
+0.6978
+0.5661
+0.0576
+0.0310
+DCRS CI
+0.8449
+0.8056
+0.3960
+0.2078
+0.8485
+0.8170
+0.3861
+0.1875
+0.7413
+0.5709
+0.0595
+0.0312
+3rd ranked cat
+NFM
+0.8344
+0.8046
+0.6665
+0.3350
+0.8172
+0.7926
+0.4156
+0.1969
+0.6931
+0.5910
+0.0554
+0.0241
+DecRS
+0.8381
+0.8062
+0.6743
+0.3423
+0.8231
+0.7968
+0.4458
+0.2137
+0.7044
+0.5920
+0.0548
+0.0233
+DCRS CI
+0.8419
+0.8055
+0.6873
+0.3463
+0.8264
+0.8014
+0.4794
+0.2291
+0.7176
+0.6162
+0.0591
+0.0245
+Table 3: Recommendation accuracy of disentangled category-independent representation
+on category-specifc testing data.
+can accurately retrieve those positively interacted items in the testing dataset from the
+item pool, which includes all items that the user did not interact with in the training
+dataset. For MMR and DPP, because they only re-rank the recommended items generated
+by NFM, a specifically created item pool that contains top-200 items of NFM is used. We
+adopted AUC (Fawcett, 2006) and UAUC (Zhou et al., 2018) as metrics to evaluate the first
+perspective. Basically, UAUC is a micro-average version of AUC, measuring the goodness
+of intra-user recommendation by averaging AUC over users. Besides, we followed previous
+work (Yan et al., 2014; Zhou et al., 2018) to use the RelaImpr metric to measure the relative
+improvement over the base NFM model on UAUC. For a random guesser, the value of AUC
+is 0.5, and thus RelaImpr is defined as:
+RelaImpr =
+� UAUC(mesured model) − 0.5
+UAUC(base NFM model) − 0.5 − 1
+�
+× 100%.
+12
+
+To evaluate the second perspective of recommendation accuracy, we adopted Recall@K
+and NDCG@K for the purpose. Regarding recommendation diversity, we used two widely-
+adopted metrics: (1) Category coverage (CC@K), which is the ratio between number of
+categories covered by top-K recommendations and the total number of categories in dataset;
+(2) Category entropy (CE@K), which is the entropy of category distribution in top-K rec-
+ommendations. Higher CC@K and CE@K suggest more diverse top-K recommendations.
+Table 2 shows the experiment results of all algorithms. We cannot report AUC, UAUC
+and RelaImpr for PD GAN, since it directly recommends a set of items. For MMR and
+DPP, we can only report UAUC and RelaImpr since it is hard to find an appropriate way
+to merge the re-ranked list of different users to calculate AUC. Based on the results, we can
+observe that:
+• Although Unawareness, MMR, DPP, PD GAN and DGCN promoted more diverse
+recommendations with higher CE@K and CC@K, their recommendation accuracy
+dropped a lot, indicating their failure to handle accuracy-diversity dilemma.
+• IPS did not consistently outperform the base NFM model in recommendation diversity
+or accuracy, due to the inaccurate estimation and high variance of propensity scores.
+• At most time, especially on ML 1M and ML 10M100K dataset, DecRS improved
+both recommendation accuracy and diversity, since it could avoid many less-relevant
+or low-quality items from the dominant categories being recommended. However, its
+improvement was not larger than our proposed DCRS.
+• Our proposed DCRS effectively improved both recommendation accuracy and diver-
+sity on all three datasets compared to the base NFM model. One can observe on
+all datasets, DCRS achieved the highest recommendation accuracy in all metrics,
+and generated more diverse recommendations than the base NFM model with higher
+CC@K and CE@K. This implies that DCRS tends to generate diverse recommenda-
+tions the users will prefer, rather than solely pursuing diversity regardless of recom-
+mendation accuracy. Moreover, compared to DCRS, the recommendation accuracy
+of DCRS CI dropped on all three datasets, confirming the importance of modeling
+users’ categorical preference.
+3.3 Predicting Users’ In-Category Preferences
+We dive deeper to investigate why DCRS can make accurate and diversified recommen-
+dations. Based on our design, the disentanglement shields the users’ preference on item
+categories from their preference on items within the category when learning the user-item
+representations. As a result, the user-item representations learnt by DCRS should better
+predict a user’s interest within item category, compared to those did not consider this as-
+pect. Thus we inspect whether the disentangled category-independent representation (i.e.,
+{h⊥C
+u,i }) can distinguish less relevant (or low-quality) items from relevant (or high-quality)
+items more accurately within a given category of items.
+We split the testing dataset according to item categories, and evaluated all algorithms
+on each category-specific testing dataset separately. On all three of our evaluation datasets,
+an item may relate to multiple categories. For example, the movie “Toy Story (1995)” in
+13
+
+ML-1M dataset is related to three categories: “Animation”, “Children’s”, and “Comedy”.
+Here, we split the testing dataset according to each unique combination of related categories.
+Then given one unique combination of categories, we traversed the testing dataset and only
+kept user-item interactions where the interacted item is associated with the same category
+combination. To ensure the reliability of the evaluation results, on each dataset, we only
+evaluated the algorithms on the top-3 most popular categories.
+In this experimental setting, we only need to evaluate DCRS CI, as all items are from
+the same category. Table 3 demonstrates the experiment results. Due to space limit, we
+only report results on AUC, UAUC, Recall@20 and NDCG@20, and omit baselines that
+perform worse than the base NFM model. From Table 3, we can observe that both DecRS
+and DCRS CI performed better than the base NFM model, as aligned with the results
+in Section 3.2. Moreover, DCRS CI achieved the best performance most time, implying
+that disentangled representations contribute to more accurate preference modeling within
+categories.
+3.4 Case Study
+We also use a case study to qualitatively illustrate the behavior of the proposed DCRS
+model. Figure 4 shows the distribution of categories in the interacted items in training
+and testing data of a user from ML-1M dataset, as well as the top-10 recommended items
+generated by NFM, DecRS and DCRS. One can observe from Figure 4 that: the top-10
+recommendations of NFM and DecRS model mainly fell in the “Thriller” category, which is
+the most popular in the training data of this user. Our proposed DCRS could capture the
+user’s preference over categories more thoroughly. As shown in Figure 4, the recommended
+items from DCRS did not simply concentrate to the dominant category “Thriller” as in other
+recommendation algorithms, but they successfully covered six out of ten categories that
+have a non-zero support in the user’s testing data. Moreover, DCRS could also identify the
+user’s preference on categories that the user seldom interacted with before, for example, the
+category of “Documentary”. This explains its improved recommendation diversity without
+losing recommendation accuracy.
+4. Online Deployment and A/B Test
+To further verify the effectiveness of DCRS, we deploy it on the recommendation channel of
+Toutiao app, one of the largest news recommendation platforms in China, for online A/B
+test.
+More specifically, we implemented DCRS based on the main candidate generator of
+Toutiao. Here, the main candidate generator is one of many candidate generators that pro-
+duce recommendation candidates, which are later scored and ranked by a separate ranking
+model before presenting to users (Chen et al., 2019). But the recommendation candidates
+produced by the main candidate generator account for the largest proportion of the recom-
+mendations shown to users. We then replaced the main candidate generator by DCRS in
+the experimental group, and used the prior main candidate generator in the control group.
+We adopted two key metrics: (1) Click Through Rate (CTR); (2) StayTime, to measure
+users’ satisfaction with the resulting recommendations. To accurately evaluate recommen-
+dation diversity, we only targeted items with more than 1000 impressions, because for those
+14
+
+Comedy
+Documentary
+Fantasy
+HorrorWar
+Mystery
+Western
+Adventure
+Musical
+Drama
+Children’s
+Thriller
+Animation
+Action
+Crime
+Film-Noir
+Sci-Fi
+Romance
+0.00
+0.20
+0.40
+0.60
+0.80
+1.00
+Probability
+Train
+Test
+DCRS
+NFM
+DecRS
+Figure 4: Categorical distributions of training data, testing data and top-10 recommended
+items of a sampled user.
+∆CTR
+∆StayTime
+∆E CN
+∆E CE
+∆R CN
+∆R CE
+DCRS
++0.973%
++0.062%
++0.197%
++0.111%
++2.372%
++2.276%
+Table 4: Results of online A/B test on Toutiao app.
+that appear less frequently could be introduced by some special strategies rather than the
+compared methods. We then calculated four metrics: (1) E CN: number of distinct cat-
+egories of targeted items shown to a user; (2) E CE: entropy of category distribution of
+targeted items shown to a user ; (3) R CN: number of distinct categories of targeted items
+read by a user ; (4) R CE: entropy of category distribution of targeted items read by a user.
+The A/B test was conducted for seventeen consecutive days and the average performance
+of the above metrics is reported. We report experimental results in Table 4. All reported
+results are significant with p-value < 0.05. We can observe that DCRS achieved higher CTR
+and StayTime, indicating improved users’ satisfaction. Moreover, while the improvements
+in E CN and E CE were not that large, DCRS gained huge improvements in R CN and
+R CE, implying DCRS is able to generate diverse recommendations the user will prefer.
+5. Relate work
+DCRS is closely related to two lines of existing work: (1) addressing accuracy-diversity
+dilemma in recommendations; and (2) disentangled user representation learning for general
+user modeling.
+15
+
+Addressing accuracy-diversity dilemma in recommendations. Besides recommen-
+dation accuracy, more and more research suggests other factors of recommendation quality
+also contribute to the overall user satisfaction about the system.
+Of these factors, rec-
+ommendation diversity has been shown as a critically important one (Anderson et al.,
+2020; Wilhelm et al., 2018), which however also leads to the so-called accuracy-diversity
+dilemma (Wang et al., 2021; Zheng et al., 2021a): higher accuracy often means losing di-
+versity to some extent and vice verse. One main reason is that previous solutions with
+accuracy as the primary goal tend to focus on items in the dominant categories in users’
+interaction history. In order to guarantee user satisfaction, three different types of solutions
+are proposed, namely post-processing, learning to rank, and diversified recommendation
+models.
+For the first, and most popular, type of solutions, a re-ranking or post-processing mod-
+ule is appended to a chosen recommendation model. The post-processing module takes
+recommended items as input and re-orders them to balance recommendation accuracy and
+diversity.
+Various post-processing algorithms (Ziegler et al., 2005; Qin and Zhu, 2013;
+Ashkan et al., 2015; Chen et al., 2018; Kaya and Bridge, 2019) are proposed. For example,
+Ziegler et. al. (Ziegler et al., 2005) first applied the Maximal Marginal Relevance (MMR)
+algorithm, which was used for topic diversification in search engines, to minimize redun-
+dancy among recommended items. Determinantal Point Process (DPP) has been shown as
+the most effective one (Chen et al., 2018) of all post-processing algorithms, which scores
+an entire list of items rather than every item individually for better modeling of item cor-
+relations. However, all these post-processing algorithms are separately constructed from
+the recommendation models, though their learning highly depends on the performance of
+the recommendation model. When the recommendation model fails to provide a diverse
+item list to start with, or gives pretty-low scores to diverse items, the effectiveness of the
+aforementioned post-processing algorithms will largely deteriorate. Moreover, as shown in
+our experiment results in Section 3, the aforementioned post-processing algorithms usually
+seriously sacrifice recommendation accuracy.
+Learning To Rank type solutions (Cheng et al., 2017; Wu et al., 2018a; Liu et al., 2022)
+aim to directly recommend a list of items to users, rather than selecting items one by one
+according to their prediction scores. However, this line of work often suffers from high time
+complexity, which limits its application in real world recommendation scenarios.
+Recently, several solutions are proposed to directly improve the diversity of recommen-
+dation models. Zheng et. al (Zheng et al., 2021a) proposed a diversified recommendation
+model based on Graph Convolutional Networks (GCN), with improving recommendation
+diversity as its only target. Wu et. al (Wu et al., 2019c) leveraged the GAN framework
+for diverse recommendations, where a generator tries to recommend diverse sets of items,
+and a discriminator aims to distinguish the generated recommendations from a set of items
+randomly sampled from the observed data of the target user. The most related work to
+ours is (Wang et al., 2021), where the authors studied the problem of lack of diversity in
+recommendations from a casual perspective, and proposed DecRS to alleviate the prob-
+lem. Experiments demonstrate the advantage of our proposed DCRS over these solutions
+in improving recommendation accuracy and diversity. A recent work (Lin et al., 2022) also
+tried to diversify recommendations in relevant recommendation scenario, where the diversi-
+fication is conducted regarding multiple item aspects such that relevance and diversity are
+16
+
+adaptively balanced among different item aspects. However, when only one item aspect is
+considered, e.g., the item category in this paper, their algorithm degenerates to the MMR
+algorithm.
+Disentangled user representations.
+Learning disentangled user representations has
+drawn increasing attention in recent years. A family of solutions are based on Variational
+Auto-Encoder (VAE) to force each dimension of learnt representations to focus on different
+latent factors (Ma et al., 2019; Xie et al., 2021; Nema et al., 2021). However, such a disen-
+tanglement is implicit and therefore one cannot associate the disentangled representation
+with the specific attributes of interest. Zheng et. al (Zheng et al., 2021b) proposed DICE
+to learn representations where user interest and conformity are structurally disentangled
+via direct supervision from cause-specific data. However, in our problem, we cannot access
+users’ preferences over item categories explicitly, thus are not able to get any direct supervi-
+sion about it. Chen et. al. (Chen et al., 2022) proposed to disentangle item representations
+to address popularity bias, by requiring the two disentangled item representations to be
+orthogonal. In our solution, we disentangle a user’s preference over an item into category
+dependent and independent segments. Both segments relate to the user and thus they do
+not need to be orthogonal to each other.
+6. Conclusion
+In this paper, we propose a new principle that the diversification of recommendations should
+be performed within a user’s preferred categories, such that improved recommendation
+diversity can be achieved without sacrificing recommendation accuracy.
+We realize this
+principle via a general framework, named DCRS, to disentangle a user’s category dependent
+and independent preference in the learnt representations. We evaluate DCRS through both
+offline experiments on three widely-used benchmark datasets for recommendation and online
+A/B test on Toutiao, one of the largest news recommendation platforms in China. We
+demonstrate DCRS can provide more accurate and diversified recommendations. Via in-
+depth analysis and case studies, we find that the benefit of DCRS is introduced by: (1) it
+can capture a user’s diverse preference in historical interactions more thoroughly; and (2)
+it can rank items in the same category more accurately.
+In this work, we took a static view of users’ preferences over items and item categories.
+But numerous studies have demonstrated that users’ preferences evolve over time Wu et al.
+(2018b, 2019b). It is interesting to study the problem of recommendation diversification
+in an interactive manner over time. Moreover, currently we only recommend one item a
+time to a user. It is interesting to study how to generate a list of diverse recommendations,
+where diversity should be optimized within and across recommendation lists.
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf,len=1098
+page_content='Disentangled Representation for Diversified  Recommendations  Xiaoying Zhang  zhangxiaoying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='xy@bytedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='com  AI Lab, Bytedance Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Hongning Wang  hw5x@virginia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='edu  Department of Computer Science  University of Virginia, USA  Hang Li  lihang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='lh@bytedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='com  AI Lab, Bytedance Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Abstract  Accuracy and diversity have long been considered to be two conflicting goals for recom-  mendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We point out, however, that as the diversity is typically measured by certain  pre-selected item attributes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', category as the most popularly employed one, improved  diversity can be achieved without sacrificing recommendation accuracy, as long as the di-  versification respects the user’s preference about the pre-selected attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' This calls for a  fine-grained understanding of a user’s preferences over items, where one needs to recognize  the user’s choice is driven by the quality of the item itself, or the pre-selected attributes of  the item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  In this work, we focus on diversity defined on item categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We propose a general di-  versification framework agnostic to the choice of recommendation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Our solution  disentangles the learnt user representation in the recommendation module into category-  independent and category-dependent components to differentiate a user’s preference over  items from two orthogonal perspectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Experimental results on three benchmark datasets  and online A/B test demonstrate the effectiveness of our solution in improving both rec-  ommendation accuracy and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' In-depth analysis suggests that the improvement is  due to our improved modeling of users’ categorical preferences and refined ranking within  item categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Keywords:  Recommender system, recommendation diversity, disentangled user repre-  sentation  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Introduction  Recommender systems learn users’ interests from historical observations (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', their clicks,  bookmarked or purchased items, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=')  so as to identify the items that best suit users’  preferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' The success of recommender system in enhancing user experience and boosting  platform utility has been witnessed in a number of scenarios including e-commerce (Zhou  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' He and Chua, 2017), online news recommendation (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2019a) and  streaming services (Covington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Recommendation accuracy, which measures whether a recommendation model can rec-  ommend items that users will like, serves as the dominant target or even the only target  in most previous work (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' He and Chua, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Guo  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Covington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Various complicated models (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Guo  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Covington et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2016) have been proposed for higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' While recom-  mendation accuracy has been shown to be closely related to user satisfaction, it is never  1  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='05492v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='IR]  13 Jan 2023  Figure 1: Illustration of recommendation accuracy and diversity optimization in different  recommendation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  the only rule of thumb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Recent work found the recommendation diversity, which measures  the dissimilarity among recommended items regarding certain pre-selected item attributes  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', item category) also plays an important role in the overall user experience (Wilhelm  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Kapoor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' For example, even if a user is a fan of  basketball, he/she can still get bored with recommendations only about basketball videos  or news, which increases the risk of user attrition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Following previous work (Steck, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021a), we fo-  cus on diversity defined on item categories in this paper and aim to address the so-called  accuracy-diversity dilemma (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' On one hand, recommendation models  with accuracy as their primary target often lose diversity to some extent, due to overly  emphasizing items in the dominant categories in a user’s interaction history (Steck, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Figure 1(a) illustrates this issue with an example in movie recommenda-  tion, where 70% of the movies watched by a user are action movies, which leads 90% of the  system’s recommendations to fall in the action movie category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Worse still, because of the  feedback loop (Chaney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2018), the emphasis on the dominant categories in the system’s  recommendations will be further intensified when the user follows the recommendations,  causing further decreased recommendation diversity and issues like filter bubbles (Nguyen  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2014) and echo chambers (Ge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' On the other hand, simply diversifying  recommendations over all item categories without considering the user’s categorical prefer-  ence hurts the accuracy of generated recommendations (Wilhelm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Ziegler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=',  2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Qin and Zhu, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=" As shown in Figure 1(b), although the  recommendation list is diverse by covering all four categories, negative feedback is more  2  Action movie  Romancemovie  Children's movie  Documentary  User browsing history  Diversifying across  Diversifying within the  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content="7  distribution  all item categories  user's preferred categories  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='1  Feedback loop  Accuracy-targeted  recommendation models  KIDS  X  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='9  distribution  X  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='1  User feedback  (a)  (b)  (c)likely on the categories where the user interacted less often or negative feedback already  prevailed, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', children’s movies and documentary movies respectively in this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Clearly one should not recklessly increase diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' For categories the user is less likely  to be interested in, the risk of making a bad recommendation overweights the benefit of  increased diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Thus, this paper focuses on conducting diversification only among  item categories that the user prefers, suggesting the possibility to improve recommendation  diversity without sacrificing recommendation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Figure 1(c) gives an example rec-  ommendation list following such a strategy, where the recommended items mainly fall in  action and romance movies, the two preferred categories inferred from the user’s interaction  history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' This strategy requires the recommendation model to clearly distinguish whether the  user’s positive/negative feedback is due to the item’s category or other category-independent  features of the item (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', the item’s own quality), which was ignored by previous recom-  mendation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  In this paper, we propose a general and model-agnostic framework to disentangle a  user’s category-dependent and category-independent preferences for an accurate and diver-  sified recommender system (DCRS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Specifically, DCRS takes a user’s preference over an  item as a product of: (1) the user’s preference over the item’s category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' and (2) the user’s  preference over category-independent features of the item, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', the item’s quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Such  disentanglement suggests a hierarchical decision making process by the user: If a user has  a strong preference over a particular category of items, he/she may still enjoy items of this  category, even though their qualities are not perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' However, if the probability that a  user likes a category is low, only items of high quality in this category could have a chance  to be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' The disentanglement ensures items of the same quality, but in different  categories that a user prefers similarly, have equal probabilities to be recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' It  naturally avoids overly recommending items from the dominant categories in the user’s in-  teraction history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' The main challenge therefore lies in how to disentangle a user’s preference  regarding the aforementioned two orthogonal perspectives, given his/her preference over the  item categories is not observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' This makes naive solutions like using different supervision  signals to separately train users’ representations (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021b), or separating items’  feature vectors into category dependent and independent segments, ineffective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  DCRS is agnostic to the choice of recommendation module, which is supposed to learn  informative representations of users and items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' In particular, DCRS adopts a discriminator  to disentangle the learnt representation into category-independent and category-dependent  segments respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' The recommendation module and discriminator are learnt simulta-  neously to ensure the effectiveness of disentangled representation learning for accurate and  diverse recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' To evaluate the proposed DCRS solution, we conduct both of-  fline experiments on three benchmark datasets and online A/B test on Toutiao app, one of  the largest news recommendation platforms in China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Experiment results demonstrate that  DCRS can successfully recommend diverse items that users prefer, and thus improve both  recommendation accuracy and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' In-depth analysis and case studies suggest strong  evidence showing: (1) the disentangled category-independent representation from DCRS  can distinguish the user’s preference within category more accurately;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' and (2) DCRS can  capture a user’s diverse preferences in historical interactions more thoroughly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' All codes  and data can be found in https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='com/Xiaoyinggit/DCRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='git.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Overall, our contribution of this work is as follows:  3  We demonstrate that accuracy and diversity are not conflicting goals for recommen-  dation, as long as the diversification respects the user’s categorical preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  To capture a user’s latent preferences on item categories more accurately, our pro-  posed DCRS disentangles the user’s preference into category-dependent and category-  independent components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Experiments on three benchmark datasets and online A/B test demonstrate the effec-  tiveness of DCRS in improving both recommendation accuracy and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' In-depth  analysis further demonstrates the improvement comes from more accurate modeling  of the user’s preference both over and within categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Framework  In this section, we describe how the proposed DCRS solution disentangles a user’s category  dependent and independent preferences to simultaneously improve recommendation accu-  racy and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' For the ease of illustration, we first briefly describe a general architecture  which covers almost all popularly used recommendation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We then depict how to  smoothly integrate DCRS into such a general architecture to diversify its recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='1 Preliminary: A General Recommendation Architecture  In a recommendation task, we are given a user behavior dataset X that contains interactions  between N users and M items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' The interaction between user u and item i is represented  as a tuple (u, i, yu,i) ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Here yu,i ∈ {0, 1} denotes user u’s feedback to item i, where  yu,i = 1 denotes positive feedback (e,g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', a click or a positive rating), and yu,i = 0 denotes  negative feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Generally speaking, a recommendation model will first learn a user-item  representation to capture the user’s preference over the item:  hu,i = f(u, i, θ) ∈ Rd,  (1)  where θ denotes a set of trainable parameters in the recommendation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Various  architectures (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' He and Chua, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Guo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2017)  have been proposed to implement f(u, i, θ), ranging from the simple matrix factorization  algorithm (Mnih and Salakhutdinov, 2007) that directly takes the element-wise product of  user and item embeddings to form the representation, to complex architectures such as the  bi-interaction layer in NFM (He and Chua, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Let ˆpu,i denote the probability that user  u gives positive feedback to item i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' The goal of the recommendation model is to use the  learnt user-item representation to estimate ˆpu,i, either by directly summing up elements in  hu,i as in matrix factorization, or through a learnable projection layer as follows:  ˆpu,i = P (Yu,i = 1|u, i) = σ  �  W ⊤hu,i  �  ,  (2)  where Yu,i is a random variable representing the feedback from user u on item i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' W ∈ Rd×1  is the learnable weight vector of the projection layer, and σ(·) is the sigmoid function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' The  parameters of the recommendation model are then optimized by minimizing the following  4  loss:  L(X, θ, W ) =  1  |X|  �  (u,i,yu,i)∈X  Lrec(yu,i, ˆpu,i),  (3)  where Lrec(·, ·) represents the chosen loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Various loss functions have been explored  in literature, inlcuding cross entropy loss, Mean Squared Error (MSE) and BPR loss (Rendle  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' In this work, we will use the cross entropy loss by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='2 Disentangle Category Dependent and Independent Representations  We consider a user’s feedback on an item as a mixture reflecting his/her preference over  the item’s category and category-independent properties, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', the item’s intrinsic quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  As shown in Figure 2, the first action movie that receives positive feedback can very likely  be caused by the user’s strong preference over the category of action movies, while his/her  positive feedback on the second romance movie is more likely to be caused by its high quality  that makes up the low probability that the user likes romance movies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' In order to diversify  the recommendations with respect to a user’s preferred categories, the recommendation  model needs to clearly distinguish the effect of item category and other category-independent  properties on a user’s decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' To make our method description general enough  to cover situations where an item can associate with multiple categories, we take item i’s  category as the set that contains all categories that the item relates to, and denote it as ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  For example, assume there are three categories {c1, c2, c3} in a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' If item i is related  to the first category, then ti = {c1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' And if item i is associated with the first two categories,  then ti = {c1, c2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  We propose to disentangle a user’s preference over an item into two parts :  Category-dependent preference: it captures the user’s preference over the item’s  category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Category-independent preference: it depicts how category-independent features  affect the user’s preference about the item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Such a disentanglement can be explained through a probabilistic view about the generation  of user u’s feedback on item i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Let Y C  u,i denote the binary random variable indicating user  u’s feedback on item i’s category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We have the following,  P(Yu,i = 1|u, i) = P(Yu,i = 1, Y C  u,i = 1|u, i)  (4a)  = P(Yu,i = 1|u, i, Y C  u,i = 1)P(Y C  u,i = 1|u, i)  (4b)  = P(Yu,i = 1|u, i, Y C  u,i = 1)P(Y C  u,i = 1|u, ti)  (4c)  In particular, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (4a) is due to the assumption that user u gives positive feedback to item i  only if user u likes item i’s category, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', P(Yu,i = 1, Y C  u,i = 1|u, i) = 1 and P(Yu,i = 1, Y C  u,i =  0|u, i) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (4b) follows the chain rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' And Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (4c) is because Y C  u,i only depends on the  item’s category, instead of specific items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  The first term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (4c) depicts how likely user u will give positive feedback to item  i when he/she is interested in item i’s category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' and the second term models how likely  user u is interested in item i’s category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Given that user u likes the category of item i, the  5  Figure 2: Hierarchical decision making process of DCRS framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Each feedback is de-  termined by: (1) the user’s preference over the item’s category;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' and (2) the user’s  preference over category-independent features of the item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  probability in the first term only depends on the category-independent features of item i,  such as item i’s quality, price, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Thus, under the disentangled user-item representations,  we can compute the first term by the probability P(Y ⊥C  u,i  = 1|u, i), which depicts user u’s  preference over item i driven by the category-independent features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Thus, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (4c) can be  rewritten as:  P(Yu,i = 1|u, i) = P(Y ⊥C  u,i  = 1|u, i)P(Y C  u,i = 1|u, ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  (5)  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (5) depicts a hierarchical decision making process illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' If user u likes  item i’s category with a higher probability P(Y C  u,i = 1|u, ti), he/she may still enjoy item  i even though item i’s quality is not perfect, indicated by a lower P(Y ⊥C  u,i  = 1|u, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' For  example, the positive feedback of the first action movie in Figure 2 is generated under such  a scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Meanwhile, if there is only a small probability that user u would be interested in  item i’s category (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', low P(Y C  u,i = 1|u, ti)), item i must be of high quality to get positive  feedback, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', high P(Y ⊥C  u,i  = 1|u, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' The positive feedback on the second romance movie  in Figure 2 is a good example of this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (5) also suggests why disentanglement makes recommendations diversified within a  user’s preferred categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Assume there are two categories c1 and c2 on which the user  has similar preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Instead of recommending more items from the dominant category  (either c1 or c2), via the disentanglement in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (5), items of the same quality within c1 and  c2 will have an equal chance to be recommended, thus diversifying the recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content="  6  Action movie  Romancemovie  Probability that the user  recommendation list  The user's preference  likeeachcategory  within category  Probability  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='9  Probability  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='3  Items  Itemcategory  Probability  ItemsUnfortunately, both terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (5) cannot be learnt via direct supervision signals, since  neither user u’s feedback on item i’s category nor feedback driven by category-independent  features of item i can be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Classical solutions would appeal to Expectation Maxi-  mization type algorithms (Dempster et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 1977) to estimate the two terms in an iterative  manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' However, given modern recommendation algorithms are usually realized via com-  plex deep neural networks, posterior inference becomes cumbersome and also leads to slow  convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Instead, DCRS implements Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (5) by simultaneously learning two disentan-  gled representations for estimating the two terms separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Specifically, DCRS learns two  disentangled representations by:  ��  h⊥C  u,i  �⊤ ,  �  hC  u,i  �⊤�⊤  = f(u, i, θ) ∈ R2d,  (6)  where h⊥C  u,i ∈ Rd aims to capture user u’s preference over category-independent features to  estimate P(Y ⊥C  u,i  = 1|u, i), and hC  u,i ∈ Rd depicts user u’s preference over item i’s category  ti, aiming to estimate P(Y C  u,i = 1|u, ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Simply splitting item i’s feature vector into two parts, even with separate networks,  cannot ensure complete disentanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Instead, in addition to requiring the learnt the  representations to best capture the user’s preference, we employ an adversarial discriminator  that enforces the learnt h⊥C  u,i and hC  u,i to be category-independent and category-dependent  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Discriminator Module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' The discriminator D(·) acts as a category classifier, which takes  one segment of disentangled representation, such as hC  u,i or h⊥C  u,i , as input, and aims to  predict the category of item i (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', ti).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' However, it is hard for the discriminator to directly  predict ti, since ti can take 2K-1 values, where K is the number of unique categories available  in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' For ease of learning, we represent ti by a vector over K unique categories,  denoted as ˜ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Again, assume there are three categories {c1, c2, c3}, if t1 = {c1}, then  ˜ti = [1, 0, 0]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' And if t1 = {c1, c2}, then ˜ti = [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='5, 0]⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Specifically, when relevance  between item i and each associated category can be measured (Pu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2020), a more  accurate ˜ti can be achieved by making the j-th element of ˜ti proportional to the relevance  between item i and the j-th category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Otherwise, ˜ti can be simply assumed to be evenly  distributed among related categories, which is also the default setting in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  The discriminator then takes hC  u,i or h⊥C  u,i as input to predict ˜ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' In our experiments, the  discriminator D(·) is implemented via a fully connected layer, and it should enforce the  following:  Given hC  u,i is closely related to item i’s category, the discriminator should predict ˜ti  accurately based on hC  u,i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', the following loss should be minimized:  min LC  D(u, i) = LCE  �  D(hC  u,i),˜ti  �  ,  where LCE is the cross entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Given h⊥C  u,i is independent from item category, h⊥C  u,i should fool the discriminator by  maximizing the classification loss:  max L⊥C  D (u, i) = LCE  �  D(h⊥C  u,i ),˜ti  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  7  Figure 3: The architecture of DCRS, which disentangles the user u’s preference on item  i into category-dependent segment hC  u,i and category-independent segment h⊥C  u,i  for diverse and accurate recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  We leverage a Gradient Reverse Layer (GRL) (Ganin and Lempitsky, 2015) to implement  above requirements due to its simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' More specifically, we insert a Gradient Reverse  Layer between h⊥C  u,i and the discriminator, as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' During back propagation,  the gradients for minimizing the discriminator loss ∂L⊥C  D (u,i)  ∂h⊥C  u,i  flow backward through the dis-  criminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' After the GRL, the gradients will be reversed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', becoming − ∂L⊥C  D (u,i)  ∂h⊥C  u,i  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Thus,  we perform gradient descent on parameters of the discriminator for accurately predicting  item i’s category, while performing gradient ascent on h⊥C  u,i , so that h⊥C  u,i cannot predict  item i’s category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Learning category-independent representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' h⊥C  u,i should be optimized under two  objectives: (1) it can accurately estimate the first term P(Y ⊥C  u,i  = 1|u, i) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (5) by:  ˆp⊥C  u,i = P  �  Y ⊥C  u,i  = 1|u, i  �  = σ  �  W1⊤h⊥C  u,i  �  ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  (7)  and (2) it needs to be independent from item categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Thus we minimize the following  loss for its learning:  Lrec  �  ˆp⊥C  u,i , yu,i  �  − λL⊥C  D (u, i)  (8)  where the two terms optimize two distinct objectives respectively, and λ is a hyper-parameter  that controls the strength of category-independent constraint on h⊥C  u,i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Learning category-dependent representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' While user u’s preference on item i’s  category is unobservable, P(Y C  u,i = 1|u, ti) can be estimated by fixing the learnt category-  independent representation h⊥C  u,i and estimating the overall probability that user u gives  8  aLrec  aL(u, i)  Oh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  u,i  User/item  Encoder  L(u,i)  G  feature  R  LC (u, i)  2  L  aLrec  a (u, i)  L(u, i)  Discriminator  Oht?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Loss  sye  u,ipositive feedback to item i:  ˆpu,i = P(Yu,i = 1|u, i) = σ  �  W2⊤  �stop gradient(h⊥C  u,i )  hC  u,i  ��  , W2 ∈ R2d×1  (9)  where stop gradient  �  h⊥C  u,i  �  implies that h⊥C  u,i will not be updated by this prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' In  other words, given the learnt user u’s preference over category-independent features of item  i, only user u’s preference over item i’s category is optimized to accurately predict the  overall feedback of user u to item i, by minimizing the loss:  Lrec (ˆpu,i, yu,i) + λLC  D(u, i),  (10)  where the second loss forces hC  u,i to predict item i’s category accurately with λ representing  the strength of the constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Overall, combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (8) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (10), given a user behavior dataset X, DCRS learns  a disentangled recommendation model as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (5) by minimizing the following loss:  L(X, θ, W1, W2) = 1  |X|  �  (u,i,yu,i)∈X  Lrec (ˆpu,i, yu,i)  + Lrec  �  ˆp⊥C  u,i , yu,i  �  − λL⊥C  D (u, i) + λLC  D(u, i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' At the inference stage, we leverage ˆpu,i in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (9) as the predicted preference of  user u over item i to rank items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We adopt ˆpu,i in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (9) since it considers both the category  dependent and independent preference of the user, while ˆp⊥C  u,i in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (7) only captures the  user’s preference over category-independent features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Offline Experiments  In this section, we conduct experiments on several public offline datasets to demonstrate  the effectiveness of DCRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We mainly investigate from two perspectives:  How does the proposed DCRS perform in terms of recommendation accuracy and  diversity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Can the disentangled category-independent representation accurately distinguish a  user’s preference within item categories?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  A case study is also conducted to illustrate the effectiveness of the proposed DCRS more  explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='1 Experimental Settings  Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We use three widely-used datasets under different recommendation scenarios for  evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  ML-1M1: This dataset contains 1 million ratings from 6040 users on 3883 movies  from the online movie recommendation service MovieLens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' It also contains rich user  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' https://grouplens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='org/datasets/movielens/1m/  9  Table 1: Statistics of Three Datasets  Dataset  #Users  #Items  #Interactions  #Group  ML-1M  6040  3883  1000209  18  ML-10M  69878  10680  10000047  19  Amazon-Books  22929  33130  1178117  141  features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', age, gender, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=') and movie features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', titles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We encode user and  movie features following previous work (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We  take yu,i = 1, if user u gives item i a rating greater than 3, otherwise yu,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  ML-10M2: This dataset is also from MovieLens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' It contains 10 million ratings from  69878 users on 10680 movies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Similarly, we take yu,i = 1, if user u gives item i a rating  greater than 3, otherwise yu,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Amazon-Books3: This dataset contains reviews and metadata of books from Ama-  zon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' To ensure data quality, we only keep categories that link to more than 20 books  with 141 categories, and adopt the 20-core settings (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', discarding  users and books with less than 20 interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' To make the number of positive and  negative samples balanced, we take yu,i = 1, if user u gives item i a rating greater  than 4, otherwise yu,i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  The statistics of the three datasets are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  On each dataset, we also randomly sampled items that the user did not interact with  as negative instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We then sorted the user-item interactions by timestamps, and split  them into training, validation, and testing datasets with the ratio of 80%, 10%, and 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' The proposed DCRS is a general and model-agnostic framework to disentangle  category dependent and independent representations for accurate and diverse recommenda-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' In this paper, we instantiated it with Neural Factorization Machine (NFM) (He and  Chua, 2017), one representative recommendation model that has been widely used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' NFM  was also taken as the backbone model in several closely related work for diversified recom-  mendations (Grgic-Hlaca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We compared DCRS with the  following algorithms that have different focuses on recommendation diversity and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  NFM (He and Chua, 2017): The state-of-the-art recommendation model serving  as the backbone model of DCRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Unawareness (Grgic-Hlaca et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2016): It also takes NFM as the backbone  model and tries to improve diversity by directly removing categorical features of items  from model input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  IPS (Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2020): It is a state-of-the-art technique of improving diversity by  boosting item categories that a user interacted with less often, while suppressing the  dominant categories in the user’s interaction history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Specifically, it takes the category  distribution in a user’s historical interactions as propensity scores to reweigh items of  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' https://grouplens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='org/datasets/movielens/10m/  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' https://jmcauley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='edu/data/amazon/  10  this category during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Propensity clipping (Saito et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2020) is also employed  to reduce the variance with clipping threshold searched in {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='001, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='05,  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  MMR (Carbonell and Goldstein, 1998): One of the state-of-art post-processing  methods for diversified recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' It re-ranks the recommended items gener-  ated by NFM by a greedy strategy to reduce redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  DPP (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2018): An effective post-processing method for diversified rec-  ommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' It selects a diverse set of items from the recommended items generated  by NFM by balancing the relevance of items and their similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  PD GAN (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2019c): A recent work that leverages the generative adver-  sarial networks (GAN) framework to generate diverse and relevant recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Its discriminator aims to distinguish the generated diverse set of items by its generator  from the ground-truth sets randomly sampled from the observed data of the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  DGCN (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021a): A recent work that leverages rebalanced neighbor  discovering, category-boosted negative sampling and adversarial learning on top of  Graph Convolutional Networks (GCN) for diversified recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  DecRS (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021): A recent work for alleviating the bias that previous  recommendation models over-recommend items of the dominant categories in a user’s  interaction history from a causal view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' It aims at improving both recommendation  accuracy and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  DCRS CI: A variant of DCRS that leverages ˆp⊥C  u,i in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (7) for item ranking without  considering the user’s preference over categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Its comparison with DCRS CI can  reveal the importance of modeling users’ categorical preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Implementation Details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Following previous work (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' He and Chua,  2017), we set the embedding size of user/item features to 64 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', d = 64), and used  AdaGrad (Duchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2011) for optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We used grid search to select the hyper-  parameters based on the model’s performance on validation dataset: the learning rate was  searched in {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='005, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='05};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' the normalization coefficient was searched in {0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='2};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  the dropout ratio was searched in {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='5};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' λ for controlling strength of category  independent and dependent constraints was searched in {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='01, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='05, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='5, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' For base-  line algorithms, when evaluating on the dataset the algorithms were also evaluated in their  original papers, we adopted the recommended hyperparameters from the original paper;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  otherwise we performed a similar grid search as above with the search range following the  original paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='2 Performance on Recommendation Accuracy & Diversity  We first evaluate all algorithms in terms of recommendation accuracy and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We evaluate the accuracy of a recommendation model from two  perspectives: (1) Whether the model can rank positively interacted items of a user before  those negatively interacted ones accurately in the testing dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (2) Whether the model  11  Dataset  Method  AUC  UAUC  RelaImpr  R@10  NDCG@10  CE@10  CC@10  R@20  NDCG@20  CE@20  CC@20  ML 1M  NFM  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='8461  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
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+page_content='0245  Table 3: Recommendation accuracy of disentangled category-independent representation  on category-specifc testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  can accurately retrieve those positively interacted items in the testing dataset from the  item pool, which includes all items that the user did not interact with in the training  dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' For MMR and DPP, because they only re-rank the recommended items generated  by NFM, a specifically created item pool that contains top-200 items of NFM is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We  adopted AUC (Fawcett, 2006) and UAUC (Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2018) as metrics to evaluate the first  perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Basically, UAUC is a micro-average version of AUC, measuring the goodness  of intra-user recommendation by averaging AUC over users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Besides, we followed previous  work (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2018) to use the RelaImpr metric to measure the relative  improvement over the base NFM model on UAUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' For a random guesser, the value of AUC  is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='5, and thus RelaImpr is defined as:  RelaImpr =  � UAUC(mesured model) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='5  UAUC(base NFM model) − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='5 − 1  �  × 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  12  To evaluate the second perspective of recommendation accuracy, we adopted Recall@K  and NDCG@K for the purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Regarding recommendation diversity, we used two widely-  adopted metrics: (1) Category coverage (CC@K), which is the ratio between number of  categories covered by top-K recommendations and the total number of categories in dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  (2) Category entropy (CE@K), which is the entropy of category distribution in top-K rec-  ommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Higher CC@K and CE@K suggest more diverse top-K recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Table 2 shows the experiment results of all algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We cannot report AUC, UAUC  and RelaImpr for PD GAN, since it directly recommends a set of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' For MMR and  DPP, we can only report UAUC and RelaImpr since it is hard to find an appropriate way  to merge the re-ranked list of different users to calculate AUC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Based on the results, we can  observe that:  Although Unawareness, MMR, DPP, PD GAN and DGCN promoted more diverse  recommendations with higher CE@K and CC@K, their recommendation accuracy  dropped a lot, indicating their failure to handle accuracy-diversity dilemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  IPS did not consistently outperform the base NFM model in recommendation diversity  or accuracy, due to the inaccurate estimation and high variance of propensity scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  At most time, especially on ML 1M and ML 10M100K dataset, DecRS improved  both recommendation accuracy and diversity, since it could avoid many less-relevant  or low-quality items from the dominant categories being recommended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' However, its  improvement was not larger than our proposed DCRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Our proposed DCRS effectively improved both recommendation accuracy and diver-  sity on all three datasets compared to the base NFM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' One can observe on  all datasets, DCRS achieved the highest recommendation accuracy in all metrics,  and generated more diverse recommendations than the base NFM model with higher  CC@K and CE@K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' This implies that DCRS tends to generate diverse recommenda-  tions the users will prefer, rather than solely pursuing diversity regardless of recom-  mendation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Moreover, compared to DCRS, the recommendation accuracy  of DCRS CI dropped on all three datasets, confirming the importance of modeling  users’ categorical preference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='3 Predicting Users’ In-Category Preferences  We dive deeper to investigate why DCRS can make accurate and diversified recommen-  dations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Based on our design, the disentanglement shields the users’ preference on item  categories from their preference on items within the category when learning the user-item  representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' As a result, the user-item representations learnt by DCRS should better  predict a user’s interest within item category, compared to those did not consider this as-  pect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Thus we inspect whether the disentangled category-independent representation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=',  {h⊥C  u,i }) can distinguish less relevant (or low-quality) items from relevant (or high-quality)  items more accurately within a given category of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  We split the testing dataset according to item categories, and evaluated all algorithms  on each category-specific testing dataset separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' On all three of our evaluation datasets,  an item may relate to multiple categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' For example, the movie “Toy Story (1995)” in  13  ML-1M dataset is related to three categories: “Animation”, “Children’s”, and “Comedy”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Here, we split the testing dataset according to each unique combination of related categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Then given one unique combination of categories, we traversed the testing dataset and only  kept user-item interactions where the interacted item is associated with the same category  combination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' To ensure the reliability of the evaluation results, on each dataset, we only  evaluated the algorithms on the top-3 most popular categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  In this experimental setting, we only need to evaluate DCRS CI, as all items are from  the same category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Table 3 demonstrates the experiment results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Due to space limit, we  only report results on AUC, UAUC, Recall@20 and NDCG@20, and omit baselines that  perform worse than the base NFM model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' From Table 3, we can observe that both DecRS  and DCRS CI performed better than the base NFM model, as aligned with the results  in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Moreover, DCRS CI achieved the best performance most time, implying  that disentangled representations contribute to more accurate preference modeling within  categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='4 Case Study  We also use a case study to qualitatively illustrate the behavior of the proposed DCRS  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Figure 4 shows the distribution of categories in the interacted items in training  and testing data of a user from ML-1M dataset, as well as the top-10 recommended items  generated by NFM, DecRS and DCRS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' One can observe from Figure 4 that: the top-10  recommendations of NFM and DecRS model mainly fell in the “Thriller” category, which is  the most popular in the training data of this user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Our proposed DCRS could capture the  user’s preference over categories more thoroughly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' As shown in Figure 4, the recommended  items from DCRS did not simply concentrate to the dominant category “Thriller” as in other  recommendation algorithms, but they successfully covered six out of ten categories that  have a non-zero support in the user’s testing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Moreover, DCRS could also identify the  user’s preference on categories that the user seldom interacted with before, for example, the  category of “Documentary”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' This explains its improved recommendation diversity without  losing recommendation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Online Deployment and A/B Test  To further verify the effectiveness of DCRS, we deploy it on the recommendation channel of  Toutiao app, one of the largest news recommendation platforms in China, for online A/B  test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  More specifically, we implemented DCRS based on the main candidate generator of  Toutiao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Here, the main candidate generator is one of many candidate generators that pro-  duce recommendation candidates, which are later scored and ranked by a separate ranking  model before presenting to users (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' But the recommendation candidates  produced by the main candidate generator account for the largest proportion of the recom-  mendations shown to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We then replaced the main candidate generator by DCRS in  the experimental group, and used the prior main candidate generator in the control group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  We adopted two key metrics: (1) Click Through Rate (CTR);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (2) StayTime, to measure  users’ satisfaction with the resulting recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' To accurately evaluate recommen-  dation diversity, we only targeted items with more than 1000 impressions, because for those  14  Comedy  Documentary  Fantasy  HorrorWar  Mystery  Western  Adventure  Musical  Drama  Children’s  Thriller  Animation  Action  Crime  Film-Noir  Sci-Fi  Romance  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='80  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='00  Probability  Train  Test  DCRS  NFM  DecRS  Figure 4: Categorical distributions of training data, testing data and top-10 recommended  items of a sampled user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  ∆CTR  ∆StayTime  ∆E CN  ∆E CE  ∆R CN  ∆R CE  DCRS  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='973%  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='062%  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='197%  +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='111%  +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='372%  +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='276%  Table 4: Results of online A/B test on Toutiao app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  that appear less frequently could be introduced by some special strategies rather than the  compared methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We then calculated four metrics: (1) E CN: number of distinct cat-  egories of targeted items shown to a user;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (2) E CE: entropy of category distribution of  targeted items shown to a user ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (3) R CN: number of distinct categories of targeted items  read by a user ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (4) R CE: entropy of category distribution of targeted items read by a user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  The A/B test was conducted for seventeen consecutive days and the average performance  of the above metrics is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We report experimental results in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' All reported  results are significant with p-value < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We can observe that DCRS achieved higher CTR  and StayTime, indicating improved users’ satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Moreover, while the improvements  in E CN and E CE were not that large, DCRS gained huge improvements in R CN and  R CE, implying DCRS is able to generate diverse recommendations the user will prefer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Relate work  DCRS is closely related to two lines of existing work: (1) addressing accuracy-diversity  dilemma in recommendations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' and (2) disentangled user representation learning for general  user modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  15  Addressing accuracy-diversity dilemma in recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Besides recommen-  dation accuracy, more and more research suggests other factors of recommendation quality  also contribute to the overall user satisfaction about the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Of these factors, rec-  ommendation diversity has been shown as a critically important one (Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=',  2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Wilhelm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2018), which however also leads to the so-called accuracy-diversity  dilemma (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021a): higher accuracy often means losing di-  versity to some extent and vice verse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' One main reason is that previous solutions with  accuracy as the primary goal tend to focus on items in the dominant categories in users’  interaction history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' In order to guarantee user satisfaction, three different types of solutions  are proposed, namely post-processing, learning to rank, and diversified recommendation  models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  For the first, and most popular, type of solutions, a re-ranking or post-processing mod-  ule is appended to a chosen recommendation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' The post-processing module takes  recommended items as input and re-orders them to balance recommendation accuracy and  diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Various post-processing algorithms (Ziegler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Qin and Zhu, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Ashkan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Kaya and Bridge, 2019) are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' For example,  Ziegler et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (Ziegler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2005) first applied the Maximal Marginal Relevance (MMR)  algorithm, which was used for topic diversification in search engines, to minimize redun-  dancy among recommended items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Determinantal Point Process (DPP) has been shown as  the most effective one (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2018) of all post-processing algorithms, which scores  an entire list of items rather than every item individually for better modeling of item cor-  relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' However, all these post-processing algorithms are separately constructed from  the recommendation models, though their learning highly depends on the performance of  the recommendation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' When the recommendation model fails to provide a diverse  item list to start with, or gives pretty-low scores to diverse items, the effectiveness of the  aforementioned post-processing algorithms will largely deteriorate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Moreover, as shown in  our experiment results in Section 3, the aforementioned post-processing algorithms usually  seriously sacrifice recommendation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Learning To Rank type solutions (Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2022)  aim to directly recommend a list of items to users, rather than selecting items one by one  according to their prediction scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' However, this line of work often suffers from high time  complexity, which limits its application in real world recommendation scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Recently, several solutions are proposed to directly improve the diversity of recommen-  dation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Zheng et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' al (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021a) proposed a diversified recommendation  model based on Graph Convolutional Networks (GCN), with improving recommendation  diversity as its only target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Wu et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' al (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2019c) leveraged the GAN framework  for diverse recommendations, where a generator tries to recommend diverse sets of items,  and a discriminator aims to distinguish the generated recommendations from a set of items  randomly sampled from the observed data of the target user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' The most related work to  ours is (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021), where the authors studied the problem of lack of diversity in  recommendations from a casual perspective, and proposed DecRS to alleviate the prob-  lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Experiments demonstrate the advantage of our proposed DCRS over these solutions  in improving recommendation accuracy and diversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' A recent work (Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2022) also  tried to diversify recommendations in relevant recommendation scenario, where the diversi-  fication is conducted regarding multiple item aspects such that relevance and diversity are  16  adaptively balanced among different item aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' However, when only one item aspect is  considered, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', the item category in this paper, their algorithm degenerates to the MMR  algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Disentangled user representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  Learning disentangled user representations has  drawn increasing attention in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' A family of solutions are based on Variational  Auto-Encoder (VAE) to force each dimension of learnt representations to focus on different  latent factors (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Nema et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' However, such a disen-  tanglement is implicit and therefore one cannot associate the disentangled representation  with the specific attributes of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Zheng et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' al (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2021b) proposed DICE  to learn representations where user interest and conformity are structurally disentangled  via direct supervision from cause-specific data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' However, in our problem, we cannot access  users’ preferences over item categories explicitly, thus are not able to get any direct supervi-  sion about it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Chen et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=', 2022) proposed to disentangle item representations  to address popularity bias, by requiring the two disentangled item representations to be  orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' In our solution, we disentangle a user’s preference over an item into category  dependent and independent segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Both segments relate to the user and thus they do  not need to be orthogonal to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Conclusion  In this paper, we propose a new principle that the diversification of recommendations should  be performed within a user’s preferred categories, such that improved recommendation  diversity can be achieved without sacrificing recommendation accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  We realize this  principle via a general framework, named DCRS, to disentangle a user’s category dependent  and independent preference in the learnt representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We evaluate DCRS through both  offline experiments on three widely-used benchmark datasets for recommendation and online  A/B test on Toutiao, one of the largest news recommendation platforms in China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' We  demonstrate DCRS can provide more accurate and diversified recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Via in-  depth analysis and case studies, we find that the benefit of DCRS is introduced by: (1) it  can capture a user’s diverse preference in historical interactions more thoroughly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' and (2)  it can rank items in the same category more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  In this work, we took a static view of users’ preferences over items and item categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  But numerous studies have demonstrated that users’ preferences evolve over time Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content='  (2018b, 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' It is interesting to study the problem of recommendation diversification  in an interactive manner over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' Moreover, currently we only recommend one item a  time to a user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
+page_content=' It is interesting to study how to generate a list of diverse recommendations,  where diversity should be optimized within and across recommendation lists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zNE5T4oBgHgl3EQfNw6V/content/2301.05492v1.pdf'}
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diff --git a/zdFQT4oBgHgl3EQfCTXT/content/tmp_files/2301.13230v1.pdf.txt b/zdFQT4oBgHgl3EQfCTXT/content/tmp_files/2301.13230v1.pdf.txt
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+Astronomy & Astrophysics manuscript no. seleff
+©ESO 2023
+February 1, 2023
+Strong lensing selection effects
+Alessandro Sonnenfeld1, 2, Shun-Sheng Li2, Giulia Despali3, Anowar J. Shajib4, 5, 6, and Edward N. Taylor7
+1 Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
+e-mail: sonnenfeld@sjtu.edu.cn
+2 Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands
+3 Institut für Theoretische Astrophysik, Zentrum für Astronomie, Heidelberg Universität, Albert-Ueberle-Str. 2, 69120, Heidelberg,
+Germany
+4 Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637, USA
+5 Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA
+6 NHFP Einstein Fellow
+7 Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn 3122, Australia
+ABSTRACT
+Context. Strong lenses are a biased subset of the general population of galaxies.
+Aims. The goal of this work is to quantify how lens galaxies and lensed sources differ from their parent distribution, namely the strong
+lensing bias.
+Methods. We first studied how the strong lensing cross-section varies as a function of lens and source properties. Then, we simulated
+strong lensing surveys with data similar to that expected for Euclid and measured the strong lensing bias in different scenarios. We
+focused particularly on two quantities: the stellar population synthesis mismatch parameter, αsps, defined as the ratio between the true
+stellar mass of a galaxy and the stellar mass obtained from photometry, and the central dark matter mass at fixed stellar mass and size.
+Results. Strong lens galaxies are biased towards larger stellar masses, smaller half-mass radii and larger dark matter masses. The
+amplitude of the bias depends on the intrinsic scatter in the mass-related parameters of the galaxy population and on the completeness
+in Einstein radius of the lens sample. For values of the scatter that are consistent with observed scaling relations and a minimum
+detectable Einstein radius of 0.5′′, the strong lensing bias in αsps is 10%, while that in the central dark matter mass is 5%. The bias
+has little dependence on the properties of the source population: samples of galaxy-galaxy lenses and galaxy-quasar lenses that probe
+the same Einstein radius distribution are biased in a very similar way. Quadruply imaged quasar lenses, however, are biased towards
+higher ellipticity galaxies.
+Conclusions. Given current uncertainties, strong lensing observations can be used directly to improve our current knowledge of the
+inner structure of galaxies, without the need to correct for selection effects.
+Key words. Gravitational lensing: strong
+1. Introduction
+Strong gravitational lensing is a powerful tool for studying
+galaxies and cosmology. Strong lenses have been used to probe
+the mass structure of massive galaxies (Auger et al. 2010; Oguri
+et al. 2014; Sonnenfeld et al. 2015; Shajib et al. 2021), to detect
+substructure (Vegetti et al. 2012; Hezaveh et al. 2016; Nieren-
+berg et al. 2020), to carry out detailed studies of magnified star-
+forming galaxies (Jones et al. 2013), and to measure the expan-
+sion rate of the universe with time delays (see Treu & Marshall
+2016 for a review).
+Strong lenses, however, are a biased subset of the general
+population of galaxies and background sources. A necessary
+condition for a galaxy to act as a strong lens with respect to a
+given source is that its projected surface mass density Σ(θ) must
+be larger than the critical surface mass density for lensing Σcr
+at at least one position θ (Schneider et al. 1992). This condi-
+tion excludes objects with a very diffuse mass distribution from
+the population of lenses. In general, galaxies with a higher con-
+centration of mass are more likely to be strong lenses, and are
+therefore over-represented in lens samples. The distribution of
+strongly lensed sources can also be biased with respect to the
+general population of background galaxies: for instance, lensing
+magnification allows the detection of fainter objects with respect
+to the field.
+In general, the probability distribution PSL of a sample of
+strong lenses from a given survey with selection criterion S is
+given by (Sonnenfeld 2022):
+PSL(ψg, ψs|S ) ∝ Pg(ψg)Ps(ψs)Psel(ψg, ψs|S ).
+(1)
+In the equation above, ψg is the set of parameters describing
+the properties of foreground galaxies that are relevant for lens-
+ing, such as their redshift and mass distribution; ψs is the set of
+parameters describing background sources; Pg and Ps describe
+the parent distribution of foreground galaxies and background
+sources in the absence of lensing; and Psel describes the proba-
+bility of selecting a lens-source system with parameters ψg and
+ψs given the criterion S used to define a lens. This last factor
+takes into account both physical effects, that is whether a lens
+with parameters ψg produces a strongly lensed image of a source
+with parameters ψs, and survey selection effects, that is whether
+such an image can be detected and recognised as a strong lens.
+The left-hand side of Equation 1 is directly accessible from
+strong lensing observations. If the main goal of a lensing survey
+is to characterise the properties of the strong lens population,
+then it can be accomplished by directly analysing this term. For
+Article number, page 1 of 22
+arXiv:2301.13230v1  [astro-ph.GA]  30 Jan 2023
+
+A&A proofs: manuscript no. seleff
+many applications of strong lensing, however, the aim is to con-
+strain the properties of the general galaxy or source population,
+Pg and Ps, which are coupled in a nontrivial way via the lens se-
+lection probability Psel. In order to obtain an unbiased estimate of
+either Pg or Ps, then, it is necessary to invert Equation 1. In prin-
+ciple, this can be done with a Bayesian hierarchical formalism
+(Sonnenfeld 2022), but knowledge of the lens selection proba-
+bility Psel is required. This factor can be written as the following
+product:
+Psel(ψg, ψs|S ) = Pdet(ψg, ψs)Pfind(ψg, ψs|S ),
+(2)
+where Pdet is the probability of detecting a strong lensing event,
+and Pfind(ψg, ψs|S, det) is the probability of correctly classifying
+it as such1. The detection probability Pdet can be obtained via
+simulation. The main challenge is characterising Pfind: in most
+of the existing strong lensing surveys, the process of determining
+whether a system is included in a strong lens sample is typically
+a combination of several cuts, usually involving a nontrivial vi-
+sual selection step.
+For the above reasons, the problem of inverting Equation 1
+is a difficult one to tackle exactly. Few studies have attempted
+to explicitly account for strong lensing selection effects, usually
+by making ad-hoc simplifying assumptions (Sonnenfeld et al.
+2015; Oldham et al. 2017; Sonnenfeld et al. 2019). Whether it is
+necessary to invert Equation 1, however, depends on the severity
+of the strong lensing bias that needs to be corrected and on the
+accuracy requirements on the key quantities of interest.
+In this paper we aim to quantify the strength of the strong
+lensing bias on a series of foreground galaxy and background
+source parameters. In particular, we aim to determine how strong
+lenses differ from the parent population of foreground galaxies
+and background sources in terms of a) the radial mass structure
+of the lenses (i.e. their stellar and dark matter mass density pro-
+files); b) the ellipticity of the lenses; c) the size-magnitude distri-
+bution of the lensed sources. The answer to this question depends
+on 1) how the lens detection probability Pdet varies as a function
+of galaxy and source properties, 2) the efficiency of a survey at
+correctly classifying detected strong lenses (i.e. Pfind), and on 3)
+the shape of the galaxy and source parameters distribution Pg
+and Ps. To understand this third point we can imagine the limit-
+ing case in which both Pg and Ps are Dirac delta functions (i.e.
+all lenses and sources are identical): in this limit, PSL simply re-
+duces to the product PgPs up to a scaling constant, corresponding
+to a case in which the lensing bias is none.
+Mandelbaum et al. (2009) carried out a thorough study of
+point 1): they quantified how the properties of a lens galaxy de-
+termine its probability of creating a lensing event with a point
+source. In this work we revisited the Mandelbaum et al. (2009)
+study, expanding it to the extended source case. We simulated in-
+dividual lenses and examined how the lens detection probability
+varies with lens and source properties. In addition, we addressed
+points 2) and 3) as well: we simulated large populations of strong
+lenses using empirical models, we simulated the lens detection
+and finding phase, and quantified the lensing bias under various
+scenarios. We explored how the results change as a function of
+the efficiency of a lens survey at discovering small image sepa-
+ration lenses, and of the scatter in mass parameters at fixed light.
+Finally, we addressed the question of how different are
+galaxy-galaxy lens samples from sets of galaxy-quasar lenses.
+This last point is relevant for time-delay cosmography studies, in
+1 Sonnenfeld (2022) implicitly assumed Pfind ≡ 1, therefore Pdet and
+Psel can be used interchangeably in the context of that work.
+which measurements of the time-delay between the multiple im-
+ages of a strongly lensed quasar are used to constrain the expan-
+sion rate of the Universe. Galaxy-galaxy lenses can in principle
+be used to help break some of the model degeneracies affecting
+these measurements (Birrer & Treu 2021), but any difference be-
+tween the two lens classes can introduce biases, if not corrected
+for. With this study we aim to quantify this bias.
+The structure of this work is the following. In Section 2 we
+introduce the basics of gravitational lensing. In Section 3 we
+study individual lens systems. In Section 4 we describe our sim-
+ulations of lens surveys. In Section 5 we show the results of our
+analysis on the simulated lens survey data. We discuss the results
+in Section 6 and draw conclusions in Section 7.
+The Python code used for the simulation and analysis of the
+lens sample can be found in a dedicated section of a GitHub
+repository2.
+2. Strong lensing basics
+The lensing properties of an object with respect to a source de-
+pend solely on its dimensionless surface mass density distribu-
+tion κ(θ) (also referred to as the convergence). This is the ratio
+between the surface mass density and the critical surface mass
+density for lensing:
+κ(θ) = Σ(θ)
+Σcr
+.
+(3)
+The latter quantity is defined as
+Σcr =
+c2Ds
+4πGDdDds
+,
+(4)
+where c is the speed of light, G the gravitational constant, and
+Dd, Ds, and Dds are the angular diameter distances between the
+observer and the lens, the observer and the source, and the lens
+and the source, respectively.
+Given a source at angular position β, images of it form at the
+positions θ that are solutions of the lens equation
+β = θ − α(θ),
+(5)
+where α is the deflection angle of the lens. This can be expressed
+in terms of the dimensionless surface mass density by means of
+the following integral over the whole sky:
+α(θ) = 1
+π
+�
+R2 d2θ′κ(θ′) θ − θ′
+|θ − θ′|2 .
+(6)
+The images of the background source are in general magnified
+in total flux and in size, while preserving the original surface
+brightness.
+2.1. The axisymmetric lens
+In the special case of an axisymmetric lens we can simplify the
+notation by considering a single coordinate axis with origin at
+the centre of the lens. We label the components of the image
+position, source position and deflection angle along this axis as
+θ, β and α, respectively. The lens equation for an axisymmetric
+lens then becomes
+β = θ − α(θ),
+(7)
+2 https://github.com/astrosonnen/strong_lensing_tools/
+papers/selection_effects
+Article number, page 2 of 22
+
+Sonnenfeld et al.: Strong lensing selection effects
+and the expression for the deflection angle reduces to
+α(θ) = 2
+θ
+� θ
+0
+dθ′κ(θ′)θ′.
+(8)
+This can also be expressed in terms of the total projected mass
+enclosed within a circle with angular radius equal to θ:
+α(θ) = 1
+πθ
+M(< θ)
+ΣcrD2
+d
+.
+(9)
+A very important quantity for describing the strength of a
+strong lens is the Einstein radius, θEin. For an axisymmetric lens,
+this is the radius corresponding to the solution of Equation 7 for a
+source placed at the same angular position as the lens centre (β =
+0). The circle with radius equal to θEin is known as the tangential
+critical curve. Images that form there have infinite magnification
+in the tangential direction. It can be shown that θEin satisfies the
+following condition,
+¯κ(< θEin) = 1,
+(10)
+where ¯κ(< θ) is the average surface mass density within a radius
+equal to θ:
+¯κ(< θ) ≡ 2
+θ2
+� θ
+0
+dθ′κ(θ′)θ′.
+(11)
+Axisymmetric lenses of the kind considered in this work typ-
+ically produce either one or three images of a point source. Fig-
+ure 1 helps to visualise this property. Plotted in Figure 1 is the
+quantity θ − α(θ) as a function of the position in the image plane
+θ, for a few lens models. According to Equation 7, images form
+at the locations where this quantity equals the position of the
+source. Therefore, given a source position β, the number of im-
+ages and their location can be determined by drawing a horizon-
+tal line at the value β on the vertical axis, and finding the points
+where this line intersects the θ − α(θ) curve.
+For small values of β (for sources close to the lens centre),
+the number of images that are produced is three: the source is
+strongly lensed. For large values of β, instead, only one image is
+formed. The value of β where the transition occurs is known as
+the radial caustic, which is marked by the horizontal dotted line
+in Figure 1. As can be seen from Figure 1, a source at this loca-
+tion is mapped to the stationary point of the θ − α(θ) curve. That
+location on the image plane is known as the radial critical curve.
+Images that form there have a formally infinite magnification in
+the radial direction3.
+2.2. The elliptical lens
+In this paper we focus mostly on lens galaxies with elliptical
+isodensity contours. Given a surface mass density profile Σ(R),
+a lens with an elliptical mass distribution can be obtained by
+replacing the radial coordinate with the circularised radius:
+R →
+�
+qx2 + y2
+q ,
+(12)
+where x and y are Cartesian axes centred on the lens centre, with
+x pointing towards the major axis, and where q is the minor-to-
+major axis ratio.
+3 The slope of the θ − α(θ) curve is the inverse of the radial magnifica-
+tion. This can be understood by taking the derivative of Equation 7 with
+respect to θ.
+θ − α(θ)
+θ
+θE
+−θE
+fDM = 0.5, γDM = 1.0
+fDM = 0.5, γDM = 1.5
+fDM = 0.5, γDM = 2.0
+fDM = 0.2, γDM = 1.5
+fDM = 0.8, γDM = 1.5
+Fig. 1. The lens equation of axisymmetric lenses. Solid lines: right-
+hand side of Equation 7 for five lenses with different density profiles.
+Given a source at position β, its lensed images form at the values of
+θ where the solid line intersects a horizontal line located at β on the
+vertical axis. Dotted lines: positions of the radial caustics. Sources lo-
+cated within the radial caustic of a given lens produce three images. The
+lens models used in this simulations consist of a stellar component and
+a dark matter halo, as described in section 3.1. Their density profile is
+plotted in Figure 4.
+Figure 2 shows the source-plane caustics of elliptical lenses
+with different values of the axis ratio. We used the software
+Glafic (Oguri 2021) to obtain the caustic curves. The outermost
+curves are radial caustics, while the inner ones are tangential
+ones. The most striking difference with respect to the axisym-
+metric case (blue curves in Figure 2) is the fact that the tangential
+caustic is transformed from a point into a diamond-like curve.
+Sources located within the diamond produce five images (one of
+which is usually highly de-magnified). Sources that lie in the re-
+gion enclosed between the two caustics are imaged three times.
+Sources outside the radial caustic are imaged only once, as in the
+axisymmetric case. The fact that the number of images changes
+by two at a caustic crossing is a general feature of gravitational
+lenses with no singularities (Schneider et al. 1992).
+2.3. Lensing event definition: point sources
+In order to compute the probability of a lensing event we must
+provide an exact definition for it. A necessary condition for a
+lens-source system to qualify as a strong lens is the presence of
+multiple images of the source. As we showed above, this requires
+the source to lie within the region enclosed by the radial caustic.
+In order to recognise a strong lens in a real survey, however, it is
+not sufficient for multiple images to exist: they must be detected.
+For this reason, given the detection limit for a point source, mlim,
+we define as strong lensing event any lens-source system with at
+least two images brighter than mlim. This is the same definition
+used by (van de Ven et al. 2009), upon which the Mandelbaum
+et al. (2009) work is based. Labelling with m2 the magnitude of
+the second-brightest image, then, in the absence of photometric
+Article number, page 3 of 22
+
+A&A proofs: manuscript no. seleff
+−0.8−0.6−0.4−0.2 0.0 0.2 0.4 0.6 0.8
+βx/θEin
+−0.8
+−0.6
+−0.4
+−0.2
+0.0
+0.2
+0.4
+0.6
+0.8
+βy/θEin
+Major axis
+q = 1.0
+q = 0.9
+q = 0.8
+q = 0.7
+q = 0.6
+Fig. 2. Caustics of lenses with fixed radial structure and different el-
+lipticity. Source-plane angular coordinates are in units of the Einstein
+radius. The outer curve is the radial caustic, while the inner diamond is
+the tangential caustic. Point sources located outside the caustic are not
+strongly lensed. Sources that lie in the region enclosed between the two
+caustics produce three images, while sources inside the tangential caus-
+tic produce five images. The lens model adopted for this experiment
+consists of a stellar component and a dark matter halo, as described in
+3.1. The two components have the same ellipticity.
+noise the lens detection probability Pdet becomes
+Pdet(ψg, ψs) =
+� 1
+if m2(ψg, ψs) < mlim
+0
+otherwise
+.
+(13)
+2.4. Lensing event definition: extended sources
+Defining a strong lensing event in the case of an extended source
+is less straightforward. In principle, we should require parts of
+the source to be multiply imaged. In practice, it is not always
+easy to determine whether a lensed image contains multiple im-
+ages or not. This is because, when the source is extended, some
+of the images can be blended together. In real strong lensing sur-
+veys, it is common to find lens candidates in which the lensed
+source consists of only a single arc. In those cases it is diffi-
+cult to establish whether the arc is a set of blending images or
+not, and the decision of including such systems in a strong lens
+sample is often arbitrary. Here we adopt the following working
+definition: an extended source is strongly lensed if the number
+of surface brightness peaks that are detected is larger than its
+intrinsic number of peaks in the absence of lensing.
+We explain how this definition applies in practice with a few
+examples. For simplicity, we focus on the case of a source with
+a single surface brightness peak, such as a Sérsic profile. All
+of the sources considered in this work belong to this family of
+surface brightness models. We adopt the following procedure to
+determine the number of detected peaks. Given the observed im-
+age of a lensed source, we define its footprint as the ensemble
+of pixels where the source is detected with S/N > 2. The foot-
+print is in general composed of multiple disconnected regions,
+corresponding to the different images. In order to only include
+images that can be clearly identified, we add the condition that
+the integrated S/N of each separate region must be S/N > 10.
+This condition has the effect of removing from the source foot-
+print any isolated region consisting only of a very small number
+of pixels. In a real-world application, it would be very hard to
+classify such marginal detections as images. If, after applying
+this cut, the source footprint is spread over multiple separated
+regions, then the system is classified as a lens. If the source foot-
+print consists instead of a single region, we iteratively increase
+the surface brightness threshold used to define the footprint and
+count the number of isolated regions with S/N > 10. The maxi-
+mum number of detected regions defines the number of detected
+peaks.
+Figure 3 shows a few examples of how this criterion can be
+used to classify lenses. The first column shows the caustic struc-
+ture and source position. The second column shows an image
+of the lensed source. The third column shows the source foot-
+print defined with the procedure described above. Pink pixels
+correspond to the largest footprint that maximises the number of
+detected images, while purple pixels belong to the 2σ detection
+footprint. In the first, second and sixth example, these two foot-
+prints coincide. In the third, fourth and fifth example, instead,
+the 2σ detection footprint consists of a single region, but an in-
+crease of the surface brightness threshold leads to the detection
+of additional images.
+Strictly speaking, our lens definition criterion fails for a per-
+fect Einstein ring. In order to cover such a scenario, we also clas-
+sify as strongly lensed any sources that produce a footprint with
+a hole.
+We point out that, although our lens definition relies on peak
+detections, it is not necessary for the peak of the source surface
+brightness to lie within the caustics in order for the source to be
+strongly lensed. This can be observed in the fourth example of
+Figure 3: although the source centroid lies outside of the caustic
+(as shown in the first column), the outskirts of the source overlap
+with the lens, and therefore multiple images are produced.
+We can identify three different regimes in strong lensing of
+extended sources, depending on the relative size of the source
+and of the lens caustics. In the limit of a very small source, the
+image configurations that are produced are qualitatively similar
+to those that can be obtained in the point-source case. When the
+source and caustic size are comparable, the multiple images tend
+to be blended into arcs or rings. Most, if not all, of the galaxy-
+scale strong lenses known belong to these two categories. How-
+ever, there is a third regime in strong lensing, corresponding to
+the case in which the source size is much bigger than the caus-
+tics, such as in the fifth example of Figure 3. In this regime, the
+overall size and total flux of the source are roughly preserved,
+and the lens produces only a relatively minor perturbation on
+a localised region of the image. Strong lenses of this kind can
+be difficult to detect, especially if the region of the image sub-
+ject to strong lensing overlaps with the light from the foreground
+galaxy. The ultimate limiting factor to the detection of strong
+lenses in the large source size regime, however, is the ability to
+spatially resolve the multiple images. This limit is set by the size
+of the point spread function (PSF).
+All of the examples that we considered were based on
+sources with a single surface brightness peak. In general this
+is not necessarily the case, and the intrinsic number of surface
+brightness peaks of a strongly lensed source is not known a pri-
+ori. When dealing with a real lens candidate, applying our lens
+definition criterion requires showing that the observed number
+of surface brightness peaks can be reproduced with a lens model
+Article number, page 4 of 22
+
+Sonnenfeld et al.: Strong lensing selection effects
+Caustics
+Source
+Lens
+2σ detection
+Max. # images
+Not a lens
+Lens
+Lens
+Lens
+Not a lens
+Fig. 3. Criterion used to classify lensed images of extended sources.
+Six examples. First column: caustics (red curves) and source position
+(blue circle). The radius of the circle corresponds to the radius at which
+the surface brightness is equal to 2σ the sky background rms fluctuation
+for a single pixel. In other words, the blue circle delimits the area of the
+source that can be detected. Second column: mock image of the lensed
+source, with added noise. Third column: footprint of the source. The
+purple footprint is obtained with a 2σ detection criterion. The coral re-
+gion is the largest footprint with the highest number of detected images.
+in which the source has a smaller number of peaks. In samples
+of lenses defined via visual inspection this process is typically
+done implicitly, by identifying multiply imaged blobs that be-
+long to the same source element.
+To our knowledge, we are the first to propose a surface
+brightness peak-based definition of a strong lensing event. A
+popular alternative definition of a strong lens is one based on
+magnification: only images that are magnified by more than
+a given threshold are considered as strongly lensed (see e.g.
+Hilbert et al. 2007). The problem with a definition of this kind
+is that magnification cannot be determined unambiguously from
+observations, unless the intrinsic properties of the lensed source
+are known from other means (e.g. if the source is a standard can-
+dle or a standard ruler). Because of the mass-sheet degeneracy
+(Falco et al. 1985), it is possible to vary the magnification of a
+lensed image while keeping its observed properties fixed: this
+could lead to the paradox of two identical-looking lenses that
+are classified differently on the basis of the underlying magnifi-
+cation. Although we could still use a magnification-based defi-
+nition for the sake of carrying out our experiments, it would then
+be difficult to apply our results to real data. Our definition of a
+strong lensing event, instead, is robust with respect to the mass-
+sheet degeneracy.
+3. Individual lenses
+In this section we study how the probability of a strong lensing
+event varies as a function of lens and source properties. In order
+to do so, it is useful to introduce the concept of strong lensing
+cross-section. Given a foreground galaxy with parameters ψg,
+a background source with parameters ψs, and a criterion S to
+define a strong lensing event, the strong lensing cross-section is
+defined as (Sonnenfeld 2022):
+σSL =
+�
+R2 dβPdet(ψg, ψ(−β)
+s
+, β|S ),
+(14)
+where β is the position of the background source, ψ(−β)
+s
+is the
+ensemble of source parameter except for the position, and the
+integral is carried out over the whole sky. The definition above
+is valid for both a point source and an extended source: although
+there is no unique way of defining the position of an extended
+source, the integral over the sky ensures that the result is inde-
+pendent of how the source position is defined. In the limit of low
+density of background sources, which is satisfied in all practi-
+cal cases, the probability of a lensing event is proportional to
+σSL. The lensing cross-section defined via Equation 14 depends
+solely on the lens detection probability Pdet and does not take
+into account whether the lens can be correctly classified by a
+lens finder. This separate selection step is captured by the term
+Pfind in Equation 2. In this section we consider exclusively the
+detectability of a lens, and therefore focus only on σSL. In Sec-
+tion 4, when considering specific lens survey simulations, we
+introduce Pfind.
+We compute σSL in a series of different scenarios of increas-
+ing complexity. The model family adopted to describe the radial
+density profile of the lenses is the same in all of our experiments.
+We describe this in section 3.1. In section 3.2 we show calcula-
+tions of the strong lensing cross-section in the case of axisym-
+metric lenses and point sources. In section 3.3 we generalise the
+lens geometry to elliptical, while in section 3.4 we replace point
+sources with extended sources.
+3.1. Lens density profile
+In this work we focus on massive early-type galaxies as lenses,
+as these make up the vast majority of known strong lenses. We
+describe their mass distribution with a model consisting of two
+concentric components, one describing the baryons and one for
+Article number, page 5 of 22
+
+A&A proofs: manuscript no. seleff
+the dark matter. We assume that the baryonic component con-
+sists entirely of stars, thereby neglecting gas, which is known to
+contribute very little to the mass of early-type galaxies in the in-
+ner regions that are probed by strong lensing. We then assume
+that the stars follow a Sérsic profile, with projected surface mass
+density given by
+Σ(R) = Σ0 exp
+�������−b
+� R
+Re
+�1/n�������.
+(15)
+In the above equation,
+Σ0 =
+M∗b2n
+2πnR2eΓ(2n),
+(16)
+M∗ is the total mass, Re is the radius enclosing half of the total
+mass, n is the Sérsic index, Γ is the incomplete Gamma function,
+and b is given by (Ciotti & Bertin 1999)
+b(n) ≈ 2n − 1
+3 +
+4
+405n +
+46
+25515n2 + O(n−3).
+(17)
+Throughout this paper we indicate with θe the angular size of the
+half-light radius.
+We fix the Sérsic index of the lenses to n = 4, correspond-
+ing to a de Vaucouleurs model. Although early-type galaxies are
+often described with a free Sérsic index, a de Vaucouleurs pro-
+file is able to reproduce their surface brightness distribution to
+a few percent in the radial range 1kpc < R < 30kpc (see e.g.
+Sonnenfeld 2020), which is the region that is most relevant for
+strong lensing. Finally, we assume that the light distribution of
+the stellar component follows its mass distribution exactly. That
+is, we do not allow for the presence of gradients in the stellar
+mass-to-light ratio. In section 6.5 we discuss qualitatively what
+the implications of relaxing this assumption would be.
+We describe the dark matter halo with a generalised Navarro,
+Frenk & White (gNFW) profile. We first define it by its three-
+dimensional distribution, which for a spherically symmetric pro-
+file is
+ρ(r) =
+ρ0
+(r/rs)γDM (1 + r/rs)3−γDM .
+(18)
+The parameter γDM is the inner density slope, ρ0 is a normal-
+isation parameter, while rs is the scale radius. The logarithmic
+slope of the density profile transitions from −γDM to −3 at a ra-
+dius r ≈ rs. The projected surface mass density of a gNFW pro-
+file can be expressed in terms of the following integral (Wyithe
+et al. 2001):
+Σ(R) = 2rsρ0
+� R
+rs
+�1−γDM � π/2
+0
+dx sin x(sin x + R/rs)γDM−3.
+(19)
+Figure 4 shows the dimensionless surface mass density pro-
+file of Sérsic + gNFW models with various values of the inner
+dark matter density slope γDM and of the fraction of projected
+dark matter mass within the half-light radius, fDM. All of the pro-
+files have a dark matter scale radius equal to ten times Re, and are
+normalised in such a way that the Einstein radius is equal to the
+half-mass radius of the stellar component (these assumptions are
+dropped later). Two main features emerge from Figure 4. First,
+the baryons generally dominate the total density in the inner re-
+gions (θ < θEin), while the dark matter is the main component at
+large radii. Second, models with different dark matter fractions
+and inner dark matter slopes can conspire to produce very sim-
+ilar total density profiles. This is the case, for example, of the
+( fDM = 0.5, γDM = 1.5) and the (fDM = 0.2, γDM = 2.0) models
+(green and blue lines in Figure 4).
+10−1
+100
+101
+θ/θEin
+10−1
+100
+101
+κ(θ)
+fDM = 0.5, γDM = 1.0
+fDM = 0.5, γDM = 1.5
+fDM = 0.5, γDM = 2.0
+fDM = 0.2, γDM = 1.5
+fDM = 0.8, γDM = 1.5
+Fig. 4. Dimensionless surface mass density profile of Sérsic + gNFW
+composite models. The Sérsic index of the baryonic component is fixed
+to n = 4 and the scale radius of the dark matter component is fixed to ten
+times the half-light radius. All of the profiles are normalised in such a
+way that the Einstein radius is equal to the half-light radius. Solid lines:
+total density profile. Dotted lines: dark matter density profile. Dashed
+lines: baryonic density profile. The blue, orange and green dashed lines
+are identical, as they correspond to profiles with the same fraction of
+baryonic mass within the half-light radius.
+3.2. Axisymmetric lenses, point sources
+Axisymmetric lenses with a density profile of the kind intro-
+duced in section 3.1 can produce either one or three images of
+a point source. This can be seen in Figure 1, which shows the
+lens equation for various values of the dark matter fraction and
+the inner dark matter slope. All of the lenses shown in this figure
+have the same Einstein radius, which is equal in size to the half-
+mass radius of the stellar component. The radius of the radial
+caustic, marked by the dotted lines in Figure 1, is a strong func-
+tion of the lens properties: it is largest in lenses with a smaller
+dark matter fraction or steeper dark matter slopes. As a result,
+the source plane area that is subject to strong lensing is an even
+stronger function of these properties, since it scales with the
+square of the radial caustic radius. In order to compute the lens-
+ing cross-section, however, we must take into account the mag-
+nification, because that determines whether multiple images can
+be detected or not.
+Figure 5 shows the magnification of the secondary image as
+a function of source position. The secondary image is the one
+located in the region between the radial and tangential critical
+curves, opposite to the source with respect to the lens centre. In
+most practical cases this is the second brightest image. As Fig-
+ure 5 shows, the magnification is very large for sources close
+to the lens centre (small values of β), decreases with increasing
+source position and then increases in close proximity to the ra-
+dial caustic. While for the model with fDM = 0.8 (purple curve)
+the magnification is above unity everywhere, other lens mod-
+els can produce highly de-magnified secondary images for large
+values of β. Depending on the intrinsic brightness of the lensed
+source, these images may or may not be detected.
+Using the definition of Equation 14, we computed the lens-
+ing cross-section of a set of axisymmetric lenses, with respect
+to point sources with different brightnesses. In particular, we
+considered model lenses with fixed Einstein radius, with angu-
+lar half-mass radius θe fixed to θEin, and varying values of the
+Article number, page 6 of 22
+
+Sonnenfeld et al.: Strong lensing selection effects
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+β/θEin
+10−1
+100
+101
+|µ2|
+fDM = 0.5, γDM = 1.0
+fDM = 0.5, γDM = 1.5
+fDM = 0.5, γDM = 2.0
+fDM = 0.2, γDM = 1.5
+fDM = 0.8, γDM = 1.5
+Fig. 5. Magnification of the secondary image as a function of the source
+position. The lenses are axisymmetric composite models. Their density
+profile is plotted in Figure 4. Vertical dotted lines mark the position of
+the radial caustic.
+dark matter fraction and dark matter inner slope. The results are
+shown in Figure 6. Each line corresponds to a source with a given
+intrinsic (i.e. unlensed) magnitude, ms. The difference between
+this magnitude and the limiting magnitude of the survey is indi-
+cated as follows:
+∆m ≡ ms − mlim.
+(20)
+We can see a clear trend between ∆m and the lensing cross-
+section, in both panels of Figure 6: σSL is larger for brighter
+sources. The trend saturates below a certain ∆m, for sufficiently
+small values of γDM or for sufficiently large values of fDM. In
+these cases the lensing cross-section coincides with the full area
+enclosed within the caustic, and further increasing the source
+brightness does not result in an increased value of σSL.
+At fixed source brightness, trends with the dark matter inner
+slope or dark matter fraction are generally weak. The lines of
+Figure 6, however, have been computed by keeping the Einstein
+radius fixed while varying γDM or fDM. This is achieved by ad-
+justing other properties of the lens, such as the total mass of the
+baryonic or the dark matter component (see Figure 4). In prac-
+tice, when varying one ingredient of the lens density profile, the
+Einstein radius varies in response. To get a complete view of how
+the lensing cross-section depends on different lens properties, we
+also computed how σSL responds in absolute terms by varying
+one lens parameter at a time. Figure 7 shows σSL as a function of
+stellar mass, half-light radius, inner dark matter slope, and pro-
+jected dark matter mass enclosed within an aperture of 5 kpc,
+MDM,5.
+The lensing cross-section increases with increasing stellar
+mass and dark matter mass, decreases with increasing Re for
+bright sources, while is only a weak function of γDM. The lack of
+a clear trend between σSL and γDM appears to be in contradiction
+with the result of Mandelbaum et al. (2009), who found a strong
+positive correlation between σSL and γDM. The origin of this dis-
+crepancy lies in the different ways in which γDM is varied in the
+0.5
+1.0
+1.5
+2.0
+γDM
+10−2
+10−1
+100
+σSL/(πθ2
+Ein)
+∆ms = −2.0
+fDM = 0.5
+0.2
+0.4
+0.6
+0.8
+fDM
+∆ms = −2.0
+γDM = 1.5
+Fig. 6. Strong lensing cross-section of an axisymmetric lens and a point
+source. The lens galaxy is a composite model, introduced in section 3.1,
+with angular half-light radius equal to the Einstein radius. The left panel
+shows the effect of varying the slope γDM for fixed fDM; conversely, the
+right panel shows variations as a function of fDM with γDM fixed. The
+system is defined as a strong lens if at least two images are detected.
+Different lines correspond to the difference ∆m between the source mag-
+nitude and the survey detection limit for a point source.
+two experiments. While we varied γDM at fixed MDM,5, Mandel-
+baum et al. (2009) kept fixed the virial mass of the dark matter
+halo. At fixed virial mass, increasing the inner dark matter slope
+results in a correspondingly larger dark matter mass in the inner
+regions, which naturally results in a larger lensing cross-section.
+Figure 6 and Figure 7 show that the trends between lens
+properties and the strong lensing cross-section can be differ-
+ent for sources with different brightnesses. The net effect in a
+strong lensing survey is an average over the source population,
+weighted by the source luminosity function. This implies that
+surveys that target different families of sources, with different lu-
+minosity functions (for instance, galaxies or quasars), can have
+different strong lensing biases. We explore this possibility in
+Section 5.
+3.3. Elliptical lenses, point sources
+We measured σSL for lenses with a fixed radial density profile
+and different ellipticities, with respect to point sources of differ-
+ent brightnesses. In particular, we set fDM = 0.5, γDM = 1.5,
+rs = 10Re, θEin = θe, and set the ellipticities of both the bary-
+onic and dark matter components to be the same, with the same
+orientation of the major axis. This is the lens model used to pro-
+duce the caustics plot of Figure 2. Figure 2 suggests that the size
+of the source plane area subject to strong lensing, the area en-
+closed within the outermost caustic, does not vary strongly with
+the ellipticity of the lens. Therefore we expect the strong lensing
+cross-section to be a weak function of ellipticity.
+We carried out the computation of σSL by means of simu-
+lation: we generated a large number of point sources randomly
+distributed over a given area that includes the caustic, then used
+Glafic to solve the lens equation, find the number of images and
+their magnification. We then measured the fraction of sources
+that are strongly lensed according to the criterion of Equation 13
+and multiplied this value by the area over which sources are lo-
+cated. The resulting σSL is shown in Figure 8 as a function of the
+minor-to-major axis ratio q.
+For bright sources the strong lensing cross-section is approx-
+imately constant with axis ratio. This is because, as pointed out
+earlier, in the bright source regime the cross-section is deter-
+mined by the area enclosed within the radial caustic, which does
+not vary much with lens ellipticity. For faint sources we observe
+Article number, page 7 of 22
+
+A&A proofs: manuscript no. seleff
+11.0 11.2 11.4 11.6 11.8 12.0
+log M∗
+10−1
+100
+101
+σSL (arcsec2)
+πθ2
+Ein
+0.6
+0.8
+1.0
+1.2
+log Re
+0.5
+1.0
+1.5
+2.0
+γDM
+10.6 10.8 11.0 11.2 11.4
+log MDM,5
+Fig. 7. Absolute value of the strong lensing cross-section as a function of various lens properties. The reference lens is a galaxy at z = 0.3 with
+log M∗ = 11.5, Re = 7 kpc, γDM = 1.5, log MDM,5 = 11.0, rs = 100 kpc, and a source at z = 1.5 In each panel, only one property of the lens is
+varied, as indicated on the label of the horizontal axis. Each curve corresponds to a different value of ∆m, in accordance with Figure 6. The dashed
+line in each panel shows the quantity πθ2
+Ein.
+0.6
+0.8
+1.0
+q
+10−3
+10−2
+10−1
+100
+σSL/(πθ2
+Ein)
+∆ms = −1.0
+∆ms = −2.0
+Fig. 8. Point source strong lensing cross-section as a function of lens
+axis ratio. Solid lines: cross-section based on the lens event definition
+of Equation 13. Dashed lines: cross-section based on the detection of
+four images (quad cross-section). Lines of different colour correspond
+to sources of different intrinsic magnitude. The dashed blue line, cor-
+responding to the brightest source magnitude, overlaps completely with
+the dashed orange line. The dashed purple line is zero: very faint sources
+cannot produce any detectable quads. The parameters of the lens den-
+sity profile are fDM = 0.5, γDM = 1.5, rs = 10Re, θe = θEin. The baryonic
+and dark matter components have the same ellipticity and direction of
+the major axis.
+a larger variation with q, with a factor of two difference between
+the largest and smallest value of σSL at fixed source brightness.
+Most of the sources that result in detectable lenses produce
+two detectable images. These are sources that are located in the
+region enclosed between the radial and the tangential caustic. If
+the source is located within the tangential caustic, however, four
+detectable images are usually created. Lenses with four visible
+images, usually referred to as quad lenses, are sometimes given
+a high priority in certain lensing studies, because they offer more
+constraints compared to double lenses. For instance, quads make
+up the majority of the lenses used so far in time-delay studies
+(Millon et al. 2020). For this reason we also computed an alter-
+native lensing cross-section, in which the definition of a lensing
+event requires the detection of four images, instead of two. This
+is plotted in Figure 8 with dashed lines. The cross-section for
+quads is a strong function of lens ellipticity, for bright sources.
+This is a consequence of the fact that the area enclosed within the
+tangential caustic, which is where a source needs to be in order
+to produce four images, increases with increased lens ellipticity,
+as Figure 2 shows. For sources that are intrinsically fainter than
+the detection limit, however, the quad cross-section is extremely
+small, regardless of ellipticity.
+3.4. Elliptical lenses, extended sources
+In the case of an extended source, the complexity of the problem
+is increased due to the addition of a series of features: the source
+surface brightness distribution, with its radial profile, shape and
+orientation, and the PSF. Moreover, as we discussed in section
+2.4, there are different regimes in strong lensing of extended
+sources, depending on the relative size of the source and the
+caustics of the lens. For this reason, we split our analysis into
+two parts. First, we explore the small source size regime, where
+the source size is comparable to or smaller than the lens caustics.
+Then, we consider cases in which the source size is bigger than
+the lens caustics, which we refer to as the large source regime.
+3.4.1. Small source sizes
+For the sake of reducing the dimensionality of the analysis, we
+focused on circularly symmetric sources. We also fixed the sur-
+face brightness profile to an exponential disk (i.e. a Sérsic model
+with n = 1). We then took a lens model with the same parameters
+used in section 3.3 and a minor-to-major axis ratio of q = 0.7.
+We simulated a large number of images of extended sources
+with Glafic and measured the fraction of them that results in
+a strong lens, according to the definition of section 2.4. For the
+small source size experiment, we used pixels with a size equal
+to 1/20θEin and convolved the images with a Moffat PSF with a
+FWHM of two pixels and a β parameter of 5.0. We also assumed
+that the background noise is an uncorrelated Gaussian field. We
+carried out experiments with sources with different values of the
+total unlensed flux, f, and half-light radius, θe,s. The results are
+shown in Figure 9. The total flux of the source, indicated in the
+Article number, page 8 of 22
+
+Sonnenfeld et al.: Strong lensing selection effects
+10−1
+100
+θe,s/θEin
+10−1
+100
+σSL/(πθ2
+Ein)
+log f = 1.0
+log f = 1.5
+log f = 2.0
+Bright point source
+Fig. 9. Strong lensing cross-section of an extended source. Small source
+size regime (θe,s < θEin). The cross-section is plotted as a function of the
+ratio between the half-light radius of the source and the Einstein radius
+of the lens. The lens model is fixed to be the same of Figure 8, with
+axis ratio q = 0.7. The source is a circular exponential profile. Each
+solid line corresponds to a different value of the total unlensed flux of
+the source. The flux f is expressed in terms of the background noise
+rms fluctuation measured over an area equal to θ2
+Ein. The vertical dashed
+lines correspond to the maximum size for which a galaxy with a given
+flux can be detected in the absence of lensing. The horizontal dotted
+line indicates the cross-section for a very bright point source (i.e. the
+area enclosed within the caustics).
+legend, is measured in units of the sky background rms fluctua-
+tion within an area equal to the square of the Einstein radius.
+As in the point source case, the lensing cross-section in-
+creases with increasing total flux, at fixed source size. At fixed
+flux, σSL stays approximately constant with increasing source
+size until a given value, then drops rapidly for larger sizes. From
+a qualitative point of view, this behaviour can be observed also
+in the absence of lensing: increasing the size of a galaxy while
+keeping its flux fixed lowers its average surface brightness. If
+the surface brightness drops below the sky rms fluctuation level,
+then it becomes very difficult to detect it. In order to determine
+whether there are lensing-specific features in the σSL − θe,s rela-
+tion of Figure 9, we computed, for each source flux, the maxi-
+mum half-light radius for which it can be detected in the absence
+of lensing. We used the same criterion as that of section 2.4 to
+define a detection: we defined the source footprint as the ensem-
+ble of pixels that are 2σ above the background and required the
+total signal-to-noise ratio within the footprint to be larger than
+ten. The resulting limiting sizes are shown as vertical lines in
+Figure 9. For each given total flux, the non-lensing size limit
+is similar to the value of θe,s at which the lensing cross-section
+drops.
+This result suggests that, to first approximation, lensing does
+not introduce a strong selection in source size. While perhaps
+surprising, this follows from the fact that gravitational lensing
+preserves surface brightness: a source that can be detected in the
+absence of lensing will produce images with the same surface
+brightness when lensed, which can be detected as well. In or-
+der to classify a source as strongly lensed, however, we require
+that multiple images are observed. Only sources that lie within
+a well-defined region give rise to a strong lensing configuration.
+If part of the source extends outside of this region, then only a
+fraction of its flux contributes to creating a set of strongly lensed
+images. This lowers the signal-to-noise ratio of the multiple im-
+ages compared to the point source case. The result is that σSL
+starts to decrease with increasing source size at values of θe,s
+that are smaller than the no-lensing detection limit, as observed
+in Figure 9.
+At the brightest flux explored in the experiment (green line in
+Figure 9), the lensing cross-section at small source sizes is larger
+than the area enclosed within the radial caustic (black dotted line
+in Figure 9). This is because, when the source is very bright,
+it can give rise to multiple images even while its centroid lies
+outside of the radial caustic, as long as its surface brightness
+distribution extends into it. This also explains why σSL increases
+with increasing source size, before dropping to zero: the more
+extended the source, the farther away from the lens centre it can
+be while still producing multiple images.
+3.4.2. Large source sizes
+For the large source size case we fixed the surface brightness
+profile of the source and varied the Einstein radius of the lens.
+In particular, we set the source half-light radius to 20 pixels and
+adjusted its total flux in such a way that the 2σ detection foot-
+print in the absence of lensing extends out to the half-light ra-
+dius. Then, starting from the lens model used in the previous sec-
+tion, we progressively increased the critical surface mass density
+to reduce the Einstein radius down to values comparable to the
+pixel size. Figure 10 shows the resulting strong lensing cross-
+section as a function of the ratio between lens Einstein radius
+and source half-light radius. For values of the Einstein radius
+close to the size of the PSF, the lensing cross-section (blue solid
+line) is very small: this is because the perturbations caused by
+lensing are not well resolved. For larger values, σSL stays ap-
+proximately constant, around values that are comparable to the
+source size (horizontal dotted line) and much larger than the area
+enclosed within the caustics (red dashed line). We conclude that,
+in the large source size regime, the main factor that determines
+the lensing cross-section is the area of the background source,
+provided that the Einstein radius of the lens is larger than the
+PSF.
+4. Lens populations simulations
+In the previous section we showed how the lensing cross-section,
+which is closely related to the lens detection probability Pdet,
+varies as a function of lens and source properties. From here
+on we focus on the effect that those trends have on popula-
+tions of lenses. We addressed this question by simulating popu-
+lations of foreground galaxies and background sources, selecting
+strong lenses among them, and comparing the properties of the
+strong lens sample with the parent galaxy population. We sim-
+ulated a lens-based search (as opposed to a source-based one),
+in which strongly lensed images are searched among a stellar
+mass-selected sample of galaxies. Our simulations are based on
+empirical models, in which existing observations of the baryonic
+component of galaxies are complemented with a set of assump-
+tions on the mass distribution of the lenses. In section 4.1 we
+explain how we built our foreground galaxy sample, while in
+section 4.2 we describe the simulation of the background source
+population. In section 4.3 we describe how our mock observa-
+tions of lenses are generated. In section 4.4 we apply a further
+selection step, based on the angular size of the Einstein radius.
+Article number, page 9 of 22
+
+A&A proofs: manuscript no. seleff
+10−1
+100
+θEin/θs,e
+10−2
+10−1
+100
+σSL/(πθ2
+s,e)
+Cross-section
+Caustic area
+Source area
+PSF FWHM
+Fig. 10.
+Strong lensing cross-section of an extended source. Large
+source size regime (θe,s ≳ θEin). The cross-section is plotted as a function
+of the ratio between the Einstein radius of the lens and the half-light ra-
+dius of the source. The source model is fixed to a circular Sérsic profile
+with n = 1, detected out to the half-light radius. The source half-light
+radius measures 20 pixels, and the PSF has an FWHM of two pixels.
+The lens model is varied: starting from the model of Figure 9, the crit-
+ical surface mass density is increased to produce progressively smaller
+values of θEin. Solid line: lensing cross-section, in units of the footprint
+area of the source in the absence of lensing, as a function of the ratio be-
+tween the lens Einstein radius and the source half-light radius. Dashed
+line: area enclosed within the lens caustics. Dotted line: value of the
+cross-section corresponding to the area of the source. Dash-dotted line:
+FWHM of the PSF.
+4.1. Foreground galaxies
+Our foreground galaxy population consists of a volume-limited
+sample of early-type galaxies, complete above a minimum ob-
+served stellar mass of 1011M⊙. We chose to focus on early-type
+galaxies because many strong lensing surveys have preferen-
+tially targeted this class of objects in their lens-finding phase
+(e.g. Bolton et al. 2006; Gavazzi et al. 2012; Sonnenfeld et al.
+2018). This, in turn, had a dual motivation: first, early-type
+galaxies are among the most massive objects in the Universe,
+and therefore they are more likely to be lenses; second, their
+smooth surface brightness distribution and red colour makes it
+easier to detect arcs from strongly lensed star forming galaxies
+around them.
+We describe lenses with elliptical versions of the two-
+component model introduced in section 3.1. In the following
+sections we explain how their parameters are generated.
+4.1.1. Stellar mass and redshift distribution
+We generated lens galaxies over a finite redshift range, 0.1 < z <
+0.7. We chose these lower and upper limits because the value of
+the critical surface mass density Σcs becomes very large outside
+of this range, and the fraction of galaxies that act as strong lenses
+drops substantially as a result.
+We drew stellar masses from the stellar mass function of qui-
+escent galaxies measured by Muzzin et al. (2013). In particular,
+we used the following comoving number density distribution
+Φ(M(obs)
+∗
+) = Φ∗
+������
+M(obs)
+∗
+M∗∗
+������
+α
+exp
+�������− M(obs)
+∗
+M∗∗
+�������.
+(21)
+We set Φ∗ = 1.009×10−3 Mpc−3, α = −0.92 and log M∗
+∗ = 11.21:
+these are the best-fit values measured by Muzzin et al. (2013).
+We refer to the stellar masses drawn from this distribution
+as the observed stellar masses, M(obs)
+∗
+, as opposed to the true
+stellar masses. The observed stellar mass is meant to represent
+an estimate of M∗ based on stellar population synthesis mod-
+elling, which is the method used by Muzzin et al. (2013) to
+measure the galaxy stellar mass function. Stellar population syn-
+thesis measurements, however, are subject to systematic uncer-
+tainties, since they have not been calibrated on galaxies whose
+stellar mass is known by other means. We quantify the discrep-
+ancy between the observed and true stellar mass by means of the
+stellar population synthesis mismatch parameter αsps, defined as
+follows:
+αsps ≡
+M∗
+M(obs)
+∗
+.
+(22)
+The most important source of systematic uncertainty in the mea-
+surement of the stellar mass is the assumption of the stellar initial
+mass function (IMF). Muzzin et al. (2013) assumed a Kroupa
+IMF (Kroupa 2001) to derive their measurements. This means
+that, in the absence of other systematic effects, a galaxy with a
+Kroupa IMF has a value of αsps = 1.
+For each galaxy in the sample, we randomly drew a value of
+log αsps from the following distribution:
+P(log αsps) ∼ N(0.1, σ2
+α),
+(23)
+where the notation N(µ, σ2) indicates a Gaussian with mean µ
+and variance σ2. We set the mean of log αsps to 0.1, as this is
+an intermediate value among estimates of αsps from the litera-
+ture (Conroy & van Dokkum 2012; Cappellari et al. 2013; Smith
+et al. 2015; Sonnenfeld et al. 2015, 2019). We adopted a few dif-
+ferent values for the scatter σα. We explain in Section 4.1.4 how
+these were chosen.
+4.1.2. Stellar mass density distribution
+We described the surface mass density distribution of each
+galaxy as an elliptical de Vaucouleurs profile (a Sérsic profile
+with n = 4). Given the observed stellar mass of a galaxy, we
+assigned a half-mass radius by drawing it from the following
+distribution in log Re:
+P(log Re) ∼ N(1.20 + 0.63(log M(obs)
+∗
+− 11.4), 0.142).
+(24)
+Then, we assigned an axis ratio q by drawing it from the follow-
+ing beta distribution:
+P(q) ∝ qα−1(1 − q)β−1,
+(25)
+with α = 6.28 and β = 2.05. The choice for these distributions
+was motivated by observations of a sample of early-type galax-
+ies. In Appendix A we explain how this sample was defined and
+how the coefficients of Equation 24 and Equation 25 were deter-
+mined.
+4.1.3. Dark matter distribution
+We modelled the dark matter distribution of each lens galaxy
+with an elliptical gNFW halo (see section 3.1 for its definition).
+The parameters of the radial density profile were assigned as fol-
+lows. Given the stellar mass of a lens, we first determined the
+value of its halo virial mass, Mh. We defined this as the mass en-
+closed within a spherical shell with average density equal to 200
+Article number, page 10 of 22
+
+Sonnenfeld et al.: Strong lensing selection effects
+times the critical density of the Universe. Using current weak
+lensing constraints on the halo mass of elliptical galaxies as a
+reference (Sonnenfeld et al. 2022), we drew halo masses from
+the following distribution:
+P(log Mh) ∼ N(13.0 + 1.0(log M∗ − 11.5), σ2
+h),
+(26)
+which is a Gaussian in log Mh with a mean that scales with stellar
+mass and scatter σh. The intrinsic scatter in halo mass is not
+well constrained observationally, therefore we ran simulations
+with different values of σh, as for the stellar population synthesis
+mismatch parameter.
+To determine the density profile of the halo we relied on a
+theoretically motivated model that takes into account the effect
+of baryons on the dark matter. We assumed the dark matter dis-
+tribution to be initially described by an NFW profile with a con-
+centration4 of five, then used the prescription of Cautun et al.
+(2020) to model the response of the halo to the infall of baryons.
+In the Cautun et al. (2020) model the halo response is approx-
+imated with an analytical function that depends on the present
+stellar mass distribution, and typically results in a more concen-
+trated and steeper density profile compared to the original NFW
+model. Finally, we fitted a gNFW profile to the surface mass den-
+sity of the contracted halo. By doing this, we were able to fully
+describe the dark matter density profile with three parameters:
+Mh, γDM and rs. The dark matter density profile defined in this
+way is determined uniquely by the halo mass, the stellar mass
+and the half-light radius (the more concentrated the stellar dis-
+tribution, the stronger the halo response and the steeper the dark
+matter density profile). In principle we could have allowed for
+additional degrees of freedom, for instance by relaxing the as-
+sumption of a fixed initial halo concentration. In practice, as we
+explain in section 6.5, our main results are not affected by this
+choice.
+Given the radial profile of the dark matter halo, we obtained
+an elliptical version of it by applying a transformation of the kind
+of Equation 12 to its projected surface mass density. We assumed
+that the axis ratio and orientation of the halo is the same as that
+of the stellar component.
+4.1.4. Intrinsic scatter
+The distribution in stellar mass, halo mass and dark matter inner
+slope of our sample of simulated foreground galaxies depends
+on parameters describing the intrinsic scatter in these properties,
+namely σsps and σh. Direct observational constraints on these
+quantities are poor. However, we can derive upper limits on them
+on the basis of observed scaling relations.
+Early-type galaxies lie on the stellar mass fundamental plane,
+a scaling relation between stellar mass, half-light radius and cen-
+tral velocity dispersion (Hyde & Bernardi 2009; de Graaff et al.
+2021):
+σe ∝ M(obs)βσ
+∗
+Rξσ
+e .
+(27)
+The existence of a fundamental plane relation is a consequence
+of the virial theorem: the velocity dispersion of a galaxy in dy-
+namical equilibrium is directly related to the 3-dimensional mass
+distribution in its inner regions, which is typically dominated by
+the stellar component. At fixed observed stellar mass distribu-
+tion, however, the central velocity dispersion can vary depend-
+ing on the stellar population synthesis mismatch parameter, on
+4 The concentration is the ratio between the halo virial radius and the
+scale radius rs.
+Table 1. Intrinsic scatter scenarios
+Model name
+σsps
+σh
+Predicted FP
+scatter
+Fiducial
+0.08
+0.20
+0.034
+Low scatter
+0.05
+0.10
+0.021
+High scatter
+0.10
+0.30
+0.043
+Notes. Adopted values of the intrinsic scatter parameters σsps and σh in
+different simulations. For each set of values, the fourth column indicates
+the scatter around the fundamental plane predicted via the spherical
+Jeans analysis of Appendix B. The observed fundamental plane scat-
+ter is 0.035.
+the dark matter mass and on the dark matter density profile. This
+gives rise to a spread in the values of the velocity dispersion
+given M(obs)
+∗
+and Re. Therefore, we can use the observed scatter
+in velocity dispersion to put an upper limit on the intrinsic scat-
+ter in the stellar population synthesis mismatch parameter, halo
+mass and dark matter slope.
+We used Jeans modelling for this purpose. We generated
+samples of early-type galaxies with the recipes described above
+and with different values of σsps and σh. Using the spherical
+Jeans equation under the assumption of isotropic orbits, we pre-
+dicted the central velocity dispersion of each galaxy in the sam-
+ple. Then, we fitted a fundamental plane relation to this mock
+sample and measured the predicted scatter in velocity dispersion
+at fixed M(obs)
+∗
+and Re. We then varied σsps and σh to match the
+observed and the predicted scatter. We did not attempt to match
+the other parameters of the fundamental plane (i.e. the constant
+of proportionality of Equation 27 and the power-law indices βσ
+and ξσ), because these are sensitive to the orbital anisotropy of
+the galaxies, which we are asserting to be zero. The details of
+this procedure are given in Appendix B.
+We settled on three different sets of intrinsic scatter param-
+eters, as indicated in Table 1. We label them the fiducial, the
+low-scatter and the high-scatter scenarios. The high-scatter sce-
+nario is ruled out both by the fundamental plane and by weak
+lensing constraints (Sonnenfeld et al. 2022). Moreover, our dy-
+namical model is quite simplistic, as it neglects the effects of or-
+bital anisotropy and departures from spherical symmetry, which
+are additional sources of scatter. Nevertheless, we use it in our
+experiment in order to obtain a more conservative upper limit on
+the amplitude of the strong lensing bias.
+4.2. Background sources
+4.2.1. Galaxies
+Our background galaxy population is taken from the surfs-based
+KiDS-Legacy-Like Simulation (SKiLLS) input catalogue, a hy-
+brid simulation catalogue integrating cosmological simulation
+with high-quality imaging observations (Li et al. 2022). The cos-
+mological simulation is obtained from the Synthetic UniveRses
+For Surveys (surfs) simulations, a set of N-body simulations
+from Elahi et al. (2018). The galaxy properties, including the star
+formation history and the metallicity history, are from an open-
+source semi-analytic model named Shark5 (Lagos et al. 2018).
+The original photometry is drawn from a stellar population syn-
+thesis technique using stellar synthesis libraries with physically
+motivated dust attenuation and re-emission models (Robotham
+et al. 2020). Li et al. (2022) further applied an empirical cor-
+5 https://github.com/ICRAR/shark
+Article number, page 11 of 22
+
+A&A proofs: manuscript no. seleff
+rection to the original synthetic photometry to better agree with
+the COSMOS2015 observations (Laigle et al. 2016). The galaxy
+morphology is described by a Sérsic profile with three parame-
+ters: the half-light radius in angular units θe,s, the Sérsic index
+ns, and the axis ratio qs. These structural parameters are learned
+from the imaging data obtained with the Advanced Camera for
+Surveys (ACS) instrument on the Hubble Space Telescope (Grif-
+fith et al. 2012). We refer to Li et al. (2022) for details on the
+learning algorithm and validation.
+The complete SKiLLS catalogue contains ∼108 deg2 of
+galaxies with redshift up to 2.5 and r-band apparent magnitude
+down to 27. We applied a lower limit to the source redshift, by
+selecting only sources with zs > 0.8. This ensures that all of
+the sources lie behind all of the lenses. A similar cut could be
+applied to a real survey using photometric redshifts, to reduce
+the incidence of false positives (e.g. arc-like features physically
+associated with the lens galaxy) in the lens finding phase. The
+resulting number density of sources is 70 arcmin−2. We approx-
+imated their spatial distribution as uniform in the sky, that is we
+neglected clustering of the sources.
+4.2.2. Quasars
+We described the population of background quasars with the fol-
+lowing double power-law luminosity function in the rest-frame
+UV absolute magnitude M:
+Φ(M, zqso) =
+Φ(M∗)
+100.4(αQ+1)(M−M∗) + 100.4(βQ+1)(M−M∗) .
+(28)
+Following Manti et al. (2017), we set αQ = −1.35 (faint-end
+slope), βQ = −3.23 (bright-end slope), and adopted a redshift-
+evolving normalisation
+log Φ∗ = −6.0991 + 0.0209zqso + 0.0171z2
+qso,
+(29)
+and characteristic magnitude
+M∗ = −22.5216 − 1.6510zqso + 0.2869z2
+qso.
+(30)
+Given the redshift and rest-frame UV luminosity of a quasar, we
+then computed the apparent magnitude in the observed i−band,
+mqso, using a quasar spectral template6 built from optical and
+near-infrared spectra obtained by Vanden Berk et al. (2001);
+Glikman et al. (2006).
+For the sake of consistency with the population of back-
+ground extended sources, we limited the redshift distribution of
+quasars to the range 0.8 < zqso < 2.5. We then truncated the
+distribution in mqso at two magnitudes fainter than the detection
+limit (which is specified in section 4.3). Finally, we randomly
+placed quasars in the source plane with a projected number den-
+sity of 70 arcmin−2. This is a much larger number density than
+observed in the real universe, but we are allowed to do so because
+we are not interested in predicting the absolute number of strong
+lenses, so this is a legitimate choice. The advantage of boosting
+the number density of quasars is that it allows us to produce a
+large number of lenses without the need for generating too big a
+population of foreground galaxies.
+4.3. Observations
+For each of the three intrinsic scatter scenarios, we drew a popu-
+lation of foreground galaxies covering 1000 square degrees. The
+6 https://archive.stsci.edu/hlsps/reference-atlases/
+cdbs/grid/comp_qso/
+expectation value of the number of galaxies in the corresponding
+volume, given the redshift and stellar mass cuts described in sec-
+tion 4.1.1, is around 300 000. We approximated the foreground
+galaxies as isolated: when determining whether a galaxy acts as
+a strong lens, we only modelled the contribution to the lensing
+signal from the galaxy itself, and neglected that of the environ-
+ment. We discuss the possible implications of this approximation
+in Section 6. For each lens, we determined its caustics relative to
+the highest source redshift, zs = 2.5. We then placed sources
+randomly behind the lens. If at least one source fell within a cir-
+cular region enclosing the caustics, we proceeded to compute its
+lensed images.
+For the simulation with extended sources, we produced im-
+ages with properties similar to those expected for the Euclid
+Wide survey (Euclid Collaboration et al. 2022). We used a pixel
+size of 0.1′′ and we applied a Moffat PSF with an FWHM of
+0.2′′ and a β parameter of 5.0. Finally, we assumed a background
+noise level such that an extended source with half-light radius
+0.5′′ and an apparent magnitude in the absence of lensing of
+ms = 25 is detected with S/N = 10. We then applied the peak
+detection-based lens selection criterion introduced in section 2.4
+to find the strong lenses.
+The fiducial scatter simulation with extended background
+sources produced a sample of 2113 lenses, corresponding to a
+number density of 2.1 deg−2. This is about a factor of five smaller
+than the number density predicted by Collett (2015) for the Eu-
+clid survey. This is a result of differences in the description of the
+source population, in the criteria used to define a strong lensing
+event, and in the redshift and stellar mass cuts that we applied to
+define the foreground galaxy population.
+For the lensed quasars we did not simulate pixel-level data,
+but simply computed the observed magnitudes of the multiple
+images. Following the definition of section 2.3, we included in
+the sample of strong lenses only systems with at least two im-
+ages brighter than a limiting magnitude mlim. We set mlim = 23.3,
+which corresponds to the 10σ detection limit of the LSST in a
+single visit (Oguri & Marshall 2010). This is motivated by the
+fact that the quasar lenses are meant to simulate a sample as-
+sembled for the purpose of carrying out time delay measure-
+ments, which in turn require combining single-visit detections
+over many epochs. The scenario that we are simulating, then, is
+that of a lens search in a Euclid-like survey, followed-up with
+LSST time-domain observations.
+The resulting number of quasar lenses in our simulation with
+the fiducial scatter is 1621. This number is meaningless, given
+that the simulation was created with an unrealistically large num-
+ber density of quasars. More interesting is the relative number of
+quad lenses with respect to the total, which is about 9%. This is
+a slightly smaller value than the fraction of quads predicted by
+Oguri & Marshall (2010). The reason for this discrepancy lies in
+the differences between the lens mass models in the two simula-
+tions.
+4.4. Lens finding probability
+Figure 11 shows the distribution in Einstein radius of the simu-
+lated lens samples. In all cases, the distribution peaks at θEin ≈
+0.7′′. Although all of the lenses in these samples are detected,
+it does not necessarily follow that they would all be included in
+a strong lensing study. There can be a few reasons for exclud-
+ing certain lenses from a sample. One is the low accuracy of
+lens finders: current automated lens finding algorithms tend to
+produce lens candidates samples with low purity (see e.g. Son-
+nenfeld et al. 2018; Petrillo et al. 2019; Savary et al. 2022). Such
+Article number, page 12 of 22
+
+Sonnenfeld et al.: Strong lensing selection effects
+0
+1
+2
+3
+θEin
+0
+50
+100
+150
+200
+250
+N
+Fiducial
+High scatter
+Low scatter
+Quasars (all)
+Quasars (quads)
+Fig. 11.
+Einstein radius distribution of the simulated lens samples.
+These are: galaxy-galaxy lenses in the fiducial, high scatter and low
+scatter scenarios; galaxy-quasar lenses with fiducial scatter, consider-
+ing all lenses or only lenses that produce four images.
+samples are then visually inspected, and only those candidates
+that can be clearly distinguished from false-positives are kept.
+This visual inspection step tends to disfavour lenses with a small
+image separation, because of the contamination from the light of
+the lens galaxy.
+Another possible reason for refining a sample of lens can-
+didates is the availability of redshift measurements. Redshifts
+of both the lens and the source are needed in order to convert
+a lens model into a measurement of mass. When working with
+large samples of lenses, obtaining spectroscopic measurements
+is not a viable option, and photometric redshifts are a necessity.
+Measuring the photometric redshift of a strongly lensed source,
+however, is challenging, especially when the Einstein radius is
+small and the source light is blended with the light from the lens
+(Langeroodi et al. 2023).
+Both of these scenarios can result in samples that are incom-
+plete below a certain value of the Einstein radius. We simulate
+this situation via the following Einstein radius-dependent lens
+finding probability:
+Pfind(θEin|S, det) =
+�
+1
+if θEin > θEin,min
+0
+otherwise
+.
+(31)
+In words, all lenses with Einstein radius larger than θEin,min are
+included in the sample, while all those with smaller Einstein ra-
+dius are excluded. We refer to θEin,min as the completeness limit:
+our simulated lens samples are complete down to θEin = θEin,min.
+We explored scenarios with different values of θEin,min. As the
+next section shows, the larger the minimum Einstein radius, the
+higher the strong lensing bias.
+5. Results
+In this section we present the results of the lens population
+simulations. Section 5.1 shows the results of the galaxy-galaxy
+lens experiment, while section 5.2 focuses on the population of
+galaxy-quasar lenses. Given the number of parameters that are
+needed to describe our model, providing a complete characteri-
+sation of the strong lensing bias is a problem with relatively high
+dimensionality, and is beyond the scope of this paper. For the
+sake of conciseness, we focus instead on the quantities that we
+consider most important. Nevertheless, the output of our simu-
+lations is available online7. We encourage readers who are inter-
+ested in studying aspects of the strong lensing bias that are not
+covered in this section to download our data and analyse them
+directly.
+5.1. Galaxy-galaxy lenses
+In this section we show the results of the experiments with pop-
+ulations of galaxy-galaxy lenses. We first present the results in
+a qualitative way, then proceed to quantify the amplitude of the
+strong lensing bias in various quantities of interest.
+Figure 12 shows the distribution in the parameters of the
+foreground galaxies of the lens systems, compared to those of
+the parent population, for the fiducial scatter scenario with two
+different values of the minimum Einstein radius: 0.5′′ and 1.0′′.
+A completeness limit of 0.5′′ is close to what can currently be
+achieved via visual inspection of high-resolution space-based
+images (Garvin et al. 2022), while the 1.0′′ limit can be seen
+as a more conservative case.
+The most striking difference between the samples is in the
+stellar mass: strong lensing selects preferentially galaxies with
+larger values of M(obs)
+∗
+. Lenses tend to also have a larger halo
+mass, a smaller half-light radius at fixed stellar mass, and a larger
+stellar population synthesis mismatch parameter. The distribu-
+tion in ellipticity and inner dark matter slope of the lenses instead
+look very similar to that of the parent population. Additionally,
+we can see that the amplitude of the strong lensing bias appears
+to be always larger in the lens sample with the more restrictive
+selection on Einstein radius. We quantify the amplitude of these
+biases later in this section.
+Figure 13 shows the distribution in parameters describing the
+background source population, for the same simulations of Fig-
+ure 12. Additionally, Figure 13 shows the subset of the parent
+population that consists of detectable sources. These are back-
+ground galaxies that, in the absence of lensing, can be detected
+according to the same criterion used for the lensed sources (i.e.
+the S/N over their 2σ footprint is larger than ten). The detection
+limit of the survey is at ms ≈ 25 (the actual limit varies depend-
+ing on the surface brightness distribution parameters). Because
+our simulated background source population extends to much
+fainter magnitudes, the distribution of detectable sources differs
+substantially from that of the parent population. We can then
+consider two different strong lensing bias definitions: one that
+quantifies the difference in lensed source properties with respect
+to the parent population, and one that describes the difference
+with respect to the detectable source population. We are mostly
+interested in the second definition.
+Strong lensing tends to preferentially select sources at higher
+redshift, especially in the more restrictive case with θEin > 1.0′′.
+This is because, at fixed lens properties, increasing the source
+redshift lowers the critical surface mass density and, conse-
+quently, increases the size of the caustics and the Einstein ra-
+dius. The distribution in the magnitude of the lensed sources is
+also very different from that of the parent distribution, as it drops
+rapidly for values larger than ms ≈ 25. Interestingly, however,
+there does not seem to be a large difference with respect to the
+distribution of detectable sources in the absence of lensing. This
+result can appear to be somewhat counter-intuitive: lensing mag-
+nification should allow the detection of sources that are intrinsi-
+cally fainter than the detection limit. To some extent, this is the
+7 https://github.com/astrosonnen/strong_lensing_tools/
+papers/selection_effects
+Article number, page 13 of 22
+
+A&A proofs: manuscript no. seleff
+General population
+Lenses, θEin > 0.5′′
+Lenses, θEin > 1.0′′
+1.2
+1.4
+1.6
+1.8
+γDM
+11.0
+11.5
+12.0
+log M (obs)
+∗
+0.0
+0.1
+0.2
+log αSPS
+0.5
+1.0
+1.5
+log Re
+0.5
+1.0
+q
+12
+14
+log Mh
+0.2
+0.4
+0.6
+zg
+1.21.41.61.8
+γDM
+11.0
+11.5
+12.0
+log M (obs)
+∗
+0.0
+0.1
+0.2
+log αSPS
+0.5
+1.0
+1.5
+log Re
+0.5
+1.0
+q
+12
+14
+log Mh
+Fig. 12. Comparison between the properties of lens samples and the parent population. Distribution in foreground galaxy parameters. Filled
+contours: distribution of the parent sample. Green solid lines: distribution of the lenses with Einstein radius larger than 0.5′′. Red solid lines:
+distribution of the lenses with Einstein radius larger than 1.0′′.
+case: the distribution of lensed sources shows a slight excess of
+fainter galaxies compared to the unlensed case. However, as we
+quantify later in this section, the difference is far from large. The
+reason for this behaviour lies in the fact that, in the detection of
+both lensed and unlensed sources, the most important quantity
+is surface brightness, which is preserved by gravitational lens-
+ing when the source is larger than the PSF size. As we showed
+in section 3.4, the lensing cross-section drops to zero once the
+surface brightness of the source reaches a value that would make
+it undetectable in the absence of lensing. For this reason, also
+the distribution in half-light radius is very similar between the
+lensed sources and the detectable source population.
+Qualitatively, the results shown in Figure 12 and Figure 13
+match our expectations from Section 3. In the rest of this section
+we quantify the lensing bias. We present the results in three dif-
+ferent parts. First, we focus on the properties of the lens galax-
+ies that can be observed directly. These are quantities that can
+be derived from photometry and spectroscopy with minimal as-
+sumptions: the lens redshift, the observed stellar mass, the half-
+light radius and the axis ratio8. Any bias in these quantities can
+8 Strictly speaking, the axis ratio in mass is not necessarily observable,
+but in the context of our simulations it is, since light and mass have the
+same ellipticity.
+be determined relatively easily in a real strong lens survey. In
+the second part, we consider the parameters related to the mass
+distribution: the stellar population synthesis mismatch parame-
+ter and the dark matter distribution parameters. Determining the
+bias on these parameters is much more difficult, but these are
+quantities of great interest from a galaxy science point of view.
+In the third part, we focus on source distribution parameters.
+5.1.1. Bias in observable lens parameters
+Figure 14 shows the median redshift, median M(obs)
+∗
+, median size
+for a given M(obs)
+∗
+, and median axis ratio of various lens popula-
+tion simulations, as a function of the minimum Einstein radius.
+We defined the median size at fixed M(obs)
+∗
+by fitting the follow-
+ing mass-size relation to the Re − M(obs)
+∗
+distribution:
+log Re ∼ µR,0 + βR(log M(obs)
+∗
+− 11.4).
+(32)
+The quantity shown in the third panel of Figure 14 is the param-
+eter µR,0, which is the average log Re at an observed stellar mass
+of log M(obs)
+∗
+= 11.4. The horizontal dashed line in each panel
+shows the value of the parent population: the larger the distance
+Article number, page 14 of 22
+
+Sonnenfeld et al.: Strong lensing selection effects
+General population
+Detectable w/o lensing
+Lenses, θEin > 0.5′′
+Lenses, θEin > 1.0′′
+0.0
+0.5
+1.0
+qs
+22
+24
+26
+28
+ms
+0.0
+0.5
+1.0
+θs,e
+0
+2
+4
+ns
+1
+2
+zs
+0.0
+0.5
+1.0
+qs
+22
+24
+26
+28
+ms
+0.0
+0.5
+1.0
+θs,e
+0
+2
+4
+ns
+Fig. 13. Comparison between the properties of lens samples and the parent population. Distribution in background source parameters. Filled
+contours: distribution of the parent sample. Black solid lines: distribution of the detectable sources. Green solid lines: distribution of the lenses
+with Einstein radius larger than 0.5′′. Red solid lines: distribution of the lenses with Einstein radius larger than 1.0′′.
+between the curve of a simulation and this line, the higher the
+strong lensing bias.
+As was already visible in Figure 12, there is a clear bias to-
+wards lower redshift, higher stellar mass and smaller sizes, with
+the bias becoming stronger for more restrictive cuts in Einstein
+radius. Biases in stellar mass and size are stronger for the simu-
+lation with low scatter in αsps and Mh. This can be explained as
+follows. For a given value of the observed stellar mass, strong
+lensing selection favours galaxies with a larger αsps or Mh, or
+with a smaller half-light radius. Since the galaxy stellar mass
+function of the parent population is steep, lenses tend to have a
+relatively small M(obs)
+∗
+and large values of αsps or Mh for their
+observed stellar mass (this is shown more clearly in the next sec-
+tion). If the intrinsic scatter in αsps and Mh is low, however, the
+number of galaxies with a small M(obs)
+∗
+and a large stellar or halo
+mass is greatly reduced. Only galaxies with a large M(obs)
+∗
+or a
+small size can therefore act as strong lenses. This is an interest-
+ing result, because it suggests that it is in principle possible to
+use the strong lensing bias on M(obs)
+∗
+or Re, which is observable,
+as a way to constrain the amplitude of the intrinsic scatter in the
+mass parameters, which is poorly known.
+5.1.2. Bias in lens mass parameters
+Figure 15 shows the median of the distribution in various mass-
+related quantities, as a function of the minimum Einstein radius.
+The top panel is the median stellar population synthesis mis-
+match parameter. In all simulations, strong lenses are biased to-
+wards larger values than the parent population. The bias is larger
+the higher the intrinsic scatter in αsps and Mh, and increases with
+increasing θEin,min. For the fiducial model, the bias on αsps can be
+as small as 0.03 dex (7%), if no cut on Einstein radius is applied.
+However, it can rise up to 0.09 dex (23%) when considering only
+lenses with θEin > 2′′. For comparison, we also show two refer-
+ence values of αsps, corresponding to a Chabrier IMF (Chabrier
+2003) and a Salpeter IMF (Salpeter 1955). These values roughly
+bracket the current systematic uncertainty on αsps.
+The second panel of Figure 15 shows the median halo mass
+of galaxies with an observed stellar mass of log M(obs)
+∗
+= 11.4.
+We measured this quantity by fitting the following relation to the
+Mh − M(obs)
+∗
+distribution,
+log Mh ∼ µh,0 + βh(log M(obs)
+∗
+− 11.4),
+(33)
+and taking the resulting value of the parameter µh,0. Similarly to
+the αsps case, the strong lenses are biased towards larger values
+Article number, page 15 of 22
+
+A&A proofs: manuscript no. seleff
+0.30
+0.35
+0.40
+0.45
+0.50
+Median zg
+11.2
+11.4
+11.6
+11.8
+Median log M(obs)
+∗
+0.8
+0.9
+1.0
+1.1
+1.2
+Median log Re
+at log M(obs)
+∗
+= 11.4
+Parent pop.
+Fiducial
+High scatter
+Low scatter
+0.0
+0.5
+1.0
+1.5
+2.0
+Minimum θEin
+0.72
+0.74
+0.76
+0.78
+0.80
+0.82
+0.84
+Median qg
+Fig. 14. Bias on lens observable properties as a function of the mini-
+mum Einstein radius. First panel: median redshift. Second panel: me-
+dian observed stellar mass. Third panel: median half-light radius at an
+observed mass of log M(obs)
+∗
+= 11.4. Fourth panel: median axis ratio. In
+each panel, the black dashed line indicates the value of the parent popu-
+lation. Error bars indicate the standard deviation of the mean of the lens
+sample.
+of the halo mass, with the bias being larger for higher-scatter
+simulations and more restrictive cuts on θEin. In the fiducial scat-
+ter scenario with a completeness limit of θEin,min = 1.0′′, the halo
+masses of lenses are on average 0.16 dex larger than those of
+their parent population.
+The third panel of Figure 15 shows the median projected dark
+matter mass enclosed within an aperture of 5 kpc, MDM,5, at fixed
+observed stellar mass and half-light radius. We obtained this
+quantity by fitting the following relation to the MDM,5−M(obs)
+∗
+−Re
+distribution:
+log MDM,5 ∼ µDM,0 + βDM(log M(obs)
+∗
+− 11.4) + ξDM(log Re − 1.2).
+(34)
+Figure 15 shows the value of µDM,0. The bias on this quantity
+is qualitatively similar to that on the total halo mass, but much
+smaller in amplitude. There are two reasons for this. First, the
+dependence of MDM,5 on Mh is shallower than linear. This is be-
+cause, as the virial mass increases, the virial radius increases as
+well: the extra mass is spread over a larger volume, and therefore
+the mass within the inner region does not increase proportionally.
+Second, while µh,0 is the halo mass at fixed stellar mass, µDM,0
+is measured at fixed half-light radius as well. The density profile
+of the dark matter halos in our simulation have a dependence on
+galaxy size: the response of dark matter to baryons is stronger
+for more concentrated stellar distributions (see section 4.1.3).
+By capturing this dependence in the model of Equation 35, the
+residual scatter in MDM,5 around the mean is reduced, and so is
+the strong lensing bias. This, however, is a minor effect: we re-
+peated the analysis while setting the dependence on size to zero
+and found minimal differences on the derived values of µDM,0. In
+the fiducial scenario, the bias on MDM,5 is as small as 0.02 dex
+for θEin,min < 0.5′′, and is negligible in the low-scatter scenario.
+In the fourth panel of Figure 15 we show the average inner
+dark matter slope at fixed observed stellar mass and half-light
+radius, which we measured by fitting the following model
+γDM ∼ µγ,0 + βγ(log M(obs)
+∗
+− 11.4) + ξγ(log Re − 1.2),
+(35)
+and taking the resulting value of µγ,0. Different simulations show
+different trends in the average γDM. These are the results of un-
+derlying correlations between the dark matter density profile and
+the stellar and halo mass. In our model, galaxies of a given size
+with a larger stellar mass have a steeper dark matter slope, be-
+cause the response of the dark matter to the infall of baryons
+is stronger. Vice-versa, galaxies with a larger halo mass have a
+shallower density slope. At fixed M(obs)
+∗
+and Re, strong lenses
+have both a larger stellar mass (because their αsps is larger) and
+a larger halo mass. In the simulation with low scatter, the bias
+in stellar mass is more important, therefore γDM has a positive
+bias. In the simulation with large scatter, the bias in halo mass
+dominates, and therefore γDM is negatively biased.
+5.1.3. Bias in source parameters
+Figure 16 shows the lensing bias in source-related parameters,
+with respect to the population of sources that are detectable with-
+out lensing. As previously seen in Figure 13, lensed sources are
+biased towards higher redshift (top panel of Figure 16), with the
+trend being larger for more restrictive cuts on the Einstein ra-
+dius. The second panel shows the bias in the source magnitude.
+This bias is negative for small values of θEin,min, that is, lensed
+sources tend to be brighter than their field counterparts. Naively
+one might have expected the opposite trend, as strong lensing
+magnification allows the detection of sources that are intrinsi-
+cally fainter than the detection limit. However, as the analysis
+of Section 3 clearly shows, the lensing cross-section is always
+Article number, page 16 of 22
+
+Sonnenfeld et al.: Strong lensing selection effects
+larger for brighter sources, and that explains the sign of the bias.
+At the same time, the median does not capture the whole picture
+of the bias in ms. For instance, in the fiducial simulation with
+no cut on the Einstein radius, the 90%-ile of the ms distribution
+is 26.05, while that of the population of detectable sources is
+0.14 mag brighter: indeed, strong lensing allows the detection of
+fainter sources.
+The third and fourth panels of Figure 13 show the bias
+in half-light radius and Sérsic index, respectively, for sources
+within a magnitude bin centred on ms = 25 and with a 0.4 mag
+width. We detected no clear sign of bias. Finally, the fifth panel
+shows the bias in source axis ratio. Also in this case no obvious
+sign of bias was detected, except in the largest values of θEin,min.
+5.2. Galaxy-quasar lenses
+The main goal of the experiment with lensed quasars is to check
+whether there are any differences in the strong lensing bias with
+respect to the extended source case, at fixed properties of the
+foreground galaxy population. For this reason, we only ran sim-
+ulations with lensed quasars in the fiducial scatter scenario and
+compared the results with those from the extended source simu-
+lation. We are also interested in understanding how the subset of
+quad lenses differs from the entire population of quasar lenses,
+therefore we also analysed that subsample on its own.
+Figure 17 shows the strong lensing bias as a function of min-
+imum Einstein radius in the following quantities: stellar popu-
+lation synthesis mismatch parameter, halo mass at fixed stellar
+mass, enclosed dark matter mass at fixed stellar mass and half-
+light radius, and axis ratio. The biases of the population of lensed
+quasars (red curves) are very similar to that of lensed galaxies
+(blue curves), especially for values of θEin,min < 1′′, When con-
+sidering only quad lenses, however, there are some differences,
+the most remarkable of which is the bias in the axis ratio: quad
+lenses tend to be on average galaxies with a higher ellipticity.
+This trend was expected, given the results of section 3.2: the
+higher the ellipticity, the larger the area enclosed within the inner
+caustic, which is where a source needs to lie in order to produce
+four or more images. Quad lenses tend to also have a slightly
+larger halo mass at fixed stellar mass, for θEin,min < 1′′.
+6. Discussion
+6.1. Key results
+In order for a galaxy-source pair to be included as a strong lens
+in a survey, three conditions must be met. First of all, at least
+part of the source must be multiply imaged. Second, the multiple
+images must be detectable. Third, the lens must be recognised as
+such. Each one of these conditions introduces a bias with respect
+to the parent population of foreground galaxies and background
+sources. Together, they define the lens selection probability term
+Psel of Equation 1. The first two points are intrinsic to a strong
+lensing survey and constitute an unavoidable source of bias. The
+best case scenario occurs when the efficiency of including a de-
+tected strong lens in a survey is always one, and the sample is
+100% complete. In this case, the third condition does not intro-
+duce any further selection and the strong lensing bias is min-
+imised. We explored this scenario in Section 5 when setting the
+minimum Einstein radius to zero. If only lenses with Einstein ra-
+dius larger than a given threshold are selected, however, the bias
+generally increases.
+The strong lensing bias affects all quantities that are related
+to the mass distribution of the lens, as well as the redshifts of
+lens and source and the source surface brightness parameters.
+Some of these quantities, such as the observed stellar mass and
+half-light radius of the lenses, can be directly measured, and it
+is straightforward to quantify their lensing bias. Other quanti-
+ties, however, such as the stellar population synthesis mismatch
+parameter or the dark matter content, are difficult to obtain via
+traditional, non-lensing, observations. Our simulations are par-
+ticularly useful to quantify the bias on these properties.
+In the fiducial scatter scenario, the one consistent with the
+observed scatter around the fundamental plane, the bias on αsps
+varies from 0.03 dex, when no restrictions on the lens Einstein
+radius are applied, to 0.09 dex, corresponding to the extreme
+case in which only lenses with θEin > 2.0′′ are selected (see Fig-
+ure 15). A reasonable value for the minimum Einstein radius in
+a space-based survey like Euclid or the Chinese Space Station
+Telescope (CSST) is θEin,min = 0.5′′. In this case, the bias on
+αsps is slightly smaller than 0.04 dex. The current systematic un-
+certainty on αsps is 0.2 − 0.3 dex: this is roughly the difference
+in measurements of the stellar mass of a galaxy obtained with a
+Chabrier or a Salpeter IMF. Compared with this uncertainty, the
+amplitude of the strong lensing bias on αsps is small: strong lens-
+ing observations can be used directly to discriminate between
+these two alternative choices of IMF, without the need to correct
+for selection effects. Such a goal could be reached with a sam-
+ple size of a thousand lenses and a statistical study of the kind
+proposed by Sonnenfeld & Cautun (2021).
+Strong lenses are also biased towards larger halo masses.
+Nevertheless, when focusing on the dark matter content in the in-
+ner regions, the amplitude of the strong lensing bias is relatively
+small, especially when controlling for the stellar distribution of
+the lens galaxies. For instance, at fixed observed stellar mass and
+half-light radius, the bias on the projected dark matter mass en-
+closed within 5 kpc, MDM,5, is only a few percent in the fidu-
+cial scatter scenario with θEin,min < 1.0′′. This means that strong
+lenses can indeed be used to understand the inner dark matter
+distribution of galaxies, as long as the dependence of the dark
+matter distribution on the properties of the baryonic component
+is accurately modelled (for example, by following the approach
+of Sonnenfeld & Cautun 2021).
+We also looked at the bias on source-related parameters.
+Strong lensing causes the luminosity function of background
+sources to be broadened, compared to the distribution of de-
+tectable sources. On the one hand, it preferentially selects
+brighter sources, because the lensing cross-section increases
+with increasing source brightness. On the other hand, it allows
+for the detection of sources that are intrinsically fainter than the
+detection limit in the absence of lensing. Interestingly, we did not
+find any significant bias on the source size, at fixed magnitude.
+This result follows from the fact that our lens detection criterion
+relies on a surface brightness threshold, and surface brightness
+is preserved by lensing. However, it appears to be in contradic-
+tion with the work of Oldham et al. (2017), who argued that their
+strong lens sample selected preferentially compact sources. The
+origin of this discrepancy probably lies in the differences be-
+tween the criteria used to define a strong lens in the two stud-
+ies. The Oldham et al. (2017) sample was selected primarily via
+spectroscopy, by looking for signatures of two galaxies at differ-
+ent redshifts in the Sloan Digital Sky Survey (SDSS York et al.
+2000) data. SDSS spectra were taken in fibres with a 1.5′′ ra-
+dius. Lensed galaxies that are comparable in size to this scale,
+or larger, are less likely to be detected, because part of their flux
+extends outside of the fibre. Compact galaxies with a large over-
+all magnification, instead, are more likely to be detected (see the
+discussion in section 5.3 of Oldham et al. 2017). We refer the
+Article number, page 17 of 22
+
+A&A proofs: manuscript no. seleff
+reader to Arneson et al. (2012) for a thorough study of selection
+effects associated with spectroscopy.
+There is also an indirect way in which strong lensing could
+preferentially probe more compact sources than possible in the
+field. Sources that are smaller than the size of the PSF can be eas-
+ily confused as stars in our own Galaxy. Since these stars cannot
+be strongly lensed into multiple images, the detection of a strong
+lens automatically confirms the extragalactic nature of a lensed
+source. This type of selection effect could explain, for example,
+the detection of an apparent outlier in the magnitude-size rela-
+tion of Lyman-break galaxy by means of strong lensing, by Jae-
+lani et al. (2020). It is, however, improper to refer to this effect as
+strong lensing bias, since the star-galaxy separation is a bias that
+primarily affects field observations. Our analysis showed that the
+tendency to select compact sources is not a general feature of
+strong lens samples.
+6.2. Combining lensed quasars with lensed galaxies
+Our experiments were also useful for understanding the possi-
+ble biases that might incur when combining information from
+samples of galaxy-galaxy lenses with samples of galaxy-quasar
+lenses. On the one hand the biases in the mass-related quantities
+(αsps and MDM,5) are very similar in simulations with the same
+foreground galaxy population and different background sources
+(see Figure 17). On the other hand, these biases are a function
+of the completeness limit. In order to use a sample of strong
+lenses as a prior for another sample, then, it is important to make
+sure that 1) the parent population of foreground galaxies among
+which lenses are searched for is the same in both surveys, and
+2) the two surveys can probe the same distribution in Einstein
+radius.
+The above argument applies to samples of quasars selected
+regardless of the number of multiple images. When dealing with
+quad lenses, however, the situation is more complicated. First of
+all, because our experiment revealed differences in the bias on
+the dark matter distribution between quads and the entire sam-
+ple of lensed quasars. Second, because quad lenses tend to have
+a preferentially higher ellipticity. The ellipticity is important in
+the context of stellar dynamical analyses, which are often used in
+combination with strong lensing to constrain lens mass param-
+eters (see e.g. Yıldırım et al. 2020). In order to correctly inter-
+pret stellar dynamics data, assumptions on the three-dimensional
+structure of a lens must be made. This is directly related to the
+projected ellipticity: galaxies with an axis ratio close to one tend
+to be preferentially elongated in the line-of-sight direction, and
+vice-versa. When using stellar dynamics-based mass measure-
+ments to inform the properties of a sample of quad lenses, there-
+fore, it is important to take into account possible biases due to
+the different three-dimensional structure of the two samples.
+6.3. The importance of the source population properties
+Strictly speaking, the results shown in Section 5 apply only to
+samples of lenses and background sources with the same prop-
+erties as our simulations. However, the comparison of section
+5.2 shows that, at fixed foreground galaxy population, the strong
+lensing bias for the sample of extended sources is indistinguish-
+able from that of the lensed quasar population, at least for values
+of θEin,min < 1.0′′. This result suggests that the dependence of
+the strong lensing bias on the details of the background source
+population is very weak, as long as no additional selection on
+the image configuration is applied (e.g. by selecting only quad
+lenses). This is not a coincidence, but is a consequence of the
+fact that the dependence of the strong lensing cross-section on
+the lens parameters is a weak function of source magnitude9, es-
+pecially for magnitudes close to the detection limit (see section
+3.2). We can then conclude that the details of the properties of
+the background source population play a secondary role in de-
+termining the strong lensing bias.
+6.4. Mitigation strategies
+As we discussed throughout this paper, the amplitude of the
+strong lensing bias depends on the intrinsic scatter in the mass
+parameters of the foreground galaxy population. One way to
+minimise the bias, therefore, is to identify scaling relations be-
+tween observable quantities and mass-related properties that can
+account for part of the scatter. Describing the inner dark matter
+distribution as a function of stellar mass and half-light radius, as
+we did in section 5.1, is the first step in this direction: the bias
+in MDM,5 at fixed M∗ and Re is smaller than the overall shift in
+the median MDM,5 of the lens population (albeit by a marginal
+amount, as we pointed out earlier). This description could be
+extended by including the central velocity dispersion as an addi-
+tional control parameter. The velocity dispersion is directly re-
+lated to the mass distribution of the lens galaxy, which means
+that, for example, the distribution in αsps at fixed velocity disper-
+sion should be narrower than its global distribution marginalised
+over the whole population. The central velocity dispersion, how-
+ever, is very sensitive to the orbital anisotropy, to the three-
+dimensional structure, and to gradients in stellar mass-to-light
+ratio, which are not well known. Therefore it is difficult to quan-
+titatively estimate the benefit of including it in the description of
+the lensing bias.
+If one wishes to directly account for the strong lensing bias,
+the formally correct procedure is to explicitly model all of the se-
+lection steps in a Bayesian hierarchical formalism, as explained
+by Sonnenfeld (2022). Although this can be computationally
+challenging, machine learning can offer an efficient alternative
+(Legin et al. 2022). In order for either of these approaches to
+work, however, it is essential that the lens selection procedure
+can be simulated. This, in turn, requires having an objective def-
+inition of a strong lensing event. A peak-based definition such as
+that introduced in section 2.4 could be used for this purpose.
+Nevertheless, it can still be difficult to fully forward model
+strong lensing selection if visual inspection by humans is in-
+volved in the definition of the lens sample. In that case, a pos-
+sible alternative is to approximate the lens finding probability
+Pfind. For example, we can make the assumption that Pfind de-
+pends purely on the Einstein radius. This is a reasonable assump-
+tion, as long as the lens finding procedure does not selectively
+pick lenses with different image configurations depending on
+their Einstein radius. The dependence of Pfind on θEin could then
+be described empirically and inferred during the analysis. This
+is essentially the approach adopted by Sonnenfeld et al. (2019)
+in the analysis of strong and weak lensing data from the Hyper
+Suprime-Cam Survey.
+6.5. Limitations of our analysis
+The simulations on which our analysis is based are as complex
+as required by the goal of the analysis itself, which is to estimate
+the amplitude of the strong lensing bias in a few key quantities.
+9 The cross-section itself is a strong function of source magnitude, but
+the trends between σSL and the lens parameters are not.
+Article number, page 18 of 22
+
+Sonnenfeld et al.: Strong lensing selection effects
+We did, however, make some simplifying assumptions. One such
+assumption consisted in neglecting the contribution from line-of-
+sight structure and the environment to the lensing signal, which
+typically introduce an external shear and convergence. The effect
+of including external shear in a lens model is similar to that of
+changing the ellipticity of the lens. External convergence mim-
+ics the effect of adding or removing a constant sheet of invisible
+mass, producing an effect similar to varying the distribution of
+dark matter. Typical values of the external shear and external
+convergence are |γ| < 0.1 and |κext| < 0.1 (Millon et al. 2020):
+these values are small compared to the typical ellipticities and
+dark matter fractions of our simulated lenses. Therefore, while
+including them might modify slightly the amplitude of the strong
+lensing bias in the axis ratio and dark matter distribution param-
+eters, the conclusions of our analysis would not be affected.
+We also assumed that the dark matter density profile of the
+galaxies is completely determined by the halo mass and by the
+stellar mass distribution (see section 4.1.3). In reality we expect
+there to be a range of density profiles, for example as a result of
+halos having a nonzero scatter in their initial (i.e. before bary-
+onic infall) concentration parameter. We could in principle carry
+out an experiment with such an additional source of variation in
+the dark matter profile. Adding a scatter in concentration while
+keeping the halo mass scatter parameter σh fixed results in a
+larger spread in the inner dark matter distribution, which would
+increase the amplitude of the lensing bias. However, the value
+of σh that we chose for our fiducial experiment was tuned on
+the basis of the predicted scatter in velocity dispersion around
+the fundamental plane. That scatter would also be increased by
+adding flexibility to the dark matter density profile. Then, in or-
+der to be consistent with the analysis carried out in this paper,
+the value of σh would need to be lowered accordingly. This, in
+turn, would reduce the amplitude of the strong lensing bias and
+introduce a bias on the halo concentration. Since lensing and dy-
+namics are sensitive to the mass distribution at comparable scales
+(the half-light radius and Einstein radius are similar for most of
+the lenses), we expect these two effects to cancel out to first ap-
+proximation. The lensing bias on quantities directly related to
+the inner mass distribution, such as αsps and MDM,5, would be
+left roughly unchanged. The main effect of adding scatter to the
+concentration is to reduce the correlation between the total halo
+mass and the dark matter mass enclosed within the Einstein ra-
+dius. This would then reduce the amplitude of the strong lensing
+bias on the halo mass.
+Another simplifying assumption that we made was that of
+adopting a constant mass-to-light ratio for the stellar compo-
+nent. Massive galaxies are known to have colour gradients, the
+main effect of which is to cause the stellar half-mass radius to
+be smaller than the half-light radius (Szomoru et al. 2013; Suess
+et al. 2019). Since, as shown in Section 5, strong lensing bias
+tends to select galaxies with a more compact stellar distribution,
+it would also preferentially select galaxies with a steeper mass-
+to-light ratio gradient. However, testing for the amplitude of the
+strong lensing bias on the mass-to-light ratio gradient is beyond
+the scope of this paper.
+7. Conclusions
+Strong lensing is a very active line of research, but a compre-
+hensive understanding of the selection effects associated with it
+has so far been lacking. This work takes a major step towards
+filling that knowledge gap. After a thorough investigation, we
+learned several lessons regarding the strong lensing bias in pho-
+tometrically selected lens samples. The following are the most
+important ones.
+1. The strong lensing cross-section increases primarily with in-
+creasing lens mass, with decreasing half-mass radius, and
+with increasing source brightness. At fixed stellar distribu-
+tion and fixed dark matter mass enclosed within a given aper-
+ture, varying the inner dark matter slope has little impact
+on the strong lensing cross-section (as long as the aperture
+within which the dark matter mass is normalised is compa-
+rable to the Einstein radius).
+2. The strong lensing cross-section has little dependence on the
+size of the source, if this is smaller than the Einstein radius
+and if it is detectable in the absence of lensing. Sources with
+a surface brightness that is too low to be detected are still
+undetected when strongly lensed.
+3. Lens galaxies tend to be more massive and more compact
+than their non-lens counterparts. Their redshift distribution
+is also modified with respect to the general galaxy popula-
+tion. At fixed observed stellar mass (i.e. inferred by means
+of stellar population synthesis), lens galaxies have a larger
+intrinsic stellar mass (i.e. a larger stellar population synthe-
+sis mismatch parameter αsps) and a larger dark matter halo
+mass. At fixed stellar mass and size, lens galaxies are still
+biased towards a larger dark matter content.
+4. The amplitude of the strong lensing bias depends on how
+broad is the distribution of the parameters describing the lens
+population. Important quantities in this regard are the intrin-
+sic scatter in the stellar population synthesis mismatch pa-
+rameter, σsps, and in the dark matter halo mass, σh. Increas-
+ing the values of σsps and σh results in a stronger bias on
+αsps and on the dark matter mass, and a weaker bias on the
+observed stellar mass and half-light radius. This implies that,
+in principle, we could constrain the intrinsic scatter param-
+eters, which are currently poorly known, by measuring the
+amplitude of the strong lensing bias on M(obs)
+∗
+and Re, which
+is easily observable.
+5. The strong lensing bias varies depending on the complete-
+ness of the lens sample as a function of Einstein radius. Sur-
+veys that can discover lenses with a smaller Einstein radius
+have a smaller associated strong lensing bias in all quantities.
+6. Under reasonable assumptions on the intrinsic scatter pa-
+rameters, for a Euclid-like survey that is complete down
+to θEin = 0.5′′ the bias on αsps is smaller than 0.04 dex
+(10%). This bias is much smaller than the current systematic
+uncertainty on the stellar population synthesis-based stellar
+masses. Therefore, strong lensing measurements could be
+used directly to calibrate stellar mass measurements of mas-
+sive galaxies to 10% accuracy, without the need to correct
+for selection effects. Under the same assumptions, the strong
+lensing bias on the average halo mass at fixed stellar mass is
+0.07 dex, while that on the inner dark matter distribution at
+fixed stellar mass and size is 0.02 dex.
+7. Strong lensing selection broadens the magnitude distribution
+of background sources, compared to the population of ob-
+jects that are detectable without lensing. At the same time,
+we did not find any evidence for a bias in the size distribu-
+tion of background sources, at fixed magnitude.
+8. Simulations with lensed quasars in place of extended sources
+showed that the amplitude of the strong lensing bias in the
+lens-related parameters has very little sensitivity on the de-
+tails of the source population. This result has positive impli-
+cations for time-delay lensing studies: it means that informa-
+tion from a sample of galaxy-galaxy lenses can be used as
+Article number, page 19 of 22
+
+A&A proofs: manuscript no. seleff
+a prior on the properties of a set of galaxy-quasar lenses, as
+long as the two samples are probing the same range in Ein-
+stein radius and lens observable properties.
+9. Samples of quad lenses are biased towards galaxies with
+larger ellipticity, which implies that their three-dimensional
+structure is also biased. This means that particular care must
+be taken when stellar dynamics measurements obtained on
+galaxy-galaxy lenses is used to inform the properties of
+quads.
+In conclusion, strong lensing selection introduces unavoid-
+able biases in the properties of the lens galaxy and background
+source populations. Biases that affect observable properties, such
+as the redshift and the light distribution of the lens, can be easily
+quantified. Biases on mass-related quantities, such as the stellar
+mass-to-light ratio or the dark matter distribution, are more dif-
+ficult to measure directly and must be modelled by taking into
+account selection effects. Designing strong lensing surveys with
+clearly-defined and easily-modellable selection criteria would
+help greatly in this task.
+Acknowledgements. The collaboration leading to this work was initiated at the
+2022 Lorentz Center workshop “Bridging gaps between dynamical probes of
+galaxies”. AS and SSL are supported by NOVA, the Netherlands Research
+School for Astronomy. Support for this work was provided by NASA through the
+NASA Hubble Fellowship grant HST-HF2-51492 awarded to AJS by the Space
+Telescope Science Institute, which is operated by the Association of Universities
+for Research in Astronomy, Inc., for NASA, under contract NAS5-26555.
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+Wyithe, J. S. B., Turner, E. L., & Spergel, D. N. 2001, ApJ, 555, 504
+Yıldırım, A., Suyu, S. H., & Halkola, A. 2020, MNRAS, 493, 4783
+York, D. G., Adelman, J., Anderson, John E., J., et al. 2000, AJ, 120, 1579
+Article number, page 20 of 22
+
+Sonnenfeld et al.: Strong lensing selection effects
+0.0
+0.1
+0.2
+Median log αSPS
+Chabrier IMF
+Salpeter IMF
+13.0
+13.2
+13.4
+13.6
+µh,0 (Mean log Mh
+at fixed M(obs)
+∗
+)
+Parent pop.
+Fiducial
+High scatter
+Low scatter
+11.0
+11.1
+11.2
+µDM,0 (Mean log MDM,5
+at fixed M(obs)
+∗
+, Re)
+0.0
+0.5
+1.0
+1.5
+2.0
+Minimum θEin
+1.35
+1.40
+1.45
+µγ,0 (Mean γDM,5
+at fixed M(obs)
+∗
+, Re)
+Fig. 15.
+Bias on lens mass properties as a function of the mini-
+mum Einstein radius. First panel: median log αsps. Dotted lines indi-
+cate values of αsps corresponding to a Chabrier and a Salpeter IMF
+(αsps = 1 corresponds to a Kroupa IMF). Second panel: median Mh
+at log M(obs)
+∗
+= 11.4. Third panel: median MDM,5 at log M(obs)
+∗
+= 11.4
+and log Re = 1.2. Fourth panel: median γDM at log M(obs)
+∗
+= 11.4 and
+log Re = 1.2. In each panel, the dashed line indicates the value of the
+parent population. Error bars indicate the standard deviation of the mean
+of the lens sample.
+1.4
+1.6
+1.8
+2.0
+Median z
+24.8
+25.0
+25.2
+25.4
+25.6
+Median ms
+0.2
+0.3
+Median θs
+at ms = 25 (′′)
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+Median n
+at ms = 25
+Detectable pop.
+Fiducial
+High scatter
+Low scatter
+0.0
+0.5
+1.0
+1.5
+2.0
+Minimum θEin
+0.44
+0.46
+0.48
+0.50
+0.52
+0.54
+0.56
+Median q
+Fig. 16. Bias on source properties as a function of the minimum Ein-
+stein radius. First panel: median source redshift. Second panel: median
+source magnitude. Third panel: median half-light radius in the magni-
+tude bin 24.8 < ms < 25.2. Fourth panel: median Sérsic index in the
+magnitude bin 24.8 < ms < 25.2. Fifth panel: median axis ratio. In each
+panel, the dashed line indicates the value of the population of detectable
+sources. Error bars indicate the standard deviation of the mean of either
+the full sample (first, second and fifth panel) or the bin (third and fourth
+panel).
+Article number, page 21 of 22
+
+A&A proofs: manuscript no. seleff
+−0.05
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+Median log αSPS
+Chabrier IMF
+Salpeter IMF
+13.0
+13.2
+13.4
+µh,0 (Mean log Mh
+at fixed M(obs)
+∗
+)
+11.0
+11.1
+11.2
+11.3
+µDM,0 (Mean log MDM,5
+at fixed M(obs)
+∗
+, Re)
+Parent pop.
+Extended sources
+Quasars (all)
+Quasars (quads)
+0.0
+0.5
+1.0
+1.5
+2.0
+Minimum θEin
+0.50
+0.55
+0.60
+0.65
+0.70
+0.75
+0.80
+Median q
+Fig. 17.
+Bias in the population of quasar lenses (all and quads
+only), compared to the extended source simulation. First panel: median
+log αsps. Second panel: median Mh at log M∗ = 11.5. Third panel: me-
+dian MDM,5 at log M∗ = 11.5 and log Re = 1.2. Fourth panel: median
+lens axis ratio. In each panel, the dashed line indicates the value of the
+parent population. Error bars indicate the standard deviation of the mean
+of the lens sample.
+Article number, page 22 of 22
+
+Sonnenfeld et al.: Strong lensing selection effects
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Axis ratio q
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+P(q)
+Observed
+Model
+Fig. A.1. Distribution in axis ratio of a sample of early-type galax-
+ies. The sample consists of 8078 galaxies from the SDSS, selected by
+means of cuts in redshift, colour and Sérsic index as explained in the
+text. Measurements of the axis ratio are taken from the de Vaucouleurs
+model fits of Meert et al. (2015). The model curve is a beta distribution,
+Equation A.2, with α = 6.28 and β = 2.05.
+Appendix A: Lens galaxy surface brightness
+distribution
+To assign half-light radii and ellipticities to the simulated lenses
+we relied on observations of a sample of early-type galaxies
+selected from the Sloan Digital Sky Survey (SDSS York et al.
+2000). This sample was selected as follows. Starting from the
+SDSS spectroscopic sample, we defined a narrow redshift slice
+around z = 0.2. Then we applied a selection in colour, by choos-
+ing objects with g − r > 1.2, and on Sérsic index, by selecting
+only galaxies with n > 2.5. We used the Sérsic fit measurements
+by Meert et al. (2015) for this purpose. These cuts produced a
+sample of 8078 galaxies. We then focused on the r−band de Vau-
+couleurs model-based photometric measurements of Meert et al.
+(2015). Using the r−band total flux from the de Vaucouleurs
+model and the stellar mass-to-light ratio estimates of Mendel
+et al. (2014), we obtained measurements of M(obs)
+∗
+. Finally, we
+fitted the stellar mass-size relation with the following model:
+log Re ∼ N(µR,0 + βR(log M(obs)
+∗
+− 11.4), σ2
+R).
+(A.1)
+We obtained µR,0 = 1.20, βR = 0.63 and σR = 0.14.
+We then proceeded to fit for the axis ratio distribution of the
+same sample of galaxies. Figure A.1 shows a histogram of the
+observed distribution. We fitted this with a beta distribution:
+P(q) ∝ qα−1(1 − q)β−1.
+(A.2)
+We obtained α = 6.28 and β = 2.05.
+Appendix B: Upper limits on the intrinsic scatter
+parameters
+We used the fundamental plane of early-type galaxies to set an
+upper limit on the intrinsic scatter parameters of the simulation:
+σsps, σh and σγ. First, we measured the fundamental plane of the
+sample of SDSS early-type galaxies introduced in Appendix A.
+We took measurements of the line-of-sight stellar velocity dis-
+persion within the SDSS spectroscopic aperture, σap, from the
+SDSS data release 16 catalogue (Ahumada et al. 2020). Then,
+we fitted the following model to the distribution of σap of the
+sample:
+P(σap) ∼ N(µσ+β(log M(obs)
+∗
+−11.4)+ξ(log Re−1.2), σσ). (B.1)
+We accounted for observational uncertainties on σap when doing
+the fit, therefore the parameter σσ describes the intrinsic scatter
+in the logarithm of the velocity dispersion, deconvolved from
+the observational scatter. In principle we should also account
+for observational uncertainties on the stellar mass measurement,
+as they too contribute to the inferred scatter in σap. In practice,
+however, it is difficult to estimate observational uncertainties on
+stellar population synthesis measurements. For this reason we
+chose not to propagate uncertainties on M(obs)
+∗
+. As a result, the
+inferred scatter parameter σσ is slightly overestimated. We ob-
+tained µσ = 2.36, βσ = 0.33, ξσ = −0.17, and σσ = 0.035.
+We then generated samples of z = 0.2 galaxies from the
+model of section 4.1 and used the spherical Jeans equation to
+predict their central velocity dispersion. The spherical Jeans
+equation is (Binney & Tremaine 1987)
+d(ρ∗σ2
+r)
+dr
++ β(r)
+r ρ∗σ2
+r = −ρ∗(r)GM(r)
+r2
+,
+(B.2)
+where ρ∗(r) is the 3-dimensional distribution of dynamical trac-
+ers (i.e. the stars), σr the radial component of the velocity dis-
+persion, β(r) the orbital anisotropy parameter, and M(r) the
+mass enclosed within a spherical shell of radius r. We assumed
+isotropic orbits (β = 0), then integrated Equation B.2 to obtain
+the seeing-convolved, surface brightness-weighted line-of-sight
+velocity dispersion within the SDSS spectroscopic aperture10.
+Finally, we fitted the fundamental plane relation of Equation B.1
+to the observed stellar mass, half-light radius and central veloc-
+ity dispersion of the mock sample. We repeated this procedure
+for different values of the intrinsic scatter parameters σsps, σh
+and σγ, then settled on the three scenarios listed in Table 1.
+10 We
+used
+Python
+code
+available
+at
+https://github.com/
+astrosonnen/spherical_jeans, for this purpose.
+Article number, page 23 of 22
+
diff --git a/zdFQT4oBgHgl3EQfCTXT/content/tmp_files/load_file.txt b/zdFQT4oBgHgl3EQfCTXT/content/tmp_files/load_file.txt
new file mode 100644
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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf,len=1869
+page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' seleff  ©ESO 2023  February 1, 2023  Strong lensing selection effects  Alessandro Sonnenfeld1, 2, Shun-Sheng Li2, Giulia Despali3, Anowar J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Shajib4, 5, 6, and Edward N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Taylor7  1 Department of Astronomy, School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China  e-mail: sonnenfeld@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='cn  2 Leiden Observatory, Leiden University, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Box 9513, 2300 RA Leiden, The Netherlands  3 Institut für Theoretische Astrophysik, Zentrum für Astronomie, Heidelberg Universität, Albert-Ueberle-Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2, 69120, Heidelberg,  Germany  4 Department of Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637, USA  5 Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA  6 NHFP Einstein Fellow  7 Centre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn 3122, Australia  ABSTRACT  Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Strong lenses are a biased subset of the general population of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The goal of this work is to quantify how lens galaxies and lensed sources differ from their parent distribution, namely the strong  lensing bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We first studied how the strong lensing cross-section varies as a function of lens and source properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Then, we simulated  strong lensing surveys with data similar to that expected for Euclid and measured the strong lensing bias in different scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We  focused particularly on two quantities: the stellar population synthesis mismatch parameter, αsps, defined as the ratio between the true  stellar mass of a galaxy and the stellar mass obtained from photometry, and the central dark matter mass at fixed stellar mass and size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Strong lens galaxies are biased towards larger stellar masses, smaller half-mass radii and larger dark matter masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The  amplitude of the bias depends on the intrinsic scatter in the mass-related parameters of the galaxy population and on the completeness  in Einstein radius of the lens sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For values of the scatter that are consistent with observed scaling relations and a minimum  detectable Einstein radius of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5′′, the strong lensing bias in αsps is 10%, while that in the central dark matter mass is 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The bias  has little dependence on the properties of the source population: samples of galaxy-galaxy lenses and galaxy-quasar lenses that probe  the same Einstein radius distribution are biased in a very similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Quadruply imaged quasar lenses, however, are biased towards  higher ellipticity galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Given current uncertainties, strong lensing observations can be used directly to improve our current knowledge of the  inner structure of galaxies, without the need to correct for selection effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Gravitational lensing: strong  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Introduction  Strong gravitational lensing is a powerful tool for studying  galaxies and cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Strong lenses have been used to probe  the mass structure of massive galaxies (Auger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Oguri  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Shajib et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2021), to detect  substructure (Vegetti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Hezaveh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Nieren-  berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2020), to carry out detailed studies of magnified star-  forming galaxies (Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2013), and to measure the expan-  sion rate of the universe with time delays (see Treu & Marshall  2016 for a review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Strong lenses, however, are a biased subset of the general  population of galaxies and background sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' A necessary  condition for a galaxy to act as a strong lens with respect to a  given source is that its projected surface mass density Σ(θ) must  be larger than the critical surface mass density for lensing Σcr  at at least one position θ (Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This condi-  tion excludes objects with a very diffuse mass distribution from  the population of lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In general, galaxies with a higher con-  centration of mass are more likely to be strong lenses, and are  therefore over-represented in lens samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The distribution of  strongly lensed sources can also be biased with respect to the  general population of background galaxies: for instance, lensing  magnification allows the detection of fainter objects with respect  to the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  In general, the probability distribution PSL of a sample of  strong lenses from a given survey with selection criterion S is  given by (Sonnenfeld 2022):  PSL(ψg, ψs|S ) ∝ Pg(ψg)Ps(ψs)Psel(ψg, ψs|S ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (1)  In the equation above, ψg is the set of parameters describing  the properties of foreground galaxies that are relevant for lens-  ing, such as their redshift and mass distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' ψs is the set of  parameters describing background sources;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Pg and Ps describe  the parent distribution of foreground galaxies and background  sources in the absence of lensing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' and Psel describes the proba-  bility of selecting a lens-source system with parameters ψg and  ψs given the criterion S used to define a lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This last factor  takes into account both physical effects, that is whether a lens  with parameters ψg produces a strongly lensed image of a source  with parameters ψs, and survey selection effects, that is whether  such an image can be detected and recognised as a strong lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  The left-hand side of Equation 1 is directly accessible from  strong lensing observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' If the main goal of a lensing survey  is to characterise the properties of the strong lens population,  then it can be accomplished by directly analysing this term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For  Article number, page 1 of 22  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='13230v1  [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='GA]  30 Jan 2023  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' seleff  many applications of strong lensing, however, the aim is to con-  strain the properties of the general galaxy or source population,  Pg and Ps, which are coupled in a nontrivial way via the lens se-  lection probability Psel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In order to obtain an unbiased estimate of  either Pg or Ps, then, it is necessary to invert Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In prin-  ciple, this can be done with a Bayesian hierarchical formalism  (Sonnenfeld 2022), but knowledge of the lens selection proba-  bility Psel is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This factor can be written as the following  product:  Psel(ψg, ψs|S ) = Pdet(ψg, ψs)Pfind(ψg, ψs|S ),  (2)  where Pdet is the probability of detecting a strong lensing event,  and Pfind(ψg, ψs|S, det) is the probability of correctly classifying  it as such1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The detection probability Pdet can be obtained via  simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The main challenge is characterising Pfind: in most  of the existing strong lensing surveys, the process of determining  whether a system is included in a strong lens sample is typically  a combination of several cuts, usually involving a nontrivial vi-  sual selection step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  For the above reasons, the problem of inverting Equation 1  is a difficult one to tackle exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Few studies have attempted  to explicitly account for strong lensing selection effects, usually  by making ad-hoc simplifying assumptions (Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Oldham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Whether it is  necessary to invert Equation 1, however, depends on the severity  of the strong lensing bias that needs to be corrected and on the  accuracy requirements on the key quantities of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  In this paper we aim to quantify the strength of the strong  lensing bias on a series of foreground galaxy and background  source parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In particular, we aim to determine how strong  lenses differ from the parent population of foreground galaxies  and background sources in terms of a) the radial mass structure  of the lenses (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' their stellar and dark matter mass density pro-  files);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' b) the ellipticity of the lenses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' c) the size-magnitude distri-  bution of the lensed sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The answer to this question depends  on 1) how the lens detection probability Pdet varies as a function  of galaxy and source properties, 2) the efficiency of a survey at  correctly classifying detected strong lenses (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Pfind), and on 3)  the shape of the galaxy and source parameters distribution Pg  and Ps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' To understand this third point we can imagine the limit-  ing case in which both Pg and Ps are Dirac delta functions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  all lenses and sources are identical): in this limit, PSL simply re-  duces to the product PgPs up to a scaling constant, corresponding  to a case in which the lensing bias is none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Mandelbaum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2009) carried out a thorough study of  point 1): they quantified how the properties of a lens galaxy de-  termine its probability of creating a lensing event with a point  source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In this work we revisited the Mandelbaum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2009)  study, expanding it to the extended source case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We simulated in-  dividual lenses and examined how the lens detection probability  varies with lens and source properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In addition, we addressed  points 2) and 3) as well: we simulated large populations of strong  lenses using empirical models, we simulated the lens detection  and finding phase, and quantified the lensing bias under various  scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We explored how the results change as a function of  the efficiency of a lens survey at discovering small image sepa-  ration lenses, and of the scatter in mass parameters at fixed light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Finally, we addressed the question of how different are  galaxy-galaxy lens samples from sets of galaxy-quasar lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  This last point is relevant for time-delay cosmography studies, in  1 Sonnenfeld (2022) implicitly assumed Pfind ≡ 1, therefore Pdet and  Psel can be used interchangeably in the context of that work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  which measurements of the time-delay between the multiple im-  ages of a strongly lensed quasar are used to constrain the expan-  sion rate of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Galaxy-galaxy lenses can in principle  be used to help break some of the model degeneracies affecting  these measurements (Birrer & Treu 2021), but any difference be-  tween the two lens classes can introduce biases, if not corrected  for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' With this study we aim to quantify this bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  The structure of this work is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In Section 2 we  introduce the basics of gravitational lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In Section 3 we  study individual lens systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In Section 4 we describe our sim-  ulations of lens surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In Section 5 we show the results of our  analysis on the simulated lens survey data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We discuss the results  in Section 6 and draw conclusions in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  The Python code used for the simulation and analysis of the  lens sample can be found in a dedicated section of a GitHub  repository2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Strong lensing basics  The lensing properties of an object with respect to a source de-  pend solely on its dimensionless surface mass density distribu-  tion κ(θ) (also referred to as the convergence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This is the ratio  between the surface mass density and the critical surface mass  density for lensing:  κ(θ) = Σ(θ)  Σcr  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (3)  The latter quantity is defined as  Σcr =  c2Ds  4πGDdDds  ,  (4)  where c is the speed of light, G the gravitational constant, and  Dd, Ds, and Dds are the angular diameter distances between the  observer and the lens, the observer and the source, and the lens  and the source, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Given a source at angular position β, images of it form at the  positions θ that are solutions of the lens equation  β = θ − α(θ),  (5)  where α is the deflection angle of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This can be expressed  in terms of the dimensionless surface mass density by means of  the following integral over the whole sky:  α(θ) = 1  π  �  R2 d2θ′κ(θ′) θ − θ′  |θ − θ′|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (6)  The images of the background source are in general magnified  in total flux and in size, while preserving the original surface  brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The axisymmetric lens  In the special case of an axisymmetric lens we can simplify the  notation by considering a single coordinate axis with origin at  the centre of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We label the components of the image  position, source position and deflection angle along this axis as  θ, β and α, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The lens equation for an axisymmetric  lens then becomes  β = θ − α(θ),  (7)  2 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='com/astrosonnen/strong_lensing_tools/  papers/selection_effects  Article number, page 2 of 22  Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' : Strong lensing selection effects  and the expression for the deflection angle reduces to  α(θ) = 2  θ  � θ  0  dθ′κ(θ′)θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (8)  This can also be expressed in terms of the total projected mass  enclosed within a circle with angular radius equal to θ:  α(θ) = 1  πθ  M(< θ)  ΣcrD2  d  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (9)  A very important quantity for describing the strength of a  strong lens is the Einstein radius, θEin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For an axisymmetric lens,  this is the radius corresponding to the solution of Equation 7 for a  source placed at the same angular position as the lens centre (β =  0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The circle with radius equal to θEin is known as the tangential  critical curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Images that form there have infinite magnification  in the tangential direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' It can be shown that θEin satisfies the  following condition,  ¯κ(< θEin) = 1,  (10)  where ¯κ(< θ) is the average surface mass density within a radius  equal to θ:  ¯κ(< θ) ≡ 2  θ2  � θ  0  dθ′κ(θ′)θ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (11)  Axisymmetric lenses of the kind considered in this work typ-  ically produce either one or three images of a point source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Fig-  ure 1 helps to visualise this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Plotted in Figure 1 is the  quantity θ − α(θ) as a function of the position in the image plane  θ, for a few lens models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' According to Equation 7, images form  at the locations where this quantity equals the position of the  source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Therefore, given a source position β, the number of im-  ages and their location can be determined by drawing a horizon-  tal line at the value β on the vertical axis, and finding the points  where this line intersects the θ − α(θ) curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  For small values of β (for sources close to the lens centre),  the number of images that are produced is three: the source is  strongly lensed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For large values of β, instead, only one image is  formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The value of β where the transition occurs is known as  the radial caustic, which is marked by the horizontal dotted line  in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' As can be seen from Figure 1, a source at this loca-  tion is mapped to the stationary point of the θ − α(θ) curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' That  location on the image plane is known as the radial critical curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Images that form there have a formally infinite magnification in  the radial direction3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The elliptical lens  In this paper we focus mostly on lens galaxies with elliptical  isodensity contours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Given a surface mass density profile Σ(R),  a lens with an elliptical mass distribution can be obtained by  replacing the radial coordinate with the circularised radius:  R →  �  qx2 + y2  q ,  (12)  where x and y are Cartesian axes centred on the lens centre, with  x pointing towards the major axis, and where q is the minor-to-  major axis ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  3 The slope of the θ − α(θ) curve is the inverse of the radial magnifica-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This can be understood by taking the derivative of Equation 7 with  respect to θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  θ − α(θ)  θ  θE  −θE  fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5, γDM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5, γDM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5, γDM = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2, γDM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8, γDM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The lens equation of axisymmetric lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Solid lines: right-  hand side of Equation 7 for five lenses with different density profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Given a source at position β, its lensed images form at the values of  θ where the solid line intersects a horizontal line located at β on the  vertical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Dotted lines: positions of the radial caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Sources lo-  cated within the radial caustic of a given lens produce three images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The  lens models used in this simulations consist of a stellar component and  a dark matter halo, as described in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Their density profile is  plotted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Figure 2 shows the source-plane caustics of elliptical lenses  with different values of the axis ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We used the software  Glafic (Oguri 2021) to obtain the caustic curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The outermost  curves are radial caustics, while the inner ones are tangential  ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The most striking difference with respect to the axisym-  metric case (blue curves in Figure 2) is the fact that the tangential  caustic is transformed from a point into a diamond-like curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Sources located within the diamond produce five images (one of  which is usually highly de-magnified).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Sources that lie in the re-  gion enclosed between the two caustics are imaged three times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Sources outside the radial caustic are imaged only once, as in the  axisymmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The fact that the number of images changes  by two at a caustic crossing is a general feature of gravitational  lenses with no singularities (Schneider et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Lensing event definition: point sources  In order to compute the probability of a lensing event we must  provide an exact definition for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' A necessary condition for a  lens-source system to qualify as a strong lens is the presence of  multiple images of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' As we showed above, this requires  the source to lie within the region enclosed by the radial caustic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  In order to recognise a strong lens in a real survey, however, it is  not sufficient for multiple images to exist: they must be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  For this reason, given the detection limit for a point source, mlim,  we define as strong lensing event any lens-source system with at  least two images brighter than mlim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This is the same definition  used by (van de Ven et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2009), upon which the Mandelbaum  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2009) work is based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Labelling with m2 the magnitude of  the second-brightest image, then, in the absence of photometric  Article number, page 3 of 22  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' seleff  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8  βx/θEin  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8  βy/θEin  Major axis  q = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='9  q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8  q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='7  q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Caustics of lenses with fixed radial structure and different el-  lipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Source-plane angular coordinates are in units of the Einstein  radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The outer curve is the radial caustic, while the inner diamond is  the tangential caustic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Point sources located outside the caustic are not  strongly lensed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Sources that lie in the region enclosed between the two  caustics produce three images, while sources inside the tangential caus-  tic produce five images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The lens model adopted for this experiment  consists of a stellar component and a dark matter halo, as described in  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The two components have the same ellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  noise the lens detection probability Pdet becomes  Pdet(ψg, ψs) =  � 1  if m2(ψg, ψs) < mlim  0  otherwise  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (13)  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Lensing event definition: extended sources  Defining a strong lensing event in the case of an extended source  is less straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In principle, we should require parts of  the source to be multiply imaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In practice, it is not always  easy to determine whether a lensed image contains multiple im-  ages or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This is because, when the source is extended, some  of the images can be blended together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In real strong lensing sur-  veys, it is common to find lens candidates in which the lensed  source consists of only a single arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In those cases it is diffi-  cult to establish whether the arc is a set of blending images or  not, and the decision of including such systems in a strong lens  sample is often arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Here we adopt the following working  definition: an extended source is strongly lensed if the number  of surface brightness peaks that are detected is larger than its  intrinsic number of peaks in the absence of lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We explain how this definition applies in practice with a few  examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For simplicity, we focus on the case of a source with  a single surface brightness peak, such as a Sérsic profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' All  of the sources considered in this work belong to this family of  surface brightness models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We adopt the following procedure to  determine the number of detected peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Given the observed im-  age of a lensed source, we define its footprint as the ensemble  of pixels where the source is detected with S/N > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The foot-  print is in general composed of multiple disconnected regions,  corresponding to the different images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In order to only include  images that can be clearly identified, we add the condition that  the integrated S/N of each separate region must be S/N > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  This condition has the effect of removing from the source foot-  print any isolated region consisting only of a very small number  of pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In a real-world application, it would be very hard to  classify such marginal detections as images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' If, after applying  this cut, the source footprint is spread over multiple separated  regions, then the system is classified as a lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' If the source foot-  print consists instead of a single region, we iteratively increase  the surface brightness threshold used to define the footprint and  count the number of isolated regions with S/N > 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The maxi-  mum number of detected regions defines the number of detected  peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Figure 3 shows a few examples of how this criterion can be  used to classify lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The first column shows the caustic struc-  ture and source position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The second column shows an image  of the lensed source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The third column shows the source foot-  print defined with the procedure described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Pink pixels  correspond to the largest footprint that maximises the number of  detected images, while purple pixels belong to the 2σ detection  footprint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In the first, second and sixth example, these two foot-  prints coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In the third, fourth and fifth example, instead,  the 2σ detection footprint consists of a single region, but an in-  crease of the surface brightness threshold leads to the detection  of additional images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Strictly speaking, our lens definition criterion fails for a per-  fect Einstein ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In order to cover such a scenario, we also clas-  sify as strongly lensed any sources that produce a footprint with  a hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We point out that, although our lens definition relies on peak  detections, it is not necessary for the peak of the source surface  brightness to lie within the caustics in order for the source to be  strongly lensed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This can be observed in the fourth example of  Figure 3: although the source centroid lies outside of the caustic  (as shown in the first column), the outskirts of the source overlap  with the lens, and therefore multiple images are produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We can identify three different regimes in strong lensing of  extended sources, depending on the relative size of the source  and of the lens caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In the limit of a very small source, the  image configurations that are produced are qualitatively similar  to those that can be obtained in the point-source case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' When the  source and caustic size are comparable, the multiple images tend  to be blended into arcs or rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Most, if not all, of the galaxy-  scale strong lenses known belong to these two categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' How-  ever, there is a third regime in strong lensing, corresponding to  the case in which the source size is much bigger than the caus-  tics, such as in the fifth example of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In this regime, the  overall size and total flux of the source are roughly preserved,  and the lens produces only a relatively minor perturbation on  a localised region of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Strong lenses of this kind can  be difficult to detect, especially if the region of the image sub-  ject to strong lensing overlaps with the light from the foreground  galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The ultimate limiting factor to the detection of strong  lenses in the large source size regime, however, is the ability to  spatially resolve the multiple images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This limit is set by the size  of the point spread function (PSF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  All of the examples that we considered were based on  sources with a single surface brightness peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In general this  is not necessarily the case, and the intrinsic number of surface  brightness peaks of a strongly lensed source is not known a pri-  ori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' When dealing with a real lens candidate, applying our lens  definition criterion requires showing that the observed number  of surface brightness peaks can be reproduced with a lens model  Article number, page 4 of 22  Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' : Strong lensing selection effects  Caustics  Source  Lens  2σ detection  Max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' # images  Not a lens  Lens  Lens  Lens  Not a lens  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Criterion used to classify lensed images of extended sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Six examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' First column: caustics (red curves) and source position  (blue circle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The radius of the circle corresponds to the radius at which  the surface brightness is equal to 2σ the sky background rms fluctuation  for a single pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In other words, the blue circle delimits the area of the  source that can be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Second column: mock image of the lensed  source, with added noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Third column: footprint of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The  purple footprint is obtained with a 2σ detection criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The coral re-  gion is the largest footprint with the highest number of detected images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  in which the source has a smaller number of peaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In samples  of lenses defined via visual inspection this process is typically  done implicitly, by identifying multiply imaged blobs that be-  long to the same source element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  To our knowledge, we are the first to propose a surface  brightness peak-based definition of a strong lensing event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' A  popular alternative definition of a strong lens is one based on  magnification: only images that are magnified by more than  a given threshold are considered as strongly lensed (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Hilbert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The problem with a definition of this kind  is that magnification cannot be determined unambiguously from  observations, unless the intrinsic properties of the lensed source  are known from other means (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' if the source is a standard can-  dle or a standard ruler).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Because of the mass-sheet degeneracy  (Falco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 1985), it is possible to vary the magnification of a  lensed image while keeping its observed properties fixed: this  could lead to the paradox of two identical-looking lenses that  are classified differently on the basis of the underlying magnifi-  cation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Although we could still use a magnification-based defi-  nition for the sake of carrying out our experiments, it would then  be difficult to apply our results to real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Our definition of a  strong lensing event, instead, is robust with respect to the mass-  sheet degeneracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Individual lenses  In this section we study how the probability of a strong lensing  event varies as a function of lens and source properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In order  to do so, it is useful to introduce the concept of strong lensing  cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Given a foreground galaxy with parameters ψg,  a background source with parameters ψs, and a criterion S to  define a strong lensing event, the strong lensing cross-section is  defined as (Sonnenfeld 2022):  σSL =  �  R2 dβPdet(ψg, ψ(−β)  s  , β|S ),  (14)  where β is the position of the background source, ψ(−β)  s  is the  ensemble of source parameter except for the position, and the  integral is carried out over the whole sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The definition above  is valid for both a point source and an extended source: although  there is no unique way of defining the position of an extended  source, the integral over the sky ensures that the result is inde-  pendent of how the source position is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In the limit of low  density of background sources, which is satisfied in all practi-  cal cases, the probability of a lensing event is proportional to  σSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The lensing cross-section defined via Equation 14 depends  solely on the lens detection probability Pdet and does not take  into account whether the lens can be correctly classified by a  lens finder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This separate selection step is captured by the term  Pfind in Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In this section we consider exclusively the  detectability of a lens, and therefore focus only on σSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In Sec-  tion 4, when considering specific lens survey simulations, we  introduce Pfind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We compute σSL in a series of different scenarios of increas-  ing complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The model family adopted to describe the radial  density profile of the lenses is the same in all of our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We describe this in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2 we show calcula-  tions of the strong lensing cross-section in the case of axisym-  metric lenses and point sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='3 we generalise the  lens geometry to elliptical, while in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4 we replace point  sources with extended sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Lens density profile  In this work we focus on massive early-type galaxies as lenses,  as these make up the vast majority of known strong lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We  describe their mass distribution with a model consisting of two  concentric components, one describing the baryons and one for  Article number, page 5 of 22  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' seleff  the dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We assume that the baryonic component con-  sists entirely of stars, thereby neglecting gas, which is known to  contribute very little to the mass of early-type galaxies in the in-  ner regions that are probed by strong lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We then assume  that the stars follow a Sérsic profile, with projected surface mass  density given by  Σ(R) = Σ0 exp  �������−b  � R  Re  �1/n�������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (15)  In the above equation,  Σ0 =  M∗b2n  2πnR2eΓ(2n),  (16)  M∗ is the total mass, Re is the radius enclosing half of the total  mass, n is the Sérsic index, Γ is the incomplete Gamma function,  and b is given by (Ciotti & Bertin 1999)  b(n) ≈ 2n − 1  3 +  4  405n +  46  25515n2 + O(n−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (17)  Throughout this paper we indicate with θe the angular size of the  half-light radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We fix the Sérsic index of the lenses to n = 4, correspond-  ing to a de Vaucouleurs model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Although early-type galaxies are  often described with a free Sérsic index, a de Vaucouleurs pro-  file is able to reproduce their surface brightness distribution to  a few percent in the radial range 1kpc < R < 30kpc (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Sonnenfeld 2020), which is the region that is most relevant for  strong lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Finally, we assume that the light distribution of  the stellar component follows its mass distribution exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' That  is, we do not allow for the presence of gradients in the stellar  mass-to-light ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5 we discuss qualitatively what  the implications of relaxing this assumption would be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We describe the dark matter halo with a generalised Navarro,  Frenk & White (gNFW) profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We first define it by its three-  dimensional distribution, which for a spherically symmetric pro-  file is  ρ(r) =  ρ0  (r/rs)γDM (1 + r/rs)3−γDM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (18)  The parameter γDM is the inner density slope, ρ0 is a normal-  isation parameter, while rs is the scale radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The logarithmic  slope of the density profile transitions from −γDM to −3 at a ra-  dius r ≈ rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The projected surface mass density of a gNFW pro-  file can be expressed in terms of the following integral (Wyithe  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2001):  Σ(R) = 2rsρ0  � R  rs  �1−γDM � π/2  0  dx sin x(sin x + R/rs)γDM−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (19)  Figure 4 shows the dimensionless surface mass density pro-  file of Sérsic + gNFW models with various values of the inner  dark matter density slope γDM and of the fraction of projected  dark matter mass within the half-light radius, fDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' All of the pro-  files have a dark matter scale radius equal to ten times Re, and are  normalised in such a way that the Einstein radius is equal to the  half-mass radius of the stellar component (these assumptions are  dropped later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Two main features emerge from Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' First,  the baryons generally dominate the total density in the inner re-  gions (θ < θEin), while the dark matter is the main component at  large radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Second, models with different dark matter fractions  and inner dark matter slopes can conspire to produce very sim-  ilar total density profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This is the case, for example, of the  ( fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5, γDM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5) and the (fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2, γDM = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0) models  (green and blue lines in Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  10−1  100  101  θ/θEin  10−1  100  101  κ(θ)  fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5, γDM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5, γDM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5, γDM = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2, γDM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8, γDM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Dimensionless surface mass density profile of Sérsic + gNFW  composite models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The Sérsic index of the baryonic component is fixed  to n = 4 and the scale radius of the dark matter component is fixed to ten  times the half-light radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' All of the profiles are normalised in such a  way that the Einstein radius is equal to the half-light radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Solid lines:  total density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Dotted lines: dark matter density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Dashed  lines: baryonic density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The blue, orange and green dashed lines  are identical, as they correspond to profiles with the same fraction of  baryonic mass within the half-light radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Axisymmetric lenses, point sources  Axisymmetric lenses with a density profile of the kind intro-  duced in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1 can produce either one or three images of  a point source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This can be seen in Figure 1, which shows the  lens equation for various values of the dark matter fraction and  the inner dark matter slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' All of the lenses shown in this figure  have the same Einstein radius, which is equal in size to the half-  mass radius of the stellar component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The radius of the radial  caustic, marked by the dotted lines in Figure 1, is a strong func-  tion of the lens properties: it is largest in lenses with a smaller  dark matter fraction or steeper dark matter slopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' As a result,  the source plane area that is subject to strong lensing is an even  stronger function of these properties, since it scales with the  square of the radial caustic radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In order to compute the lens-  ing cross-section, however, we must take into account the mag-  nification, because that determines whether multiple images can  be detected or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Figure 5 shows the magnification of the secondary image as  a function of source position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The secondary image is the one  located in the region between the radial and tangential critical  curves, opposite to the source with respect to the lens centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In  most practical cases this is the second brightest image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' As Fig-  ure 5 shows, the magnification is very large for sources close  to the lens centre (small values of β), decreases with increasing  source position and then increases in close proximity to the ra-  dial caustic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' While for the model with fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8 (purple curve)  the magnification is above unity everywhere, other lens mod-  els can produce highly de-magnified secondary images for large  values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Depending on the intrinsic brightness of the lensed  source, these images may or may not be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Using the definition of Equation 14, we computed the lens-  ing cross-section of a set of axisymmetric lenses, with respect  to point sources with different brightnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In particular, we  considered model lenses with fixed Einstein radius, with angu-  lar half-mass radius θe fixed to θEin, and varying values of the  Article number, page 6 of 22  Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' : Strong lensing selection effects  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2  β/θEin  10−1  100  101  |µ2|  fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5, γDM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5, γDM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5, γDM = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2, γDM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8, γDM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Magnification of the secondary image as a function of the source  position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The lenses are axisymmetric composite models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Their density  profile is plotted in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Vertical dotted lines mark the position of  the radial caustic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  dark matter fraction and dark matter inner slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The results are  shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Each line corresponds to a source with a given  intrinsic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' unlensed) magnitude, ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The difference between  this magnitude and the limiting magnitude of the survey is indi-  cated as follows:  ∆m ≡ ms − mlim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (20)  We can see a clear trend between ∆m and the lensing cross-  section, in both panels of Figure 6: σSL is larger for brighter  sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The trend saturates below a certain ∆m, for sufficiently  small values of γDM or for sufficiently large values of fDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In  these cases the lensing cross-section coincides with the full area  enclosed within the caustic, and further increasing the source  brightness does not result in an increased value of σSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  At fixed source brightness, trends with the dark matter inner  slope or dark matter fraction are generally weak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The lines of  Figure 6, however, have been computed by keeping the Einstein  radius fixed while varying γDM or fDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This is achieved by ad-  justing other properties of the lens, such as the total mass of the  baryonic or the dark matter component (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In prac-  tice, when varying one ingredient of the lens density profile, the  Einstein radius varies in response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' To get a complete view of how  the lensing cross-section depends on different lens properties, we  also computed how σSL responds in absolute terms by varying  one lens parameter at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Figure 7 shows σSL as a function of  stellar mass, half-light radius, inner dark matter slope, and pro-  jected dark matter mass enclosed within an aperture of 5 kpc,  MDM,5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  The lensing cross-section increases with increasing stellar  mass and dark matter mass, decreases with increasing Re for  bright sources, while is only a weak function of γDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The lack of  a clear trend between σSL and γDM appears to be in contradiction  with the result of Mandelbaum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2009), who found a strong  positive correlation between σSL and γDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The origin of this dis-  crepancy lies in the different ways in which γDM is varied in the  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  γDM  10−2  10−1  100  σSL/(πθ2  Ein)  ∆ms = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8  fDM  ∆ms = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  γDM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Strong lensing cross-section of an axisymmetric lens and a point  source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The lens galaxy is a composite model, introduced in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1,  with angular half-light radius equal to the Einstein radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The left panel  shows the effect of varying the slope γDM for fixed fDM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' conversely, the  right panel shows variations as a function of fDM with γDM fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The  system is defined as a strong lens if at least two images are detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Different lines correspond to the difference ∆m between the source mag-  nitude and the survey detection limit for a point source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  two experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' While we varied γDM at fixed MDM,5, Mandel-  baum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2009) kept fixed the virial mass of the dark matter  halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' At fixed virial mass, increasing the inner dark matter slope  results in a correspondingly larger dark matter mass in the inner  regions, which naturally results in a larger lensing cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Figure 6 and Figure 7 show that the trends between lens  properties and the strong lensing cross-section can be differ-  ent for sources with different brightnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The net effect in a  strong lensing survey is an average over the source population,  weighted by the source luminosity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This implies that  surveys that target different families of sources, with different lu-  minosity functions (for instance, galaxies or quasars), can have  different strong lensing biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We explore this possibility in  Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Elliptical lenses, point sources  We measured σSL for lenses with a fixed radial density profile  and different ellipticities, with respect to point sources of differ-  ent brightnesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In particular, we set fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5, γDM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5,  rs = 10Re, θEin = θe, and set the ellipticities of both the bary-  onic and dark matter components to be the same, with the same  orientation of the major axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This is the lens model used to pro-  duce the caustics plot of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Figure 2 suggests that the size  of the source plane area subject to strong lensing, the area en-  closed within the outermost caustic, does not vary strongly with  the ellipticity of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Therefore we expect the strong lensing  cross-section to be a weak function of ellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We carried out the computation of σSL by means of simu-  lation: we generated a large number of point sources randomly  distributed over a given area that includes the caustic, then used  Glafic to solve the lens equation, find the number of images and  their magnification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We then measured the fraction of sources  that are strongly lensed according to the criterion of Equation 13  and multiplied this value by the area over which sources are lo-  cated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The resulting σSL is shown in Figure 8 as a function of the  minor-to-major axis ratio q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  For bright sources the strong lensing cross-section is approx-  imately constant with axis ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This is because, as pointed out  earlier, in the bright source regime the cross-section is deter-  mined by the area enclosed within the radial caustic, which does  not vary much with lens ellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For faint sources we observe  Article number, page 7 of 22  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' seleff  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  log M∗  10−1  100  101  σSL (arcsec2)  πθ2  Ein  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2  log Re  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  γDM  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4  log MDM,5  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Absolute value of the strong lensing cross-section as a function of various lens properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The reference lens is a galaxy at z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='3 with  log M∗ = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5, Re = 7 kpc, γDM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5, log MDM,5 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0, rs = 100 kpc, and a source at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5 In each panel, only one property of the lens is  varied, as indicated on the label of the horizontal axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Each curve corresponds to a different value of ∆m, in accordance with Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The dashed  line in each panel shows the quantity πθ2  Ein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  q  10−3  10−2  10−1  100  σSL/(πθ2  Ein)  ∆ms = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  ∆ms = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Point source strong lensing cross-section as a function of lens  axis ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Solid lines: cross-section based on the lens event definition  of Equation 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Dashed lines: cross-section based on the detection of  four images (quad cross-section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Lines of different colour correspond  to sources of different intrinsic magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The dashed blue line, cor-  responding to the brightest source magnitude, overlaps completely with  the dashed orange line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The dashed purple line is zero: very faint sources  cannot produce any detectable quads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The parameters of the lens den-  sity profile are fDM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5, γDM = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5, rs = 10Re, θe = θEin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The baryonic  and dark matter components have the same ellipticity and direction of  the major axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  a larger variation with q, with a factor of two difference between  the largest and smallest value of σSL at fixed source brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Most of the sources that result in detectable lenses produce  two detectable images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' These are sources that are located in the  region enclosed between the radial and the tangential caustic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' If  the source is located within the tangential caustic, however, four  detectable images are usually created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Lenses with four visible  images, usually referred to as quad lenses, are sometimes given  a high priority in certain lensing studies, because they offer more  constraints compared to double lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For instance, quads make  up the majority of the lenses used so far in time-delay studies  (Millon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For this reason we also computed an alter-  native lensing cross-section, in which the definition of a lensing  event requires the detection of four images, instead of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This  is plotted in Figure 8 with dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The cross-section for  quads is a strong function of lens ellipticity, for bright sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  This is a consequence of the fact that the area enclosed within the  tangential caustic, which is where a source needs to be in order  to produce four images, increases with increased lens ellipticity,  as Figure 2 shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For sources that are intrinsically fainter than  the detection limit, however, the quad cross-section is extremely  small, regardless of ellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Elliptical lenses, extended sources  In the case of an extended source, the complexity of the problem  is increased due to the addition of a series of features: the source  surface brightness distribution, with its radial profile, shape and  orientation, and the PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Moreover, as we discussed in section  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4, there are different regimes in strong lensing of extended  sources, depending on the relative size of the source and the  caustics of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For this reason, we split our analysis into  two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' First, we explore the small source size regime, where  the source size is comparable to or smaller than the lens caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Then, we consider cases in which the source size is bigger than  the lens caustics, which we refer to as the large source regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Small source sizes  For the sake of reducing the dimensionality of the analysis, we  focused on circularly symmetric sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We also fixed the sur-  face brightness profile to an exponential disk (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' a Sérsic model  with n = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We then took a lens model with the same parameters  used in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='3 and a minor-to-major axis ratio of q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We simulated a large number of images of extended sources  with Glafic and measured the fraction of them that results in  a strong lens, according to the definition of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For the  small source size experiment, we used pixels with a size equal  to 1/20θEin and convolved the images with a Moffat PSF with a  FWHM of two pixels and a β parameter of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We also assumed  that the background noise is an uncorrelated Gaussian field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We  carried out experiments with sources with different values of the  total unlensed flux, f, and half-light radius, θe,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The results are  shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The total flux of the source, indicated in the  Article number, page 8 of 22  Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' : Strong lensing selection effects  10−1  100  θe,s/θEin  10−1  100  σSL/(πθ2  Ein)  log f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  log f = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  log f = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  Bright point source  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Strong lensing cross-section of an extended source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Small source  size regime (θe,s < θEin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The cross-section is plotted as a function of the  ratio between the half-light radius of the source and the Einstein radius  of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The lens model is fixed to be the same of Figure 8, with  axis ratio q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The source is a circular exponential profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Each  solid line corresponds to a different value of the total unlensed flux of  the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The flux f is expressed in terms of the background noise  rms fluctuation measured over an area equal to θ2  Ein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The vertical dashed  lines correspond to the maximum size for which a galaxy with a given  flux can be detected in the absence of lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The horizontal dotted  line indicates the cross-section for a very bright point source (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' the  area enclosed within the caustics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  legend, is measured in units of the sky background rms fluctua-  tion within an area equal to the square of the Einstein radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  As in the point source case, the lensing cross-section in-  creases with increasing total flux, at fixed source size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' At fixed  flux, σSL stays approximately constant with increasing source  size until a given value, then drops rapidly for larger sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' From  a qualitative point of view, this behaviour can be observed also  in the absence of lensing: increasing the size of a galaxy while  keeping its flux fixed lowers its average surface brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' If  the surface brightness drops below the sky rms fluctuation level,  then it becomes very difficult to detect it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In order to determine  whether there are lensing-specific features in the σSL − θe,s rela-  tion of Figure 9, we computed, for each source flux, the maxi-  mum half-light radius for which it can be detected in the absence  of lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We used the same criterion as that of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4 to  define a detection: we defined the source footprint as the ensem-  ble of pixels that are 2σ above the background and required the  total signal-to-noise ratio within the footprint to be larger than  ten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The resulting limiting sizes are shown as vertical lines in  Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For each given total flux, the non-lensing size limit  is similar to the value of θe,s at which the lensing cross-section  drops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  This result suggests that, to first approximation, lensing does  not introduce a strong selection in source size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' While perhaps  surprising, this follows from the fact that gravitational lensing  preserves surface brightness: a source that can be detected in the  absence of lensing will produce images with the same surface  brightness when lensed, which can be detected as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In or-  der to classify a source as strongly lensed, however, we require  that multiple images are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Only sources that lie within  a well-defined region give rise to a strong lensing configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  If part of the source extends outside of this region, then only a  fraction of its flux contributes to creating a set of strongly lensed  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This lowers the signal-to-noise ratio of the multiple im-  ages compared to the point source case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The result is that σSL  starts to decrease with increasing source size at values of θe,s  that are smaller than the no-lensing detection limit, as observed  in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  At the brightest flux explored in the experiment (green line in  Figure 9), the lensing cross-section at small source sizes is larger  than the area enclosed within the radial caustic (black dotted line  in Figure 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This is because, when the source is very bright,  it can give rise to multiple images even while its centroid lies  outside of the radial caustic, as long as its surface brightness  distribution extends into it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This also explains why σSL increases  with increasing source size, before dropping to zero: the more  extended the source, the farther away from the lens centre it can  be while still producing multiple images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Large source sizes  For the large source size case we fixed the surface brightness  profile of the source and varied the Einstein radius of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  In particular, we set the source half-light radius to 20 pixels and  adjusted its total flux in such a way that the 2σ detection foot-  print in the absence of lensing extends out to the half-light ra-  dius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Then, starting from the lens model used in the previous sec-  tion, we progressively increased the critical surface mass density  to reduce the Einstein radius down to values comparable to the  pixel size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Figure 10 shows the resulting strong lensing cross-  section as a function of the ratio between lens Einstein radius  and source half-light radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For values of the Einstein radius  close to the size of the PSF, the lensing cross-section (blue solid  line) is very small: this is because the perturbations caused by  lensing are not well resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For larger values, σSL stays ap-  proximately constant, around values that are comparable to the  source size (horizontal dotted line) and much larger than the area  enclosed within the caustics (red dashed line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We conclude that,  in the large source size regime, the main factor that determines  the lensing cross-section is the area of the background source,  provided that the Einstein radius of the lens is larger than the  PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Lens populations simulations  In the previous section we showed how the lensing cross-section,  which is closely related to the lens detection probability Pdet,  varies as a function of lens and source properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' From here  on we focus on the effect that those trends have on popula-  tions of lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We addressed this question by simulating popu-  lations of foreground galaxies and background sources, selecting  strong lenses among them, and comparing the properties of the  strong lens sample with the parent galaxy population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We sim-  ulated a lens-based search (as opposed to a source-based one),  in which strongly lensed images are searched among a stellar  mass-selected sample of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Our simulations are based on  empirical models, in which existing observations of the baryonic  component of galaxies are complemented with a set of assump-  tions on the mass distribution of the lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1 we  explain how we built our foreground galaxy sample, while in  section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2 we describe the simulation of the background source  population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='3 we describe how our mock observa-  tions of lenses are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4 we apply a further  selection step, based on the angular size of the Einstein radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Article number, page 9 of 22  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' seleff  10−1  100  θEin/θs,e  10−2  10−1  100  σSL/(πθ2  s,e)  Cross-section  Caustic area  Source area  PSF FWHM  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Strong lensing cross-section of an extended source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Large  source size regime (θe,s ≳ θEin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The cross-section is plotted as a function  of the ratio between the Einstein radius of the lens and the half-light ra-  dius of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The source model is fixed to a circular Sérsic profile  with n = 1, detected out to the half-light radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The source half-light  radius measures 20 pixels, and the PSF has an FWHM of two pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  The lens model is varied: starting from the model of Figure 9, the crit-  ical surface mass density is increased to produce progressively smaller  values of θEin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Solid line: lensing cross-section, in units of the footprint  area of the source in the absence of lensing, as a function of the ratio be-  tween the lens Einstein radius and the source half-light radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Dashed  line: area enclosed within the lens caustics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Dotted line: value of the  cross-section corresponding to the area of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Dash-dotted line:  FWHM of the PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Foreground galaxies  Our foreground galaxy population consists of a volume-limited  sample of early-type galaxies, complete above a minimum ob-  served stellar mass of 1011M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We chose to focus on early-type  galaxies because many strong lensing surveys have preferen-  tially targeted this class of objects in their lens-finding phase  (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Bolton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Gavazzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This, in turn, had a dual motivation: first, early-type  galaxies are among the most massive objects in the Universe,  and therefore they are more likely to be lenses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' second, their  smooth surface brightness distribution and red colour makes it  easier to detect arcs from strongly lensed star forming galaxies  around them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We describe lenses with elliptical versions of the two-  component model introduced in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In the following  sections we explain how their parameters are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Stellar mass and redshift distribution  We generated lens galaxies over a finite redshift range, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1 < z <  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We chose these lower and upper limits because the value of  the critical surface mass density Σcs becomes very large outside  of this range, and the fraction of galaxies that act as strong lenses  drops substantially as a result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We drew stellar masses from the stellar mass function of qui-  escent galaxies measured by Muzzin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In particular,  we used the following comoving number density distribution  Φ(M(obs)  ∗  ) = Φ∗  ������  M(obs)  ∗  M∗∗  ������  α  exp  �������− M(obs)  ∗  M∗∗  �������.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (21)  We set Φ∗ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='009×10−3 Mpc−3, α = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='92 and log M∗  ∗ = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='21:  these are the best-fit values measured by Muzzin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We refer to the stellar masses drawn from this distribution  as the observed stellar masses, M(obs)  ∗  , as opposed to the true  stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The observed stellar mass is meant to represent  an estimate of M∗ based on stellar population synthesis mod-  elling, which is the method used by Muzzin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2013) to  measure the galaxy stellar mass function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Stellar population syn-  thesis measurements, however, are subject to systematic uncer-  tainties, since they have not been calibrated on galaxies whose  stellar mass is known by other means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We quantify the discrep-  ancy between the observed and true stellar mass by means of the  stellar population synthesis mismatch parameter αsps, defined as  follows:  αsps ≡  M∗  M(obs)  ∗  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (22)  The most important source of systematic uncertainty in the mea-  surement of the stellar mass is the assumption of the stellar initial  mass function (IMF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Muzzin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2013) assumed a Kroupa  IMF (Kroupa 2001) to derive their measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This means  that, in the absence of other systematic effects, a galaxy with a  Kroupa IMF has a value of αsps = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  For each galaxy in the sample, we randomly drew a value of  log αsps from the following distribution:  P(log αsps) ∼ N(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1, σ2  α),  (23)  where the notation N(µ, σ2) indicates a Gaussian with mean µ  and variance σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We set the mean of log αsps to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1, as this is  an intermediate value among estimates of αsps from the litera-  ture (Conroy & van Dokkum 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Cappellari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Smith  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2015, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We adopted a few dif-  ferent values for the scatter σα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We explain in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4 how  these were chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Stellar mass density distribution  We described the surface mass density distribution of each  galaxy as an elliptical de Vaucouleurs profile (a Sérsic profile  with n = 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Given the observed stellar mass of a galaxy, we  assigned a half-mass radius by drawing it from the following  distribution in log Re:  P(log Re) ∼ N(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='20 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='63(log M(obs)  ∗  − 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4), 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='142).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (24)  Then, we assigned an axis ratio q by drawing it from the follow-  ing beta distribution:  P(q) ∝ qα−1(1 − q)β−1,  (25)  with α = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='28 and β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The choice for these distributions  was motivated by observations of a sample of early-type galax-  ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In Appendix A we explain how this sample was defined and  how the coefficients of Equation 24 and Equation 25 were deter-  mined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Dark matter distribution  We modelled the dark matter distribution of each lens galaxy  with an elliptical gNFW halo (see section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1 for its definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  The parameters of the radial density profile were assigned as fol-  lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Given the stellar mass of a lens, we first determined the  value of its halo virial mass, Mh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We defined this as the mass en-  closed within a spherical shell with average density equal to 200  Article number, page 10 of 22  Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' : Strong lensing selection effects  times the critical density of the Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Using current weak  lensing constraints on the halo mass of elliptical galaxies as a  reference (Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2022), we drew halo masses from  the following distribution:  P(log Mh) ∼ N(13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0(log M∗ − 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5), σ2  h),  (26)  which is a Gaussian in log Mh with a mean that scales with stellar  mass and scatter σh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The intrinsic scatter in halo mass is not  well constrained observationally, therefore we ran simulations  with different values of σh, as for the stellar population synthesis  mismatch parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  To determine the density profile of the halo we relied on a  theoretically motivated model that takes into account the effect  of baryons on the dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We assumed the dark matter dis-  tribution to be initially described by an NFW profile with a con-  centration4 of five, then used the prescription of Cautun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (2020) to model the response of the halo to the infall of baryons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  In the Cautun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2020) model the halo response is approx-  imated with an analytical function that depends on the present  stellar mass distribution, and typically results in a more concen-  trated and steeper density profile compared to the original NFW  model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Finally, we fitted a gNFW profile to the surface mass den-  sity of the contracted halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' By doing this, we were able to fully  describe the dark matter density profile with three parameters:  Mh, γDM and rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The dark matter density profile defined in this  way is determined uniquely by the halo mass, the stellar mass  and the half-light radius (the more concentrated the stellar dis-  tribution, the stronger the halo response and the steeper the dark  matter density profile).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In principle we could have allowed for  additional degrees of freedom, for instance by relaxing the as-  sumption of a fixed initial halo concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In practice, as we  explain in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5, our main results are not affected by this  choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Given the radial profile of the dark matter halo, we obtained  an elliptical version of it by applying a transformation of the kind  of Equation 12 to its projected surface mass density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We assumed  that the axis ratio and orientation of the halo is the same as that  of the stellar component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Intrinsic scatter  The distribution in stellar mass, halo mass and dark matter inner  slope of our sample of simulated foreground galaxies depends  on parameters describing the intrinsic scatter in these properties,  namely σsps and σh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Direct observational constraints on these  quantities are poor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' However, we can derive upper limits on them  on the basis of observed scaling relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Early-type galaxies lie on the stellar mass fundamental plane,  a scaling relation between stellar mass, half-light radius and cen-  tral velocity dispersion (Hyde & Bernardi 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' de Graaff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  2021):  σe ∝ M(obs)βσ  ∗  Rξσ  e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (27)  The existence of a fundamental plane relation is a consequence  of the virial theorem: the velocity dispersion of a galaxy in dy-  namical equilibrium is directly related to the 3-dimensional mass  distribution in its inner regions, which is typically dominated by  the stellar component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' At fixed observed stellar mass distribu-  tion, however, the central velocity dispersion can vary depend-  ing on the stellar population synthesis mismatch parameter, on  4 The concentration is the ratio between the halo virial radius and the  scale radius rs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Intrinsic scatter scenarios  Model name  σsps  σh  Predicted FP  scatter  Fiducial  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='034  Low scatter  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='021  High scatter  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='043  Notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Adopted values of the intrinsic scatter parameters σsps and σh in  different simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For each set of values, the fourth column indicates  the scatter around the fundamental plane predicted via the spherical  Jeans analysis of Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The observed fundamental plane scat-  ter is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  the dark matter mass and on the dark matter density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This  gives rise to a spread in the values of the velocity dispersion  given M(obs)  ∗  and Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Therefore, we can use the observed scatter  in velocity dispersion to put an upper limit on the intrinsic scat-  ter in the stellar population synthesis mismatch parameter, halo  mass and dark matter slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We used Jeans modelling for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We generated  samples of early-type galaxies with the recipes described above  and with different values of σsps and σh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Using the spherical  Jeans equation under the assumption of isotropic orbits, we pre-  dicted the central velocity dispersion of each galaxy in the sam-  ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Then, we fitted a fundamental plane relation to this mock  sample and measured the predicted scatter in velocity dispersion  at fixed M(obs)  ∗  and Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We then varied σsps and σh to match the  observed and the predicted scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We did not attempt to match  the other parameters of the fundamental plane (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' the constant  of proportionality of Equation 27 and the power-law indices βσ  and ξσ), because these are sensitive to the orbital anisotropy of  the galaxies, which we are asserting to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The details of  this procedure are given in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We settled on three different sets of intrinsic scatter param-  eters, as indicated in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We label them the fiducial, the  low-scatter and the high-scatter scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The high-scatter sce-  nario is ruled out both by the fundamental plane and by weak  lensing constraints (Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Moreover, our dy-  namical model is quite simplistic, as it neglects the effects of or-  bital anisotropy and departures from spherical symmetry, which  are additional sources of scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Nevertheless, we use it in our  experiment in order to obtain a more conservative upper limit on  the amplitude of the strong lensing bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Background sources  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Galaxies  Our background galaxy population is taken from the surfs-based  KiDS-Legacy-Like Simulation (SKiLLS) input catalogue, a hy-  brid simulation catalogue integrating cosmological simulation  with high-quality imaging observations (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The cos-  mological simulation is obtained from the Synthetic UniveRses  For Surveys (surfs) simulations, a set of N-body simulations  from Elahi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The galaxy properties, including the star  formation history and the metallicity history, are from an open-  source semi-analytic model named Shark5 (Lagos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  The original photometry is drawn from a stellar population syn-  thesis technique using stellar synthesis libraries with physically  motivated dust attenuation and re-emission models (Robotham  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2022) further applied an empirical cor-  5 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='com/ICRAR/shark  Article number, page 11 of 22  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' seleff  rection to the original synthetic photometry to better agree with  the COSMOS2015 observations (Laigle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The galaxy  morphology is described by a Sérsic profile with three parame-  ters: the half-light radius in angular units θe,s, the Sérsic index  ns, and the axis ratio qs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' These structural parameters are learned  from the imaging data obtained with the Advanced Camera for  Surveys (ACS) instrument on the Hubble Space Telescope (Grif-  fith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We refer to Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2022) for details on the  learning algorithm and validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  The complete SKiLLS catalogue contains ∼108 deg2 of  galaxies with redshift up to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5 and r-band apparent magnitude  down to 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We applied a lower limit to the source redshift, by  selecting only sources with zs > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This ensures that all of  the sources lie behind all of the lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' A similar cut could be  applied to a real survey using photometric redshifts, to reduce  the incidence of false positives (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' arc-like features physically  associated with the lens galaxy) in the lens finding phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The  resulting number density of sources is 70 arcmin−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We approx-  imated their spatial distribution as uniform in the sky, that is we  neglected clustering of the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Quasars  We described the population of background quasars with the fol-  lowing double power-law luminosity function in the rest-frame  UV absolute magnitude M:  Φ(M, zqso) =  Φ(M∗)  100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4(αQ+1)(M−M∗) + 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4(βQ+1)(M−M∗) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (28)  Following Manti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2017), we set αQ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='35 (faint-end  slope), βQ = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='23 (bright-end slope), and adopted a redshift-  evolving normalisation  log Φ∗ = −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0991 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0209zqso + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0171z2  qso,  (29)  and characteristic magnitude  M∗ = −22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5216 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6510zqso + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2869z2  qso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (30)  Given the redshift and rest-frame UV luminosity of a quasar, we  then computed the apparent magnitude in the observed i−band,  mqso, using a quasar spectral template6 built from optical and  near-infrared spectra obtained by Vanden Berk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2001);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Glikman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  For the sake of consistency with the population of back-  ground extended sources, we limited the redshift distribution of  quasars to the range 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8 < zqso < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We then truncated the  distribution in mqso at two magnitudes fainter than the detection  limit (which is specified in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Finally, we randomly  placed quasars in the source plane with a projected number den-  sity of 70 arcmin−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This is a much larger number density than  observed in the real universe, but we are allowed to do so because  we are not interested in predicting the absolute number of strong  lenses, so this is a legitimate choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The advantage of boosting  the number density of quasars is that it allows us to produce a  large number of lenses without the need for generating too big a  population of foreground galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Observations  For each of the three intrinsic scatter scenarios, we drew a popu-  lation of foreground galaxies covering 1000 square degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The  6 https://archive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='edu/hlsps/reference-atlases/  cdbs/grid/comp_qso/  expectation value of the number of galaxies in the corresponding  volume, given the redshift and stellar mass cuts described in sec-  tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1, is around 300 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We approximated the foreground  galaxies as isolated: when determining whether a galaxy acts as  a strong lens, we only modelled the contribution to the lensing  signal from the galaxy itself, and neglected that of the environ-  ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We discuss the possible implications of this approximation  in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For each lens, we determined its caustics relative to  the highest source redshift, zs = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We then placed sources  randomly behind the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' If at least one source fell within a cir-  cular region enclosing the caustics, we proceeded to compute its  lensed images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  For the simulation with extended sources, we produced im-  ages with properties similar to those expected for the Euclid  Wide survey (Euclid Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We used a pixel  size of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1′′ and we applied a Moffat PSF with an FWHM of  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2′′ and a β parameter of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Finally, we assumed a background  noise level such that an extended source with half-light radius  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5′′ and an apparent magnitude in the absence of lensing of  ms = 25 is detected with S/N = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We then applied the peak  detection-based lens selection criterion introduced in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4  to find the strong lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  The fiducial scatter simulation with extended background  sources produced a sample of 2113 lenses, corresponding to a  number density of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1 deg−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This is about a factor of five smaller  than the number density predicted by Collett (2015) for the Eu-  clid survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This is a result of differences in the description of the  source population, in the criteria used to define a strong lensing  event, and in the redshift and stellar mass cuts that we applied to  define the foreground galaxy population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  For the lensed quasars we did not simulate pixel-level data,  but simply computed the observed magnitudes of the multiple  images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Following the definition of section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='3, we included in  the sample of strong lenses only systems with at least two im-  ages brighter than a limiting magnitude mlim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We set mlim = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='3,  which corresponds to the 10σ detection limit of the LSST in a  single visit (Oguri & Marshall 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This is motivated by the  fact that the quasar lenses are meant to simulate a sample as-  sembled for the purpose of carrying out time delay measure-  ments, which in turn require combining single-visit detections  over many epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The scenario that we are simulating, then, is  that of a lens search in a Euclid-like survey, followed-up with  LSST time-domain observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  The resulting number of quasar lenses in our simulation with  the fiducial scatter is 1621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This number is meaningless, given  that the simulation was created with an unrealistically large num-  ber density of quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' More interesting is the relative number of  quad lenses with respect to the total, which is about 9%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This is  a slightly smaller value than the fraction of quads predicted by  Oguri & Marshall (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The reason for this discrepancy lies in  the differences between the lens mass models in the two simula-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Lens finding probability  Figure 11 shows the distribution in Einstein radius of the simu-  lated lens samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In all cases, the distribution peaks at θEin ≈  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='7′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Although all of the lenses in these samples are detected,  it does not necessarily follow that they would all be included in  a strong lensing study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' There can be a few reasons for exclud-  ing certain lenses from a sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' One is the low accuracy of  lens finders: current automated lens finding algorithms tend to  produce lens candidates samples with low purity (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Son-  nenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Petrillo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Savary et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Such  Article number, page 12 of 22  Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' : Strong lensing selection effects  0  1  2  3  θEin  0  50  100  150  200  250  N  Fiducial  High scatter  Low scatter  Quasars (all)  Quasars (quads)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Einstein radius distribution of the simulated lens samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  These are: galaxy-galaxy lenses in the fiducial, high scatter and low  scatter scenarios;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' galaxy-quasar lenses with fiducial scatter, consider-  ing all lenses or only lenses that produce four images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  samples are then visually inspected, and only those candidates  that can be clearly distinguished from false-positives are kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  This visual inspection step tends to disfavour lenses with a small  image separation, because of the contamination from the light of  the lens galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Another possible reason for refining a sample of lens can-  didates is the availability of redshift measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Redshifts  of both the lens and the source are needed in order to convert  a lens model into a measurement of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' When working with  large samples of lenses, obtaining spectroscopic measurements  is not a viable option, and photometric redshifts are a necessity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Measuring the photometric redshift of a strongly lensed source,  however, is challenging, especially when the Einstein radius is  small and the source light is blended with the light from the lens  (Langeroodi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Both of these scenarios can result in samples that are incom-  plete below a certain value of the Einstein radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We simulate  this situation via the following Einstein radius-dependent lens  finding probability:  Pfind(θEin|S, det) =  �  1  if θEin > θEin,min  0  otherwise  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (31)  In words, all lenses with Einstein radius larger than θEin,min are  included in the sample, while all those with smaller Einstein ra-  dius are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We refer to θEin,min as the completeness limit:  our simulated lens samples are complete down to θEin = θEin,min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We explored scenarios with different values of θEin,min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' As the  next section shows, the larger the minimum Einstein radius, the  higher the strong lensing bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Results  In this section we present the results of the lens population  simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1 shows the results of the galaxy-galaxy  lens experiment, while section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2 focuses on the population of  galaxy-quasar lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Given the number of parameters that are  needed to describe our model, providing a complete characteri-  sation of the strong lensing bias is a problem with relatively high  dimensionality, and is beyond the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For the  sake of conciseness, we focus instead on the quantities that we  consider most important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Nevertheless, the output of our simu-  lations is available online7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We encourage readers who are inter-  ested in studying aspects of the strong lensing bias that are not  covered in this section to download our data and analyse them  directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Galaxy-galaxy lenses  In this section we show the results of the experiments with pop-  ulations of galaxy-galaxy lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We first present the results in  a qualitative way, then proceed to quantify the amplitude of the  strong lensing bias in various quantities of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Figure 12 shows the distribution in the parameters of the  foreground galaxies of the lens systems, compared to those of  the parent population, for the fiducial scatter scenario with two  different values of the minimum Einstein radius: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5′′ and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  A completeness limit of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5′′ is close to what can currently be  achieved via visual inspection of high-resolution space-based  images (Garvin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2022), while the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0′′ limit can be seen  as a more conservative case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  The most striking difference between the samples is in the  stellar mass: strong lensing selects preferentially galaxies with  larger values of M(obs)  ∗  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Lenses tend to also have a larger halo  mass, a smaller half-light radius at fixed stellar mass, and a larger  stellar population synthesis mismatch parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The distribu-  tion in ellipticity and inner dark matter slope of the lenses instead  look very similar to that of the parent population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Additionally,  we can see that the amplitude of the strong lensing bias appears  to be always larger in the lens sample with the more restrictive  selection on Einstein radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We quantify the amplitude of these  biases later in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Figure 13 shows the distribution in parameters describing the  background source population, for the same simulations of Fig-  ure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Additionally, Figure 13 shows the subset of the parent  population that consists of detectable sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' These are back-  ground galaxies that, in the absence of lensing, can be detected  according to the same criterion used for the lensed sources (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  the S/N over their 2σ footprint is larger than ten).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The detection  limit of the survey is at ms ≈ 25 (the actual limit varies depend-  ing on the surface brightness distribution parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Because  our simulated background source population extends to much  fainter magnitudes, the distribution of detectable sources differs  substantially from that of the parent population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We can then  consider two different strong lensing bias definitions: one that  quantifies the difference in lensed source properties with respect  to the parent population, and one that describes the difference  with respect to the detectable source population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We are mostly  interested in the second definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Strong lensing tends to preferentially select sources at higher  redshift, especially in the more restrictive case with θEin > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  This is because, at fixed lens properties, increasing the source  redshift lowers the critical surface mass density and, conse-  quently, increases the size of the caustics and the Einstein ra-  dius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The distribution in the magnitude of the lensed sources is  also very different from that of the parent distribution, as it drops  rapidly for values larger than ms ≈ 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Interestingly, however,  there does not seem to be a large difference with respect to the  distribution of detectable sources in the absence of lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This  result can appear to be somewhat counter-intuitive: lensing mag-  nification should allow the detection of sources that are intrinsi-  cally fainter than the detection limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' To some extent, this is the  7 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='com/astrosonnen/strong_lensing_tools/  papers/selection_effects  Article number, page 13 of 22  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' seleff  General population  Lenses, θEin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5′′  Lenses, θEin > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0′′  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8  γDM  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  log M (obs)  ∗  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2  log αSPS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  log Re  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  q  12  14  log Mh  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6  zg  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8  γDM  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  log M (obs)  ∗  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2  log αSPS  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  log Re  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  q  12  14  log Mh  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Comparison between the properties of lens samples and the parent population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Distribution in foreground galaxy parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Filled  contours: distribution of the parent sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Green solid lines: distribution of the lenses with Einstein radius larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Red solid lines:  distribution of the lenses with Einstein radius larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  case: the distribution of lensed sources shows a slight excess of  fainter galaxies compared to the unlensed case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' However, as we  quantify later in this section, the difference is far from large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The  reason for this behaviour lies in the fact that, in the detection of  both lensed and unlensed sources, the most important quantity  is surface brightness, which is preserved by gravitational lens-  ing when the source is larger than the PSF size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' As we showed  in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4, the lensing cross-section drops to zero once the  surface brightness of the source reaches a value that would make  it undetectable in the absence of lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For this reason, also  the distribution in half-light radius is very similar between the  lensed sources and the detectable source population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Qualitatively, the results shown in Figure 12 and Figure 13  match our expectations from Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In the rest of this section  we quantify the lensing bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We present the results in three dif-  ferent parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' First, we focus on the properties of the lens galax-  ies that can be observed directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' These are quantities that can  be derived from photometry and spectroscopy with minimal as-  sumptions: the lens redshift, the observed stellar mass, the half-  light radius and the axis ratio8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Any bias in these quantities can  8 Strictly speaking, the axis ratio in mass is not necessarily observable,  but in the context of our simulations it is, since light and mass have the  same ellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  be determined relatively easily in a real strong lens survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In  the second part, we consider the parameters related to the mass  distribution: the stellar population synthesis mismatch parame-  ter and the dark matter distribution parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Determining the  bias on these parameters is much more difficult, but these are  quantities of great interest from a galaxy science point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  In the third part, we focus on source distribution parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Bias in observable lens parameters  Figure 14 shows the median redshift, median M(obs)  ∗  , median size  for a given M(obs)  ∗  , and median axis ratio of various lens popula-  tion simulations, as a function of the minimum Einstein radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We defined the median size at fixed M(obs)  ∗  by fitting the follow-  ing mass-size relation to the Re − M(obs)  ∗  distribution:  log Re ∼ µR,0 + βR(log M(obs)  ∗  − 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (32)  The quantity shown in the third panel of Figure 14 is the param-  eter µR,0, which is the average log Re at an observed stellar mass  of log M(obs)  ∗  = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The horizontal dashed line in each panel  shows the value of the parent population: the larger the distance  Article number, page 14 of 22  Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' : Strong lensing selection effects  General population  Detectable w/o lensing  Lenses, θEin > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5′′  Lenses, θEin > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0′′  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  qs  22  24  26  28  ms  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  θs,e  0  2  4  ns  1  2  zs  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  qs  22  24  26  28  ms  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  θs,e  0  2  4  ns  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Comparison between the properties of lens samples and the parent population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Distribution in background source parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Filled  contours: distribution of the parent sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Black solid lines: distribution of the detectable sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Green solid lines: distribution of the lenses  with Einstein radius larger than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Red solid lines: distribution of the lenses with Einstein radius larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  between the curve of a simulation and this line, the higher the  strong lensing bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  As was already visible in Figure 12, there is a clear bias to-  wards lower redshift, higher stellar mass and smaller sizes, with  the bias becoming stronger for more restrictive cuts in Einstein  radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Biases in stellar mass and size are stronger for the simu-  lation with low scatter in αsps and Mh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This can be explained as  follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For a given value of the observed stellar mass, strong  lensing selection favours galaxies with a larger αsps or Mh, or  with a smaller half-light radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Since the galaxy stellar mass  function of the parent population is steep, lenses tend to have a  relatively small M(obs)  ∗  and large values of αsps or Mh for their  observed stellar mass (this is shown more clearly in the next sec-  tion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' If the intrinsic scatter in αsps and Mh is low, however, the  number of galaxies with a small M(obs)  ∗  and a large stellar or halo  mass is greatly reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Only galaxies with a large M(obs)  ∗  or a  small size can therefore act as strong lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This is an interest-  ing result, because it suggests that it is in principle possible to  use the strong lensing bias on M(obs)  ∗  or Re, which is observable,  as a way to constrain the amplitude of the intrinsic scatter in the  mass parameters, which is poorly known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Bias in lens mass parameters  Figure 15 shows the median of the distribution in various mass-  related quantities, as a function of the minimum Einstein radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  The top panel is the median stellar population synthesis mis-  match parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In all simulations, strong lenses are biased to-  wards larger values than the parent population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The bias is larger  the higher the intrinsic scatter in αsps and Mh, and increases with  increasing θEin,min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For the fiducial model, the bias on αsps can be  as small as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='03 dex (7%), if no cut on Einstein radius is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  However, it can rise up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='09 dex (23%) when considering only  lenses with θEin > 2′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For comparison, we also show two refer-  ence values of αsps, corresponding to a Chabrier IMF (Chabrier  2003) and a Salpeter IMF (Salpeter 1955).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' These values roughly  bracket the current systematic uncertainty on αsps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  The second panel of Figure 15 shows the median halo mass  of galaxies with an observed stellar mass of log M(obs)  ∗  = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We measured this quantity by fitting the following relation to the  Mh − M(obs)  ∗  distribution,  log Mh ∼ µh,0 + βh(log M(obs)  ∗  − 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4),  (33)  and taking the resulting value of the parameter µh,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Similarly to  the αsps case, the strong lenses are biased towards larger values  Article number, page 15 of 22  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' seleff  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='30  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='35  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='40  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='45  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='50  Median zg  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8  Median log M(obs)  ∗  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='9  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2  Median log Re  at log M(obs)  ∗  = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4  Parent pop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Fiducial  High scatter  Low scatter  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  Minimum θEin  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='72  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='74  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='76  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='78  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='80  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='84  Median qg  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Bias on lens observable properties as a function of the mini-  mum Einstein radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' First panel: median redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Second panel: me-  dian observed stellar mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Third panel: median half-light radius at an  observed mass of log M(obs)  ∗  = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Fourth panel: median axis ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In  each panel, the black dashed line indicates the value of the parent popu-  lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Error bars indicate the standard deviation of the mean of the lens  sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  of the halo mass, with the bias being larger for higher-scatter  simulations and more restrictive cuts on θEin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In the fiducial scat-  ter scenario with a completeness limit of θEin,min = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0′′, the halo  masses of lenses are on average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='16 dex larger than those of  their parent population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  The third panel of Figure 15 shows the median projected dark  matter mass enclosed within an aperture of 5 kpc, MDM,5, at fixed  observed stellar mass and half-light radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We obtained this  quantity by fitting the following relation to the MDM,5−M(obs)  ∗  −Re  distribution:  log MDM,5 ∼ µDM,0 + βDM(log M(obs)  ∗  − 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4) + ξDM(log Re − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (34)  Figure 15 shows the value of µDM,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The bias on this quantity  is qualitatively similar to that on the total halo mass, but much  smaller in amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' There are two reasons for this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' First, the  dependence of MDM,5 on Mh is shallower than linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This is be-  cause, as the virial mass increases, the virial radius increases as  well: the extra mass is spread over a larger volume, and therefore  the mass within the inner region does not increase proportionally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Second, while µh,0 is the halo mass at fixed stellar mass, µDM,0  is measured at fixed half-light radius as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The density profile  of the dark matter halos in our simulation have a dependence on  galaxy size: the response of dark matter to baryons is stronger  for more concentrated stellar distributions (see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  By capturing this dependence in the model of Equation 35, the  residual scatter in MDM,5 around the mean is reduced, and so is  the strong lensing bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This, however, is a minor effect: we re-  peated the analysis while setting the dependence on size to zero  and found minimal differences on the derived values of µDM,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In  the fiducial scenario, the bias on MDM,5 is as small as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='02 dex  for θEin,min < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5′′, and is negligible in the low-scatter scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  In the fourth panel of Figure 15 we show the average inner  dark matter slope at fixed observed stellar mass and half-light  radius, which we measured by fitting the following model  γDM ∼ µγ,0 + βγ(log M(obs)  ∗  − 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4) + ξγ(log Re − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2),  (35)  and taking the resulting value of µγ,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Different simulations show  different trends in the average γDM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' These are the results of un-  derlying correlations between the dark matter density profile and  the stellar and halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In our model, galaxies of a given size  with a larger stellar mass have a steeper dark matter slope, be-  cause the response of the dark matter to the infall of baryons  is stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Vice-versa, galaxies with a larger halo mass have a  shallower density slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' At fixed M(obs)  ∗  and Re, strong lenses  have both a larger stellar mass (because their αsps is larger) and  a larger halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In the simulation with low scatter, the bias  in stellar mass is more important, therefore γDM has a positive  bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In the simulation with large scatter, the bias in halo mass  dominates, and therefore γDM is negatively biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Bias in source parameters  Figure 16 shows the lensing bias in source-related parameters,  with respect to the population of sources that are detectable with-  out lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' As previously seen in Figure 13, lensed sources are  biased towards higher redshift (top panel of Figure 16), with the  trend being larger for more restrictive cuts on the Einstein ra-  dius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The second panel shows the bias in the source magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  This bias is negative for small values of θEin,min, that is, lensed  sources tend to be brighter than their field counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Naively  one might have expected the opposite trend, as strong lensing  magnification allows the detection of sources that are intrinsi-  cally fainter than the detection limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' However, as the analysis  of Section 3 clearly shows, the lensing cross-section is always  Article number, page 16 of 22  Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' : Strong lensing selection effects  larger for brighter sources, and that explains the sign of the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  At the same time, the median does not capture the whole picture  of the bias in ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For instance, in the fiducial simulation with  no cut on the Einstein radius, the 90%-ile of the ms distribution  is 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='05, while that of the population of detectable sources is  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='14 mag brighter: indeed, strong lensing allows the detection of  fainter sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  The third and fourth panels of Figure 13 show the bias  in half-light radius and Sérsic index, respectively, for sources  within a magnitude bin centred on ms = 25 and with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4 mag  width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We detected no clear sign of bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Finally, the fifth panel  shows the bias in source axis ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Also in this case no obvious  sign of bias was detected, except in the largest values of θEin,min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Galaxy-quasar lenses  The main goal of the experiment with lensed quasars is to check  whether there are any differences in the strong lensing bias with  respect to the extended source case, at fixed properties of the  foreground galaxy population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For this reason, we only ran sim-  ulations with lensed quasars in the fiducial scatter scenario and  compared the results with those from the extended source simu-  lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We are also interested in understanding how the subset of  quad lenses differs from the entire population of quasar lenses,  therefore we also analysed that subsample on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Figure 17 shows the strong lensing bias as a function of min-  imum Einstein radius in the following quantities: stellar popu-  lation synthesis mismatch parameter, halo mass at fixed stellar  mass, enclosed dark matter mass at fixed stellar mass and half-  light radius, and axis ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The biases of the population of lensed  quasars (red curves) are very similar to that of lensed galaxies  (blue curves), especially for values of θEin,min < 1′′, When con-  sidering only quad lenses, however, there are some differences,  the most remarkable of which is the bias in the axis ratio: quad  lenses tend to be on average galaxies with a higher ellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  This trend was expected, given the results of section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2: the  higher the ellipticity, the larger the area enclosed within the inner  caustic, which is where a source needs to lie in order to produce  four or more images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Quad lenses tend to also have a slightly  larger halo mass at fixed stellar mass, for θEin,min < 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Discussion  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Key results  In order for a galaxy-source pair to be included as a strong lens  in a survey, three conditions must be met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' First of all, at least  part of the source must be multiply imaged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Second, the multiple  images must be detectable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Third, the lens must be recognised as  such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Each one of these conditions introduces a bias with respect  to the parent population of foreground galaxies and background  sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Together, they define the lens selection probability term  Psel of Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The first two points are intrinsic to a strong  lensing survey and constitute an unavoidable source of bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The  best case scenario occurs when the efficiency of including a de-  tected strong lens in a survey is always one, and the sample is  100% complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In this case, the third condition does not intro-  duce any further selection and the strong lensing bias is min-  imised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We explored this scenario in Section 5 when setting the  minimum Einstein radius to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' If only lenses with Einstein ra-  dius larger than a given threshold are selected, however, the bias  generally increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  The strong lensing bias affects all quantities that are related  to the mass distribution of the lens, as well as the redshifts of  lens and source and the source surface brightness parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Some of these quantities, such as the observed stellar mass and  half-light radius of the lenses, can be directly measured, and it  is straightforward to quantify their lensing bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Other quanti-  ties, however, such as the stellar population synthesis mismatch  parameter or the dark matter content, are difficult to obtain via  traditional, non-lensing, observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Our simulations are par-  ticularly useful to quantify the bias on these properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  In the fiducial scatter scenario, the one consistent with the  observed scatter around the fundamental plane, the bias on αsps  varies from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='03 dex, when no restrictions on the lens Einstein  radius are applied, to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='09 dex, corresponding to the extreme  case in which only lenses with θEin > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0′′ are selected (see Fig-  ure 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' A reasonable value for the minimum Einstein radius in  a space-based survey like Euclid or the Chinese Space Station  Telescope (CSST) is θEin,min = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In this case, the bias on  αsps is slightly smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='04 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The current systematic un-  certainty on αsps is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='3 dex: this is roughly the difference  in measurements of the stellar mass of a galaxy obtained with a  Chabrier or a Salpeter IMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Compared with this uncertainty, the  amplitude of the strong lensing bias on αsps is small: strong lens-  ing observations can be used directly to discriminate between  these two alternative choices of IMF, without the need to correct  for selection effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Such a goal could be reached with a sam-  ple size of a thousand lenses and a statistical study of the kind  proposed by Sonnenfeld & Cautun (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Strong lenses are also biased towards larger halo masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Nevertheless, when focusing on the dark matter content in the in-  ner regions, the amplitude of the strong lensing bias is relatively  small, especially when controlling for the stellar distribution of  the lens galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For instance, at fixed observed stellar mass and  half-light radius, the bias on the projected dark matter mass en-  closed within 5 kpc, MDM,5, is only a few percent in the fidu-  cial scatter scenario with θEin,min < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This means that strong  lenses can indeed be used to understand the inner dark matter  distribution of galaxies, as long as the dependence of the dark  matter distribution on the properties of the baryonic component  is accurately modelled (for example, by following the approach  of Sonnenfeld & Cautun 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We also looked at the bias on source-related parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Strong lensing causes the luminosity function of background  sources to be broadened, compared to the distribution of de-  tectable sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' On the one hand, it preferentially selects  brighter sources, because the lensing cross-section increases  with increasing source brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' On the other hand, it allows  for the detection of sources that are intrinsically fainter than the  detection limit in the absence of lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Interestingly, we did not  find any significant bias on the source size, at fixed magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  This result follows from the fact that our lens detection criterion  relies on a surface brightness threshold, and surface brightness  is preserved by lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' However, it appears to be in contradic-  tion with the work of Oldham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2017), who argued that their  strong lens sample selected preferentially compact sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The  origin of this discrepancy probably lies in the differences be-  tween the criteria used to define a strong lens in the two stud-  ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The Oldham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2017) sample was selected primarily via  spectroscopy, by looking for signatures of two galaxies at differ-  ent redshifts in the Sloan Digital Sky Survey (SDSS York et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  2000) data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' SDSS spectra were taken in fibres with a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5′′ ra-  dius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Lensed galaxies that are comparable in size to this scale,  or larger, are less likely to be detected, because part of their flux  extends outside of the fibre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Compact galaxies with a large over-  all magnification, instead, are more likely to be detected (see the  discussion in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='3 of Oldham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We refer the  Article number, page 17 of 22  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' seleff  reader to Arneson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2012) for a thorough study of selection  effects associated with spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  There is also an indirect way in which strong lensing could  preferentially probe more compact sources than possible in the  field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Sources that are smaller than the size of the PSF can be eas-  ily confused as stars in our own Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Since these stars cannot  be strongly lensed into multiple images, the detection of a strong  lens automatically confirms the extragalactic nature of a lensed  source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This type of selection effect could explain, for example,  the detection of an apparent outlier in the magnitude-size rela-  tion of Lyman-break galaxy by means of strong lensing, by Jae-  lani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' It is, however, improper to refer to this effect as  strong lensing bias, since the star-galaxy separation is a bias that  primarily affects field observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Our analysis showed that the  tendency to select compact sources is not a general feature of  strong lens samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Combining lensed quasars with lensed galaxies  Our experiments were also useful for understanding the possi-  ble biases that might incur when combining information from  samples of galaxy-galaxy lenses with samples of galaxy-quasar  lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' On the one hand the biases in the mass-related quantities  (αsps and MDM,5) are very similar in simulations with the same  foreground galaxy population and different background sources  (see Figure 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' On the other hand, these biases are a function  of the completeness limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In order to use a sample of strong  lenses as a prior for another sample, then, it is important to make  sure that 1) the parent population of foreground galaxies among  which lenses are searched for is the same in both surveys, and  2) the two surveys can probe the same distribution in Einstein  radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  The above argument applies to samples of quasars selected  regardless of the number of multiple images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' When dealing with  quad lenses, however, the situation is more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' First of  all, because our experiment revealed differences in the bias on  the dark matter distribution between quads and the entire sam-  ple of lensed quasars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Second, because quad lenses tend to have  a preferentially higher ellipticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The ellipticity is important in  the context of stellar dynamical analyses, which are often used in  combination with strong lensing to constrain lens mass param-  eters (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Yıldırım et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In order to correctly inter-  pret stellar dynamics data, assumptions on the three-dimensional  structure of a lens must be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This is directly related to the  projected ellipticity: galaxies with an axis ratio close to one tend  to be preferentially elongated in the line-of-sight direction, and  vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' When using stellar dynamics-based mass measure-  ments to inform the properties of a sample of quad lenses, there-  fore, it is important to take into account possible biases due to  the different three-dimensional structure of the two samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The importance of the source population properties  Strictly speaking, the results shown in Section 5 apply only to  samples of lenses and background sources with the same prop-  erties as our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' However, the comparison of section  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2 shows that, at fixed foreground galaxy population, the strong  lensing bias for the sample of extended sources is indistinguish-  able from that of the lensed quasar population, at least for values  of θEin,min < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This result suggests that the dependence of  the strong lensing bias on the details of the background source  population is very weak, as long as no additional selection on  the image configuration is applied (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' by selecting only quad  lenses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This is not a coincidence, but is a consequence of the  fact that the dependence of the strong lensing cross-section on  the lens parameters is a weak function of source magnitude9, es-  pecially for magnitudes close to the detection limit (see section  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We can then conclude that the details of the properties of  the background source population play a secondary role in de-  termining the strong lensing bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Mitigation strategies  As we discussed throughout this paper, the amplitude of the  strong lensing bias depends on the intrinsic scatter in the mass  parameters of the foreground galaxy population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' One way to  minimise the bias, therefore, is to identify scaling relations be-  tween observable quantities and mass-related properties that can  account for part of the scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Describing the inner dark matter  distribution as a function of stellar mass and half-light radius, as  we did in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1, is the first step in this direction: the bias  in MDM,5 at fixed M∗ and Re is smaller than the overall shift in  the median MDM,5 of the lens population (albeit by a marginal  amount, as we pointed out earlier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This description could be  extended by including the central velocity dispersion as an addi-  tional control parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The velocity dispersion is directly re-  lated to the mass distribution of the lens galaxy, which means  that, for example, the distribution in αsps at fixed velocity disper-  sion should be narrower than its global distribution marginalised  over the whole population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The central velocity dispersion, how-  ever, is very sensitive to the orbital anisotropy, to the three-  dimensional structure, and to gradients in stellar mass-to-light  ratio, which are not well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Therefore it is difficult to quan-  titatively estimate the benefit of including it in the description of  the lensing bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  If one wishes to directly account for the strong lensing bias,  the formally correct procedure is to explicitly model all of the se-  lection steps in a Bayesian hierarchical formalism, as explained  by Sonnenfeld (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Although this can be computationally  challenging, machine learning can offer an efficient alternative  (Legin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In order for either of these approaches to  work, however, it is essential that the lens selection procedure  can be simulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This, in turn, requires having an objective def-  inition of a strong lensing event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' A peak-based definition such as  that introduced in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4 could be used for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Nevertheless, it can still be difficult to fully forward model  strong lensing selection if visual inspection by humans is in-  volved in the definition of the lens sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In that case, a pos-  sible alternative is to approximate the lens finding probability  Pfind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For example, we can make the assumption that Pfind de-  pends purely on the Einstein radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This is a reasonable assump-  tion, as long as the lens finding procedure does not selectively  pick lenses with different image configurations depending on  their Einstein radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The dependence of Pfind on θEin could then  be described empirically and inferred during the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This  is essentially the approach adopted by Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2019)  in the analysis of strong and weak lensing data from the Hyper  Suprime-Cam Survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Limitations of our analysis  The simulations on which our analysis is based are as complex  as required by the goal of the analysis itself, which is to estimate  the amplitude of the strong lensing bias in a few key quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  9 The cross-section itself is a strong function of source magnitude, but  the trends between σSL and the lens parameters are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Article number, page 18 of 22  Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' : Strong lensing selection effects  We did, however, make some simplifying assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' One such  assumption consisted in neglecting the contribution from line-of-  sight structure and the environment to the lensing signal, which  typically introduce an external shear and convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The effect  of including external shear in a lens model is similar to that of  changing the ellipticity of the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' External convergence mim-  ics the effect of adding or removing a constant sheet of invisible  mass, producing an effect similar to varying the distribution of  dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Typical values of the external shear and external  convergence are |γ| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1 and |κext| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1 (Millon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2020):  these values are small compared to the typical ellipticities and  dark matter fractions of our simulated lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Therefore, while  including them might modify slightly the amplitude of the strong  lensing bias in the axis ratio and dark matter distribution param-  eters, the conclusions of our analysis would not be affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We also assumed that the dark matter density profile of the  galaxies is completely determined by the halo mass and by the  stellar mass distribution (see section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In reality we expect  there to be a range of density profiles, for example as a result of  halos having a nonzero scatter in their initial (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' before bary-  onic infall) concentration parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We could in principle carry  out an experiment with such an additional source of variation in  the dark matter profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Adding a scatter in concentration while  keeping the halo mass scatter parameter σh fixed results in a  larger spread in the inner dark matter distribution, which would  increase the amplitude of the lensing bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' However, the value  of σh that we chose for our fiducial experiment was tuned on  the basis of the predicted scatter in velocity dispersion around  the fundamental plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' That scatter would also be increased by  adding flexibility to the dark matter density profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Then, in or-  der to be consistent with the analysis carried out in this paper,  the value of σh would need to be lowered accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This, in  turn, would reduce the amplitude of the strong lensing bias and  introduce a bias on the halo concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Since lensing and dy-  namics are sensitive to the mass distribution at comparable scales  (the half-light radius and Einstein radius are similar for most of  the lenses), we expect these two effects to cancel out to first ap-  proximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The lensing bias on quantities directly related to  the inner mass distribution, such as αsps and MDM,5, would be  left roughly unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The main effect of adding scatter to the  concentration is to reduce the correlation between the total halo  mass and the dark matter mass enclosed within the Einstein ra-  dius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This would then reduce the amplitude of the strong lensing  bias on the halo mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Another simplifying assumption that we made was that of  adopting a constant mass-to-light ratio for the stellar compo-  nent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Massive galaxies are known to have colour gradients, the  main effect of which is to cause the stellar half-mass radius to  be smaller than the half-light radius (Szomoru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Suess  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Since, as shown in Section 5, strong lensing bias  tends to select galaxies with a more compact stellar distribution,  it would also preferentially select galaxies with a steeper mass-  to-light ratio gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' However, testing for the amplitude of the  strong lensing bias on the mass-to-light ratio gradient is beyond  the scope of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Conclusions  Strong lensing is a very active line of research, but a compre-  hensive understanding of the selection effects associated with it  has so far been lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This work takes a major step towards  filling that knowledge gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' After a thorough investigation, we  learned several lessons regarding the strong lensing bias in pho-  tometrically selected lens samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The following are the most  important ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The strong lensing cross-section increases primarily with in-  creasing lens mass, with decreasing half-mass radius, and  with increasing source brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' At fixed stellar distribu-  tion and fixed dark matter mass enclosed within a given aper-  ture, varying the inner dark matter slope has little impact  on the strong lensing cross-section (as long as the aperture  within which the dark matter mass is normalised is compa-  rable to the Einstein radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The strong lensing cross-section has little dependence on the  size of the source, if this is smaller than the Einstein radius  and if it is detectable in the absence of lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Sources with  a surface brightness that is too low to be detected are still  undetected when strongly lensed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Lens galaxies tend to be more massive and more compact  than their non-lens counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Their redshift distribution  is also modified with respect to the general galaxy popula-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' At fixed observed stellar mass (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' inferred by means  of stellar population synthesis), lens galaxies have a larger  intrinsic stellar mass (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' a larger stellar population synthe-  sis mismatch parameter αsps) and a larger dark matter halo  mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' At fixed stellar mass and size, lens galaxies are still  biased towards a larger dark matter content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The amplitude of the strong lensing bias depends on how  broad is the distribution of the parameters describing the lens  population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Important quantities in this regard are the intrin-  sic scatter in the stellar population synthesis mismatch pa-  rameter, σsps, and in the dark matter halo mass, σh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Increas-  ing the values of σsps and σh results in a stronger bias on  αsps and on the dark matter mass, and a weaker bias on the  observed stellar mass and half-light radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This implies that,  in principle, we could constrain the intrinsic scatter param-  eters, which are currently poorly known, by measuring the  amplitude of the strong lensing bias on M(obs)  ∗  and Re, which  is easily observable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The strong lensing bias varies depending on the complete-  ness of the lens sample as a function of Einstein radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Sur-  veys that can discover lenses with a smaller Einstein radius  have a smaller associated strong lensing bias in all quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Under reasonable assumptions on the intrinsic scatter pa-  rameters, for a Euclid-like survey that is complete down  to θEin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5′′ the bias on αsps is smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='04 dex  (10%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This bias is much smaller than the current systematic  uncertainty on the stellar population synthesis-based stellar  masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Therefore, strong lensing measurements could be  used directly to calibrate stellar mass measurements of mas-  sive galaxies to 10% accuracy, without the need to correct  for selection effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Under the same assumptions, the strong  lensing bias on the average halo mass at fixed stellar mass is  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='07 dex, while that on the inner dark matter distribution at  fixed stellar mass and size is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='02 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Strong lensing selection broadens the magnitude distribution  of background sources, compared to the population of ob-  jects that are detectable without lensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' At the same time,  we did not find any evidence for a bias in the size distribu-  tion of background sources, at fixed magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Simulations with lensed quasars in place of extended sources  showed that the amplitude of the strong lensing bias in the  lens-related parameters has very little sensitivity on the de-  tails of the source population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This result has positive impli-  cations for time-delay lensing studies: it means that informa-  tion from a sample of galaxy-galaxy lenses can be used as  Article number, page 19 of 22  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' seleff  a prior on the properties of a set of galaxy-quasar lenses, as  long as the two samples are probing the same range in Ein-  stein radius and lens observable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Samples of quad lenses are biased towards galaxies with  larger ellipticity, which implies that their three-dimensional  structure is also biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This means that particular care must  be taken when stellar dynamics measurements obtained on  galaxy-galaxy lenses is used to inform the properties of  quads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  In conclusion, strong lensing selection introduces unavoid-  able biases in the properties of the lens galaxy and background  source populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Biases that affect observable properties, such  as the redshift and the light distribution of the lens, can be easily  quantified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Biases on mass-related quantities, such as the stellar  mass-to-light ratio or the dark matter distribution, are more dif-  ficult to measure directly and must be modelled by taking into  account selection effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Designing strong lensing surveys with  clearly-defined and easily-modellable selection criteria would  help greatly in this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The collaboration leading to this work was initiated at the  2022 Lorentz Center workshop “Bridging gaps between dynamical probes of  galaxies”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' AS and SSL are supported by NOVA, the Netherlands Research  School for Astronomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Support for this work was provided by NASA through the  NASA Hubble Fellowship grant HST-HF2-51492 awarded to AJS by the Space  Telescope Science Institute, which is operated by the Association of Universities  for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=', for NASA, under contract NAS5-26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
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+page_content=', Richards, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=', Bauer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2001, AJ, 122, 549  Vegetti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=', Lagattuta, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=', McKean, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2012, Nature, 481, 341  Wyithe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=', Turner, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=', & Spergel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2001, ApJ, 555, 504  Yıldırım, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=', Suyu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=', & Halkola, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2020, MNRAS, 493, 4783  York, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=', Adelman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=', Anderson, John E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=', J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2000, AJ, 120, 1579  Article number, page 20 of 22  Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' : Strong lensing selection effects  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2  Median log αSPS  Chabrier IMF  Salpeter IMF  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6  µh,0 (Mean log Mh  at fixed M(obs)  ∗  )  Parent pop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Fiducial  High scatter  Low scatter  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2  µDM,0 (Mean log MDM,5  at fixed M(obs)  ∗  , Re)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  Minimum θEin  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='35  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='40  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='45  µγ,0 (Mean γDM,5  at fixed M(obs)  ∗  , Re)  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Bias on lens mass properties as a function of the mini-  mum Einstein radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' First panel: median log αsps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Dotted lines indi-  cate values of αsps corresponding to a Chabrier and a Salpeter IMF  (αsps = 1 corresponds to a Kroupa IMF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Second panel: median Mh  at log M(obs)  ∗  = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Third panel: median MDM,5 at log M(obs)  ∗  = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4  and log Re = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Fourth panel: median γDM at log M(obs)  ∗  = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4 and  log Re = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In each panel, the dashed line indicates the value of the  parent population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Error bars indicate the standard deviation of the mean  of the lens sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  Median z  24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4  25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6  Median ms  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='3  Median θs  at ms = 25 (′′)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6  Median n  at ms = 25  Detectable pop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Fiducial  High scatter  Low scatter  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
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+page_content='0  Minimum θEin  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='44  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='46  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='48  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='52  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='54  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='56  Median q  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Bias on source properties as a function of the minimum Ein-  stein radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' First panel: median source redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Second panel: median  source magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Third panel: median half-light radius in the magni-  tude bin 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8 < ms < 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Fourth panel: median Sérsic index in the  magnitude bin 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8 < ms < 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Fifth panel: median axis ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In each  panel, the dashed line indicates the value of the population of detectable  sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Error bars indicate the standard deviation of the mean of either  the full sample (first, second and fifth panel) or the bin (third and fourth  panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Article number, page 21 of 22  A&A proofs: manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' seleff  −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='05  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='15  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='20  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='25  Median log αSPS  Chabrier IMF  Salpeter IMF  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2  13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4  µh,0 (Mean log Mh  at fixed M(obs)  ∗  )  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
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+page_content='3  µDM,0 (Mean log MDM,5  at fixed M(obs)  ∗  , Re)  Parent pop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Extended sources  Quasars (all)  Quasars (quads)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  Minimum θEin  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='50  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='55  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='60  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='65  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='70  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='75  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='80  Median q  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Bias in the population of quasar lenses (all and quads  only), compared to the extended source simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' First panel: median  log αsps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Second panel: median Mh at log M∗ = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Third panel: me-  dian MDM,5 at log M∗ = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5 and log Re = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Fourth panel: median  lens axis ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In each panel, the dashed line indicates the value of the  parent population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Error bars indicate the standard deviation of the mean  of the lens sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Article number, page 22 of 22  Sonnenfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' : Strong lensing selection effects  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  Axis ratio q  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='0  P(q)  Observed  Model  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Distribution in axis ratio of a sample of early-type galax-  ies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The sample consists of 8078 galaxies from the SDSS, selected by  means of cuts in redshift, colour and Sérsic index as explained in the  text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Measurements of the axis ratio are taken from the de Vaucouleurs  model fits of Meert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The model curve is a beta distribution,  Equation A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2, with α = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='28 and β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Appendix A: Lens galaxy surface brightness  distribution  To assign half-light radii and ellipticities to the simulated lenses  we relied on observations of a sample of early-type galaxies  selected from the Sloan Digital Sky Survey (SDSS York et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' This sample was selected as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Starting from the  SDSS spectroscopic sample, we defined a narrow redshift slice  around z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Then we applied a selection in colour, by choos-  ing objects with g − r > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2, and on Sérsic index, by selecting  only galaxies with n > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We used the Sérsic fit measurements  by Meert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2015) for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' These cuts produced a  sample of 8078 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We then focused on the r−band de Vau-  couleurs model-based photometric measurements of Meert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Using the r−band total flux from the de Vaucouleurs  model and the stellar mass-to-light ratio estimates of Mendel  et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (2014), we obtained measurements of M(obs)  ∗  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Finally, we  fitted the stellar mass-size relation with the following model:  log Re ∼ N(µR,0 + βR(log M(obs)  ∗  − 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4), σ2  R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1)  We obtained µR,0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='20, βR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='63 and σR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We then proceeded to fit for the axis ratio distribution of the  same sample of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Figure A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1 shows a histogram of the  observed distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We fitted this with a beta distribution:  P(q) ∝ qα−1(1 − q)β−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2)  We obtained α = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='28 and β = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Appendix B: Upper limits on the intrinsic scatter  parameters  We used the fundamental plane of early-type galaxies to set an  upper limit on the intrinsic scatter parameters of the simulation:  σsps, σh and σγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' First, we measured the fundamental plane of the  sample of SDSS early-type galaxies introduced in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We took measurements of the line-of-sight stellar velocity dis-  persion within the SDSS spectroscopic aperture, σap, from the  SDSS data release 16 catalogue (Ahumada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' Then,  we fitted the following model to the distribution of σap of the  sample:  P(σap) ∼ N(µσ+β(log M(obs)  ∗  −11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='4)+ξ(log Re−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2), σσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1)  We accounted for observational uncertainties on σap when doing  the fit, therefore the parameter σσ describes the intrinsic scatter  in the logarithm of the velocity dispersion, deconvolved from  the observational scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In principle we should also account  for observational uncertainties on the stellar mass measurement,  as they too contribute to the inferred scatter in σap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' In practice,  however, it is difficult to estimate observational uncertainties on  stellar population synthesis measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' For this reason we  chose not to propagate uncertainties on M(obs)  ∗  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' As a result, the  inferred scatter parameter σσ is slightly overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We ob-  tained µσ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='36, βσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='33, ξσ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='17, and σσ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  We then generated samples of z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2 galaxies from the  model of section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1 and used the spherical Jeans equation to  predict their central velocity dispersion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' The spherical Jeans  equation is (Binney & Tremaine 1987)  d(ρ∗σ2  r)  dr  + β(r)  r ρ∗σ2  r = −ρ∗(r)GM(r)  r2  ,  (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2)  where ρ∗(r) is the 3-dimensional distribution of dynamical trac-  ers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' the stars), σr the radial component of the velocity dis-  persion, β(r) the orbital anisotropy parameter, and M(r) the  mass enclosed within a spherical shell of radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We assumed  isotropic orbits (β = 0), then integrated Equation B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='2 to obtain  the seeing-convolved, surface brightness-weighted line-of-sight  velocity dispersion within the SDSS spectroscopic aperture10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Finally, we fitted the fundamental plane relation of Equation B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='1  to the observed stellar mass, half-light radius and central veloc-  ity dispersion of the mock sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content=' We repeated this procedure  for different values of the intrinsic scatter parameters σsps, σh  and σγ, then settled on the three scenarios listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  10 We  used  Python  code  available  at  https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='com/  astrosonnen/spherical_jeans, for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}
+page_content='  Article number, page 23 of 22' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/zdFQT4oBgHgl3EQfCTXT/content/2301.13230v1.pdf'}